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+{"text":"\\section{INTRODUCTION AND MOTIVATION} \nReactivity estimates are an important facet of nuclear criticality safety.  Currently, reactivity cannot be directly measured and is instead inferred from the prompt neutron decay constant, $\\alpha$.  The value of $\\alpha$ is estimated using Rossi-alpha measurements that are predicated on measuring the time difference between neutron detections~\\cite{uhrig}.  Rossi-alpha measurements are traditionally conducted with $^3$He detectors, which use polyethylene to moderate neutrons to improve the detection efficiency.  Because neutrons take time to moderate in the polyethylene, timing properties change (the decay of prompt neutrons and neutron slowing down time are convolved) and information can be lost.  Organic scintillation detectors can detect neutrons directly and are fast compared to $^3$He systems.  In this work, a fast plutonium assembly is simultaneously measured by $^3$He detectors and organic scintillators, and the detection systems are compared.\n\n\\section{THE ROSSI-ALPHA METHOD}\nIn a Rossi-alpha measurement, neutron detection times are recorded, the time differences between detections are calculated (see Fig.~\\ref{fig:RA}), and a Rossi-alpha histogram of the time differences is constructed~\\cite{uhrig,Feynman44_1,hansen}.  The histogram is fit and $\\alpha$ is obtained from the fit parameters.  Traditionally, the histogram is fit with a single-exponential-plus-constant model \\cite{ornbro}.  Recent work has developed a double-exponential-plus-constant-model~\\cite{mikwa_2exp} that is more suitable than the single exponential model for measurements of reflected assemblies.  The two-exponential model estimates a second parameter: $\\ell_{ctd}$, which describes the mean time a neutron spends in the reflector prior to detection.  Sample Rossi-alpha histograms from a $^3$He detector measuring plutonium are shown in Fig.~\\ref{fig:sample_RA}.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.45\\linewidth]{Rossi_alpha}\n\t\\caption{Time difference calculation for Rossi-alpha measurements.}\n\t\\label{fig:RA} \n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.87\\linewidth]{sample_fits}\n\t\\caption{Rossi-alpha plot with one- and two-exp fits on linear (a) and semilog (b) scales.}\n\t\\label{fig:sample_RA}\n\\end{figure}\n\\section{ASSEMBLY SPECIFICATIONS AND EXPERIMENTAL SETUP}\n\\subsection{Assembly Specifications}\nIn this work, the bottom layer of the Comet critical assembly -- lead-moderated, copper-reflected plutonium (93 wt\\% $^{239}$Pu) -- was measured.  A 3D rendering of the assembly is shown in Fig.~\\ref{fig:3D}; the layout of the bottom layer of copper or plutonium boxes is shown in Fig.~\\ref{fig:bottom_layer}, and a sample plutonium box is shown in Fig.~\\ref{fig:box} ~\\cite{joetta_PHYSOR}.  The total mass of plutonium was approximately 15 kg.  \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.45\\linewidth]{3D_rendering}\n\t\\caption{3D rendering of the Comet critical assembly~\\cite{joetta_PHYSOR}.}\n\t\\label{fig:3D}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.7\\linewidth]{bottom_layer_layout}\n\t\t\\caption{Bottom layer box layout~\\cite{joetta_PHYSOR}.}\n\t\t\\label{fig:bottom_layer}\n\t\\end{minipage}%\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.6\\linewidth]{sample_box}\n\t\t\\caption{Photo of a plutonium box \\cite{joetta_PHYSOR}.}\n\t\t\\label{fig:box}\n\t\\end{minipage}\n\\end{figure}\n\\subsection{Simulation of the Assembly}\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.45\\linewidth]{sample_TiBTaF}\n\t\\caption{Sample plot of the time-binned surface tally (F1) used to estimate the Rossi-alpha.}\n\t\\label{fig:TiBTaF}\n\\end{figure}\nTo estimate the prompt neutron decay constant, $\\alpha$, the measurement was simulated using MCNP6\\textregistered.  The KCODE option estimated $k_\\text{eff} \\approx 0.624$.  To determine $\\alpha$, surface (F1) and point-detector (F5) tallies were time-binned, and the tails (linear on a semilog plot) were fit.  A sample time-bin tail-fit plot is shown in Fig.~\\ref{fig:TiBTaF} and $\\alpha = 52.3\\pm2.5$ ns.  The uncertainty comes from the fit uncertainty.  \n\\subsection{Experimental Setup and Detection System Details}\nIn the measurement of the assembly, two organic scintillator arrays (OSCARs) and one Neutron Multiplicity $^3$He Array Detector (NoMAD) were used. An OSCAR comprises 12 5.08 cm $\\times$ 5.08 cm diameter \\textit{trans}-stilbene organic scintillators coupled to photomultiplier tubes~\\cite{stilbene,stilbene2}.  The NoMAD is similar to the MC-15 detection system \\cite{mc15_manual}, comprising 15 $^3$He detectors embedded in a polyethylene matrix.  The systems were placed 50 cm from the edge of the assembly; a schematic is shown in Fig.~\\ref{fig:schematics} and a photo of the systems side-by-side is shown in Fig.~\\ref{fig:photo}.  For this work, only 21 of the 24 OSCAR detectors were operational.  Based on neutron detection rates, the NoMAD (in the given configuration) is 3.34 times more efficient than the OSCARs.      \n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{detector_layout}\n\t\t\\caption{Schematic of detector placement.}\n\t\t\\label{fig:schematics}\n\t\\end{minipage}%\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{detection_systems}\n\t\t\\caption{Photo of detection systems.}\n\t\t\\label{fig:photo}\n\t\\end{minipage}\n\\end{figure}\n\\section{DATA ANALYSIS}\n\\subsection{Data Analysis for the $^3$He-based NoMAD System}\\label{sec:DA_NoMAD}\nThe output from measurement and preliminary data analysis is a list of detection times.  The Rossi-alpha histogram is created using type-I binning (illustrated in Fig.~\\ref{fig:RA})~\\cite{hansen}.  In theory, Rossi-alpha histograms peak at a time difference of 0 s; however, measurement considerations such as dead time and time of flight cause the peak to occur later.  Suppose the max occurs in bin $b$.  To mitigate the measurement considerations at short time differences, the first $2b$ bins are discarded.  For some comparison purposes in this work, the histograms are integral normalized and constant-subtracted by taking the mean of the last points in the tail.  A sample, resultant Rossi-alpha histogram for the NoMAD is shown in Fig.~\\ref{fig:NoMAD_RA}.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{He3_fits}\n\t\\caption{Rossi-alpha histogram with fits from the NoMAD system.}\n\t\\label{fig:NoMAD_RA}\n\\end{figure}\n\\subsection{Data Analysis for the Organic Scintillator-based OSCAR System}\nThe output from measurement and preliminary data analysis is a list of detection times, total pulse integrals, and tail integrals.  Because organic scintillators are sensitive to both neutrons and photons, pulse shape discrimination (PSD) is used to discriminate the pulses.  The PSD is done for each detector and is both time and energy dependent; a sample PSD plot is shown in Fig.~\\ref{fig:PSD}.  The PSD analysis results in three sets of data: neutron pulses, photon pulses, and pulses to discard (due to, for example, pulse pileup).  Currently, gamma-ray Rossi-alpha is not considered; however, the photon pulses are still needed to correct for timing offsets. The OSCAR system is sensitive to time of flight and offsets due to electronics.  To correct for offsets, the photon-photon coincidence peak (shown in Fig.~\\ref{fig:PP_offset}, present from prompt fission photons) is created for all detectors relative to one detector.  If the peak is not centered about zero, all times in the neutron and photon pulse lists are subsequently shifted by a constant. Once the list of neutron detection times is corrected, the Rossi-alpha analysis is the same as that for the NoMAD system (see Section~\\ref{sec:DA_NoMAD}); the resultant Rossi-alpha plot is shown in Fig.~\\ref{fig:OSCAR_RA}.  \n\\begin{figure}[H]\n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{PSD}\n\t\t\\caption{Sample PSD plot.}\n\t\t\\label{fig:PSD}\n\t\\end{minipage}%\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=.9\\linewidth]{PP_offset}\n\t\t\\caption{Sample photon-photon coincidence plot.}\n\t\t\\label{fig:PP_offset}\n\t\\end{minipage}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{OSCAR_RA}\n\t\\caption{Rossi-alpha histogram from the OSCAR system.}\n\t\\label{fig:OSCAR_RA}\n\\end{figure}\n\\section{RESULTS AND DISCUSSION}\nUnnormalized, non-constant-subtracted Rossi-alpha histograms generated from two hours of data for each detection system are shown in Fig.~\\ref{fig:accidentals}.  The OSCAR has fewer accidentals than that of the NoMAD: the constant value of the tail for the NoMAD is 95\\% of the maximum value, whereas the constant value of the tail for the OSCAR is only 0.7\\% of the maximum value.  In some cases, the high proportion of the accidentals in the case of the NoMAD may obscure the second exponential.  Obscuring the second exponential would reduce the fit model to a single exponential fit; however, since the parameters of interest are a linear combination of the two exponentials, $\\alpha$ and $\\ell_{ctd}$ cannot be determined.\n\nFit metrics plotted as a function of measurement time (and bin width for the NoMAD) are shown in Fig.~\\ref{fig:fit_metrics}.  The root mean square error (RMSE) is normalized by the asymptotic values of the respective data series such that the y-axis is a measure of convergence.  It takes the OSCAR less than 30 minutes to be within 50\\% of its asymptotic value, while it takes the NoMAD approximates 120 minutes (note that RMSE is fairly independent of the bin width, as expected).  It takes the OSCAR less than 20 minutes to achieve an $R^2$ value greater than 0.90, whereas the the NoMAD with 2 $\\mu$s bins requires approximately 70 minutes.  The NoMAD's $R^2$ convergence could be improved by increasing the bin widths; however, 2 $\\mu$s bin widths are already large compared to the time-decay constant (52.3 $\\pm$ 2.5 ns) the NoMAD is trying to observe.  Reducing the bin widths to increase sensitivity to the physical phenomenon the system is trying to measure results in increases in the time is takes the NoMAD to achieve $R^2 > 0.90$; bin widths of 1 $\\mu$s require 140 minutes and bin widths of 500 ns require 280 minutes (the relationship is approximately linear).   \n\nFrom simulation, the ``true\" value of $\\alpha$ for the assembly is taken to be 52.3 $\\pm$ 2.5 ns.  Fitting the OSCAR data with a two exponential, $\\alpha$ is estimated to be 47.4 $\\pm$ 2.0 ns.  The error is 9.37\\% and, qualitatively, the values are similar since the $1.09\\sigma$-confidence intervals overlap.  The NoMAD estimate of $\\alpha$ is $\\approx 37$  $\\mu$s.  The NoMAD has a known slowing down time of 35-40 $\\mu$s and, because $\\alpha\\ll 35$ $\\mu$s, the NoMAD is likely only sensitive to the neutron moderation time. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{accidentals}\n\t\\caption{Unnormalized, non-constant-subtracted Rossi-alpha histograms.}\n\t\\label{fig:accidentals}\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{convergence}\n\t\\caption{Fit metrics as a function of measurement time for the NoMAD at different bin widths and the OSCAR.}\n\t\\label{fig:fit_metrics}\n\\end{figure}\n\n\\section{CONCLUSIONS AND FUTURE WORK}\nIn this work, the organic scintillator array (OSCAR), comprising 21 total operational \\textit{trans}-stilbene detectors, and the Neutron Multiplicity $^3$He Array Detector (NoMAD), comprising 15 $^3$He tubes embedded in a polyethylene matrix, simultaneously measured 15 kg of plutonium (93 wt\\% $^{239}$Pu) moderated by lead and reflected by copper with $k_\\text{eff} = 0.624$ and $\\alpha = $ 52.3$\\pm$2.5 ns.  It was found that the OSCAR converged on its estimate of $\\alpha$ faster than the NoMAD, which translates to reduced procedural and operational costs in practical implementation.  The convergence needs to be investigated further for assemblies where $\\alpha$ is much larger ($\\alpha\\propto 10-100s$ of $\\mu$s). Because neutrons are moderated in the polyethylene matrix of the NoMAD (and moderation is not inherent to the OSCAR), the OSCAR is an inherently faster detection system.  The entire Rossi-alpha histogram (reset time) is less than 100 ns for the OSCAR (1 ns bins), whereas 100 ns is the clock tick length for the NoMAD.  Therefore, for fast assemblies ($\\alpha \\propto 1-100s$ of ns), it is more suitable to use the OSCAR that estimated the true $\\alpha$ within 1.09 standard deviations and an error of 9.37\\% (on the order of uncertainty in nuclear data).  Larger accidental contributions are more likely to wash out time information; the NoMAD has a large accidental contribution and the OSCAR has a negligible accidental contribution. Future work involves determining when each system is more suitable to a given measurement.  Furthermore, gamma-ray and mixed-particle Rossi-alpha will be investigated with the organic scintillators.  Work will also be done with more measurements to validate organic scintillator-based Rossi-alpha measurements.\n\n\\section*{ACKNOWLEDGEMENTS}\n\nThis work is supported by the National Science Foundation Graduate Research Fellowship under Grand No. DGE-1256260, by the Consortium for Verification Technology under Department of Energy National Nuclear Security Administration award number DENA0002534, and by the DOE Nuclear Criticality Safety Program, funded and managed by the National Nuclear Security Administration for the Department of Energy.  Any opinion, findings, and conclusion or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or other funding organizations.\n\n\n\n\\setlength{\\baselineskip}{12pt}\n\\bibliographystyle{physor}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\nCataclysmic variables (CVs) are interacting binary systems in which a low-mass star---usually a red dwarf---overfills its Roche lobe and transfers mass onto a white dwarf (WD).  \\citet{warner} and \\citet{hellier}  offer excellent overviews of these intriguing systems. In a subset of CVs known as polars, the exceptionally strong magnetic field ($\\sim$ tens of MG) of the WD synchronizes the WD's spin period with the orbital period of the binary (see \\citet{cropper} for a comprehensive review of polars specifically). The accretion stream from the secondary star follows a ballistic trajectory toward the WD until the magnetic pressure matches the stream's ram pressure. When this occurs, a threading region forms in which the accretion stream couples onto the WD's magnetic field lines, and the captured material is then channeled onto one or more accretion regions near the WD's magnetic poles. The impact of the stream creates a shock in which the plasma is heated to X-ray-emitting temperatures, so polars can be significantly brighter in X-ray wavelengths than ordinary non-magnetic CVs. In addition to X-rays, the accretion region produces polarized cyclotron emission in the optical and in the infrared, the detection of which is a defining characteristic of polars.\n\nEclipses of the WD have provided great insight into polars. Because a polar has no accretion disk, an eclipsing polar will generally exhibit a two-step eclipse: a very sharp eclipse of the compact ($\\sim$ white dwarf radius) cyclotron-emitting region, followed by a much more gradual eclipse of the extended accretion stream (see, {\\it e.g.}, \\citet{harrop-allin} for an eclipse-mapping study of HU Aqr). When the accretion rate is high, the WD photosphere makes only a modest contribution to the overall optical flux, overshadowed by the two accretion-powered components mentioned above. \n\nEclipsing polars also make it possible to determine the orientation of the magnetic axis with respect to the secondary.  In HU Aqr, the orientation of the dominant magnetic pole leads the line of centers of the binary by about 45$^{\\circ}$ \\citep{harrop-allin}, while in DP Leo, another eclipsing polar, the equilibrium orientation leads the line of centers by 7$^{\\circ} \\pm 3^{\\circ}$ but with a long-term oscillation with an amplitude of $\\sim25^{\\circ}$ \\citep{beuermann}.\n\nIn at least four polars,\\footnote{In addition to the subject of this study (V1432 Aql), three other polars are incontrovertibly asynchronous: BY Cam, V1500 Cyg, and CD Ind. At the time of writing, there are at least two candidate systems: V4633 Sgr \\citep{lipkin} and CP Pup \\citep{bianchini}.} the WD's spin period differs from the orbital period by as much as several percent. In these asynchronous polars, the WD's magnetic field is gradually synchronizing the spin period with the orbital period on timescales of centuries. For example, \\citet{ss91} detected a derivative in the WD spin period in V1500 Cyg and estimated that the system would approach resynchronization about 150 years after the publication of their study. \n\nBecause the prototype asynchronous polar, V1500 Cyg, was almost certainly desynchronized during its 1975 nova eruption, the canonical view is that these systems are byproducts of nova eruptions which break the synchronous rotation by causing the primary to lose mass and to interact with the secondary \\citep{ssl}. However, \\citet{warner02} combined the fraction of asynchronous systems among all known polars with their estimated synchronization timescales and estimated an unexpectedly short nova recurrence time of a few thousand years for polars---far more rapid than the expected recurrence time of $\\sim1 \\times 10^5$ years. Every aspect of Warner's deduction ought to be explored, including the possibility of an additional channel for desynchronizing polars, selection effects that might alter the fraction of asynchronous polars, and methods of calculating the synchronization time scale. \n\nInterestingly, in each of the four confirmed asynchronous polars, the threading process is inefficient in comparison to fully synchronous systems. In synchronous systems, the accretion stream is fully captured not long after it leaves the L1 point, well before it can travel around the WD \\citep[e.g.][]{schwope97}. In none of the asynchronous systems is this efficient threading seen. For example, Doppler tomography by \\citet{schwope} of V1432 Aql showed an azimuthally extended accretion curtain, a finding which is possible only if the accretion stream can travel significantly around the WD. X-ray observations of V1432 Aql also indicate that the accretion stream travels most of the way around the WD before it is fully threaded onto the magnetic field lines \\citep{mukai}. Likewise, in the other three systems, there is mounting evidence that the accretion flow can significantly extend around the WD. In CD Ind, the accretion stream appears to thread onto the same magnetic field line throughout the beat cycle, requiring that the stream be able to travel around the WD \\citep{ramsay}. With regard to V1500 Cyg, \\citet{ss91} argued that the smooth sinusoidal variation of the polarization curve was consistent with the infalling stream forming a thin accretion ring around the WD. More recently, \\citet{litvinchova} detected evidence that this accretion ring is fragmented, periodically reducing the irradiation of the donor star by the hot WD. In the remaining system, BY Cam, Doppler tomograms show that the accretion curtain extends over $\\sim180^\\circ$ in azimuth around the WD, requiring a similar extent of the accretion stream \\citep{schwarz}. Although a sample size of four is small, it is remarkable that in each of the confirmed asynchronous polars, the threading process is so inefficient that the accretion stream can travel much of the way around the WD. \n\n\\section{V1432 Aql}\n\nV1432 Aql (= RX J1940.1-1025) is the only known eclipsing, asynchronous polar and was identified as such by \\citet{patterson} and \\citet{friedrich}. There are two stable periodicities in optical photometry of V1432 Aql. The first (12116 seconds) is the orbital period, which is easily measured from the timings of the eclipses of the WD by the secondary. Initially, the nature of the eclipses was unclear; \\citet{patterson} argued that the secondary was the occulting body, but \\citet{watson} contended that a dense portion of the accretion stream was the culprit. Much of the confusion was attributable to the presence of residual emission lines and X-rays throughout the eclipses, as well as the variable eclipse depth. Since X-rays in polars originate on or just above the WD's surface, the apparent X-ray signal throughout the eclipse was inconsistent with occultations by the donor star. Additionally, there was considerable scatter in the eclipse timings, and the system's eclipse light curves did not show the rapid ingresses and egresses characteristic of synchronous polars \\citep{watson}. However, \\citet{mukai} resolved the dispute with high-quality X-ray observations which showed that the donor actually eclipses the WD and that the residual X-ray flux previously attributed to V1432 Aql was actually contamination from a nearby Seyfert galaxy.\n\nThe second periodicity ($\\sim12150$ seconds) is the spin modulation of the WD. In optical photometry, this periodicity manifests itself in several ways. In particular, at $\\phi_{sp} = 0.0$, the WD is occulted by material accreting onto one of the magnetic poles, producing a broad ``spin minimum'' \\citep{friedrich}. Analyses of the spin minima have revealed several fascinating insights into V1432 Aql. For example, \\citet{gs97} undertook an O$-$C study of the timing residuals of the spin minima and managed to detect a decrease in the WD spin period, indicating that the system is resynchronizing itself. They also measured a cyclical variation in the timings of the spin minima, caused by (1) a longitudinal offset between the magnetic pole and its corresponding accretion region on the WD's surface and (2) the accretion stream threading onto different magnetic field lines throughout the spin-orbit beat period  ($P^{-1}_{beat} = |P^{-1}_{orb}-P^{-1}_{sp}|$). Using these timings and a dipole accretion model, the authors managed to constrain the combined effect of the threading radius and the colatitude of the magnetic axis on the WD, but they could not constrain these parameters individually. \\citet{staubert03} applied the methodology of \\citet{gs97} to a larger dataset and refined the results of the earlier paper.\n\nA critical concept which emerges from the literature is the beat period between the spin and orbital periods. The beat period is simply the amount of time that it takes for the WD (and its magnetic field) to rotate once as seen from the perspective of the donor star. As \\citet{gs97} first demonstrated, the accretion stream will interact with different magnetic field lines as the system progresses through its beat period, a foundational principle which informs our analysis throughout this paper.\n\nV1432 Aql is especially suitable for long-term study because its long-term brightness has remained constant not only in our own observations but also in data from the American Association of Variable Star Observers\\footnote{www.aavso.org} dating back to 2002. Similarly, the Catalina Sky Survey \\citep{drake} does not show any low states in the system since coverage of V1432 Aql began in 2005. While many polars alternate unpredictably between bright and faint states due to changes in the mass-transfer rate, V1432 Aql has not been observed to do so.\n\nWe supplement these previous studies by reporting the detection of stable periodicities in both the residual eclipse flux and the O$-$C timing residuals of the eclipses. These phenomena occur at the beat period, and we use a model to show that our observations are consistent with a threading radius whose position with respect to the WD varies throughout the beat cycle.\n\nIn response to this study's observational findings, one of us (DB) followed up by analyzing a different set of observations obtained by the Center for Backyard Astrophysics\\footnote{http:\/\/cbastro.org\/} over a much longer timespan. His group's analysis provides confirmation of the residual-flux and timing variations described in this paper while also reporting additional beat-cycle-related phenomena \\citep{boyd}. \n\n\\section{Observations}\n\n\\begin{figure}\n\n\t\\includegraphics[width=0.45\\textwidth]{sample-eclipses-models}\n\t\n\\caption{Two representative eclipses of V1432 Aql. The data represented in black were obtained at $\\phi_{beat} = 0.89$, and the data in gray at $\\phi_{beat} = 0.54$. The solid lines are the best-fit polynomials for each dataset. The polynomials satisfactorily model the asymmetric eclipses while smoothing noisy, possibly spurious features in the light curves.}\n\\label{sample-eclipses}\n\\end{figure}\n\nAs part of a twenty-eight-month effort to study V1432 Aql's behavior at different beat phases, six of us (CL, RM, RC, KCM, TC, and DS) obtained unfiltered, time-resolved photometry using the University of Notre Dame's 28-cm Schmidt-Cassegrain telescope and SBIG ST-8XME CCD camera between July 2012 and July 2014. The exposure time was 30~seconds for each individual image, with an overhead time of 8~seconds per image. A total of 76 light curves, consisting of over 17,500 individual measurements, were obtained with this instrument. These observations constitute the bulk of our dataset, and their uniformity avoids the introduction of errors caused by combining unfiltered observations from different telescope-CCD combinations. Because of their homogeneity, we use these data for all three parts of our analysis: studying the eclipse O-C variations, measuring the mid-eclipse magnitude, and for constructing phase plots of the system at different beat phases.\n\nWe also obtained a number of light curves with other telescopes, but since these instruments have different spectral responses, we only used this supplemental data to explore eclipse O$-$C variations. CL obtained four unfiltered time series in July 2014 using the University of Notre Dame's 80-cm Sarah L. Krizmanich Telescope and two more with Wesleyan University's 60-cm Perkin Telescope in September 2014. The data obtained with the Krizmanich and Perkin Telescopes have much higher time resolution (exposure times between 5 and 7 seconds, each with a $\\sim$3-second readout time, for a total cadence of 10 seconds or less), facilitating the study of the rapid variability during the eclipses. In addition, MC, JU, DB, and LM respectively used a 40-cm Schmidt-Cassegrain and QSI-516 CCD camera with a Johnson $V$ filter, a 23-cm Schmidt-Cassegrain and QSI-583ws CCD camera, a 25-cm Newtonian with an unfiltered SXV-H9 CCD camera, and a 28-cm Schmidt-Cassegrain equipped with an STT-1603 CCD camera. With the exception of LM, who used 45-second exposures, each of them used an exposure time of 60 seconds.\n\nTo compensate for light-travel delays caused by Earth's orbital motion, the timestamp for each observation was corrected to the BJD (TDB) standard \\citep{eastman}.\n\nWith unfiltered photometry of a CV, it is possible to infer the approximate $V$-band magnitude of the CV by selecting a same-color comparison star and using its $V$ magnitude when calculating the magnitude of the CV. Since polars tend to be quite blue, we relied upon AAVSO field photometry to select two relatively comparison blue stars;\\footnote{These stars are labeled 117 and 120 in AAVSO chart 13643GMF, and they have $B-V$ colors of 0.20 and 0.43, respectively, according to the APASS photometric survey \\citep{APASS}.} we utilized these comparison stars for all photometry used in the analyses of mid-eclipse magnitude and the spin modulation at different beat phases.\n\nOne of the most obvious phenomena in the photometry is the highly variable magnitude of the system at mid-eclipse, which ranges from $V\\sim$ 16.0 to $V\\sim$ 17.5. Different eclipses also displayed strikingly different morphologies, and in Figure~\\ref{sample-eclipses}, we plot two eclipses which are representative of this variation. Such behavior is plainly at odds with normal eclipsing polars, which almost invariably have very abrupt ingresses and egresses since most of the flux originates in a small---and thus rapidly eclipsed---area on the WD \\citep[e.g.][]{harrop-allin}. V1432 Aql's gradual ingresses and egresses indicate that its flux originates in an extended region, and in this regard, its eclipses bear a superficial resemblance to those of CVs with accretion disks.\n\nWe measured both the time of minimum eclipse flux and the magnitude at mid-eclipse by fitting a fifth-order polynomial to each eclipse (see Table~\\ref{eclipse_timings}). Figure~\\ref{sample-eclipses} demonstrates the adequacy of the fit by plotting two eclipse light curves, each fitted with a fifth-order polynomial. Since the system's eclipses are frequently asymmetric, the time of minimum flux is not necessarily the midpoint between ingress and egress. Indeed, several eclipses were W-shaped, with two distinct minima. For these eclipses, we report the time of the deepest minimum. One particularly remarkable eclipse, observed on JD 2456843 and discussed in Section~\\ref{application_of_model}, had two minima of equal depth, so we report both times.\n\nAdditionally, we detected a number of spin minima. Since previous studies of the spin minima \\citep[e.g.][]{gs97} have measured the timing of each spin minimum by locating its vertical axis of symmetry, we fit a second-order polynomial to each spin minimum on the assumption that the minimum of this parabola will roughly approximate the vertical axis of symmetry. While a higher-order polynomial would do a better job of modeling the often-asymmetric spin minima, using the second-order polynomial increases the compatibility of our timings with those presented in other works.\n\nWe list in Table~\\ref{spin_timings} the timings of all clearly-detected spin minima. A number of spin minima were ill-defined or had multiple mimima of comparable depth, and in those instances, we did not report a timing because it was impossible to objectively identify the middle of the spin minimum.\n\n\n\\begin{table}\n\t\\centering\n\t\\begin{minipage}{\\textwidth}\n\t\\caption{Observed Times of Minimum Eclipse Flux \\label{eclipse_timings}}\n\n\n\n\t\\begin{tabular}{cccccc}\n\t\\hline\n\tBJD\\footnote{$2456000+$} & $\\phi_{beat}$ &$\\phi_{sp}$& BJD& $\\phi_{beat}$\n\t & $\\phi_{sp}$\\\\\n\t\\hline\n\n117.75353(52)&0.67&0.46&531.58818(26)&0.44&0.71\\\\\n121.67928(60)&0.73&0.39&534.67318(47)&0.49&0.66\\\\\n129.67477(51)&0.87&0.28&538.59720(41)&0.55&0.58\\\\\n129.81416(50)&0.87&0.27&539.57897(30)&0.57&0.56\\\\\n131.49702(40)&0.90&0.24&539.71935(49)&0.57&0.56\\\\\n132.47891(47)&0.91&0.23&540.70141(35)&0.58&0.55\\\\\n133.46253(73)&0.93&0.22&545.60953(54)&0.66&0.47\\\\\n134.44280(55)&0.94&0.20&546.59071(53)&0.68&0.45\\\\\n138.51071(58)&0.01&0.14&548.69482(44)&0.71&0.42\\\\\n145.80112(28)&0.13&0.01&549.67629(38)&0.73&0.40\\\\\n162.77047(40)&0.41&0.73&558.65164(39)&0.88&0.26\\\\\n175.67045(38)&0.62&0.51&559.63391(37)&0.89&0.24\\\\\n180.57845(40)&0.70&0.43&560.61564(49)&0.91&0.23\\\\\n180.71866(44)&0.70&0.43&562.58006(36)&0.94&0.21\\\\\n181.70019(43)&0.72&0.41&565.66515(61)&0.99&0.16\\\\\n182.68111(44)&0.74&0.39&566.64823(74)&0.01&0.15\\\\\n194.60329(44)&0.93&0.21&567.62862(61)&0.02&0.12\\\\\n428.79371(28)&0.76&0.37&573.65832(71)&0.12&0.02\\\\\n431.87824(79)&0.81&0.31&574.63842(48)&0.14&1.00\\\\\n447.86614(30)&0.07&0.06&575.61915(43)&0.15&0.97\\\\\n451.79366(39)&0.14&0.00&576.60209(40)&0.17&0.97\\\\\n460.76848(41)&0.28&0.85&577.58372(37)&0.18&0.95\\\\\n462.73257(35)&0.32&0.83&579.54661(33)&0.22&0.92\\\\\n463.71538(66)&0.33&0.82&580.52958(50)&0.23&0.91\\\\\n477.73569(51)&0.56&0.57&593.57155(31)&0.44&0.70\\\\\n484.74744(37)&0.68&0.45&594.55383(40)&0.46&0.69\\\\\n484.74771(71)&0.68&0.46&600.58074(45)&0.56&0.57\\\\\n484.88740(65)&0.68&0.45&787.79453(54)&0.58&0.54\\\\\n485.72961(41)&0.69&0.44&799.85494(46)&0.78&0.35\\\\\n485.72974(50)&0.69&0.44&801.81835(58)&0.81&0.32\\\\\n486.71075(55)&0.71&0.42&813.73962(81)&0.00&0.14\\\\\n486.85131(39)&0.71&0.42&814.72096(43)&0.02&0.12\\\\\n486.85137(40)&0.71&0.42&815.70217(53)&0.03&0.10\\\\\n487.69245(51)&0.72&0.41&815.8433(12)&0.04&0.10\\\\\n487.69267(51)&0.72&0.41&822.85472(77)&0.15&0.99\\\\\n488.81394(44)&0.74&0.39&842.76944(26)&0.47&0.68\\\\\n490.77791(51)&0.77&0.36&842.76957(49)&0.47&0.68\\\\\n503.67970(72)&0.98&0.15&843.74858(55)&0.48&0.65\\\\\n506.62498(99)&0.03&0.10&843.75227(55)&0.48&0.67\\\\\n506.76450(79)&0.03&0.10&843.7524(12)&0.48&0.67\\\\\n508.72870(47)&0.07&0.07&847.67489(53)&0.55&0.58\\\\\n510.69232(55)&0.10&0.04&847.8138(15)&0.55&0.57\\\\\n515.60006(51)&0.18&0.96&848.65653(31)&0.56&0.56\\\\\n528.64310(38)&0.39&0.76&849.77851(33)&0.58&0.55\\\\\n529.62302(21)&0.40&0.73&903.77121(45)&0.45&0.70\\\\\n529.76522(39)&0.41&0.74&904.61311(18)&0.46&0.69\\\\\n530.60621(49)&0.42&0.72&905.59470(33)&0.48&0.67\\\\\n\n\n\n\t\\hline\n\t\\end{tabular}\n\n\n  \n\t\\end{minipage}\n\\end{table}\n\n\n\\section{Analysis}\n\n\\subsection{Orbital, Spin and Beat Ephemerides}\\label{ephem}\n\n\t\\begin{figure}\n\n\t\\includegraphics[width=0.45\\textwidth]{minima-O-C}\n\t\\caption{O$-$C timing residuals for the spin minima as a function of $\\phi_{beat}$. The black dataset represents the new timings which we report in Table~\\ref{spin_timings}, while the gray datapoints are from previously published studies as described in the text. The data are repeated for clarity. Our lack of timings from $0.0 < \\phi_{beat} < 0.5$ is a consequence of the weakness of the spin minima during this half of the beat cycle.}\n\t\\label{minima-timings}\n\t\n\t\\end{figure}\n\nWe used $\\chi^{2}$ minimization to determine the best-fit ephemerides for the spin and orbital periods using our data in conjunction with the published optical eclipse and spin-minima timings in \\citet{patterson}, \\citet{gs97}, \\citet{staubert03}, and \\citet{mukai}. Some of the timings from these studies lacked uncertainties; for those observations, we adopted the average uncertainty of all measurements which did have error estimates. Furthermore, both \\citet{abb06} and \\citet{b12} have made their photometry of V1432 Aql available electronically, and while their time resolution was too low for inclusion in our eclipse analysis, it was adequate for measuring the spin minima. In the interest of uniformity of analysis, we measured the spin minima in the \\citet{abb06} and \\citet{b12} datasets ourselves instead of using their published timings.\\footnote{The original preprint of this paper used the timings reported in \\citet{b12} without reanalyzing their photometry. Using our timing measurements of their spin minima resulted in a significantly lower values of values of $\\chi^{2}_{red}$ for our spin ephemerides in Section~\\ref{spin_ephemerides}.}\n\n\\subsubsection{Orbital Ephemeris}\n\nThe best-fit linear eclipse ephemeris is \\begin{equation} T_{ecl}[HJD] = T_{0, ecl} + P_{orb}E_{ecl},\\end{equation} with $T_{0, ecl} = 2454289.51352 \\pm 0.00004$ and  $P_{orb} = 0.1402347644 \\pm 0.0000000018$ d. Even though our timestamps use the BJD standard, we report our epochs using the slightly less accurate HJD standard because the previously published data use HJD. We find no evidence of a period derivative in the orbital ephemeris, but both \\citet{b12} and \\citet{boyd} have reported quadratic orbital ephemerides. The latter paper had a larger dataset than the one used in this study, so our non-detection of an orbital period derivative does not necessarily contradict those claims.\n\n\\subsubsection{Spin Ephemeris}\\label{spin_ephemerides}\n\nThe spin ephemeris of \\citet{b12} fits our data very well, and we offer only a modestly refined cubic spin ephemeris of \\begin{equation} T_{min, sp}[HJD] = T_{0, sp} + P_{sp,0}E_{sp} + \\frac{\\dot{P}}{2}E^{2}_{sp} + \\frac{\\ddot{P}}{6}E^{3}_{sp},\\end{equation} where $T_{min}$ is the midpoint of the spin minimum, $T_{0, sp} =  2449638.3278 (\\pm 0.001), P_{sp,0} = 0.14062835 (\\pm 0.00000022)$ d, $\\dot{P}\/2 = -8.10 (\\pm 0.10) \\times 10^{-10}$ d cycle$^{-2}$, and $\\ddot{P}\/6 =  -8.5 (\\pm 1.4) \\times 10^{-16}$ d cycle$^{-3}$. The uncertainties on these parameters were determined by bootstrapping the data. We do not have enough observations to meaningfully search for higher-order period derivatives like those reported by \\citet{boyd}, but these values are within the error bounds of those reported by \\citet{b12}. \n\nWhile a polynomial fit accurately models the existing data, $P_{sp}$ will likely approach $P_{orb}$ asymptotically over the synchronization timescale (P. Garnavich, private communication). If this is correct, then $\\dot{P}$ is probably proportional to the difference between $P_{sp}$ and $P_{orb}$ so that \\begin{equation}\\dot{P} \\equiv \\frac{dP_{sp}}{dE_{sp}} = k(P_{sp} - P_{orb}).\\label{pdot_exp}\\end{equation} Integrating the solution to this differential equation yields an ephemeris of \\begin{equation} T_{min, sp} = \\frac{P_{sp, 0} - P_{orb}}{k}(e^{kE_{sp}} - 1) + P_{orb}E_{sp} + T_{0, sp}, \\end{equation} where $P_{orb}$ is the measured value and the three free parameters are $k = -4.205 (\\pm0.008) \\times 10^{-6}$ cycles$^{-1}$, $P_{sp, 0} = 0.14062863 (\\pm0.00000008)$ d, and $T_{0} = 2449638.3277 (\\pm0.0010)$. \n\nAlthough $\\chi^{2}_{red} = 2.9$ for both the cubic ephemeris and the exponential ephemeris, both of these ephemerides  neglect the cyclical shifts in the location of the accretion spot first reported by \\citet{gs97}. To illustrate the effect of these variations on the quality of our fit, Figure~\\ref{minima-timings} plots the residuals from the cubic ephemeris as a function of beat phase. Because this particular variation is not an actual change in the spin period, we did not attempt to incorporate it into our ephemerides. Unless a spin ephemeris were to take into account these variations and their $\\sim$1000-second peak-to-peak amplitude, it would be difficult to achieve a significantly lower $\\chi^{2}_{red}$.\n\nWith this caveat in mind, the comparable values of $\\chi^{2}_{red}$ for each ephemeris lead us to conclude that they model the data equally well as could be expected. Though we use the cubic ephemeris for the sake of simplicity when calculating the beat phase, the exponential spin ephemeris is at least grounded in a physical theory of the resynchronization process. Moreover, in principle, the only parameter which should need to be updated in the future is the constant $k$. By contrast, a polynomial ephemeris could require an ungainly number of terms in order to attain a satisfactory fit.\n\n\\subsubsection{Beat Ephemeris}\n\nBecause there are several non-trivial steps in calculating the system's beat phase, the beat ephemeris is too unwieldy to list here. Nevertheless, to facilitate future studies, we have written a Python script which calculates the system's beat phase at a user-specified Heliocentric Julian Date using the procedure outlined in Appendix~\\ref{beatphase}. Additionally, it calculates future dates at which the system will reach a user-specified beat phase. This script is available for download as supplemental online material and may also be obtained via e-mail from CL.\n\n\\subsubsection{Synchronization Timescale}\n\nAs defined by \\citet{ss91}, a first-order approximation of an asynchronous polar's synchronization timescale is given by \\begin{equation} \\tau_{s} = \\frac{P_{orb} - P_{sp}}{\\dot{P}}.\\label{timescale-formula} \\end{equation} If one assumes rather unrealistically that $\\dot{P}$ will remain constant until resynchronization, this formula provides a very rough estimate of when resynchronization will occur. If Equation~\\ref{pdot_exp} is substituted for $\\dot{P}$ in Equation~\\ref{timescale-formula}, this equation simplifies to $\\tau_{s} = -k^{-1}$. Since $k$ is essentially a decay rate, this formula yields the amount of time necessary for the initial value (in this context, the asynchronism at $T_0$, given by $P_{spin,0} - P_{orb}$) to be reduced by a factor of $e^{-1}$. Because $-k^{-1} = 237700$ spin cycles, $\\tau_{s} = 71.5\\pm0.4$ years with respect to August 2014, so in the year $\\sim$2086, the predicted spin period would be $\\sim$12128.8 seconds, fully 12.5 seconds longer than $P_{orb}$. While this estimate of $\\tau_{s}$ is obviously not an estimate of when resynchronization will actually occur, it is slightly less than the values in \\citet{gs97} and \\citet{abb06} and considerably less than \\citet{staubert03}.\n\nIt is unclear how long an exponential spin ephemeris might remain valid, but if ours were to hold true indefinitely, it predicts that $P_{sp}$ will approach $P_{orb}$ to within one second in the year $\\sim2320$ and to within 0.1 seconds in $\\sim2750$. These are not synchronization timescales as defined by \\citet{ss91}, but in the case of an exponential ephemeris, they provide a more realistic manner of extrapolating when the system might approach resynchronization. The inferred $\\sim$300-year timespan necessary just to attain $P_{sp} - P_{orb} < 1$ seconds is longer than the $\\sim$100-year timescales in \\citet{gs97} and \\citet{abb06}, but it is within the error bounds of the $\\sim$200-year synchronization timescale announced in \\citet{staubert03}. An important disclaimer with these synchronization timescales is that the orbital period may be decreasing, as claimed by both \\citet{b12} and \\citet{boyd}. Since V1432 Aql's WD is spinning up, a decreasing orbital period would presumably lengthen the resynchronization timescale.\n\nIf asynchronous polars do resynchronize asymptotically, it would suggest that a number of supposedly synchronous polars are very slightly asynchronous, with beat periods of months, years, or even decades. Unless they were closely observed for extended periods of time, these polars might be misclassified as being synchronous, so the true fraction of polars which are asynchronous might actually be higher than is currently believed. If correct, this result would be relevant in any examination of the problem of the unreasonably short nova-recurrence time in polars \\citep{warner02}. On one hand, a greater proportion of polars which are asynchronous would imply an even faster recurrence time, but on the other hand, an asymptotic approach to synchronism would also prolong the resynchronization process---and thus, the recurrence time. We leave it to a future work to more fully explicate these matters, but clearly, it will be important to independently confirm our exponential ephemeris, to resolve the possibility of an orbital-period derivative in V1432 Aql, and to determine if the other asynchronous systems also show evidence of asymptotic resynchronization.\n\n\\subsection{Variations in Eclipse O$-$C} \\label{O-C}\n\n\\begin{figure*}\n\n\t\\begin{subfigure}{\n\t\\includegraphics[width=0.5\\textwidth]{eclipse-timings-power}\n\t\\includegraphics[width=0.5\\textwidth]{eclipse-timings-waveform}}\n\t\\end{subfigure}\n\t\n\\caption{From left to right: the power spectrum of the timing residuals of the combined dataset described in section~\\ref{O-C}, and the waveform of the combined dataset when phased at the beat period. Black data points represent our data as listed in Table~\\ref{eclipse_timings}, while gray data points indicate previously published data as described in Section~\\ref{O-C}.}\n\\label{timing}\n\\end{figure*}\n\n\\subsubsection{Periodicity}\n\nIn a conference abstract, \\citet{gs99} first reported the discovery of a 200-second O$-$C shift in V1432 Aql's eclipse timings. We followed up on this periodicity by performing an O$-$C analysis on all eclipse timings listed in Table~\\ref{eclipse_timings}. We calculated both the O$-$C timing residual and the beat cycle count ($C_{beat}$; see Appendix~\\ref{beatphase}) for each eclipse and then used the analysis-of-variance (ANOVA) technique \\citep{anova} to generate several periodograms, with $C_{beat}$ serving as the abscissa.\n\nThe first periodogram used all of the eclipse timings reported in Table~\\ref{eclipse_timings}, and it showed a moderately strong signal at 1.00$\\pm$0.02 cycles per beat period, with the folded eclipse timings exhibiting a sawtooth waveform. We then recalculated the power spectrum after adding previously published optical eclipse timings by \\citet{patterson} and \\citet{watson} to the dataset. \n\n\\begin{table}\n\t\\centering\n\n\t\\begin{minipage}{\\textwidth}\n\t\\caption{Observed Times of Spin Minima\\label{spin_timings}}\n\t\\begin{tabular}{ccc|ccc}\n\t\\hline\n\tBJD\\footnote{2456000+} & $\\phi_{beat}$ & $\\phi_{orb}$ & BJD & $\\phi_{beat}$ & $\\phi_{orb}$\\\\\n\t\\hline\n\n117.8317(18)&0.64&0.55&486.7913(24)&0.69&0.57\\\\\n119.6574(22)&0.67&0.57&486.7925(18)&0.69&0.58\\\\\n121.6249(21)&0.70&0.60&487.7752(21)&0.70&0.58\\\\\n129.7710(26)&0.84&0.69&488.7581(29)&0.72&0.59\\\\\n131.4553(26)&0.87&0.70&490.7272(15)&0.75&0.63\\\\\n132.4406(31)&0.88&0.73&528.6812(16)&0.37&0.28\\\\\n162.6717(14)&0.38&0.3&534.7212(18)&0.47&0.35\\\\\n175.6023(18)&0.59&0.51&539.6384(12)&0.55&0.41\\\\\n180.6599(21)&0.68&0.57&540.6261(45)&0.57&0.46\\\\\n181.6459(22)&0.69&0.61&546.6710(19)&0.66&0.56\\\\\n182.6293(23)&0.71&0.62&549.6188(16)&0.71&0.58\\\\\n194.5668(21)&0.90&0.74&558.6081(23)&0.86&0.69\\\\\n431.8292(22)&0.79&0.64&560.5729(23)&0.89&0.70\\\\\n484.6855(25)&0.65&0.55&593.6136(18)&0.43&0.31\\\\\n484.8263(22)&0.66&0.55&594.5979(16)&0.44&0.32\\\\\n485.6698(17)&0.67&0.57&607.5310(19)&0.65&0.55\\\\\n485.8085(21)&0.67&0.56&&&\\\\\n\n\n\t\\hline\n\t\\end{tabular}\n\n\\end{minipage}\n\n\\end{table} \n\nThe combined dataset consists of 133 measurements spanning a total of 138 beat cycles.  The strongest signal is at the beat period (1.001$\\pm$0.002 cycles per beat period), and its waveform consists of an abrupt 240-second shift in the timing variations near $\\phi_{beat}\\sim0.5$, which is when the residual eclipse flux is strongest (see Section~\\ref{flux-periodicity}). Both the periodogram and waveform are shown in Figure~\\ref{timing}. Between $\\sim0.5 < \\phi_{beat} < \\sim 0.85$, the eclipses occur $\\sim$120 seconds early, but after $\\phi_{beat} \\sim 0.85$, the eclipses begin occurring later, and by $\\phi_{beat} \\sim 1.0$, the eclipses are occurring $\\sim$120 seconds late. Although the 240-second O$-$C jump at $\\phi_{beat}\\sim0.5$ is the most obvious feature in the O$-$C plot, there is a 120-second jump towards earlier eclipses at $\\phi_{beat}\\sim0.0$. Considering the gradual eclipse ingresses and egresses, the WD must be surrounded by an extended emission region, so these eclipse timings track the centroid of emission rather than the actual position of the WD.\n\n\\subsubsection{Description of Model} \\label{description_of_model}\n\nGiven the asynchronous nature of the system and the ability of the stream to travel most of the way around the WD \\citep{mukai}, we hypothesize that cyclical changes in the location of the threading region are responsible for the O$-$C variation. In an asynchronous system, the position of the threading region can vary because the WD rotates with respect to the accretion stream, causing the amount of magnetic pressure at a given point along the stream to vary during the beat period. Threading occurs when the magnetic pressure ($\\propto r^{-6}$) balances the stream's ram pressure  ($\\propto v^{2}$). For a magnetic dipole, the magnetic flux density $B$ has a radial dependence of $\\propto r^{-3}$, but with an additional dependence on the magnetic latitude; the magnetic pressure will be even greater by a factor of 4 near a magnetic pole as opposed to the magnetic equator. An additional consideration is that the stream's diameter is large enough that the magnetic pressure varies appreciably across the stream's cross section \\citep{mukai88}.\n\nKM modeled this scenario using a program which predicts times of eclipse ingresses and egresses of a point given its $x, y$, and $z$ coordinates within the corotating frame of the binary. The physical parameters used in the program are $P_{orb} = 3.365664$ h (measured), $M_{WD}$ = 0.88M$_{\\odot}$, $M_{donor}$ = 0.31M$_{\\odot}$, $R_{donor}$ = $2.47 \\times 10^{10}$ cm, $i = 76.8^{\\circ}$, and binary separation $a = 8.4 \\times 10^{10}$ cm \\citep{mukai}. The code treats the donor star as a sphere for simplicity, but since we do not attempt to comprehensively model the system in this paper, the errors introduced by this approximation should be minimal. For instance, as a result of this approximation, we had to decrease $i$ by 0.9$^{\\circ}$ compared to the value from \\citet{mukai} in order to reproduce the observed eclipse length.\n\nWe first calculated the ballistic trajectory of the accretion stream and arbitrarily selected four candidate threading regions along the stream (P1, P2, P3, and P4) under the assumption that the stream will follow its ballistic trajectory until captured by the magnetic field \\citep{mukai88}. The eclipse-prediction program then returned the phases of ingress and egress for each of the four points given their $x$ and $y$ coordinates within the corotating frame of the binary. We selected these four points arbitrarily in order to demonstrate the effects that a changing threading region would have on eclipse O$-$C timings; we do not claim that threading necessarily occurs at these positions or that this process is confined to a discrete point in the $x,y$ plane. Figure~\\ref{model} shows a schematic diagram of this model.\n\n\\begin{figure}\n\n\t\\includegraphics[width=0.5\\textwidth]{model}\n\t\\caption{A schematic diagram of the system as used in our model, viewed from above the binary rest frame. The WD is rest at the origin, and the black curved line is the accretion stream trajectory, which originates at the L1 point near the right edge of the diagram. P1, P2, P3, and P4 are illustrative threading regions, and the cross indicates the location of the stream's closest approach to the WD. Since $P_{sp} > P_{orb}$, the WD rotates clockwise in this figure.}\n\t\\label{model}\n\t\n\t\\end{figure}\n\n\\begin{figure*}\n\n\t\\includegraphics[width=1\\textwidth]{diagram}\n\t\n\\caption{A sketch indicating the general positions of the accretion spots at different beat phases as seen from the donor star. The black crosses represent accretion spots visible from the donor, and the vertical line is the WD's spin axis. Section~\\ref{orientation} explains how we inferred the positions of the two magnetic poles.}\n\\label{diagram}\n\\end{figure*}\n\nOnce threading occurs, the captured material will follow the WD's magnetic field lines until it accretes onto the WD. To simulate the magnetically channeled portion of the stream, we assumed that captured material travels in a straight line in the $x,y$ plane from the threading region to the WD while curving in the $z$ direction, where $z$ is the elevation above or below the $x,y$ plane. This is another simplification since the magnetic portion of the stream might be curved in the $x,y$ plane, but presumably, this approximation is reasonable. Since the magnetic field lines will lift the captured material out of the orbital plane, we calculated the $x,y$ coordinates of the midpoint between each threading region and the WD and computed its ingress and egress phases at several different values of $z$. We reiterate that this is not a comprehensive model, but as we explain shortly, it is sufficiently robust to offer an explanation for the observed O$-$C variations.\n\n\\subsubsection{Orientation of the Poles} \\label{orientation}\n\nBefore this model is applied to the observations, it is helpful to determine the orientations of the poles at different points in the beat cycle. We assume that there are two magnetic poles which are roughly opposite each other on the WD \\citep{mukai}. Since $i \\neq 90^{\\circ}$, one hemisphere of the WD is preferentially tilted toward Earth, and we refer to the magnetic pole in that hemisphere as the upper pole. The lower pole is the magnetic pole in the hemisphere which is less favorably viewed from Earth. In isolation, our observations do not unambiguously distinguish between these two poles, but since the midpoint of the spin minimum ({\\it i.e.},  $\\phi_{sp} = 0.0$) corresponds with the transit of the accretion region across the meridian of the WD \\citep[e.g.][]{staubert03}, we can estimate when the poles face the donor star. When $\\phi_{beat} \\sim 0.15$, the spin minimum coincides with the orbital eclipse, so one of the poles is approximately oriented towards the secondary at that beat phase. At $\\phi_{beat} \\sim 0.65$, the spin minimum occurs at an orbital phase of $\\sim$0.5, indicating that this pole is roughly facing the P3 region at that beat phase. But the question remains: Is this the upper pole, or the lower one?\n\n\\citet{mukai} relied upon X-ray observations of eclipse ingresses and egresses to differentiate between the upper and lower poles (see their Figure~15 and the accompanying text). While the accretion spots have not been identified in optical photometry, they are the system's dominant X-ray sources, so they produce steep, rapid X-ray eclipses \\citep{mukai}. The authors took advantage of the fact that since $P_{sp} > P_{orb}$,  the accretion spots will increasingly lag  behind the orbital motion of the donor star with each subsequent orbit. Consequently, the orientation of the accretion regions with respect to the donor star will continuously change across the beat cycle. When viewed throughout the beat period at the phase of eclipse, the accretion spots appear to slowly move across the face of the WD, thereby causing detectable changes in the times of X-ray ingress and egress. \n\nCritically, at some point during the beat cycle, each accretion region will have rotated out of view at the phase of eclipse, resulting in a jump in either the ingress or egress timings, depending on which pole has disappeared. The \\citet{mukai} model predicts that when the upper pole is aimed in the general direction of P4, the X-ray egresses will undergo a shift to later phases as the upper polecap rotates behind the left limb of the WD as seen at egress (see their Figure~15). Likewise, the disappearance of the lower pole behind the left limb at the phase of ingress results in a shift toward later phases in the ingress timings. Based on data in Table~5 of \\citet{mukai}, the egress jump occurs near $\\phi_{beat}\\sim0.9$, so at that beat phase, the upper pole should be pointed toward the P3-P4 region. The egress jump is more distinct than the ingress jump, so we base our identification of the poles on the egress jump only.\n\nOur identification of the upper and lower poles is an inference and should not be viewed as a definitive claim. For our method to be reliable, it would be necessary for the accretion geometry to repeat itself almost perfectly in both 1998 (when \\citet{mukai} observed) and the 28-month span from 2012-2014 when we observed V1432 Aql. Even though the accretion geometry does seem to repeat itself on a timescale of two decades (see, {\\it e.g.}, Section~\\ref{spin}), this may not always be the case, as is evidenced by an apparent discontinuity in the timings of the of the spin minima in 2002 \\citep{boyd}. If the accretion rate during our observations was different than it was in 1998, there would be changes in the location and size of the X-ray-emitting accretion regions \\citep{mukai88}. Moreover, \\citet{mukai} cautioned that their model was a simplification because the accretion geometry was poorly constrained. For example, they noted that their model did not account for the offset between the accretion region and the corresponding magnetic pole.\n\nIf the upper pole is aimed towards P3-P4 near $\\phi_{beat}\\sim0.9$, then the upper pole would face the donor at $\\phi_{beat} \\sim 0.65$ since the WD appears to rotate clockwise as seen from the donor. Thus, the lower pole is likely pointed in the general direction of the donor star near $\\phi_{beat} \\sim 0.15$. We provide a sketch of the system in Figure~\\ref{diagram} which shows the inferred positions of the polecaps throughout the beat cycle. \n\n\n\\subsubsection{Application of Model}\\label{application_of_model}\n\n\n\\begin{figure}\n\n\t\n\t\\includegraphics[width=0.45\\textwidth]{krizmanich-eclipse1}\n\t\\includegraphics[width=0.45\\textwidth]{krizmanich-eclipse2}\n\t\n\\caption{Two eclipses observed on consecutive nights with the 80-cm Krizmanich Telescope. Note the different vertical scale for the two panels. The vertical dashed lines indicate the expected phases of the WD's ingress and egress. On the first night (Panel A), the eclipse is very deep and begins with the WD's disappearance, but on the second night (Panel B), the eclipse starts before the occultation of the WD. These light curves are consistent with the appearance of a new threading region near P3-P4 in our model, indicating that this process requires less than 24 hours to take place.}\n\\label{shift}\n\\end{figure}\n\nEven though the four P points were arbitrarily selected, the results of the eclipse-timing program provide testable predictions concerning the O$-$C variations. In our model, the emission from the accretion curtain and the threading region result in a moving centroid which is responsible for an O$-$C shift with a half-amplitude of about $\\pm$120 seconds (see Fig.~\\ref{O-C}). When the centroid of emission is in the $+y$ region in Figure~\\ref{model}, the O$-$C would be positive, and if it were in the $-y$ half of the plot, the O$-$C would be negative. According to calculations using the model, eclipses of point sources at P1, P2, P3, and P4 would result in O$-$C values of 289 seconds, 204 seconds, 0 seconds, and $-$533 seconds, respectively. As for the midpoints between each of those four points and the WD, the O$-$C values would be 122 seconds, 103 seconds, 0 seconds, and $-$289 seconds for the P1, P2, P3, and P4 midpoints, respectively. The O$-$C values for the midpoints have a negligible dependence on the height above the orbital plane (provided that the secondary can still eclipse that point). Since the actual O$-$C variation does not exceed $\\pm$120 seconds, it is clear that the actual O$-$C timings are inconsistent with a centroid near P1, P2, and P4. However, centroids near the midpoints for P1, P2, and P3 would be consistent with the observed O$-$C timings.\n\nIt makes sense that the centroid of the emission region would have a less dramatic O$-$C value than the candidate threading points. Because we expect that the magnetically-channeled part of the stream travels from the threading region to the WD, the light from this accretion curtain would shift the projected centroid of emission towards the WD. In addition, since the threading region likely subtends a wide azimuthal range, the ability of the projected centroid to deviate dramatically from the WD's position would be limited. With these considerations in mind, the consistency of the theoretical O$-$C values for the P1, P2, and P3 midpoints with the observed O$-$C variations indicates that our model offers a plausible explanation of the O$-$C timings.\n\nThe sudden jump to early eclipses near $\\phi_{beat} \\sim 0.5$ occurs when the inferred orientation of the lower pole is toward the general direction of P3-P4. We surmise that the increased magnetic pressure on that part of the stream is able to balance the decreasing ram pressure, resulting in a luminous threading region. Since the P3-P4 vicinity is in the $-y$ half of Figure~\\ref{model}, an emission region there would result in an early ingress. In all likelihood, the centroid of that threading region does not approach P4 or its midpoint because the theoretical O$-$C values do not agree with the observed values. However, a centroid closer to P3 would result in a less-early eclipse which would be more consistent with the observations.\n\nAs the WD slowly rotates clockwise in Figure~\\ref{model}, the corresponding changes in the magnetic pressure along the stream's ballistic trajectory would move the position of the threading region within the binary rest frame, and the eclipses would gradually shift to later phases. Half a beat cycle after the  $\\phi_{beat} \\sim 0.5$ jump in O$-$C timings, the lower pole would be oriented in the general direction of P2 and the upper pole towards P4. As the upper pole's magnetic pressure increases on the stream in the P3-P4 vicinity, a new threading region would form there, producing the O$-$C jump observed near $\\phi_{beat} \\sim 0.0$. In short, our model predicts the two distinct O$-$C jumps and explains why they are from late eclipses to earlier eclipses.\n\nOur observations provide circumstantial evidence of the brief, simultaneous presence of two separate emission regions as the system undergoes its O$-$C jump near $\\phi_{beat} \\sim 0.5$ during one beat cycle in July 2014. On JD 2456842, less than one day before the O$-$C jump, the time of minimum eclipse flux had an O$-$C of $\\sim$140 seconds, but on the very next night, there were two distinct minima within the same eclipse. Separated by a prominent increase in brightness, one minimum had an O$-$C of $-80$ seconds, while the other had an O$-$C of 240 seconds, consistent with the presence of discrete emission regions in the $-y$ and $+y$ halves of the plot in Figure~\\ref{model}. Moreover, assuming a WD eclipse duration of 700 seconds \\citep{mukai} centered upon orbital phase 0.0, the optical eclipse on the first night commenced when the donor occulted the WD, implying a lack of emission in the $-y$ region. However, the egress of that eclipse continued well after the reappearance of the WD, as one would expect if there were considerable emission in the $+y$ area. Indeed, a centroid of emission near the P1 midpoint would account for the observed O$-$C value. On the ensuing night, by contrast, the eclipse began before the disappearance of the WD, and ended almost exactly when the WD reappeared. The implication of these two light curves is that within a 24-hour span between $\\phi_{beat} \\sim 0.47-0.48$, the locations of the emission regions changed dramatically. Figure~\\ref{shift} shows these light curves and indicates in both of them the times of anticipated WD ingress and egress. Further observations are necessary to determine whether this behavior recurs during each beat cycle.\n\n\\begin{figure*}\n\n\t\\begin{subfigure}{\n\t\\includegraphics[width=0.5\\textwidth]{flux-power}\n\t\\includegraphics[width=0.5\\textwidth]{flux-waveform}}\n\t\\end{subfigure}\n\t\n\\caption{The power spectrum of the residual flux and a phase plot showing the waveform of the signal at the beat period. Spanning 11.8 beat cycles, these plots use only the observations made with the 28-cm Notre Dame telescope. The double-wave sinusoid in the phase plot is meant to assist with visualizing the data and does not represent an actual theoretical model of the system. }\n\\label{eclipses}\n\\end{figure*}\n\n\\subsubsection{Implications of Findings}\n\nOur hypothesis that the location of the threading radius is variable has ramifications for previous works. In particular, \\citet{gs97} and \\citet{staubert03} used the timing residuals of the spin minima to track the accretion spot as it traced an ellipse around one of the magnetic poles. One of their assumptions was that the threading radius is constant, but this is inconsistent with the conclusions we infer from our observations and model of the system. A variable threading radius would change the size and shape of the path of the accretion spot \\citep{mukai88}---and therefore, of the waveform of the spin minima timings used in those studies to constrain the accretion geometry.\n\nAdditionally, the agreement between the model and our observations provides compelling evidence which substantiates previous claims (see Section~\\ref{intro}) that the accretion stream in V1432 Aql is able to travel around the WD, as is also observed in the other asynchronous polars. The inefficient threading in asynchronous systems could be indicative of a relatively weak magnetic field or a high mass-transfer rate. For example, \\citet{schwarz} found that if the accretion rate in the asynchronous polar BY Cam were 10-20 times higher than normal accretion rates in polars, the stream could punch deeply enough into the WD's magnetosphere to reproduce the observed azimuthal extent of the accretion curtain. Although it is at least conceivable that the asynchronism itself causes the inefficient threading, it is not immediately apparent why this would be so when $P_{sp}$ and $P_{orb}$ are so close to each other.\n\nRegarding the possibility of a high mass-transfer rate, previous works \\citep[e.g.,][]{kps88} have proposed that irradiation by a nova can temporarily induce an elevated mass-transfer rate which persists for many decades after the eruption has ended. In line with this theory, \\citet{bklyn} proposed that CVs with consistently elevated mass-transfer rates---specifically, nova-like and ER UMa systems---exist fleetingly while the donor star cools after having been extensively irradiated by a nova. If all asynchronous polars are recent novae, as is commonly believed, this theory would predict that the same nova which desynchronizes the system also triggers a sustained, heightened mass-transfer rate as a result of irradiation. The increased ram pressure of the accretion stream would enable it to penetrate deeply into the WD's magnetosphere, thereby offering a plausible explanation as to why all four confirmed asynchronous polars show strong observational evidence of inefficient threading. However, this would not resolve the problem of the short nova-recurrence time in polars \\citep[][ discussed in Section~\\ref{intro}]{warner02}.\n\n\\subsection{Variations in the Residual Eclipse Flux}\n\n\\subsubsection{Periodicity} \\label{flux-periodicity}\n\nThe WD is invisible during eclipse, leaving two possible causes for the variation in residual eclipse flux: the donor star and the accretion stream. The magnetic field lines of the WD can carry captured material above the orbital plane of the system, so depending on projection effects, some of the accretion flow could remain visible throughout the WD's eclipse. Therefore, as the accretion flow threads onto different magnetic field lines throughout the beat period, the resulting variations in the accretion flow's trajectory could cause the residual eclipse flux to vary as a function of $\\phi_{beat}$. \n\nAfter we calculated the beat cycle count ($C_{beat}$) for each eclipse observation, we generated a power spectrum using the ANOVA method with $C_{beat}$ as the abscissa and the minimum magnitude as the ordinate. For this particular periodogram, we used only the 71 eclipses observed with the 28-cm Notre Dame telescope due to the difficulty of combining unfiltered data obtained with different equipment. The strongest signal in the resulting power spectrum has a frequency of $0.998 \\pm0.012$ cycles per beat period. Figure~\\ref{eclipses} shows both the periodogram and the corresponding phase plot, with two unequal maxima per beat cycle.\n\nWhile a double-wave sinusoid provides an excellent overall fit to the residual-flux variations, the observed mid-eclipse magnitude deviated strongly from the double sinusoid near $\\phi_{beat} \\sim 0.47$ in at least three beat cycles.\\footnote{While there are sporadic departures from the double-sinusoid, none is as dramatic as the behavior near $\\phi_{beat} \\sim 0.47$ or shows evidence of persistence across multiple beat cycles.} Two eclipses observed on consecutive nights in high-cadence photometry with the 80-cm Krizmanich telescope provide the best example of this variation. On JD 2456842, the system plummeted to $V\\sim17.8$ during an eclipse ($\\phi_{beat} = 0.469$) near the expected time of maximum residual flux. But just 24 hours later, the mid-eclipse magnitude had surged to $V\\sim16.2$ ($\\phi_{beat} = 0.485$), which was the approximate brightness predicted by the double-sinusoid fit. Furthermore, the eclipse light curve from the second night exhibited intricate structure which had not been present during the previous night's eclipse. These light curves were shown in Figure~\\ref{shift}. Comparably deep eclipses near $\\phi_{beat}\\sim0.47$ were observed during two additional beat cycles (one in 2013 and another in 2014), so there is at least some evidence that the residual flux might be consistently lower near this beat phase. Unfortunately, gaps in our data coverage make it impossible to ascertain whether the mid-eclipse magnitude always fluctuates near $\\phi_{beat} \\sim 0.47$, so confirmation of this enigmatic variation is necessary.\n\n\\subsubsection{Application of Model}\n\nWe propose that the overall variation in mid-eclipse flux is the signature of an accretion curtain whose vertical extent varies as a function of the threading radius. When the threading region is farther from the WD, the stream can couple onto magnetic field lines which achieve such a high altitude above the orbital plane that the donor star cannot fully eclipse them. By contrast, when the threading region is closer to the WD, the corresponding magnetic field lines are more compact, producing a smaller accretion curtain which the donor occults more fully. The schematic diagram in Figure~\\ref{flux-diagram} offers a visualization of this scenario.\n\nWhile it is conceivable that the residual flux variation is caused by material within the orbital plane, the available evidence disfavors this possibility. In particular, \\citet{ss01} saw no diminution in the strength of high-excitation UV emission lines during an eclipse with considerable residual flux at $\\phi_{beat} = 0.58$. If these emission lines originated within the orbital plane, they would have faded during the eclipse. Furthermore, if the source of the residual flux were in the orbital plane, the eclipse width would likely correlate with the mid-eclipse magnitude. The eclipses with high levels of residual flux would be long, while the deeper eclipses would be short. We do not see this pattern in our data, and Figure~12 in \\citet{boyd} does not show such a correlation, either.\n\n\\begin{figure}\n\n\t\\centerline{\\includegraphics[width=0.45\\textwidth]{flux-sketch-1}}\n\t\\par\n\t\\centerline{\\includegraphics[width=0.45\\textwidth]{flux-sketch-2}}\t\n\n\\caption{Two schematic diagrams providing a simplified illustration of our explanation for the residual flux variations at mid-eclipse. In both panels, the captured material travels in both directions along an illustrative magnetic field line. The secondary is the gray sphere eclipsing the WD, and the threading point is shown as a large $+$. The inclination of the magnetic axis with respect to the rotational axis was arbitrarily chosen as 30$^{\\circ}$. The portion of the magnetic stream which travels upward and which is visible at mideclipse is highlighted. The threading point in Panel A is near P4, and its threading radius is 3.6 times larger than that of the threading point in Panel B, when the threading point is near the stream's closest approach to the WD.}\n\n\\label{flux-diagram}\n\\end{figure}\n\nOur model from Section~\\ref{description_of_model} predicts that the threading radius will vary by a factor of $\\sim3.6$ between P4 and the stream's point of closest approach to the WD. (We reiterate that since these points are meant to be illustrative, this is not necessarily the actual variation in the threading radius.) The upshot is that at P3, threading would take place significantly deeper in the WD's magnetosphere than it would at P4. Moreover, since the predicted threading radius would be largest near an O$-$C jump, this hypothesis predicts that the amount of residual flux would be greatest near those jumps and lowest between them, as is observed in a comparison of Figures~\\ref{timing}~and~\\ref{eclipses}. In the case of a magnetic stream originating from a threading region between P2-P4, the midpoint of the stream would be visible if it achieves a minimum altitude of $z \\sim 0.08a$ above the orbital plane, where $a$ is the binary separation. At P4, this is only one-quarter the predicted threading radius, but at P2 and P3, this is three-quarters of the predicted threading radius. \n\nThis hypothesis also explains why some spectra of V1432 Aql during mid-eclipse show intense emission lines \\citep[e.g.][]{watson, ss01}, while others show only weak emission \\citep[e.g.][]{patterson}. For each of these previously published spectroscopic observations, we calculated $\\phi_{beat}$ and found that the ones showing strong emission lines were obtained when the predicted residual flux was near one of its maxima in Figure~\\ref{eclipses}. By contrast, the spectra containing weak emission were obtained when the expected residual flux was approaching one of its minima. If our hypothesis is correct, then the variation in the emission lines is simply the result of the changing visibility of the accretion curtain during eclipse. \\citet{watson} suggested a somewhat related scenario to account for the presence of emission lines throughout the eclipse, but they disfavored this possibility largely because of the apparent residual flux at X-ray wavelengths. (As mentioned previously, \\citet{mukai} later demonstrated that the residual X-ray flux was contamination from a nearby galaxy.)\n\nAn excellent way to test our theory would be to obtain Doppler tomograms near the times of maximum and minimum residual eclipse flux. \\citet{schwarz} showed that this technique is capable of revealing the azimuthal extent of the accretion curtain in BY Cam, and it would likely prove to be equally effective with V1432 Aql.\n\nWe do not have enough data to consider why the residual flux can vary by as much as $\\sim$1.5 mag in one day near the expected time of maximum residual flux. Knowing whether the residual flux is always low near $\\phi_{beat} = 0.47$ would be a necessary first step in this analysis.\n\n\\subsection{The Dependence of the Spin Modulation on Beat Phase}\\label{spin}\n\nAs the WD slowly spins with respect to the secondary, the accretion stream will couple to different magnetic field lines, meaning that the spin modulation will gradually change throughout the beat cycle. To explore this variation, we constructed non-overlapping, binned phase plots of the spin modulation in ten equal segments of the beat cycle ({\\it e.g.}, between $0.00 < \\phi_{beat} < 0.10$). As with the residual-eclipse-flux measurements, we used only the data obtained with the Notre Dame 28-cm telescope in order to avoid errors stemming from the different unfiltered spectral responses of multiple telescope-CCD combinations. In an effort to prevent eclipse observations from contaminating the spin modulation, we excluded all observations obtained between orbital phases 0.94 and 1.06. We then calculated the beat phase for all remaining observations and used only those observations which fell into the desired segment of the beat cycle. We used a bin width of 0.01$P_{sp}$, and we did not calculate bins if they consisted of fewer than five individual observations.\n\nFigure~\\ref{spin-waveform} shows these ten phase plots, and several features are particularly striking. For example, the spin minimum near spin phase 0.0 is highly variable. Conspicuous between $0.5 < \\phi_{beat} < 1.0$, it becomes feeble and ill-defined for most of the other half of the beat cycle. Sometimes, the spin minimum is quite smooth and symmetric, as it is between $0.7 < \\phi_{beat} < 0.8$, but it is highly asymmetric in other parts of the beat cycle, such as $0.5 < \\phi_{beat} < 0.6$. Additionally, there is a striking difference between the phase plots immediately before and after the O$-$C jump near $\\phi_{beat} \\sim 0.5$, as one would expect if the O$-$C jump marks a drastic change in the accretion geometry.\n\nThere is also a stable photometric maximum near spin phase $\\sim0.6$ which is visible for most of the beat cycle, though its strength is quite variable. We refer to this feature as the primary spin maximum, but it is not as prominent as the spin minimum. Its behavior is unremarkable.\n\n\\begin{figure*}\n\t\\centering\n  \t\\begin{tabular}{cc}\n    \\includegraphics[width=.5\\textwidth]{spin-phase-05.eps} &\n    \\includegraphics[width=.5\\textwidth]{spin-phase-55.eps} \\\\\n    \\includegraphics[width=.5\\textwidth]{spin-phase-15.eps} &\n    \\includegraphics[width=.5\\textwidth]{spin-phase-65.eps}   \\\\\n    \\includegraphics[width=.5\\textwidth]{spin-phase-25.eps} &\n    \\includegraphics[width=.5\\textwidth]{spin-phase-75.eps} \\\\\n    \\includegraphics[width=.5\\textwidth]{spin-phase-35.eps} &\n    \\includegraphics[width=.5\\textwidth]{spin-phase-85.eps}   \\\\\n    \\includegraphics[width=.5\\textwidth]{spin-phase-45.eps} &\n    \\includegraphics[width=.5\\textwidth]{spin-phase-95.eps} \\\\\n \\end{tabular}\n \\caption{Binned phase plots of the spin modulation at different beat phases, with each bin representing 0.01 spin cycles. Gaps in the light curves are due to eclipses. The second spin maximum ($\\phi_{sp}\\sim0.4$) is strongest in panel C.}\n\\label{spin-waveform}\n\\end{figure*}\n\nInterestingly, there is another, much stronger photometric maximum  at $\\phi_{sp} \\sim 0.4$ which is visible only between $0.0 < \\phi_{beat} < 0.5$. Since this feature shares the WD's spin period, we refer to it as the second spin maximum. The second spin maximum can be exceptionally prominent in photometry, attaining a peak brightness of $V \\sim 14.1$ in several of our light curves---which is the brightest that we have observed V1432 Aql to be. When visible, the second spin maximum precedes the primary spin maximum by $\\sim 0.2$ phase units. It begins to emerge near  $\\phi_{beat} \\sim 0.0$, and gradually strengthens until it peaks between between $0.2 < \\phi_{beat} < 0.3$. It then weakens considerably as $\\phi_{beat}$ approaches 0.5, and after the O$-$C jump near $\\phi_{beat} \\sim 0.5$, the second spin maximum is replaced by a dip in the light curve.\n\nAlthough the second spin maximum consistently appears between $0.0 < \\phi_{beat} < 0.5$, it vanished in a matter of hours on JD 2456842 ($\\phi_{beat} \\sim 0.47$), only to reappear the next night. On the first night, our observations covered two spin cycles, and while the second spin maximum was obvious in the first cycle, it had disappeared by the second. Just 24 hours later, it was again visible in two successive spin cycles. This unexpected behavior coincides with the approximate beat phase at which we would expect the dominant threading region to shift to the P3-P4 region in our model. Nevertheless, our lack of observations near this beat phase precludes a more rigorous examination of this particular variation.\n\nThe second spin maximum is very apparent in some previously published light curves of V1432 Aql from as far back as two decades ago. For example, \\citet{watson} presented light curves of V1432 Aql obtained in 1993 which showcase the gradual growth of the second spin maximum (see Panels B-G of their Figure~2). Using our method of determining the beat phase, we extrapolate a beat phase of 0.96 for the light curve shown in their Panel B and a beat phase of 0.12 for the light curve in their Panel G. The increasing strength of the second spin maximum in their light curves agrees with the behavior that we observed at those beat phases (see our Figure~\\ref{spin-waveform}). Likewise, Figure~1 in \\citet{patterson} shows the second spin maximum at the expected beat phases. These considerations suggest that the second spin maximum is a stable, recurring feature in optical photometry of V1432 Aql.\n\nThe overall predictability of the second spin maximum does not answer the more fundamental question of what causes it. One possibility is that it is the result of an elevated accretion rate on one pole for half of the beat cycle. The apparent gap between the two spin maxima, therefore, might simply be the consequence of an absorption dip superimposed on the photometric maximum or a cyclotron beaming effect, splitting the spin maximum into two.\n\nA more interesting scenario is that the second spin maximum could be the optical counterpart to the possible third polecap detected by \\citet{rana} in X-ray and polarimetric data. In that study, \\citet{rana} detected three distinct maxima in X-ray light curves as well as negative circular polarization at spin phase 0.45, which is the approximate spin phase of the second spin maximum in optical photometry. They also measured positive circular polarization at spin phases 0.1 and 0.7, which correspond with the spin minimum and the primary spin maximum, respectively. Quite fortuitously, the authors obtained their polarimetric observations within several days of the photometric detection of the second spin maximum by \\citet{patterson}. Thus, it is reasonable to conclude that the circular polarization feature near spin phase 0.45 is related to the second spin maximum, consistent with a third accreting polecap. \n\nThe conclusions of \\citet{rana}, coupled with our identification of a second spin maximum, suggest that V1432 Aql might have at least three accreting polecaps---and therefore, a complex magnetic field. However, the available evidence is inconclusive, and follow-up polarimetry across the beat cycle could clarify the ambiguity concerning the WD's magnetic field structure.\n\n\\section{Conclusion}\n\nWe have presented the results of a two-year photometric study of V1432 Aql's beat cycle. We have confirmed and analyzed the eclipse O$-$C variations first reported by \\citet{gs99}, and we found that the residual mid-eclipse flux is modulated at the system's beat period. We interpret these variations as evidence that the threading region's location within the binary rest frame varies appreciably as a function of beat phase. Doppler tomography of the system at different beat phases could reveal any changes in the azimuthal extent of the accretion curtain, thereby providing a direct observational test of our model of the system.\n\nOur observations provide circumstantial evidence that the mid-eclipse magnitude undergoes high-amplitude variations on a timescale of less than a day near $\\phi_{beat} \\sim0.47$, deviating strongly from the expected brightness at that beat phase. In the most remarkable example of this variation, the mid-eclipse magnitude varied by $\\sim$1.5 mag in two eclipses observed just 24 hours apart. Whereas the first eclipse was deep and smooth, the second eclipse was shallow and W-shaped, with two distinct minima. Similar variations in residual flux were observed in two other beat cycles, providing at least some evidence that this behavior might be recurrent. Still, additional photometric observations are necessary to confirm the $\\phi_{beat}\\sim0.47$ fluctuations in mid-eclipse magnitude. Amateur astronomers are ideally suited to undertake such an investigation, especially when one considers that our residual-flux analysis utilized a small telescope and commercially available CCD camera. Moreover, observers with larger telescopes could also obtain relatively high-cadence photometry to study whether double-minima eclipses consistently appear near this beat phase.\n\nIn addition, we report a second photometric spin maximum which appears for only about half of the beat cycle. This phenomenon might be evidence of a complex magnetic field, but a careful polarimetric study of the beat cycle would be necessary to investigate this possibility in additional detail.\n\nWe also offer updated ephemerides of the orbital and spin periods (see Sec.~\\ref{ephem}), as well as a Python script which calculates V1432 Aql's beat phase at a given time and which also predicts when the system will reach a user-specified beat phase. An exponential spin ephemeris models the data as well as a polynomial ephemeris and is consistent with an asymptotic approach of the spin period toward the orbital period. According to the exponential ephemeris, the rate of change of the spin period is proportional to the level of asynchronism in the system; consequently, if the exponential ephemeris were to remain valid indefinitely, the resynchronization process in V1432 Aql would take considerably longer than previous estimates.\n\nFinally, while a comprehensive theoretical model of V1432 Aql is beyond the scope of this paper, such an analysis could refine our description of the system and shed additional light on V1432 Aql's unusual threading mechanisms.\n\n\n\\section*{Acknowledgments}\n\nWe thank Peter Garnavich and Joe Patterson for their helpful comments, as well as the anonymous referee, whose suggestions greatly improved the paper.\n\nThis study made use of observations in the AAVSO International Database, which consists of variable star observations contributed by a worldwide network of observers.\n\nThe Sarah L. Krizmanich Telescope was generously donated to the University of Notre Dame in memory of its namesake by the Krizmanich family. This is the first publication to make use of data obtained with this instrument.\n\nDB, MC, and JU participate in the Center for Backyard Astrophysics collaboration, which utilizes a global team of professional and amateur astronomers to study cataclysmic variable stars.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStatistical models with short range interactions on two-dimensional (2D) regular lattices exhibit no spontaneously symmetry \nbreaking at finite temperature, if the symmetry in local degrees of freedom is continuous~\\cite{Mermin_Wagner}. \nThe classical ferromagnetic XY model \nis a typical example, which has $O(2)$ symmetry, where the thermal average of the magnetization is zero at finite temperature. \nAn introduction of discrete nature to local degrees of freedom then induces an order-disorder transition in low temperature,\nwhere the universality class is dependent on the type of discretization. \nThe $q$-state clock model, which has $Z_q^{~}$ symmetry, is a well-known discrete analogue of the XY model. \nFor the case of $q \\le 4$, the clock model exhibits a second-order phase transition described by unitary minimal series of conformal field theory (CFT). \nIf $q > 4 $, the clock model has an intermediate critical phase between the high-temperature disordered phase and low-temperature ordered phase~\\cite{Elitzur, Nomura, Ortiz, Kumano}, where transitions to the critical phase are of Berezinskii-Kosterlitz-Thouless (BKT) type~\\cite{B1, B2, KT}. \nAs $q$ increases, the low-temperature ordered phase shrinks, and the $O(2)$ symmetry is finally recovered \nin the limit $q \\rightarrow \\infty$.\n\nDiscretization of the classical Heisenberg model, which has $O(3)$ symmetry, \nis not straightforward, in the sense that there is no established route of \ntaking continuous-symmetry limit. A possible manner of discretization is to introduce the polyhedral anisotropies, such as tetrahedral, cubic, \noctahedral, icosahedral, and dodecahedral ones, which correspond to the discrete subgroups of the $O(3)$ symmetry group. \nLet us consider the discrete vector-spin models, where on each lattice site there is a unit \nvector spin that can point to vertices of a polyhedron. The tetrahedron model \ncan be mapped to the four-state Potts model~\\cite{wu}. For the octahedron model, presence of weak first-order phase transition is \nsuggested by Patrascioiu and Seiler~\\cite{Patrascioiu}, and afterward is numerically confirmed~\\cite{Krcmar}. The cube model can be mapped to \nthree decoupled Ising models. \nPatrascioiu {\\it et al} reported a second-order transition for the icosahedron \nand dodecahedron models, respectively, which have 12 and 20 local degrees of freedom~\\cite{Patrascioiu, Patrascioiu2, Patrascioiu3}. \nFor the icosahedron model,  the estimated transition temperatures is $1 \/ T_{\\rm c}^{~} = 1.802\\pm0.001$ and its critical indices are $\\nu \\sim 1.7$ and $\\gamma \\sim 3.0$, which are inconsistent with the minimal series of CFT. \nBy contrast, Surungan {\\it et al} gave another estimation $\\nu \\simeq 1.31$ for the same transition temperature\\cite{Surungan}.\nHowever, the system size of Monte Carlo simulations in provious works may be too small to conclude the universality of the icosahedron model.\nFinally, a possibility of an intermediate phase is suggested for the dodecahedron model in ~Refs. [\\onlinecite{Patrascioiu2}] and [\\onlinecite{Patrascioiu3}], whereas a solo second-order transition is suggested in Ref.~[\\onlinecite{Surungan}].\n\n\nIn this article, we focus on the critical behavior of the icosahedron model.\nWe calculate magnetization, effective correlation length and entanglement entropy in the bulk limit by means of the corner-transfer-matrix renormalization group (CTMRG) method~\\cite{ctmrg1, ctmrg2}, which is based on Baxter's corner-transfer matrix (CTM) scheme~\\cite{Baxter1, Baxter2, Baxter3}. \nAn advantage of the CTMRG method is that we can treat sufficiently large system size to obtain the conventional bulk physical quantities. \nActually, the system size of CTM in this work is up to $10^4 \\times 10^4$ sites, which can be viewed as a bulk limit in comparison with (effective) correlation length of the system.\nInstead, CTMRG results are strongly dependent on $m$, the number of states kept for the block-spin variables, near the transition point. \nNevertheless, this $m$-dependence of CTMRG results provides a powerful tool of the scaling analysis with respect to $m$~\\cite{fes1, tagliacozzo, pollmann, pivru}, the formulation of which is similar to the conventional finite-size scaling analysis~\\cite{Fisher, Barber}. \nThe $m$-scaling analysis actually extracts the presence of the second-order phase transition with the critical exponents $\\nu = 1.62\\pm0.02$ and $\\beta = 0.12\\pm0.01$.\nAnother interesting point on the CTMRG  approach is that the classical analogue of the entanglement entropy~\\cite{entent} can be straightforwardly calculated through a reduced density matrix constructed from CTMs.\nThe $m$-dependence analysis of the entanglement entropy also yields the central charge $c = 1.90\\pm0.02$, which cannot be explained by the minimal series of CFT.\n\nThis article is organized as follows. In the next section, we introduce the icosahedron model, and briefly explain its tensor-network representation and CTMRG method. \nWe first show the temperature dependence of the magnetization to capture the nature of the phase transition. \nIn Section~III, we apply the finite-$m$ scaling to the effective correlation length, magnetization, and the entanglement entropy. \nTransition temperature, critical exponents, and the central charge are estimated in detail.\nThe results are summarized in the last section.\n\n\n\\section{Icosahedron model}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{Fig_1.eps}\n\\caption{\n(a) Numbering of the vertices of the icosahedron. \n(b) Local Boltzmann weight in Eq.~(2) defined for a `black' plaquette, \nand its tensor representation. \n}\n\\label{Fig_1}\n\\end{figure}\n\nLet us consider the icosahedron model, which is a discrete analog of the classical Heisenberg model.\nOn each site of the square lattice, there is a vector spin ${\\bm v}^{(p)}_{~}\\!$ of unit length, which points to one of the vertices of the icosahedron, shown in Fig.~1 (a), where  $p$ is the index of vertices running from 1 to 12.\nFigure 1 (b) shows four vector spins ${\\bm v}^{(p)}_{~}\\!$, ${\\bm v}^{(q)}_{~}\\!$, ${\\bm v}^{(r)}_{~}\\!$, \nand ${\\bm v}^{(s)}_{~}\\!$, around a `black' plaquette, where we have introduced the \nchess-board pattern on the lattice. We have omitted the lattice index of these\nspins, since they can be formally distinguished by $p$, $q$, $r$, and $s$, which represent the direction of the spins. \nNeighboring spins have Heisenberg-like interaction, which is represented by the inner product between them. \nThus, the local energy around the plaquette in Fig.~1 (b) is written as\n\\begin{eqnarray}\nh_{pqrs}^{~} = - J && \\left( \n{\\bm v}^{(p)}_{~} \\! \\cdot {\\bm v}^{(q)}_{~} + \n{\\bm v}^{(q)}_{~} \\! \\cdot {\\bm v}^{(r)}_{~} \\right. \\nonumber\\\\\n&& +  \\left.\n{\\bm v}^{(r)}_{~} \\! \\cdot {\\bm v}^{(s)}_{~} + \n{\\bm v}^{(s)}_{~} \\! \\cdot {\\bm v}^{(p)}_{~}\n\\right) \\, .\n\\label{Eq_1}\n\\end{eqnarray}\nIn the following, we assume that coupling constant is spatially uniform and ferromagnetic $J > 0$. \n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig_2.eps}\n\\caption{Icosahedron model on the diagonal lattice, where $W$ on each `black' plaquette represents local Boltzmann weight\nof Eq.~(2). The partition function can be represented by a tensor-network on the square lattice. \nThe dashed lines show the division of the system into the quadrants corresponding to CTMs.}\n\\label{Fig_2}\n\\end{figure}\n\nWe represent the partition function of the system in the form of a vertex model, which can be \nregarded as a two-dimensional tensor network. For each `black' plaquette on the chess-board pattern introduced to the square \nlattice, we assign the local Boltzmann weight\n\\begin{equation} \nW_{pqrs}^{~} = \\exp\\biggl[ \\frac{h_{pqrs}^{~}}{T} \\biggr] \\, ,\n\\label{Eq_2}\n\\end{equation}\nwhere $T$ denotes the temperature in the unit of Boltzmann constant. \nNote that the vertex weight $W_{pqrs}^{~}$ is invariant under cyclic rotations of the indices. \nThroughout this article we choose $J$ as the unit of energy. As shown in Fig.~1 (b), the weight $W_{pqrs}^{~}$ is \nnaturally interpreted as the four-leg tensor, and thus the partition function can be represented as a \ncontraction among tensors, as schematically drawn on the right side panel of Fig.~2. \n\nIn Baxter's CTM formulation, the whole lattice is divided into four quadrants~\\cite{Baxter1, Baxter2, Baxter3}, as shown in Fig.~2. The partition function of a square-shaped\nfinite-size lattice is expressed by a trace of the fourth power of CTMs\n\\begin{equation}\nZ = {\\rm Tr} \\, C^4_{~} \\, ,\n\\label{Eq_3}\n\\end{equation}\nwhere $C$ denotes the CTM.\nNote that each matrix element of $C$ corresponds to the partition function of the quadrant where the spin configurations along the row and  column edges are specified. \nWe numerically obtain $Z$ by means of the CTMRG method~\\cite{ctmrg1, ctmrg2}, \nwhere the area of CTM is increased iteratively by repeating the system-size extension and renormalization group (RG) transformation. \nThen, the matrix dimension of $C$ is truncated with cutoff dimension $m$, and under an appropriate normalization, $C$ converges to its bulk limit after a sufficient number of iterations, even if we assume a fixed boundary condition.  \nAll the numerical data shown in this article are obtained after such convergence. \nThe numerical precision of CTMRG results are controlled by the cutoff $m$ for the singular value spectrum $\\{\\lambda_i\\}$ of CTMs with a truncation error $\\epsilon(m) = 1-\\sum_{i=1}^m \\lambda_i^4$. \nThe universal distribution of the spectrum  \\cite{OHA, cftdistribution} suggests that the asymptotic behavior of $\\epsilon(m)$ could be model independent.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig_3.eps}\n\\caption{(Color online) Temperature dependence of magnetization $M$ for several \n $m$. The inset: magnified view in the region $0.54 \\leq T \\leq 0.59$. }\n\\label{Fig_3}\n\\end{figure}\nIn practical computations, we assume the fixed boundary condition, where all the \nspins are pointing to the direction ${\\bm v}^{(1)}_{~}\\!$ on the boundary of the system.\nWe define an order parameter as the magnetization $M$ at the center of the system\n\\begin{equation}\nM = \\frac{1}{Z} \\, \\sum^{12}_{s = 1} \\, \\left( {\\bm v}^{(1)}_{~} \\! \\cdot {\\bm v}^{(s)}_{~} \\, {\\rm Tr}'_{~} \n\\bigl[  C^4_{~} \\bigr] \\right) \\, ,\n\\label{Eq_4}\n\\end{equation}\nwhere ${\\bm v}^{(s)}_{~}\\!$ is the vector spin at the center, and \n${\\rm Tr}'_{~}\\!$ represents partial trace except for ${\\bm v}^{(s)}_{~}\\!$.\nFigure 3 shows the temperature dependence of the magnetization $M$ calculated with\n$m = 100$, $200$, $300$, $400$, and $500$. \nThe magnetization is well converged with respect to $m$ for $T < 0.55$ or $T> 0.57$, and the result supports emergence of the ordered phase in low-temperature\nregion as reported by Patrascioiu {\\it et al}~\\cite{Patrascioiu, Patrascioiu2, Patrascioiu3}.\nAs shown in the inset, however, the curve of $M$ has the shoulder structure exhibiting the strong $m$ dependence in the region $0.55 <T < 0.57$.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig_4.eps}\n\\caption{(Color online) Probability $P^{(s)}_{~}\\!$ of finding ${\\bm v}^{(s)}_{~}$ at the center of the system with $m=500$. Plus marks and green circles denote $P^{(1)}_{~}\\!$ and $P^{(12)}_{~}\\!$, respectively. Blue squares denote $P^{(s)}_{~}\\!$ for $s = 2, 3, 4, 5$, and $6$, where these probabilities are the same. \nRed crosses denote $P^{(s)}_{~}\\!$ for $s = 7, 8, 9, 10$ \nand $11$. }\n\\label{Fig_4}\n\\end{figure}\n\nIn order to see the nature of the observed shoulder structure in $M$, we calculate the probability $P^{(s)}_{~}\\!$ of finding ${\\bm v}^{(s)}_{~}\\!$ at the center of the system. Figure 4 shows the temperature dependence of $P^{(s)}_{~}\\!$ with $m=500$. \nIn the region $T < 0.55$ the probability $P^{(1)}_{~}\\!$ is dominant. \nAround $T \\sim 0.56$, the value of $P^{(s)}_{~}\\!$ for $s = 2, 3, 4, 5$, and $6$ are comparable to $P^{(1)}_{~}\\!$, and the sum\n$P^{(2)}_{~}\\! + P^{(3)}_{~}\\! + P^{(4)}_{~}\\! + P^{(5)}_{~}\\! + P^{(6)}_{~}\\!$ is larger than $P^{(1)}_{~}\\!$. \nSuch a marginal behavior might suggest a possibility of an intermediate (or floating) critical phase, like the two succeeding phase transitions in \nclock-like models~\\cite{tobochnik, challa, yamagata, tomita, hwang, brito, baek1, baek2, kumano, \nKromar2016}. \nWe perform the scaling analysis with respect to $m$ to clarify the nature of the phase transition in the next section.\n\n\n\\section{Scaling analysis}\n\nAs described above, the calculated results of the magnetization $M$ exhibit the finite-$m$ dependence near the transition point. \nIn general, the cutoff dimension $m$ for the CTM introduces an effective correlation length in the critical region~\\cite{Tsang, Liu}, which corresponds to a regularization for the infrared divergence.\n By controlling the cutoff $m$, we can systematically analyze the critical behavior in the vicinity of the critical point, which we call the finite-$m$ scaling~\\cite{fes1, tagliacozzo, pollmann, pivru}, which shares many aspects in common with the finite-size scaling analysis~\\cite{Fisher, Barber}. \n\nIn the scaling analysis, one generally assumes that an observable $A$ with the scaling dimension $x_A^{~}$ \nobeys the following scaling function,\n\\begin{equation}\nA( b, t ) = b^{x_A^{~} \/ \\nu}_{~} \\, f_A^{~} \\left( b^{1 \/ \\nu}_{~} \\, t \\right) \\, ,\n\\label{Eq_5}\n\\end{equation}\nwhere $t = T \/ T_c - 1$ is the scaled temperature and $b$ is a characteristic length scale intrinsic to the \nsystem, which basically corresponds to the correlation length. In finite-size scaling analysis, $b$ is replaced \nby the linear dimension of system $\\ell$ and then, the scaling function $f_A^{~}$ is extracted by a systematical \ncontrol of $\\ell$. Note that the asymptotic forms $f_A( y )^{~} \\sim y^{-x_A}_{~}$ for $y \\gg 1$ and \n$f_A( y )^{~} \\sim {const.}$ for $y \\to 0 $ are also assumed in Eq.~(5), in order to reproduce the proper \nscaling law in the bulk limit $\\ell \\rightarrow \\infty$. \n\nFor the finite-$m$ scaling in CTMRG, meanwhile, we introduce a well-controllable length scale through the cutoff dimension $m$, instead of $\\ell$. \nAfter sufficient number of iterations in CTMRG, we have a renormalized row-to-row transfer matrix. \nWe can then define an effective correlation length\n\\begin{equation}\n\\xi( m, t ) = \\Bigl[ \\ln\\bigl( \\zeta_1^{~} \/ \\zeta_2^{~} \\bigr) \\Bigr]^{-1}_{~} \\, ,\n\\label{Eq_6}\n\\end{equation}\nwhere $\\zeta_1^{~}$ and $\\zeta_2^{~}$ are the largest and second-largest eigenvalues of the renormalized row-to-row transfer matrix. \nNote that the unit of length is set as the lattice constant. \nAn essential point is that the following scaling relation can be assumed, \n\\begin{equation}\n\\xi( m, t ) \\sim m^{\\kappa}_{~} \\, g\\left( m^{\\kappa \/ \\nu}_{~} \\, t \\right) \\, , \n\\label{Eq_7}\n\\end{equation}\nwith the asymptotic forms $g( y ) \\sim | y |^{- \\nu}_{~}$ for $y \\gg 1$ and $g( y ) \\sim {const.}$ for $y \\to 0$.\nEach limit yields the behavior $\\xi( m, t ) \\sim t^{- \\nu}_{~}$ for a finite $t$ under the condition $m^{\\kappa}_{~} \\gg t^{- \\nu}_{~}$, and $\\xi( m, t ) \\sim m^{\\kappa}_{~}$ for a finite $m$ under $m^{\\kappa}_{~} \\ll t^{- \\nu}_{~}$~\\cite{fes1, tagliacozzo}. Note that $\\kappa$ is an independent scaling dimension, which is characteristic to the matrix-product-state (MPS) description of the eigenvector of the row-to-row transfer matrix. Combining $b \\sim \\xi( m, t )$ and Eq.~(5), \nwe obtain the finite-$m$ scaling formula as\n\\begin{equation}\nA( m, t ) = m^{x_A^{~} \\kappa \/ \\nu}_{~} \\chi_A^{~}\\left( m^{\\kappa \/ \\nu}_{~} \\, t \\right) \\, ,\n\\label{Eq_8}\n\\end{equation}\nwhere $\\chi_A^{~}$ is a new scaling function satisfying $\\chi_A^{~}( y ) \\sim | y |^{- x_A^{~}}_{~}$ for \n$y \\gg 1$. For a finite $t$ under the condition $m^{\\kappa \/ \\nu}_{~} \\, t \\gg 1$, Eq.~(8) reproduces \n$A( m, t ) \\sim | t |^{- x_A^{~}}_{~}$, while for a finite $m$ under $m^{\\kappa \/ \\nu}_{~} \\, t \\ll 1$, \nEq.~(8) gives $A( m, t ) \\sim m^{- \\kappa x_A^{~} \/ \\nu}_{~}$.\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{Fig_5.eps}\n\\caption{(Color online) Finite-$m$ scalings for \n(a) correlation length in Eq.~(6), \n(b) magnetization $M$ in Eq.~(4), and \n(c) entanglement entropy in Eq.~(10).}\n\\label{Fig_5}\n\\end{figure}\n\nWe apply the scaling analysis to several quantities calculated by CTMRG, with help of the Bayesian analysis for fitting~\\cite{Harada}. \nWe consider the temperature region \n$0.520 \\leq T \\leq 0.619$ for $m=100, 200, 300, 400$, and $500$ in the following scaling analysis. \nWe first apply the analysis to $\\xi( m, t )$ in Eq.~(6) and estimate critical temperature $T_{\\rm c}^{~}$, exponents $\\kappa$ and $\\nu$. Figure 5 (a) shows the scaling plot of $\\xi( m, t )$ with the best fit values, $T_{\\rm c}^{~} = 0.555048(43)$, $\\nu = 1.617(13)$, and $\\kappa = 0.8983(17)$, where all data collapse on the scaling function $g$ in Eq.~(7).\nThe fitting errors in the Bayesian analysis are shown in the round brackets.\nIf we use the data for $200 \\leq m \\leq 500$, we obtain \n$T_{\\rm c}^{~} = 0.554940(42)$, $\\nu = 1.623(13)$, and $\\kappa = 0.8830(19)$. Comparing these two fitting results, we adopt $T_{\\rm c}^{~} = 0.5550\\pm0.0001$, \n$\\nu = 1.62\\pm0.02$, and $\\kappa = 0.89\\pm0.02$.\nThis result of $T_c$ is consistent with the values $T_{\\rm c}^{~} \\simeq 0.555$ reported by both Patrascioiu {\\it et al}~\\cite{Patrascioiu, Patrascioiu2, Patrascioiu3} and Surungan {\\it et al}\\cite{Surungan}.\nWhile, the critical exponent $\\nu$ is consistent with the value $\\nu = 1.7^{+0.3}_{-0.1}$ in Refs.\\cite{Patrascioiu, Patrascioiu2, Patrascioiu3}, but has a discrepancy from $\\nu=1.31\\pm0.01$ in Ref.~\\cite{Surungan}.\n\nOn the basis of the above $T_{\\rm c}^{~}$, $\\nu$, and $\\kappa$, moreover, we perform the finite $m$ scaling analysis for the magnetization $M$ shown in Fig.~3.\nA particular point is that the shoulder structure in the inset of Fig. 3  directly reflects on the scaling function of Fig. 5(b).\nMoreover, such shoulder structures of the scaling functions are consistently observed in Figs. 5(a) and (c).\nThese behaviors imply that the transition of the icosahedron model is described by a solo second-order transition, unlike to the clock models of $q>4$ where the intermediate critical region emerges.\nUsing the Basian fitting, then, we obtain $\\beta = 0.1293(27)$ for $m=100 \\sim 500$ and $\\beta = 0.1234(33)$ for $m=200 \\sim 500$. \nTaking into account the discrepancy, we adopt $\\beta = 0.12\\pm0.01$. \nWe however think that this value should be improved in further extensive calculations.\n\n\n\nIn order to obtain additional information for the scaling universality, we calculate the classical analogue of the entanglement entropy. \nThe concept of entanglement can be introduced to two-dimensional statistical models through the quantum-classical correspondence~\\cite{fradkin, Trotter, Suzuki1, Suzuki2}.  \nThen, an essential point is that the fourth power of CTM, which appears in Eqs.~(3) and (4), can be interpreted as a density matrix of the corresponding one-dimensional quantum system~\\cite{HU2014}. \nFrom the normalized density matrix\n\\begin{equation}\n\\rho = \\frac{C^4_{~}}{Z} \\, ,\n\\label{Eq_9}\n\\end{equation}\nwe obtain the classical analogue of the entanglement entropy, in the form of Von Neumann entropy~\\cite{vnent1, vnent2}\n\\begin{equation}\nS_{\\rm E}^{~} = - {\\rm Tr} \\, \\rho  \\ln \\, \\rho \\, .\n\\label{Eq_10}\n\\end{equation}\n\nIn the context of CTMRG, the following relation\n\\begin{equation}\nS_{\\rm E}^{~}( m, t ) \\sim \\frac{c}{6} \\, \\ln \\, \\xi( m, t ) + const.~,\n\\label{Eq_11}\n\\end{equation}\nis satisfied around the criticality~\\cite{Vidal, Calabrese}, where $c$ is the central charge. \nTaking the exponential of both sides of this equation, and substituting Eq.~(7), we obtain\n\\begin{eqnarray}\ne^{S_{\\rm E}^{~}}_{~} \\sim  a \\Bigl[ \\xi( m, t ) \\Bigr]^{c\/6}_{~} \n&=& \\, a \\Bigl[ m^{\\kappa}_{~} \\, g\\bigl( m^{\\kappa \/ \\nu}_{~} \\, t \\bigr) \\Bigr]^{c\/6}_{~} \\nonumber \\\\\n&=& \\, m^{c \\kappa \/ 6}_{~} \\, {\\tilde g}\\bigl( m^{\\kappa \/ \\nu}_{~} \\, t \\bigr) \\, ,\n\\end{eqnarray}\nwhere $a$ is a non-universal constant, and ${\\tilde g} \\equiv ag^{c\/6}$. \nThus the critical exponent for $e^{S_{\\rm E}}_{~}$ is identified as $c \\nu \/ 6$. \n\nUsing $T_{\\rm c}^{~}$, $\\kappa$ and $\\nu$ previously obtained by the finite-$m$ scaling for $\\xi( m, t )$, we can estimate the central charge $c$. \nFigure 5 (c) shows the scaling plot of Eq.~(12) for the data of $m = 100, 200, 300, 400$, and $500$.\nThe central charge is estimated as $c = 1.894(12)$. \nIf we exclude the case $m = 100$ for the scaling analysis, we obtain $c = 1.900(15)$. \nConsidering the discrepancy between the above values of $c$, we adopt $c = 1.90\\pm0.02$.\n\nHere, it should be noted that this value is consistent with the relation\n\\begin{equation}\n\\kappa = \\frac{6}{ c\\bigl( \\sqrt{12 \/ c} \\, + 1 \\bigr) }\\, ,\n\\label{pollmann}\n\\end{equation}\nwhich is derived from the MPS description of one-dimensional critical quantum system.~\\cite{pollmann}\nSubstituting $c = 1.90$ and $\\kappa = 0.89$ to Eq. (\\ref{pollmann}), we actually have $6 \/ \\{ c( \\sqrt{12 \/ c}+1) \\} - \\kappa = 0.009$, which provides a complemental check of the finite-$m$ scaling in CTMRG.\n\n\\section{Summary and discussion}\n\nWe have investigated the phase transition and its critical properties of the icosahedron model on a square lattice, where the local vector spin has twelve degrees of freedom. \nWe have calculated the magnetization, the effective correlation length, and the classical analogue of the entanglement entropy by means of the CTMRG method. \nThe CTMRG results are strongly dependent on $m$, which is the cutoff dimension of CTMs, near the critical point.\nWe have then performed the finite-$m$ scaling analysis and  found that the all numerical data can be well fitted with the scaling functions including the shoulder structures.\nWe have thus confirmed that the icosahedron model exhibits the second-order phase transition at $T_{\\rm c}=0.5550\\pm0.0001$, below which the icosahedral symmetry is broken to a five-fold axial symmetry.\nAlso, the scaling exponents are estimated as $\\nu = 1.62\\pm0.02$,  $\\kappa = 0.89\\pm0.02$, and $\\beta=0.12\\pm0.01$. \nFrom the relation between entanglement entropy and the effective correlation length, moreover, we have extracted the central charge as $c = 1.90\\pm0.02$, which cannot be described by the minimal series of CFT.\nTo clarify the mechanism  of such a non-trivial critical behavior in the icosahedron model is an important future issue. \n\n\nOur original motivation was from the systematical analysis of the continuous-symmetry limit toward the $O( 3 )$ Heisenberg spin.\nIn this sense, the next target is the dodecahedron model having twenty local degrees of freedom, which requires massive parallelized computations of CTMRG. \nIn addition,  it is an interesting problem to introduce the XY-like uniaxial anisotropy to the icosahedron and dodecahedron models;\nA crossover of universality between the icosahedron\/dodecahedron model and the clock models can be expected, where the shoulder structures of the scaling functions may play an essential role.\n\n\n\n\n\\section{Acknowledgment}\n\nThis research was  partially supported by Grants-in-Aid for Scientific Research under Grant No. 25800221, 26400387, 17H02931, and 17K14359 from JSPS and by VEGA 2\/0130\/15 and APVV-16-0186. \nIt  was also  supported by MEXT as ``Challenging Research on Post-K computer'' (Challenge of Basic Science: Exploring the Extremes through Multi-Physics Multi-Scale Simulations). \nThe numerical computations were performed on the K computer provided by the RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID:hp160262).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nIn light nuclei, the cluster aspect is one of the essential features, as well as the shell-model aspect.\nOwing to the coexistence of these two natures, namely, cluster and shell-model features, various structures \nappear in stable and unstable nuclei.\n\n$^{12}$C is one of the typical examples where the cluster and shell-model aspects coexist.\nThe ground state of $^{12}$C is known to have mainly a shell-model feature of \nthe $p_{3\/2}$ subshell closed configuration, whereas\nthe well-developed 3$\\alpha$-cluster structures appear in excited states.\nIn the theoretical works on the 3$\\alpha$-cluster structures,\\cite{Horiuchi_OCM_74,Uegaki_12C_77,Kamimura_12C_77,Descouvemont_12C_87,En'yo_12C_98,Tohsaki_12C_01,Funaki_12C_03,Neff_12C_04,En'yo_12C_07,Kurokawa_12C_07} \\ \nvarious configurations of the 3$\\alpha$-cluster structures were suggested in the excited states\nabove the 3$\\alpha$ threshold energy, for example, \nthe $\\alpha$ condensation of weakly interacting \nthree $\\alpha$ clusters in the $0^{+}_2$ state and \nthe equilateral-triangular structure of three $\\alpha$ clusters in the $3^{-}_{1}$ state.\nMoreover, a linear-chainlike (or an obtuse-angle-triangular) structure \nof three $\\alpha$ clusters in the $0^{+}_{3}$ state was suggested.\n\nCluster structures have also been found in light neutron-rich nuclei such as Be isotopes.\nIn $^{10}$Be, the low-lying states are understood in a molecular\n2$\\alpha+2n$ picture,\\cite{vonOertzen_ClusterRev_06,Itagaki_10Be_00} \\ where \ntwo $\\alpha$ cores are formed and two excess neutrons occupy molecular orbitals around the $2\\alpha$.\nIn terms of a simple shell model, $^{10}$Be is an $N=6$ nucleus, and therefore, the $p_{3\/2}$ subshell closure \neffect is also important, as well as the $2\\alpha+2n$ cluster feature at least in the ground state. \nThis means that the cluster-shell competition is essential in unstable nuclei \nas well as stable nuclei, as argued in Ref.~\\citen{Itagaki_ClusterShellCompetition_04}.\n\nFor theoretical investigations of such nuclei, it is necessary\nto describe the coexistence of shell and cluster features systematically. \nHowever, many theoretical frameworks still have deficiencies \nin describing both the shell-model and cluster structures.\nIn fact, in the case of $^{12}$C, shell models can be used to describe low-lying shell-model states but they\nusually fail to describe high-lying 3$\\alpha$-cluster states. \nOn the other hand, conventional cluster models are suitable for studying the 3$\\alpha$-cluster states,\nbut it is not easy to reproduce well the detailed properties of \nlow-lying shell-model states because $\\alpha$ cluster breaking is not incorporated\nin the cluster models.\n\nA method of antisymmetrized molecular dynamics (AMD)\\cite{En'yo_PTP_95,En'yo_AMD_95} \\ is one of the frameworks useful for overcoming this\nproblem. It was applied to $^{12}$C and \nsucceeded to describe the shell and cluster features due to the flexibility of its \nwave functions.\\cite{En'yo_12C_98,En'yo_12C_07} \\ \nMoreover, in the study of fermionic molecular dynamics (FMD), in which model wave functions \nare similar to those of AMD, the coexistence of shell and cluster features in $^{12}$C was \ndescribed successfully.\\cite{Neff_12C_04} \\ \n\nThe AMD method has also been applied to various stable and unstable nuclei, \nand it has been proved to be one of the powerful approaches of describing various structures\nsuch as cluster structures and shell-model structures.\\cite{En'yo_AMD_95,En'yo_AMD_03,En'yo_sup_01} \\ \nThere are some versions of the AMD, for example, the variation after parity and total-angular-momentum projections (VAPs),\\cite{En'yo_12C_98} \\ \nthe variation with the constraint on the quadrupole deformation \n$\\beta$ ($\\beta$ constraint AMD),\\cite{Dote_Beta-Constraint_97,Kimura_uptoMg_01,En'yo_AMD_03} \\ or the constraint on \nthe cluster distances ($d$-constraint AMD).\\cite{Taniguchi_D-Constraint_04} \\ \nIn principle, a basis AMD wave function is given by a Slater determinant of Gaussian wave packets, and\nexcited states are described by superposition of Slater determinants.\nIn practical calculations of excited states of light nuclei, it is important to prepare efficiently \nvarious cluster configurations \nincluding 2-body and 3-body clusterings as basis wave functions in the AMD framework.\nMoreover, in the study of unstable nuclei, further flexible model wave functions such as \n2-body or 3-body cluster structures with surrounding valence nucleons will be required to describe \npossible exotic cluster structures in excited states.\n\nTo study a variety of cluster structures and the coexistence of cluster and shell features\nin light unstable nuclei,\nwe propose an extended method of constraint AMD\nto describe various cluster and shell structures. That is the\ntwo-dimensional constraint with respect to the quadrupole deformation parameters, $\\beta$ and $\\gamma$,\nwhich is expected to be efficient for preparing basis wave functions with various cluster configurations.\nWe call this method $\\beta$-$\\gamma$ constraint AMD.\nWe expect shell-model structures to appear in the small $\\beta$ region, whereas\ndeveloped 2-body or 3-body cluster structures can be obtained for large $\\beta$.\nIn the large $\\beta$ region, various configurations of cluster structures may appear\ndepending on $\\beta$ and $\\gamma$. \n\nThe $\\beta$-$\\gamma$ constraint AMD may also be useful\nin the study of triaxial deformations.\nOn the other hand, in the Hartree-Fock-Bogolyubov (HFB) calculations, \nthe $\\beta$-$\\gamma$ constraint was adopted, for example, \nin Ref.~\\citen{Girod_triaxial_83}, and \nthe superposition of $\\beta$-$\\gamma$ constraint wave functions\nhas been performed recently by Bender and Heenen.\\cite{Bender_24Mg_08}.\nIt was found that the triaxiality is important to reproduce the experimental \ndata of $^{24}$Mg in the HFB calculations. \nHowever, works on triaxial calculations with the superposition are limited, and\nit is still a challenging problem.\nMoreover, such mean-field approaches are not necessarily \nsuitable for describing cluster structures.\nFor the study of cluster features, it is important to apply \nthe $\\beta$-$\\gamma$ constraint to a framework that can describe cluster structures.\n\nIn this paper, we applied the $\\beta$-$\\gamma$ constraint AMD\nto $N=6$ isotones, $^{10}$Be, $^{12}$C, $^{9}$Li, and $^{11}$B\nto check the applicability of this method.\nWe analyze the results and confirm that various structures appear as functions of \nthe deformation parameters, $\\beta$ and $\\gamma$, in the present framework. \nIn particular, we focus on the coexistence of shell and cluster features. \nFor $^{10}$Be and $^{12}$C, we also calculate the energy spectra of excited states\nby the superposition of the obtained basis wave functions and compare the results with \nthe experimental data.\nWe show that the $\\beta$-$\\gamma$ constraint AMD is useful for reproducing the energy spectra.\nA role of the $\\gamma$ degree of freedom is also discussed.\n\nThe content of this paper is as follows. \nIn \\S \\ref{framework}, we explain the framework of the $\\beta$-$\\gamma$ constraint AMD.\nThe calculated results are shown in \\S \\ref{results}.\nIn \\S \\ref{discussions}, we discuss the effect of the triaxial deformation parameter $\\gamma$.\nFinally, in \\S \\ref{summary}, a summary and an outlook are given.\n\n\n\\section{Framework of $\\beta$-$\\gamma$ constraint AMD}\\label{framework}\n\nWe adopt a method of AMD with constraint. \nThe frameworks of AMD and constraint AMD are described in detail, for example, in \nRefs.~\\citen{En'yo_AMD_95,En'yo_sup_01,En'yo_AMD_03}.\nIn this paper, we propose a two-dimensional constraint with respect to quadrupole deformation\nparameters.\n\n\\subsection{Wave function of AMD}\n\nIn the method of AMD, \na basis wave function of an $A$-nucleon system $|\\Phi \\rangle$ \nis described by a Slater determinant of single-particle wave functions $|\\varphi_{i} \\rangle$ as\n\\begin{equation}\n|\\Phi \\rangle = \\frac{1}{\\sqrt{A!}} \\det \\left\\{ |\\varphi_{1} \\rangle, \\cdots ,|\\varphi_{A} \\rangle \\right\\}.\n\\end{equation}\nThe $i$-th single-particle wave function $|\\varphi_{i} \\rangle$ consists of \nthe spatial part $|\\phi_{i} \\rangle$, spin part $|\\chi_{i} \\rangle$, and isospin part $|\\tau_{i} \\rangle$ as\n\\begin{equation}\n\t|\\varphi_{i} \\rangle = |\\phi_{i} \\rangle |\\chi_{i} \\rangle |\\tau_{i} \\rangle.\n\\end{equation}\nThe spatial part $|\\phi_{i} \\rangle$ is given by a Gaussian wave packet\nwhose center is located at $\\bm{Z}_{i}\/\\sqrt{\\nu}$ as\n\\begin{equation}\n\t\\langle \\bm{r} | \\phi_{i} \\rangle = \\left( \\frac{2\\nu}{\\pi} \\right)^{\\frac{3}{4}}\n\t\t\\exp \\left[ - \\nu \\left( \\bm{r} - \\frac{\\bm{Z}_{i}}{\\sqrt{\\nu}} \\right)^{2} \n\t\t+ \\frac{1}{2} \\bm{Z}_{i}^{2}\\right] \n\t\\label{single_particle_spatial}, \n\\end{equation}\nwhere $\\nu$ is the width parameter and is taken to be a common value for all the\nsingle-particle Gaussian wave functions in the present work.\nThe spin orientation is given by the parameter $\\bm{\\xi}_{i}$, while\nthe isospin part $|\\tau_{i} \\rangle$ is fixed to be up (proton) or down (neutron), \n\\begin{align}\n\t|\\chi_{i} \\rangle &= \\xi_{i\\uparrow} |\\uparrow \\ \\rangle + \\xi_{i\\downarrow} |\\downarrow \\ \\rangle,\\\\\n\t|\\tau_{i} \\rangle &= |p \\rangle \\ or \\ |n \\rangle.\n\\end{align}\nIn a basis wave function $|\\Phi \\rangle$, $\\{ X \\} \\equiv \\{ \\bm{Z} , \\bm{\\xi} \\} = \\{ \\bm{Z}_{1} , \\bm{\\xi}_{1} , \\bm{Z}_{2} , \\bm{\\xi}_{2} , \n\\cdots , \\bm{Z}_{A} , \\bm{\\xi}_{A} \\}$ are complex variational parameters and they \nare determined by the energy optimization using the frictional cooling method.\\cite{En'yo_sup_01,En'yo_AMD_03} \\ \nAs the variational wave function, we employ the parity-projected wave function\n\\begin{equation}\n\t|\\Phi ^{\\pm} \\rangle = P^{\\pm} |\\Phi \\rangle = \\frac{1 \\pm P}{2} |\\Phi \\rangle.\n\\end{equation}\nHere, $P$ is the parity transformation operator. \nWe perform the variation for the parity-projected energy\n$\\langle \\Phi ^{\\pm}| H |\\Phi ^{\\pm} \\rangle \/ \\langle \\Phi ^{\\pm}|\\Phi ^{\\pm} \\rangle$,\nwhere $H$ is the Hamiltonian.\nAfter the variation, we project the obtained wave function onto the \ntotal-angular-momentum eigenstate.\nIt means that the parity projection is performed before the variation, and\nthe total-angular-momentum projection is carried after the variation.\n\n\\subsection{$\\beta$-$\\gamma$ constraint}\n\nTo describe various cluster and shell-model structures that may appear \nin the ground and excited states of light nuclei,\nwe constrain the quadrupole deformation parameters, $\\beta$ and $\\gamma$, and perform the \nenergy variation with the constraints on the $\\beta$-$\\gamma$ plane.\n\nThe deformation parameters, $\\beta$ and $\\gamma$, are defined as\n\\begin{align}\n\t&\\beta \\cos \\gamma \\equiv \\frac{\\sqrt{5\\pi}}{3} \n\t\t\\frac{2\\langle z^{2} \\rangle -\\langle x^{2} \\rangle -\\langle y^{2} \\rangle }{R^{2}}, \\\\\n\t&\\beta \\sin \\gamma \\equiv \\sqrt{\\frac{5\\pi}{3}} \n\t\t\\frac{\\langle x^{2} \\rangle -\\langle y^{2} \\rangle }{R^{2}} \\label{definition_beta_gamma}, \\\\\n\t&R^{2} \\equiv \\frac{5}{3} \\left( \\langle x^{2} \\rangle + \\langle y^{2} \\rangle \n\t\t+ \\langle z^{2} \\rangle \\right).\n\\end{align}\nHere, $\\langle O \\rangle$ represents the expectation value of the operator $O$ for an intrinsic wave function $| \\Phi \\rangle$.\n$x$, $y$, and $z$ are the inertia principal axes that are chosen as\n$\\langle y^{2} \\rangle \\le \\langle x^{2} \\rangle \\le \\langle z^{2} \\rangle $ and\n$\\langle xy \\rangle = \\langle yz \\rangle = \\langle zx \\rangle =0$.\nTo satisfy the latter condition, we also impose \nthe constraints $\\langle xy \\rangle\/R^{2} = \\langle yz \\rangle\/R^{2} = \\langle zx \\rangle\/R^{2} =0$. \nTo obtain the energy minimum state under the constraint condition,\nwe add the constraint potential $V_{\\text{const}}$ to the total energy of the system\nin the energy variation. The constraint potential $V_{\\text{const}}$ is given as\n\\begin{align} \n\tV_{\\text{const}} \\equiv &\\eta_{1} \n\t\\left[ (\\beta \\cos \\gamma - \\beta_{0} \\cos \\gamma_{0})^{2} + (\\beta \\sin \\gamma - \\beta_{0} \\sin \\gamma_{0})^{2} \\right] \\notag \\\\\n\t+ &\\eta_{2} \\left[ \\left( \\frac{\\langle xy \\rangle}{R^{2}} \\right)^{2} \n\t\t+ \\left( \\frac{\\langle yz \\rangle}{R^{2}} \\right)^{2} \n\t\t+ \\left( \\frac{\\langle zx \\rangle}{R^{2}} \\right)^{2} \\right].\n\t\\label{constraint_energy}\n\\end{align}\nHere, $\\eta_{1}$ and $\\eta_{2}$ take sufficiently large values.\nAfter the variation with the constraint, we obtain the optimized wave functions\n$|\\Phi^{\\pm}(\\beta_{0}, \\gamma_{0}) \\rangle$\nfor each set of parameters, $(\\beta, \\gamma) = (\\beta_{0}, \\gamma_{0})$.\n\nIn the calculations of energy levels, \nwe superpose the total-angular-momentum projected \nwave functions $P^{J}_{MK} |\\Phi^{\\pm}(\\beta, \\gamma) \\rangle$. \nThus, the final wave function for the $J^\\pm_n$ state is given by\na linear combination of the basis wave functions as \n\\begin{equation}\n\t|\\Phi ^{J\\pm}_{n} \\rangle = \\sum_{K} \\sum_{i} f_{n}(\\beta_{i}, \\gamma_{i}, K) P^{J}_{MK} |\\Phi^{\\pm}(\\beta_{i}, \\gamma_{i}) \\rangle.\n\t\\label{dispersed_GCM}\n\\end{equation}\nThe coefficients $f_{n}(\\beta_{i}, \\gamma_{i}, K)$ are determined using the Hill-Wheeler equation\n\\begin{equation}\n\t\\delta \\left( \\langle \\Phi ^{J\\pm}_{n} | H | \\Phi ^{J\\pm}_{n} \\rangle - \n\tE_{n} \\langle \\Phi ^{J\\pm}_{n} | \\Phi ^{J\\pm}_{n} \\rangle\\right) = 0.\n\t\\label{Hill-Wheeler}\n\\end{equation}\nThis means the superposition of multiconfigurations described by \nparity and total-angular-momentum projected AMD wave functions.\nIn the limit of sufficient basis wave functions \non the $\\beta$-$\\gamma$ plane, it corresponds to the\ngenerator coordinate method (GCM) with the two-dimensional generator coordinates \nof the quadrupole deformation parameters, $\\beta$ and $\\gamma$.\n\n\\subsection{Hamiltonian and parameters}\n\nThe Hamiltonian $H$ consists of the kinetic term \nand effective two-body interactions as\n\\begin{equation}\n\tH = \\sum_{i} t_{i} - T_{\\text{G}} + \\sum_{i<j} V^{\\text{central}}_{ij} \n\t+ \\sum_{i<j} V^{\\text{spin-orbit}}_{ij} + \\sum_{i<j} V^{\\text{Coulomb}}_{ij},\n\\end{equation}\nwhere $V^{\\text{central}}_{ij}$, $V^{\\text{spin-orbit}}_{ij}$ and $V^{\\text{Coulomb}}_{ij}$ \nare the central force, spin-orbit force, and Coulomb force.\nAs the central force, we use the Volkov No.~2 interaction,\\cite{Volkov_No2_65} \\ \n\\begin{equation}\n\tV^{\\text{central}}_{ij} = \\sum_{k=1}^{2} v_{k} \\exp \\left[- \\left( \\frac{r_{ij}}{a_{k}} \\right)^{2} \\right]\n                                  (W + B P_{\\sigma} - H P_{\\tau} - M P_{\\sigma} P_{\\tau}),\n\\end{equation}\nwhere $v_{1} = -60.65$ MeV, $v_{2} = 61.14$ MeV, $a_{1} = 1.80$ fm, and $a_{2} = 1.01$ fm.\nFor the spin-orbit part, we used the spin-orbit term of the G3RS interaction,\\cite{G3RS_79} \\ which is a two-range\nGaussian with a projection operator $P(^{3} \\mbox{O})$ onto the triplet odd state,\n\\begin{align}\n\tV^{\\text{spin-orbit}}_{ij} = \\sum_{k=1}^{2} &u_{k} \\exp \\left[- \\left( \\frac{r_{ij}}{b_{k}} \\right)^{2} \\right] P(^{3} \\mbox{O}) \\bm{L} \\cdot \\bm{S}, \\\\\n\tP(^{3} \\mbox{O}) &= \\frac{1+P_{\\sigma}}{2} \\frac{1+P_{\\tau}}{2},\n\\end{align}\nwhere $b_{1} = 0.600$ fm and $b_{2} = 0.477$ fm.\n\nWe take interaction parameters, the Majorana exchange parameter $M = 0.6$ ($W = 0.4$), the Bartlett exchange parameter $B = 0.125$,\nand the Heisenberg exchange parameter $H = 0.125$ in the central force, and \n$u_{1} = -1600$ MeV and $u_{2} = 1600$ MeV in the spin-orbit force.\nAll these parameters are \nthe same as those adopted in Refs.~\\citen{Itagaki_10Be_00} and \\citen{Okabe_parameter_79}, \nexcept for a small modification in the strength of the spin-orbit force.\nThey are adjusted to reproduce \nthe $\\alpha + \\alpha$ phase shift ($M$, $W=1-M$, $a_{1}$, $a_{2}$), binding energy of the deuteron ($B=H$),\nand $\\alpha + n$ phase shift ($u_{1}=-u_{2}$, $b_{1}$, $b_{2}$).\nWe slightly modify only the strengths of the spin-orbit force \nfrom $-u_{1} = u_{2} = 2000$ MeV adopted in Ref.~\\citen{Itagaki_10Be_00}\nto  $-u_{1} = u_{2} = 1600$ MeV to fit the $0^{+}_{1}$ energy of $^{12}$C \nin the present calculations.\nBy using these parameters, \nthe deuteron binding energy in an exact calculation is 2.24 MeV, which\nis in agreement with the experimental value, 2.22 MeV.\nWhen the deuteron wave function is approximated by one AMD wave function, \nthe calculated binding energy is 0.67 MeV, which slightly underestimates the experimental \nvalue, 2.2 MeV.\nThe Coulomb force $V^{\\text{Coulomb}}_{ij}$ is approximated by a sum of seven Gaussians.\n\nFor the width parameter of\nsingle-particle Gaussian wave packets in Eq.~\\eqref{single_particle_spatial}, \nwe used the value $\\nu = 0.235$ fm$^{-2}$, which is \ndetermined from a variational calculation \nfor the ground state of $^{9}$Be in Ref.~\\citen{Okabe_parameter_79}.\n\n\n\\section{Results}\\label{results}\n\nWe applied the $\\beta$-$\\gamma$ constraint \nAMD to the $N=6$ isotones, $^{10}$Be, $^{12}$C, $^9$Li, and $^{11}$B.\nIn this section, we analyze the results of the $\\beta$-$\\gamma$ constraint\nand show that \nvarious structures including well-developed cluster ones appear as functions of \ndeformation parameters, $\\beta$ and $\\gamma$, in the present framework. \nIn particular, we focus on the coexistence of shell and cluster features. \nFor $^{10}$Be and $^{12}$C, we also show the energy spectra of the excited states\ncalculated by the superposition of the basis wave functions and compare them with the experimental data.\n\n\\subsection{Energy surfaces}\n\nFirst, we performed variational calculations with the $\\beta$-$\\gamma$ constraint\nat a total of 196 mesh points of the triangle lattice on the $\\beta$-$\\gamma$ plane.\nWe obtained energy surfaces as functions of $\\beta$ and $\\gamma$.\nThe calculated energy surfaces on the $\\beta$-$\\gamma$ plane\nbefore and after the total-angular-momentum projection for $^{10}$Be,\n$^{12}$C, $^9$Li, and $^{11}$B  are shown \nin Figs.~\\ref{10Be_energy_surface},~\\ref{12C_energy_surface},~\\ref{9Li_energy_surface}, and~\\ref{11B_energy_surface}, respectively. \nWe also show the energy curves along the axial symmetric line, \n$\\gamma=0^\\circ$ and $\\gamma=60^\\circ$, in Fig.~\\ref{symmetric_energy_surface}.\n\nIn Fig.~\\ref{10Be_energy_surface} for $^{10}$Be,\nthe left panel shows the energy of the positive-parity states before the total-angular-momentum projection,\nand the right panel shows the results \nfor $0^{+}$ states calculated using the total-angular-momentum projection after the variation.\nWe call the former the positive-parity energy surface and the latter \nthe $0^{+}$ energy surface.\nIn the positive-parity energy surface, the minimum and a local minimum \nexist in the prolate region along the $\\gamma=0^{\\circ}$ line.\nThe minimum point is at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.0)$\nin the normal deformation region,\nwhile the local minimum exists \naround $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.9, 0.0)$\nin the large deformation region.\nAfter the total-angular-momentum projection, the shape of the energy surface changes \nbecause the energy gain due to the projection tends to be large\nin the large $\\beta$ region and triaxial region compared with that in the \nspherical or axial symmetric regions.\nAs a result, the minimum point of the $0^{+}$ energy surface\nshifts to $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.55, 0.09)$ \nin the finite $\\gamma$ region. \nThe local minimum point of the $0^+$ energy surface also\nhas a finite $\\gamma$ value as $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.03, 0.04)$. \nAround the local minimum, the energy surface is rather flat. \nAs we show later, the largely deformed excited band \nappears from this region in the GCM energy spectra.\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{0Be_+.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{0Be_0+.eps}}\n\t\\caption{Energy surfaces of $^{10}$Be on the $\\beta$-$\\gamma$ plane.\n\tThe left shows the energy for the positive-parity states before the total-angular-momentum projection and \n\tthe right shows that for the $0^{+}$ states after the total-angular-momentum projection.}\n\t\\label{10Be_energy_surface}\n\\end{figure}\n\nThe calculated energy surfaces of $^{12}$C are shown in \nFig.~\\ref{12C_energy_surface}.\nThe positive-parity energy surface and the $0^{+}$ energy surface are displayed in the \nleft and right panels, respectively.\nThe minimum point of the positive-parity energy surface is at\n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.13, 0.22)$, which indicates an oblate deformation. \nIn contrast to $^{10}$Be, which energetically favors the prolate deformation for any $\\beta$, \nin the case of $^{12}$C, the prolate deformation around \n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.8, 0.0)$ is unfavored.\nThere is a peak with 7 MeV height at this point, and a valley \ngoes from the oblate energy minimum to the prolate region keeping away from this peak.\nAfter the total-angular-momentum projection, the minimum point of the $0^{+}$ energy surface becomes\n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.17)$. This indicates that\nthe deformation of the energy minimum state changes from the oblate shape to the \ntriaxial shape after the total-angular-momentum projection.\nThere is also the flat region around $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.0, 0.1)$\nin the $0^+$ energy surface.\nIn the region around $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.6, 0.0)$, \nthe $0^{+}$ energy surface has a large discontinuity \nbecause the intrinsic states in this region contain very small $0^{+}$ components.\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{12C_+.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{12C_0+.eps}}\n\t\\caption{Energy surfaces of $^{12}$C on the $\\beta$-$\\gamma$ plane.\n\tThe left shows the energy for the positive-parity states and \n\tthe right shows that for the $0^{+}$ states after the total-angular-momentum projection.}\n\t\\label{12C_energy_surface}\n\\end{figure}\n\nThe energy surfaces of $^{9}$Li are shown in Fig.~\\ref{9Li_energy_surface}.\nThe left and right panels show the negative-parity energy surface before\nthe total-angular-momentum projection\nand the $3\/2^{-}$ energy surface after the total-angular-momentum projection, respectively.\nIn the negative-parity energy surface, it is found that \nthe energy minimum state has a small deformation.\nA local minimum exists around $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.0, 0.0)$\nin the large prolate region.\nAfter the total-angular-momentum projection, \nthe minimum point of the $3\/2^{-}$ energy surface becomes\n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.38, 0.13)$.\nIn the region near $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.03, 0.04)$, \nthe energy surface is rather flat, and there exists a local minimum.\nIt is interesting that \nthe behavior of the $3\/2^{-}$ energy surface of $^{9}$Li is \nqualitatively similar to that of the $0^+$ energy surface of $^{10}$Be.\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{9Li_-.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{9Li_1.5-.eps}}\n\t\\caption{Energy surfaces of $^{9}$Li on the $\\beta$-$\\gamma$ plane.\n\tThe left shows the energy for the negative-parity states and \n\tthe right shows that for the $3\/2^{-}$ states after the total-angular-momentum projection.}\n\t\\label{9Li_energy_surface}\n\\end{figure}\n\nThe results of $^{11}$B are shown in Fig.~\\ref{11B_energy_surface}.\nThe negative-parity energy surface in the left panel shows \nthe energy minimum point at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.13, 0.13)$.\nAfter the total-angular-momentum projection, the minimum point of the $3\/2^{-}$ energy surface \nbecomes $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.33, 0.13)$ as seen in the right panel of\nFig.~\\ref{11B_energy_surface}. In the large prolate region, \na valley is found around $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.9, 0.1)$\nin a similar way to $^{12}$C.\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{11B_-.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{11B_1.5-.eps}}\n\t\\caption{Energy surfaces of $^{11}$B on the $\\beta$-$\\gamma$ plane.\n\tThe left shows the energy for the negative-parity states and \n\tthe right shows that for the $3\/2^{-}$ states after the total-angular-momentum projection.}\n\t\\label{11B_energy_surface}\n\\end{figure}\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[angle=-90,width=6cm]{symmetric_energy.eps}\n\t\\caption{Positive-parity energy curves on the symmetric axes, $\\gamma = 0^{\\circ}$ and $60^{\\circ}$ axes.\n\tThe energy of the prolate states on the $\\gamma = 0^{\\circ}$ line is displayed \n\tin positive $\\beta$ and that of the oblate states on the $\\gamma = 60^{\\circ}$ line \n\tis displayed in negative $\\beta$.}\n\t\\label{symmetric_energy_surface}\n\\end{figure}\n\n\\subsection{Structures on the $\\beta$-$\\gamma$ plane}\n\nIn this section, we discuss the intrinsic structures obtained \nusing the $\\beta$-$\\gamma$ constraint AMD, while paying attention to the cluster aspect.\n\nWe analyze the spatial configurations of the Gaussian centers $\\{\\bm{Z}_1,\\bm{Z}_2,\\cdots,\\bm{Z}_A\\}$ \nand proton and neutron density distributions of each intrinsic wave functions $|\\Phi(\\beta,\\gamma) \\rangle$.\nWe also investigate the difference \n$\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$ between the neutron density $\\Tilde{\\rho}_{n}$ and\nproton density $\\Tilde{\\rho}_{p}$ for neutron-rich nuclei to observe valence neutron behaviors.\nHere, the density $\\Tilde{\\rho}$ is that integrated along the $y$-axis as\n\\begin{align}\n\t& \\Tilde{\\rho} (x, z) \\equiv \\int dy \\rho (\\bm{r}), \\\\ \n\t& \\rho (\\bm{r}) \\equiv \\langle \\Phi (\\beta, \\gamma) | \\sum_{i} \\delta (\\bm{r} - \\bm{r}_{i}) |\\Phi (\\beta, \\gamma) \\rangle.\n\t\\label{density_distribution}\n\\end{align}\n\nThe density distributions for $^{10}$Be are illustrated in Fig.~\\ref{density_10Be}.\nFigure~\\ref{density_10Be}(a) shows the density distributions for $|\\Phi (\\beta,\\gamma) \\rangle$ at\n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.00)$, which is \nthe energy minimum point of the positive-parity energy surface.\nIn this state, two $\\alpha$ clusters are formed as seen in the dumbbell shape of the proton density.\nOn the other hand, excess neutrons are found to occupy the axial symmetric orbitals and regarded \nto be in the $\\pi_{3\/2}$-like molecular orbitals. \nIn fact, the expectation value of the squared neutron spin $\\langle S_{n}^{2} \\rangle$ is \n1.07 and is close to the value $\\langle S_{n}^{2} \\rangle=1$ \nfor the $(\\pi_{3\/2})^{2}$ configuration.\nFigure~\\ref{density_10Be}(b) shows the density distribution of the energy minimum state\nwith $(\\beta \\cos \\gamma, \\beta \\sin \\gamma)=(0.55, 0.09)$ in the $0^{+}$ energy surface.\nIt is found that two $\\alpha$ clusters develop further.\nIn this state, the density distribution of excess neutrons is not axial symmetric but\nis extended to the $x$-axis direction resulting in the axial asymmetric.\nThis means that excess neutrons occupy the $p_{x}$-like orbitals to have the finite $\\gamma$ value\ninstead of the axial symmetric molecular orbitals.\nFurthermore, $\\langle S_{n}^{2} \\rangle$ becomes small to be 0.47. \nThis indicates that excess neutrons have the larger $S_{n}=0$ component \nthan in the Fig.~\\ref{density_10Be}(a) state and is consistent with the \ndominant $(p_{x})^{2}$ configuration.\nThis indicates that, after the total-angular-momentum\nprojection, the energy minimum state has \nthe more developed 2$\\alpha$ cluster structure and more two-neutron correlations of a spin-zero pair \nthan those before the projection.\nFigures~\\ref{density_10Be}(c), (d), and (e) are the density distributions for \ntypical structures with oblate, prolate, and triaxial deformations in the large $\\beta$ region.\nInterestingly, two $\\alpha$ clusters develop well in all these states \nwith the large deformations, but differences are found in the distributions of excess neutrons.\nIn Fig.~\\ref{density_10Be}(c) for the oblate state, \nexcess neutrons distribute far from the 2$\\alpha$ core indicating a dineutron cluster.\nIn Fig.~\\ref{density_10Be}(d)  for the almost prolate state, \nexcess neutrons distribute around one of the two $\\alpha$ cluster causing the \n$^{6}$He correlation.\nIn terms of the molecular \norbital model, this state is also regarded as the molecular structure with \nthe excess neutrons in the $\\sigma_{1\/2}$-like orbitals as discussed in \nRef.~\\citen{En'yo_10Be_99}.\nFigure~\\ref{density_10Be}(e) for the triaxial state shows \ntwo $\\alpha$ clusters and a dineutron cluster with a configuration other than that in the oblate state.\nWhen we pay attention to the neutron densities of these three states, \nit is found that the neutron structure changes \nfrom the triangular configuration in the oblate region (c) to the narrow chainlike structure in the\nprolate region (d) as $\\gamma$ becomes small.\nTherefore, the structures of $^{10}$Be as functions of deformation parameters,\n$\\beta$ and $\\gamma$, are understood as follows. \nAs $\\beta$ increases on the $\\gamma = 0^{\\circ}$ axis, \na cluster structure with two $\\alpha$ clusters and surrounding two neutrons develops, \nand the excess neutron orbitals change from the $\\pi_{3\/2}$-like orbitals \nto the $\\sigma_{1\/2}$-like orbitals.\nOn the other hand, as $\\gamma$ becomes finite from $0^{\\circ}$ to\n$60^{\\circ}$, two neutrons have more the $S_{n}=0$ correlation.\nWhen the deformation is sufficiently large,\na dineutron cluster develops and drops out of the 2$\\alpha$ core.\n\n\n\\begin{figure}[tb]\n\t\\centering\n\t{\\tabcolsep=0.7mm\n\t\\begin{tabular}{cccccc}\n\t\\vspace{0.3cm}\n\t($\\beta \\cos \\gamma$, $\\beta \\sin \\gamma$) &\n\t\\hspace{-0.2cm} $\\Tilde{\\rho}_{p}$ & \\hspace{-0.2cm} $\\Tilde{\\rho}_{n}$ \n\t& \\hspace{-0.15cm}  $\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$ & \\hspace{-0.15cm}  $\\Tilde{\\rho}_{n}^{\\prime} - \\Tilde{\\rho}_{p}^{\\prime}$ \\\\\n\t\\vspace{-2.2cm}\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(a) ($0.35$, $0.00$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{0Be_10211_p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_10211_n.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_10211_n-p.eps} & \n\t\\includegraphics[width=1.8cm]{0Be_10211_n-p_xy.eps} &\n\t\\begin{minipage}{.10\\linewidth}\\vspace{-3.2cm}[1\/fm$^{2}$]\\end{minipage} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(b) ($0.55$, $0.09$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{0Be_20301_p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_20301_n.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_20301_n-p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_20301_n-p_xy.eps} &\n\t\\includegraphics[width=1.5cm,bb = 0 0 60 160,clip]{legend.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(c) ($0.28$, $0.39$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{0Be_55055_p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_55055_n.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_55055_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(d) ($1.03$, $0.04$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{0Be_15624_p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_15624_n.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_15624_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(e) ($0.70$, $0.26$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{0Be_40348_p.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_40348_n.eps} &\n\t\\includegraphics[width=1.8cm]{0Be_40348_n-p.eps} \\\\\n\t\\end{tabular}}\n\t\\caption{Density distributions of $^{10}$Be.\n\tThe proton density $\\Tilde{\\rho}_{p}$, neutron density $\\Tilde{\\rho}_{n}$, \n\tdifference between the neutron and proton densities \n\t$\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$ integrated along the $y$-axis\n\tare illustrated in the left three panels.\n\tThe density distributions of the intrinsic wave functions at \n\t(a) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.35,0.00)$,\n\t(b) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.55,0.09)$,\n\t(c) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.28,0.39)$,\n\t(d) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(1.03,0.04)$, and \n\t(e) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.70,0.26)$ on the $\\beta$-$\\gamma$ plane are shown. \n\tFor the states (a) and (b), the difference between neutron and proton densities \n\t$\\Tilde{\\rho}_{n}^{\\prime} - \\Tilde{\\rho}_{p}^{\\prime}$, which is integrated \n\talong the $z$-axis but not the $y$-axis, is illustrated in the right panels.\n\tThe size of the box is 10 $\\times$ 10 fm$^{2}$.}\n\t\\label{density_10Be}\n\\end{figure}\n\nNext, we show the density distributions of $^{12}$C in Fig.~\\ref{density_12C}.\nIn all the obtained wave functions $|\\Phi (\\beta,\\gamma) \\rangle$, \nthe proton and neutron densities are almost the same with each other.\nFigure~\\ref{density_12C}(a) is the density distribution for the minimum point \n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma)=(0.13, 0.22)$ in the positive-parity energy surface.\nIn this wave function, the centers of the single-particle Gaussian wave packets \ngather around the origin and there is no spatially developed cluster structure. \nThe expectation value of squared intrinsic spin of protons and that of neutrons are 0.45,\nwhich indicates a large component of non-3$\\alpha$ configurations.\nIn other words, this state is considered to be the shell-model-like structure \nwith the dominant $p_{3\/2}$ subshell closed component.\nFigure~\\ref{density_12C}(b) is the density distribution for the \nminimum point $(\\beta \\cos \\gamma, \\beta \\sin \\gamma)=(0.35, 0.17)$ in the $0^{+}$ energy surface.\nIn this state, an $\\alpha$ cluster begins to develop compared with the state (a) \nfor the energy minimum before the total-angular-momentum projection.\nHowever, single-particle Gaussian wave packets still gather around the origin, and this state\nis regarded as the intermediate between the shell-model-like structure and the cluster structure.\nIn the large deformation region, three $\\alpha$ clusters develop well in $^{12}$C.\nAs is expected, various configurations of 3 $\\alpha$ clusters appear, depending on the deformation\nparameters, $\\beta$ and $\\gamma$.\nFigures~\\ref{density_12C}(c), (d), and (e) are typical density distributions \nfor the oblate, prolate, and triaxial deformed states, respectively.\nIt is found that the equilateral-triangular configuration,\nlinear-chainlike structure, and obtuse-angle-triangular configuration arise\nin the oblate state (c), prolate state (d), and triaxial state (e), respectively.\nThus, on the $\\beta$-$\\gamma$ plane for $^{12}$C system, \nwe obtained various structures such as \nthe shell-model structure in the small $\\beta$ region\nand the cluster structure in the large $\\beta$ region.\nIn particular, three $\\alpha$ clusters develop as the deformation parameter $\\beta$ \nbecomes large. Various spatial configurations of three $\\alpha$ clusters such as \nthe linear chain and the equilateral-triangular structures are obtained \nas a function of the triaxiality $\\gamma$.\n\n\\begin{figure}[tb]\n\t\\centering\n\t{\\tabcolsep=0.7mm\n\t\\begin{tabular}{cccc}\n\t\\vspace{0.3cm}\n\t($\\beta \\cos \\gamma$, $\\beta \\sin \\gamma$) &\n\t\\hspace{-0.2cm} $\\Tilde{\\rho}_{p}$ & \\hspace{-0.2cm} $\\Tilde{\\rho}_{n}$ \\\\\n\t\\vspace{-2.2cm}\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(a) ($0.13$, $0.22$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{12C_35017_p.eps} &\n\t\\includegraphics[width=1.8cm]{12C_35017_n.eps} &\n\t\\begin{minipage}{.10\\linewidth}\\vspace{-3.2cm}[1\/fm$^{2}$]\\end{minipage} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(b) ($0.35$, $0.17$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{12C_30178_p.eps} &\n\t\\includegraphics[width=1.8cm]{12C_30178_n.eps} &\n\t\\includegraphics[width=1.5cm,bb = 0 0 60 160,clip]{legend.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(c) ($0.33$, $0.48$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{12C_65038_p.eps} &\n\t\\includegraphics[width=1.8cm]{12C_65038_n.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(d) ($1.13$, $0.04$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{12C_15668_p.eps} &\n\t\\includegraphics[width=1.8cm]{12C_15668_n.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(e) ($0.65$, $0.26$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{12C_40330_p.eps} &\n\t\\includegraphics[width=1.8cm]{12C_40330_n.eps} \\\\\n\t\\end{tabular}}\n\t\\caption{Density distributions of $^{12}$C.\n\tThe proton density $\\Tilde{\\rho}_{p}$ and neutron density $\\Tilde{\\rho}_{n}$ \n\tare illustrated in the left and right columns, respectively. \n\tThe density distributions of the intrinsic wave functions at \n\t(a) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.13,0.22)$,\n\t(b) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.35,0.17)$,\n\t(c) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.33,0.48)$,\n\t(d) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(1.13,0.04)$, and \n\t(e) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.65,0.26)$ on the $\\beta$-$\\gamma$ plane are shown.\n\tThe size of the box is 10 $\\times$ 10 fm$^{2}$.}\n\t\\label{density_12C}\n\\end{figure}\n\nFigure~\\ref{density_9Li} shows the density distributions of $^{9}$Li.\nFigures~\\ref{density_9Li}(a) and (b) correspond to the \nenergy minimum states before and after the total-angular-momentum\nprojection, respectively.\nFigures~\\ref{density_9Li}(c), (d), and (e) are density distributions of typical structures\nin the oblate, prolate, and triaxial states with large deformations, respectively.\nThe cluster features in $^{9}$Li are a good analogy to those in $^{10}$Be, and\ntherefore, they can be understood in a similar way as the above discussion for \n$^{10}$Be by replacing one $\\alpha$ cluster in the $^{10}$Be system to a triton cluster in $^9$Li.\nNamely, as the deformation parameter $\\beta$ increases on the $\\gamma = 0^{\\circ}$ axis, \nthe structure changes from the spherical shell-model state \nto the developed cluster structure with the $\\alpha+t$ cluster and\nsurrounding two excess neutrons in the axial symmetric molecular orbitals.\nAs the triaxiality $\\gamma$ increases from 0$^\\circ$ to\n60$^\\circ$, a pair of two neutrons drops out of the $\\alpha+t$ cluster \nand describes the two-neutron correlation.\n\n\\begin{figure}[tb]\n\t\\centering\n\t{\\tabcolsep=0.7mm\n\t\\begin{tabular}{ccccc}\n\t\\vspace{0.3cm}\n\t($\\beta \\cos \\gamma$, $\\beta \\sin \\gamma$) &\n\t\\hspace{-0.2cm} $\\Tilde{\\rho}_{p}$ & \\hspace{-0.2cm} $\\Tilde{\\rho}_{n}$ & \\hspace{-0.15cm}  $\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$ \\\\\n\t\\vspace{-2.2cm}\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(a) ($0.15$, $0.00$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{9Li_10094_p.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_10094_n.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_10094_n-p.eps} &\n\t\\begin{minipage}{.10\\linewidth}\\vspace{-3.2cm}[1\/fm$^{2}$]\\end{minipage} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(b) ($0.38$, $0.13$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{9Li_25201_p.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_25201_n.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_25201_n-p.eps} &\n\t\\includegraphics[width=1.5cm,bb = 0 0 60 160,clip]{legend.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(c) ($0.28$, $0.39$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{9Li_55058_p.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_55058_n.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_55058_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(d) ($1.03$, $0.04$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{9Li_15619_p.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_15619_n.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_15619_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(e) ($0.70$, $0.26$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{9Li_40348_p.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_40348_n.eps} &\n\t\\includegraphics[width=1.8cm]{9Li_40348_n-p.eps} \\\\\n\t\\end{tabular}}\n\t\\caption{Density distributions of $^{9}$Li.\n\tThe proton density $\\Tilde{\\rho}_{p}$, neutron density $\\Tilde{\\rho}_{n}$, \n\tand difference between the neutron and proton densities $\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$\n\tare illustrated in the left, middle, and right columns, respectively. \n\tThe density distributions of the intrinsic wave functions at \n\t(a) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.15,0.00)$,\n\t(b) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.38,0.13)$,\n\t(c) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.28,0.39)$,\n\t(d) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(1.03,0.04)$, and \n\t(e) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.70,0.26)$ on the $\\beta$-$\\gamma$ plane are shown.\n\tThe size of the box is 10 $\\times$ 10 fm$^{2}$.}\n\t\\label{density_9Li}\n\\end{figure}\n\nThe density distributions of $^{11}$B are illustrated in Fig.~\\ref{density_11B}.\nThe shell and cluster features of $^{11}$B have a good correspondence to those of $^{12}$C.\nIn the small deformation region, the neutron density has an almost spherical shape \nand is regarded as the dominant $p_{3\/2}$-shell closed configuration.\nAs the deformation parameter $\\beta$ increases, the structure changes from a shell-model state to \na developed cluster structure via the shell-cluster competition region.\nThe cluster features in the large deformation region are understood\nwith the $2\\alpha+t$ clustering in $^{11}$B instead of the 3$\\alpha$ cluster structure in $^{12}$C. \nSimilarly to the case of $^{12}$C, as a function of $\\gamma$, \nwe obtained various types of spatial configurations of three-center clusters such as \nthe equilateral-triangular structure in the oblate deformation \nand the linear-chain structure in the prolate deformation.\n\n\\begin{figure}[tb]\n\t\\centering\n\t{\\tabcolsep=0.7mm\n\t\\begin{tabular}{ccccc}\n\t\\vspace{0.3cm}\n\t($\\beta \\cos \\gamma$, $\\beta \\sin \\gamma$) &\n\t\\hspace{-0.2cm} $\\Tilde{\\rho}_{p}$ & \\hspace{-0.2cm} $\\Tilde{\\rho}_{n}$ & \\hspace{-0.15cm}  $\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$ \\\\\n\t\\vspace{-2.2cm}\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(a) ($0.13$, $0.13$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{11B_25057_p.eps} &\n\t\\includegraphics[width=1.8cm]{11B_25057_n.eps} &\n\t\\includegraphics[width=1.8cm]{11B_25057_n-p.eps} &\n\t\\begin{minipage}{.10\\linewidth}\\vspace{-3.2cm}[1\/fm$^{2}$]\\end{minipage} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(b) ($0.33$, $0.13$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{11B_25123_p.eps} &\n\t\\includegraphics[width=1.8cm]{11B_25123_n.eps} &\n\t\\includegraphics[width=1.8cm]{11B_25123_n-p.eps} &\n\t\\includegraphics[width=1.5cm,bb = 0 0 60 160,clip]{legend.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(c) ($0.33$, $0.48$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{11B_65055_p.eps} &\n\t\\includegraphics[width=1.8cm]{11B_65055_n.eps} &\n\t\\includegraphics[width=1.8cm]{11B_65055_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(d) ($1.13$, $0.04$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{11B_15661_p.eps} &\n\t\\includegraphics[width=1.8cm]{11B_15661_n.eps} &\n\t\\includegraphics[width=1.8cm]{11B_15661_n-p.eps} \\\\\n\t\\begin{minipage}{.17\\linewidth}\\vspace{-1.7cm}(e) ($0.65$, $0.26$)\\end{minipage} &\n\t\\includegraphics[width=1.8cm]{11B_40323_p.eps} &\n\t\\includegraphics[width=1.8cm]{11B_40323_n.eps} &\n\t\\includegraphics[width=1.8cm]{11B_40323_n-p.eps} \\\\\n\t\\end{tabular}}\n\t\\caption{Density distributions of $^{11}$B.\n\tThe proton density $\\Tilde{\\rho}_{p}$, neutron density $\\Tilde{\\rho}_{n}$, \n\tand difference between the neutron and proton densities $\\Tilde{\\rho}_{n} - \\Tilde{\\rho}_{p}$\n\tare illustrated in the left, middle, and right columns, respectively. \n\tThe density distributions of the intrinsic wave functions at \n\t(a) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.13,0.13)$,\n\t(b) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.33,0.13)$,\n\t(c) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.33,0.48)$,\n\t(d) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(1.13,0.04)$, and \n\t(e) $(\\beta \\cos \\gamma,\\beta \\sin \\gamma)=(0.65,0.26)$ on the $\\beta$-$\\gamma$ plane are shown.\n\tThe size of the box is 10 $\\times$ 10 fm$^{2}$.}\n\t\\label{density_11B}\n\\end{figure}\n\nWe summarize the shell and cluster features of these $N=6$ systems \nobtained by the present method of the $\\beta$-$\\gamma$ constraint AMD on the $\\beta$-$\\gamma$ plane.\nIn the small $\\beta$ region, $\\beta < 0.2 \\sim 0.3$, shell-model-like structures appear.\nIn particular, the neutron structure has the dominant $p_{3\/2}$-shell closed\nconfigurations with the spherical shape.\nIn this region, the centers of the single-particle Gaussian wave packets \ngather around the origin to keep the deformation small and \nthere are no spatially developed cluster structures. \nIn the large $\\beta$ region, cluster structures develop well.\nDepending on the $\\gamma$ parameter, various cluster configurations appear.\nThat is, in the prolate region near the $\\gamma = 0^{\\circ}$ line,\ntwo-body cluster structures or linear-chainlike structures \ndevelop well as $\\beta$ becomes large.\nIn the oblate region near the $\\gamma = 60^{\\circ}$ line, \nthree-body cluster structures with equilateral-triangular or isosceles-triangular configurations appear.\nIn the triaxial region, various triangle configurations of the three-body clusters are obtained.\n\n\\subsection{GCM results}\n\nWe superposed the wave functions obtained using the $\\beta$-$\\gamma$ constraint AMD with the GCM to calculate \nthe energy spectra of $^{10}$Be and $^{12}$C.\n\nFirst, we describe the results for $^{10}$Be.\nThe calculated binding energy (B.E.) is $59.2$ MeV, while the experimental one is 64.98 MeV.\nIn the results, we obtained many excited states above the ground state.\nWe show the calculated energy levels of $^{10}$Be in Fig.~\\ref{Energy_level_0Be_+}\nas well as the experimental levels.\\cite{Tilley_A=8-9-10_04,Freer_10Be_06,Curtis_Be_06,Bohlen_10Be_07} \\ \n\nThe calculated results reproduce well the experimental energy levels.\nWe classified the calculated states in four groups by analyzing \nthe components of the basis wave functions and intrinsic structures.\n\n\\begin{figure}[tb]\n\\centering\n\t\\includegraphics[angle=-90,width=11.2cm]{0Be_level_base_cut.eps}\n\t\\caption{Energy levels of the positive-parity states in $^{10}$Be.\n\t\tFour columns on the left are the experimental data, four columns in the middle are the two-dimensional $\\beta$-$\\gamma$ GCM results,\n\t\tand three columns on the right are the axial symmetric GCM results.\n\t\tIn the two-dimensional $\\beta$-$\\gamma$ GCM results, we classified the states into four groups.\n\t\tThe first column is the ground band, \n\t\tthe second column is the $K^{\\pi}=2^{+}$ side band of the ground band, and\n\t\tthe third column is the $0^{+}_{2}$ band.\n\t\tOther states are plotted in the fourth column.\n\t\tThe details are described in the text.}\n\\label{Energy_level_0Be_+}\n\\end{figure}\n\nThe calculated $0^{+}_{1}$, $2^{+}_{1}$, and $4^{+}_{1}$ states constitute the ground band.\nThey have the large component of the basis wave function at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.55, 0.09)$.\nFor instance, the overlap of the $0^{+}$ state with this  \nwave function is 87\\%.\nAs discussed before, in this dominant basis wave function, \ntwo $\\alpha$ clusters develop well and excess neutrons occupy $p_{x}$-like orbitals.\nThe $0^{+}_{1}$ state also has a large overlap with the wave function at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.00)$\nwith 70\\%. As already mentioned earlier, the \nwave function at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.00)$ shows the structure with\nthe $2\\alpha$ core and excess neutrons in the $\\pi_{3\/2}$-like molecular orbitals.\nTherefore, roughly speaking, the ground bands can be interpreted as the molecular orbital structure with the $2\\alpha$ core\nand excess neutrons. However, it should be pointed out that the dominant basis wave function is not the axial symmetric one \nbut the triaxially deformed state with \nthe excess neutrons in the $p_{x}$-like orbitals. This means that the two-neutron correlation is contained \nsignificantly in the ground band.\n\nThe $2^{+}_{2}$ and $3^{+}_{1}$ states construct the $K^{\\pi}=2^{+}$ side band built on the ground band.\nThese states appear because of the triaxiality of the ground band. \nNamely, the $K^{\\pi}=2^{+}$ band arises because the excess neutrons occupy the $p_{x}$-like orbitals\ndue to the two-neutron correlation.\nThis is consistent with the theoretical work by Itagaki et al. where the triaxiality in $^{10}$Be was discussed.\\cite{Itagaki_10Be_02} \\ \n\nThe $0^{+}_{2}$, $2^{+}_{3}$, and $4^{+}_{2}$ states belong to the excited $K^{\\pi}=0^{+}$ band\nwith a large prolate deformation.\nIn fact, they have a large overlap with the wave function at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.03, 0.04)$,\nfor example, as 64\\% overlap in the $0^{+}_{2}$ state.\nAs we discussed Fig.~\\ref{density_10Be}(d), this dominant wave function has the extremely large prolate deformation\ndue to the developed two $\\alpha$ clusters and excess neutrons surrounding the 2$\\alpha$ core.\n\nWe also obtained other excited states below 14 MeV plotted in the rightmost column in Fig.~\\ref{Energy_level_0Be_+}.\nThe $0^{+}_3$ state around 12 MeV is dominated by the basis wave function with developed two $\\alpha$ and dineutron clusters.\nThis might relate with the $2\\alpha$+dineutron condensate state suggested by Itagaki et al.\\cite{Itagaki_alpha_condense_07} \\ \n\nThe calculated $E2$ transition strengths are listed in Table~\\ref{E2_Radius_10Be}~(A).\nThe calculated strengths, $B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{1})=9.4$ $e^{2}$fm$^{4}$ and \n$B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{2})=1.2$ $e^{2}$fm$^{4}$, \nagree well with the experimental data, $B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{1})=10.24 \\pm 0.97$ $e^{2}$fm$^{4}$ and \n$B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{2})=0.64 \\pm 0.23$ $e^{2}$fm$^{4}$,\\cite{Tilley_A=8-9-10_04} \\ respectively.\nThe interband transition strengths between the ground band and its $K^{\\pi}=2^{+}$ side band are\n$B(E2, 2^{+}_{2} \\rightarrow 0^{+}_{1})=0.7$ $e^{2}$fm$^{4}$ and $B(E2, 2^{+}_{2} \\rightarrow 2^{+}_{2})=4.2$ $e^{2}$fm$^{4}$.\nThe origin of these interband transition strengths is \nthe recoil effect of the valence neutrons related to two $\\alpha$ clusters as discussed in Ref.~\\citen{{Itagaki_10Be_02}}.\n\nThe calculated root-mean-square radii for mass distributions are listed in Table~\\ref{E2_Radius_10Be}~(B).\nThe calculated radius of the $0^{+}_{1}$ state, 2.39 fm, is close to the \nexperimental one, $2.30 \\pm 0.02$ fm.\\cite{Ozawa_rmsRadius_01} \\ \n\n\\begin{table}[tb]\n\t\\caption{Electromagnetic transition strengths $B(E2)$ and radii of $^{10}$Be. \n\t(A)  Calculated $B(E2)$ values in the unit of $e^{2}$fm$^{4}$.\n\tThe values in the $\\beta$-$\\gamma$ GCM column are the two-dimensional $\\beta$-$\\gamma$ GCM results and \n\tthe values in the AS GCM column are the axial symmetric GCM results.\n\t(B) Root-mean-square radii for mass distributions of $^{10}$Be calculated by the \n\ttwo-dimensional $\\beta$-$\\gamma$ GCM. The unit is fm.}\n\t\\label{E2_Radius_10Be}\n\t\\begin{minipage}[t]{.55\\textwidth}\n\t\t\\centering\n\t\t(A) $B(E2)$\n\t\t\\begin{tabular}{cccc} \\hline \\hline\n\t\tTransitions & $\\beta$-$\\gamma$ GCM & AS GCM & Experiment \\\\ \\hline\n\t\t$2^{+}_{1} \\rightarrow 0^{+}_{1}$ & 9.4 & 7.8 & $10.24 \\pm 0.97$ \\\\\n\t\t$2^{+}_{1} \\rightarrow 0^{+}_{2}$ & 1.2 & 0.1 & $0.64 \\pm 0.23$ \\\\\n\t\t$2^{+}_{2} \\rightarrow 0^{+}_{1}$ & 0.7 & 0.1 & \\\\\n\t\t$2^{+}_{2} \\rightarrow 2^{+}_{1}$ & 4.2 & 0.8 & \\\\\n\t\t\\hline \\hline\n\t\t\\end{tabular}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}[t]{.40\\textwidth}\n\t\t\\centering\n\t\t(B) Root-mean-square radii\n\t\t\\begin{tabular}{ccc} \\hline \\hline\n\t\tStates & $\\beta$-$\\gamma$ GCM & Experiment \\\\ \\hline\n\t\t$0^{+}_{1}$ & 2.39 & $2.30 \\pm 0.02$ \\\\\n\t\t$0^{+}_{2}$ & 2.98 & \\\\\n\t\t$0^{+}_{3}$ & 2.96 & \\\\ \n\t\t\\hline \\hline\n\t\t\\end{tabular}\n\t\\end{minipage}\n\\end{table}\n\nThe present results of $^{10}$Be are quantitatively similar to the earlier work studied with \nthe molecular orbital model\\cite{Itagaki_10Be_00} \\ and also to that with AMD.\\cite{En'yo_10Be_99} \\ \nIt indicates that the $\\beta$-$\\gamma$ constraint AMD is a useful method of preparing\nproper basis wave functions that are sufficient to describe the ground and excited states of $^{10}$Be.\n\nTo check the advantages of the present method of the two-dimensional constraint\nand to discuss the importance of the triaxiality, \nwe also calculated the one-dimensional axial symmetric GCM in which the basis is limited to axial symmetric ones\nand compared the results with the two-dimensional GCM ones.\nThis is a GCM calculation performed in a subset of two-dimensional $\\beta$-$\\gamma$ GCM. \nThe number of superposed basis is only 40 on the $\\gamma=0^{\\circ}$ and $\\gamma=60^{\\circ}$ axes,\nwhereas 196 basis are superposed in the two-dimensional $\\beta$-$\\gamma$ GCM calculation.\nThe calculated energy levels are shown in Fig.~\\ref{Energy_level_0Be_+}.\nThe calculated B.E. is $58.6$ MeV, which is just $0.6$ MeV smaller than the $\\beta$-$\\gamma$ GCM result.\nWe found that axial symmetric basis wave functions are appropriate to reproduce the ground state.\nOn the other hand, the axial symmetric GCM fails to reproduce some excited states.\nFor example, the $2^{+}_{2}$ and $3^{+}_{1}$ states are 5.5 MeV higher than the $\\beta$-$\\gamma$ GCM result.\nFurthermore, the agreement of the calculated $B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{1})$ with the experimental result\nis worse than the $\\beta$-$\\gamma$ GCM result, as shown in Table~\\ref{E2_Radius_10Be}.\nThese failures are caused by the insufficiency of the GCM basis, which miss the triaxial ones.\nAs a result of the lack of triaxial wave functions,\ninterband transition strengths, $B(E2, 2^{+}_{2} \\rightarrow 0^{+}_{1})$ and \n$B(E2, 2^{+}_{2} \\rightarrow 2^{+}_{2})$, become small.\n\nNext, we discuss the results of $^{12}$C.\nThe calculated B.E. is $92.5$ MeV (The experimental B.E. is 92.16 MeV).\nIn Fig.~\\ref{Energy_level_12C_+}, we give the calculated energy levels of $^{12}$C \nas well as the experimental data.\\cite{Ajzenber_A=11-12_90} \\ \nFor the theoretical results, we classified the states in three groups by analyzing overlaps with the basis wave functions.\nIn Fig.~\\ref{overlap_12C_0_+_2}, \nthe squared overlaps $|\\langle 0^+| P^{J=0}_{00} | \\Phi^+ (\\beta,\\gamma) \\rangle|^2$\nof the GCM wave functions for the $0^{+}$ states with \nthe basis AMD wave functions at each point $(\\beta, \\gamma)$ are shown.\n\n\\begin{figure}[tb]\n\\centering\n\t\\includegraphics[angle=-90,width=11.2cm]{12C_level_base_cut.eps}\n\t\\caption{Energy levels of the positive-parity states in $^{12}$C.\n\t\tTwo columns on the left are the experimental data, three columns in the middle are the two-dimensional $\\beta$-$\\gamma$ GCM results,\n\t\tand three columns on the right are the axial symmetric GCM results.\n\t\tIn the two-dimensional $\\beta$-$\\gamma$ GCM results, we classified the states in three groups.\n\t\tThe first column is the ground band and\n\t\tthe second column shows the spectra of the states, which have overlap with the basis AMD wave function\n\t\tin the broad $\\gamma$ area of the large $\\beta$ region.\n\t\tThe third column is the $0^{+}_{3}$ states.\n\t\tThe details are described in the text.}\n\\label{Energy_level_12C_+}\n\\end{figure}\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{overlap_12C_0_1st.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{overlap_12C_0_2nd.eps}} \\\\\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{overlap_12C_0_3rd.eps}}\n\t\\caption{Overlaps of the GCM wave functions for the $0^{+}$ states of $^{12}$C with \n\t\teach basis AMD wave functions obtained with the $\\beta$-$\\gamma$ constraint.\n\t\tThe overlap with a certain ($\\beta,\\gamma$) point is shown in the contour map.\n\t\tThe solid lines show the $\\beta$ constraint paths, which indicate\n\t\tthe optimum $\\gamma$ of the energy minimum position for a given $\\beta$ value. }\n\\label{overlap_12C_0_+_2}\n\\end{figure}\n\nThe $0^{+}_{1}$, $2^{+}_{1}$, and $4^{+}_{1}$ states constitute the ground band. \nThey have a large overlap with the basis wave function at $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (0.35, 0.17)$,\nfor instance, as the 90\\% overlap in the $0^{+}_{1}$ state.\nAs discussed earlier, this dominant basis wave function has \nno spatial development of three $\\alpha$ clusters but the shell-model-like structure (Fig.~\\ref{density_12C}(b)). \nIt is interesting that the dominant wave function shows the triaxiality in the intrinsic deformation.\n\nThe calculated states displayed in the second column of the right side in Fig.~\\ref{Energy_level_12C_+}\nhave large overlaps with the basis AMD wave functions\nin the broad $\\gamma$ area of the large $\\beta$ region. For example, as shown in Fig.~\\ref{overlap_12C_0_+_2}, \nthe components of the $0^{+}_{2}$ state do not concentrate on some specific basis wave function but distribute\nwidely into various basis wave functions. \nIn this area of the large $\\beta$ region, the basis wave functions describe\nvarious configurations of the developed three $\\alpha$ clusters depending on $\\beta$ and $\\gamma$.\nIt means that the $0^{+}_{2}$ state is described using the linear combination of various 3$\\alpha$-cluster configurations.\nThis is analogous to the earlier works with\nthe 3$\\alpha$ GCM,\\cite{Uegaki_12C_77} \\ FMD,\\cite{Neff_12C_04} \\ and \nAMD calculations,\\cite{En'yo_12C_07} \\ \nand is also consistent with the suggestion of the $\\alpha$ condensate state.\\cite{Tohsaki_12C_01,Funaki_12C_03} \\ \nThis indicates that the present method of $\\beta$-$\\gamma$ constraint is effective for describing \nthe three-body cluster feature of the excited states described by a superposition of multiconfigurations.\nIn particular, the degree of freedom of the triaxiality $\\gamma$ is essential to describe \nthis feature. A more detailed discussion will be given in the next section.\n\nThe $0^{+}_3$ state has the largest overlap with the wave function at \n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.13, 0.04)$ with 54\\% overlap.\nAs shown in Fig.~\\ref{density_12C}(d), \nthis basis wave function has three $\\alpha$ clusters with the linear-chain-like configuration.\nThis result is consistent with the prediction of the $0^+_3$ state in the \n3$\\alpha$ model,\\cite{Uegaki_12C_77} \\ \nFMD,\\cite{Neff_12C_04} \\ and AMD calculations.\\cite{En'yo_12C_07} \\ \n\nThe calculated $E2$ transition strengths are listed in Table~\\ref{E2_Radius_12C}~(A).\nThe calculated strengths, $B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{1})=6.0$ $e^{2}$fm$^{4}$ and \n$B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{2})=1.9$ $e^{2}$fm$^{4}$, \nagree well with the experimental results, $B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{1})=7.59 \\pm 0.42$ $e^{2}$fm$^{4}$ and \n$B(E2, 2^{+}_{1} \\rightarrow 0^{+}_{2})=2.6 \\pm 0.4$ $e^{2}$fm$^{4}$,\\cite{Ajzenber_A=11-12_90} \\ respectively.\n\nThe calculated root-mean-square radii for mass distributions are listed in Table~\\ref{E2_Radius_12C}~(B).\nThe calculated value of the $0^{+}_{1}$ state, 2.31 fm, is close to the experimental value, \n$2.35 \\pm 0.02$ fm or $2.31 \\pm 0.02$ fm.\\cite{Ozawa_rmsRadius_01} \\ \n\n\\begin{table}[tb]\n\t\\caption{Electromagnetic transition strengths $B(E2)$ and radii of $^{12}$C. \n\t(A)  Calculated $B(E2)$ values in the unit of $e^{2}$fm$^{4}$.\n\tThe values in the $\\beta$-$\\gamma$ GCM column are the two-dimensional $\\beta$-$\\gamma$ GCM results and \n\tthe values in the AS GCM column are the axial symmetric GCM results.\n\t(B) Root-mean-square radii for mass distributions of $^{12}$C calculated by the \n\ttwo-dimensional $\\beta$-$\\gamma$ GCM. The unit is fm.}\n\t\\label{E2_Radius_12C}\n\t\\begin{minipage}[t]{.55\\textwidth}\n\t\t\\centering\n\t\t(A) $B(E2)$\n\t\t\\begin{tabular}{cccc} \\hline \\hline\n\t\tTransitions & $\\beta$-$\\gamma$ GCM & AS GCM & Experiment \\\\ \\hline\n\t\t$2^{+}_{1} \\rightarrow 0^{+}_{1}$ & 6.0 & 5.3 & $7.59 \\pm 0.42$ \\\\\n\t\t$2^{+}_{1} \\rightarrow 0^{+}_{2}$ & 1.9 & 1.5 & $2.6 \\pm 0.4$ \\\\ \n\t\t$2^{+}_{2} \\rightarrow 0^{+}_{1}$ & 1.1 & 0.4 & \\\\ \n\t\t$2^{+}_{2} \\rightarrow 0^{+}_{2}$ & 58 & 11 & \\\\\n\t\t\\hline \\hline\n\t\t\\end{tabular} \n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}[t]{.40\\textwidth}\n\t\t\\centering\n\t\t(B) Root-mean-square radii\n\t\t\\begin{tabular}{ccc} \\hline \\hline\n\t\tStates & $\\beta$-$\\gamma$ GCM & Experiment \\\\ \\hline\n\t\t$0^{+}_{1}$ & 2.31 & $2.35 \\pm 0.02$ \\\\\n\t\t& & $2.31 \\pm 0.02$ \\\\\n\t\t$0^{+}_{2}$ & 2.90 & \\\\\n\t\t$0^{+}_{3}$ & 3.26 & \\\\ \n\t\t\\hline \\hline\n\t\t\\end{tabular}\n\t\\end{minipage}\n\\end{table}\n\nThe present calculations describe various structures in $^{12}$C.\nThe shell-model structure appears in the low-lying states, while $3\\alpha$-cluster structures are found in\nthe excited states. \nThe shell and cluster coexistence obtained in the present results is consistent with \nthe works with the FMD\\cite{Neff_12C_04} \\ and AMD calculations.\\cite{En'yo_12C_07} \\ \nThe success of the present calculations \nis due to the fact that the $\\beta$-$\\gamma$ constraint AMD\nis effective for obtaining the basis wave functions for the ground and excited states of $^{12}$C.\n\nLet us discuss the energies calculated by the axial symmetric GCM. The energy levels \nobtained by the axial symmetric GCM calculations are shown in Fig.~\\ref{Energy_level_12C_+}.\nThe calculated B.E. is $91.6$ MeV, which is $0.9$ MeV smaller than the $\\beta$-$\\gamma$ GCM result.\nSince the difference from the $\\beta$-$\\gamma$ GCM result is small, \nthe axial symmetric calculation is sufficient to describe the ground state, as in the case of $^{10}$Be.\nOn the other hand, the level structure of excited states changes considerably.\nFor example, the $0^{+}_{2}$ and $0^{+}_{3}$ states are 2.6 and 3.4 MeV higher than the $\\beta$-$\\gamma$ GCM result, respectively.\nAdditionally, the $B(E2, 2^{+}_{2} \\rightarrow 0^{+}_{2})$ value becomes small\nas shown in Table~\\ref{E2_Radius_12C}.\nClearly, the triaxial basis has an essential role in the description of excited states.\n\n\n\\section{Discussion}\\label{discussions}\n\nIn calculations of deformed systems with the quadrupole deformation constraint,\none often adopts only one-dimensional $\\beta$ constraint\ninstead of the two-dimensional $\\beta$-$\\gamma$ constraint.\nAs mentioned earlier, various structures appear on the $\\beta$-$\\gamma$ plane and they are\nimportant basis wave functions, especially for the excited states.\nIn this section, we discuss the advantages of the present method of the two-dimensional constraint\nand the importance of the triaxiality described by the $\\gamma$ degree of freedom.\n\nThe simplest assumption for deformed systems is the axial symmetry \nas carried out in deformed mean-field calculations. In such axial symmetric calculations, \nthe deformation parameter $\\beta$ is the constraint parameter. \nThis corresponds to the $\\gamma=0^{\\circ}$ and $\\gamma=60^{\\circ}$ lines on the $\\beta$-$\\gamma$ plane.\nIn Fig.~\\ref{symmetric_energy_surface}, we have shown the energy curves \non the $\\gamma = 0^{\\circ}, 60^{\\circ}$ lines for $^9$Li, $^{10}$Be, $^{11}$B, and $^{12}$C.\nThe energy curves for the axial symmetric deformation show the double-well structure around\n$\\beta=0$. However, as we mentioned earlier, it does not mean the coexistence of oblate and prolate shapes\nbut it corresponds to one genuine minimum on the $\\beta$-$\\gamma$ plane for the ground state.\nFor example, in the case of $^{10}$Be, \ntwo local minima at $\\beta = -0.2$ and $\\beta = 0.35$ are observed in the axial symmetric \nenergy curve, as shown in Fig.~\\ref{symmetric_energy_surface}.\nHowever, only the $\\beta = 0.35$ minimum is the genuine minimum\nand the $\\beta = -0.2$ minimum comes from the projection of the energy pocket of the \ngenuine minimum at $\\gamma=0^\\circ$ onto the $\\gamma=60^\\circ$ line.\nIn the case of $^{12}$C, the local minima in the large prolate deformation around $\\beta = 1$ \nis also artificial. This is a part of the valley in the region near \n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.0, 0.1)$.\nThat is, a local minimum on the symmetric axis is sometimes \nan artifact owing to the axial symmetry restriction.\nTherefore, one should discuss carefully the energy curve of the axial symmetric \ncalculations. In other words, the $\\beta$-$\\gamma$ constraint method is helpful\nfor understanding the detailed behavior of the energy surface.\n\nWe mention again the one-dimensional axial symmetric GCM.\nNeedless to say, the basis wave functions with the triaxial deformations are missing in the \naxial symmetric calculations, and therefore, some results for the excited states change considerably \nin comparison with the two-dimensional $\\beta$-$\\gamma$ GCM calculations.\nFor example, the $K^{\\pi}=2^{+}$ side band of $^{10}$Be \ncannot be described with the axial symmetric GCM calculations as discussed earlier.\n\nAnother method of the one-dimensional constraint is \nthe $\\beta$ constraint method without the axial symmetry assumption, which is  \nusually applied to AMD calculations.\nIn this method, only the deformation parameter $\\beta$ is constrained, and the triaxiality $\\gamma$\nis a free parameter determined in the energy variation.\nSince $\\gamma$ is automatically optimized for each $\\beta$ constraint, it can take \na nonzero value if a system favors a triaxial deformation.\nThis method has been found to be useful for\ntreating triaxial deformations of low-lying states.\\cite{En'yo_AMD_03,Kimura_44Ti_06,Taniguchi_40Ca_07} \\ \nFigure~\\ref{beta_const_line} shows the paths of the $\\beta$ constraint method \non the $\\beta$-$\\gamma$ plane.\nFor reference, we also show the energy minimum point of the total-angular-momentum projected\nenergy surface with the cross points in the figure.\nThe energy curves along the paths obtained by the $\\beta$ constraint method\nare plotted as a function of $\\beta$ in Fig.~\\ref{beta_constraint_energy_surface}.\nAs seen in Fig.~\\ref{beta_const_line}, the $\\beta$ constraint path goes {\\it by} the \ncross point for the energy minimum point of the total-angular-momentum projected energy surface\nbut does not necessarily pass {\\it on} the minimum.\nThe difference in the minimum energy after the total-angular-momentum projection\nbetween the $\\beta$ constraint and $\\beta$-$\\gamma$ constraint is only 1 MeV\nin cases of $^{10}$Be, $^{12}$C, and $^{9}$Li.\nTherefore, we can say that the $\\beta$ constraint method is reasonable, at least, \nin the description of the ground state properties.\nHowever, as we explain below, the basis wave functions obtained by the \n$\\beta$ constraint method are insufficient to describe some excited states of $^{10}$Be and\n$^{12}$C, and the two-dimensional $\\beta$-$\\gamma$ constraint is found to be essential.\n\n\\begin{figure}[tb]\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{beta_const_0Be.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{beta_const_12C.eps}} \\\\\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{beta_const_9Li.eps}}\n\t\\hfill\n\t\\parbox{\\halftext}{\\includegraphics[width=6.6cm, bb=10 12 693 323, clip]{beta_const_11B.eps}}\n\t\\caption{$\\beta$-$\\gamma$ values obtained by the $\\beta$ constraint AMD.\n     \t\tThe solid lines show the $\\beta$ constraint paths that indicate\n            the optimum $\\gamma$ of the energy minimum position for a given $\\beta$ value.\n            The energy surfaces before total-angular-momentum projection calculated by\n            the $\\beta$-$\\gamma$ constraint AMD are also shown with the contour map.\n\t\t\tCross points stand for the energy minimum after the total-angular-momentum projection\n            obtained by the $\\beta$-$\\gamma$ constraint AMD.}\n\t\\label{beta_const_line}\n\\end{figure}\n\n\\begin{figure}[tb]\n\t\\centering\n\t\\includegraphics[angle=-90,width=6cm]{beta_const_energy.eps}\n\t\\caption{Energy curves calculated with $\\beta$ constraint AMD.}\n\t\\label{beta_constraint_energy_surface}\n\\end{figure}\n\nIn the case of $^{10}$Be, the prolate deformations are obtained mostly by \nthe $\\beta$ constraint, as seen in Fig.~\\ref{beta_const_line},\nwhich shows that the $\\beta$ constraint path goes along the $\\gamma = 0^{\\circ}$ axis.\nIn other words, the triaxially deformed structures in the finite $\\gamma$ region cannot be\nobtained within the $\\beta$ constraint method.\nOn the other hand, the $2^{+}_{2}$ and $3^{+}_{1}$ states in the $K^{\\pi}=2^{+}$ side band \nare obtained from the triaxially deformed basis wave functions.\nThat is, the $\\beta$ constraint method is insufficient to reproduce\nthe $K^{\\pi}=2^{+}$ side band in $^{10}$Be.\n\nIn the case of $^{12}$C, the two-dimensional $\\beta$-$\\gamma$ constraint method is\ncrucial for describing the $0^{+}_{2}$ state as follows.\nWe have shown the squared overlaps of the basis wave functions at each point $(\\beta, \\gamma)$ \nwith the GCM wave functions for the $0^{+}$ states in Fig.~\\ref{overlap_12C_0_+_2}.\nThe $0^{+}_{1}$ and $0^{+}_{3}$ states have large overlaps with the basis wave functions\nin the $(\\beta \\cos \\gamma, \\beta \\sin \\gamma) \\sim (0.35, 0.17)$ and \n$(\\beta \\cos \\gamma, \\beta \\sin \\gamma) = (1.13, 0.04)$ regions, respectively. \nThe $\\beta$ constraint line goes through these regions, and therefore,\nthe $0^{+}_{1}$ and $0^{+}_{3}$ states may be reproduced with the basis wave functions \non the $\\beta$ constraint line.\nHowever, the $0^{+}_{2}$ state has a large overlap with the basis wave functions \nin the broad $\\gamma$ area of the large $\\beta$ region, which \ncannot be approached in the one-dimensional $\\beta$ constraint method.\nIn this area, various configurations of the developed three $\\alpha$ clusters appear\ndepending on $\\beta$ and $\\gamma$.\nAs explained earlier, the linear combination of many 3$\\alpha$-cluster configurations is essential \nfor describing the $0^{+}_{2}$ state.\nThese facts indicate that the $\\beta$ constraint method is insufficient \nand the $\\beta$-$\\gamma$ constraint method is necessary to describe the $0^{+}_{2}$ state of $^{12}$C.\n\n\n\\section{Summary and outlook}\\label{summary}\n\nTo describe various cluster and shell structures in light nuclei,\nwe proposed a new method of $\\beta$-$\\gamma$ constraint AMD.   \nWe applied this method to the $N=6$ isotones, $^{10}$Be, $^{12}$C, $^{9}$Li, and $^{11}$B.\nIn the energy variation in the $\\beta$-$\\gamma$ constraint AMD,\nvarious structures appear depending on the two-dimensional constraint parameters, $\\beta$ and $\\gamma$.\nIn the small $\\beta$ region, shell-model-like structures appear.\nIn particular, the neutron structure has the dominant $p_{3\/2}$-shell closed\nconfigurations with the spherical shape.\nIn the large $\\beta$ region, cluster structures develop well.\nAs a function of the $\\gamma$ parameter, various cluster configurations appear. \nThat is, in the prolate region along the $\\gamma = 0^{\\circ}$ line,\ntwo-body cluster structures or linear-chainlike structures \ndevelop well as the deformation parameter $\\beta$ becomes large.\nIn the oblate region along the $\\gamma = 60^{\\circ}$ line, \nthree-body cluster structures with equilateral-triangular or isosceles-triangular configurations appear.\nIn the triaxial region, various triangle configurations of the three-body clusters are obtained.\n\nFor $^{10}$Be and $^{12}$C, we superposed the basis AMD wave functions obtained \nby the $\\beta$-$\\gamma$ constraint method to calculate the energy spectra.\nWe compared the present results with the experimental data. The results reproduce well the experimental\nspectra and are consistent with other theoretical studies. It was proved that \nthe $\\beta$-$\\gamma$ constraint method is useful for describing the ground and excited states of these nuclei.\nWe also mentioned the advantages of the two-dimensional $\\beta$-$\\gamma$ constraint method over the \none-dimensional constraint method.\nIt was confirmed that the degree of freedom of the triaxiality $\\gamma$ in the two-dimensional $\\beta$-$\\gamma$ constraint\nhas considerable importance to describe such excited states\nas the $K^{\\pi}=2^{+}$ side band of $^{10}$Be and the $0^{+}_{2}$ state of $^{12}$C.\n\nMoreover, in $^{9}$Li and $^{11}$B, cluster structures are expected to appear in the excited states \nfrom the similarity of the energy surface between $^{9}$Li and $^{10}$Be, and that between\n$^{11}$B and $^{12}$C. Further investigations with the GCM calculations using basis wave functions on the $\\beta$-$\\gamma$ plane\nare required to discuss the energy spectra of $^{9}$Li and $^{11}$B.\n\nThe present method proved to be very effective for describing various cluster and shell structures\nin the ground and excited states of light nuclei.\nWe have applied the $\\beta$-$\\gamma$ constraint AMD to other nuclei such as $^{14}$C,\nwhere shell-model-like features and cluster aspect are expected to coexist.\nWe will report the results in a future paper.\n\n\n\\section*{Acknowledgements}\n\nThe computational calculations of this work were performed  by supercomputers in YITP and KEK. \nThis work was supported by the YIPQS program in YITP.\nIt was also supported by the Grant-in-Aid for the Global COE Program \n``The Next Generation of Physics, Spun from Universality and Emergence\" \nfrom the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan,\nand a Grant-in-Aid from JSPS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAs the early universe cooled and expanded, it may have undergone\nseveral symmetry-breaking phase transitions. Depending on the details\nof the symmetry breaking, it is possible that topological defects\ncould have formed during such phase transitions. Probably the most\nimportant class of topological defects for the purposes of cosmology\nare cosmic strings~\\cite{Kibble:1976sj} (see\nRefs.~\\cite{Hindmarsh:1994re,Vilenkin:2000jqa,Copeland:2011dx,Hindmarsh:2011qj}\nfor reviews). These are one-dimensional defects which, in the simplest\ncase of an Abelian Higgs model, arise from the breaking of a\n$\\mathrm{U}(1)$ symmetry. The resulting cosmic strings are then\nextended Nielsen-Olesen vortex lines~\\cite{Nielsen:1973cs}. Cosmic\nstrings can also arise as fundamental objects from an underlying\nstring\ntheory~\\cite{Witten:1985fp,Sarangi:2002yt,Copeland:2003bj,Lizarraga:2016hpd}.\n\nAbelian Higgs strings have been widely studied, and their\nobservational consequences thoroughly\nexplored~\\cite{Laguna:1989hn,Vincent:1997cx,Moore:2001px,Bevis:2006mj,Bevis:2010gj}.\nSuperstrings or field theories with non-Abelian symmetries can produce\nricher physics: for example, the symmetry-breaking transition\n$\\mathrm{SU}(2) \\to Z_N$ has multiple species of strings with $N$-fold\njunctions \\cite{Vachaspati:1986cc}. Networks of strings with junctions\nhave been numerically simulated in\nRefs.~\\cite{Copeland:2005cy,Hindmarsh:2006qn,Urrestilla:2007yw}, and\nmodelled in Refs.~\\cite{Leblond:2007tf,Martins:2010ma}.\n\nIn this paper we report on the first 3-dimensional numerical\nsimulations of a network of strings in a non-abelian gauge theory, one\nwith symmetry-breaking SU(2)$\\to\nZ_2$~\\cite{Nielsen:1973cs,Hindmarsh:1985xc,deVega:1986eu,deVega:1986hm,Aryal:1987sn}.\nThis model is particularly attractive because it can be embedded\nnaturally in Grand Unified Theories (GUTs) such as SO(10)\n\\cite{Kibble:1982ae}, for which cosmic strings are themselves argued\nto be generic~\\cite{Jeannerot:2003qv}. In addition to the SO(10) case\nwith a single scale, our model permits two symmetry-breaking scales\nwith an intermediate unbroken U(1) symmetry, modelling a two-stage GUT\nsymmetry-breaking. In this case the first stage, SU(2)$\\to$U(1),\nproduces 't Hooft-Polyakov\nmonopoles~\\cite{Hooft:1974qc,Polyakov:1974ek}, and the second attaches\neach monopole to two strings, both carrying half the flux.  This\ncombination -- of a monopole trapped on a cosmic string -- is called a\nbead \\cite{Hindmarsh:1985xc}, and if many such beads exist on one\nstring then the configuration is commonly referred to as a\nnecklace~\\cite{Berezinsky:1997td}. For a review of these systems, see\nRef.~\\cite{Kibble:2015twa}.\n\nIn \\cite{Hindmarsh:2016lhy} we emphasised the importance of global\nsymmetries in the classification of the beads. In particular, there is\na $Z_2\\times Z_2$ symmetry spontaneously broken to $Z_2$ by the string\nsolutions, and beads can be viewed as the resulting kinks.  We\ndiscovered new solutions in the case where the SU(2) and U(1)\nsymmetry-breaking scales are degenerate, due to an enlarged discrete\nglobal symmetry $D_4$.  Each bead splits into two ``semipoles'', and\nthese four semipoles can annihilate only with the corresponding\nanti-semipole: in a generic configuration a semipole may not find\nitself next to its antipole.\n\nThe discrete global symmetry can be further promoted to a global O(2) symmetry,\nwhich is spontaneously broken by the string solution but not the\nvacuum.  Hence semipoles dissolve and the strings carry persistent\nglobal currents, rather like a tube of superfluid.\n\nThere is wide disagreement in the literature about how necklaces\nevolve in the early universe. The necklace network is characterised by\ntwo length scales, the average comoving monopole separation $\\xi_\\text{m}$ and\nthe average comoving string separation $\\xi_\\text{s}$, in terms of which the\nphysical energy densities $\\rho_\\text{m}$ and $\\rho_\\text{s}$ are\n\\begin{equation}\n\\rho_\\text{m} \\simeq \\frac{M_\\text{m}}{(a\\xi_\\text{m})^3},\\quad \\rho_\\text{s} \\simeq \\frac{\\mu}{(a\\xi_\\text{s})^2},\n\\label{e:rhom_rhos_def}\n\\end{equation}\nwhere $\\mu$ is the string mass per unit length when monopoles are\nabsent, $M_\\text{m}$ is the monopole mass and $a$ is the scale factor.\nNote that the mass of a monopole on a string is generally less than\nthat of a free monopole, so (\\ref{e:rhom_rhos_def}) is only an\nestimate of the extra energy due to the trapped monopoles.\n\nIn a normal string network, the string separation is proportional to\nthe horizon distance, so $\\xi_\\text{s} \\propto t$, where $t$ is conformal\ntime: this behaviour is known as scaling.  In a scaling network, all\nquantities with dimensions of length (apart from the string width)\ngrow in proportion to the horizon distance.\n \nIn a necklace, there is a new dynamically important length scale\n\\cite{Berezinsky:1997td}\n\\begin{equation}\n\\label{eq:dbvdefn}\nd_\\text{BV} = \\frac{M_\\text{m}}{\\mu}.\n\\end{equation}\nThe ratio of the monopole energy density to the string energy density\n$r = \\rho_\\text{m}\/\\rho_\\text{s}$ can be written as\n\\begin{equation}\n\\label{e:rDef}\nr = \\frac{d_\\text{BV}}{ad},\n\\end{equation}\nwhere $d = {\\xi_\\text{m}^3}\/{\\xi_\\text{s}^2}$ is the average comoving separation\nbetween monopoles along the string.  It was argued in\n\\cite{Berezinsky:1997td} that $r$ should grow, and it was supposed\nthat eventually the average monopole separation should tend to the\nstring width. With this assumption, $r$ would evolve quickly to a\nmaximum value set by the ratio of the two symmetry-breaking scales.\n\nHowever, it was argued in \\cite{BlancoPillado:2007zr} that this\npicture underestimates the effect of monopole annihilations, which act\nto reduce the number of monopoles per unit length of string $1\/d$. If\nmonopole annihilation is efficient, their average separation along the\nstring should scale, so $d \\propto t$, or equivalently $r \\sim\nd_\\text{BV}\/t_{\\rm p}$, where $t_{\\rm p} \\propto at$ is the physical time.\n\nGiven that the total density of the necklace network is\n$(1+r)\\mu\/(a\\xi_\\text{s})^2$, there is a very big difference in the two\nscenarios, and in particular the flux of ultra-high energy cosmic\nrays, $\\gamma$-rays and neutrinos coming from monopole annihilation\ndiffers by many orders of magnitude.  It is clearly important to\nsettle the issue.\n\nWe have performed a set of numerical simulations of a network of\nstrings in the SU(2)$\\to Z_2$ theory (see\nFig.~\\ref{fig:volplot}). They confirm the spontaneous formation of\nmonopoles, semipoles, and supercurrents along with the string network.\n\nWe are particularly interested in extracting the asymptotic behaviour\nof the network with time, as this is essential for extrapolating to\ncosmological times much later than the defect formation time.  Our\nresults support scaling behaviour in the total density of the necklace\nnetwork; that is, it decreases as $t^{-2}$, as does a conventional\ncosmic string network.  The string and monopole average separations\n$\\xi_\\text{s}$ and $\\xi_\\text{m}$ both grow with time, and the ratio of their energy\ndensities $r$ decreases with time.  Our simulations, while limited in\nrange, indicate that the mean comoving monopole separation along the\nstring $d$ is approximately constant, and stays the same order of\nmagnitude as its value when the strings form.  We also measure the\nroot mean square (RMS) velocities of both strings and monopoles,\nfinding that both relax to a values of about 0.5. If the string mass\nscale is the same as the monopole mass scale, the RMS semipole\nvelocity is a little higher than the string RMS velocity, indicating\nsome relative motion.\n\nNecklaces with a constant comoving monopole separation are a new\npossibility, which has not been considered before.  In the conclusions\nwe briefly discuss how such a network would alter the predictions for\nimportant observational signals.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{{visit-0.25-0.025-necklaces-nocredit}.jpeg}\n\\includegraphics[width=0.3\\textwidth]{{visit-kappa2-nocredit}.jpeg}\n\\includegraphics[width=0.3\\textwidth]{{visit-kappa1-nocredit}.jpeg}\n\\end{center}\n\\caption{\\label{fig:volplot} Views of a small $360^3$ simulation for\n  three different parameter choices at $t\\approx 240$. The two fields\n  $\\Phi_1$ and $\\Phi_2$ have blue and green shading respectively.  At\n  left, a simulation with $m_1^2 = 0.25$ and $m_2^2 = 0.025$ (contours\n  shown with $\\mathrm{Tr}\\,\\Phi_1^2 = 0.2$ and $\\mathrm{Tr}\\,\\Phi_2^2\n  = 0.04$), giving rise to monopoles (blue) as beads on strings\n  (green). The other two images show a system with $m_1^2 = m_2^2 =\n  0.25$ (contours shown with $\\mathrm{Tr}\\,\\Phi^2 = 0.2$ for both\n  $\\Phi_1$ and $\\Phi_2$). In the centre, $\\kappa=2$, showing the\n  semipoles at the boundaries between the two colours. At right,\n  $\\kappa=1$, and the theory has a continuous global symmetry, meaning\n  that the boundaries between the colours have no extra energy.}\n\\end{figure*}\n\n\\section{The model and its static solutions}\n\\label{s:ModSol}\nWe study the $\\mathrm{SU}(2)$ Georgi-Glashow model with two Higgs\nfields in a spatially flat Robertson-Walker metric.  In comoving\ncoordinates and conformal time, and with scale factor $a$, the action\nis\n\\begin{align}\n\\mathcal{S} &=& \\int d^4x \\left( -\\frac{1}{4}  F_{\\mu\\nu}^a F^{\\mu\\nu a} + a^2 \\sum_n \\mathrm{Tr} \\; [D_\\mu,\\Phi_n][D^\\mu,\\Phi_n] \\right. \\nonumber\\\\\n&& \\left. - a^4V(\\Phi_1, \\Phi_2) \\right)\n\\label{e:SU2HLag}\n\\end{align}\nwhere $D_\\mu = \\partial_\\mu + i g A_\\mu$ is the covariant derivative,\n$A_\\mu = A_\\mu^a \\tau^a$, and $\\tau^a = \\sigma^a\/2$ where $\\sigma^a$\nis a Pauli matrix. The Higgs fields $\\Phi_n$, $n=1,2$, are in the\nadjoint representation, $\\Phi_n = \\phi_n^a \\tau^a$. Spacetime indices\nhave been raised with the Minkowski metric with mostly negative\nsignature.\n\nThe potential is\n\\begin{multline}\n\\label{e:ScaPot}\nV(\\Phi_1, \\Phi_2) =  m_1^2 \\mathrm{Tr} \\; \\Phi_1^2 + \\lambda (\\mathrm{Tr} \\;\n\\Phi_1^2)^2 + m_2^2 \\mathrm{Tr} \\; \\Phi_2^2   \n+ \\lambda (\\mathrm{Tr} \\; \\Phi_2^2)^2  \\\\\n+ \\kappa (\\mathrm{Tr} \\; \\Phi_1 \\Phi_2)^2,\n\\end{multline}\nwith $\\lambda$ and $\\kappa$ positive.\nOne could add a $\\mathrm{Tr}\\Phi_1^2 \\mathrm{Tr}\\Phi_2^2$ term, and have \nseparate quartic couplings for the fields. However, this would not alter \nthe important dynamical features of the necklace network.\n\nThe directions of the vevs are perpendicular, because of the\n$(\\mathrm{Tr} \\; \\Phi_1 \\Phi_2)^2$ term in the potential.  The system\ntherefore undergoes two symmetry-breaking phase transitions,\n$\\mathrm{SU}(2) \\to \\mathrm{U}(1) \\to Z_2$.  The vacuum expectation\nvalues of the two adjoint scalar fields are given by $\\mathrm{Tr} \\;\n\\Phi_{1,2}^2 = \\left| m_{1,2}^2 \\right|\/2 \\lambda$, or $v_{1,2}^2 =\n\\left| m_{1,2}^2 \\right|\/ \\lambda$. The scalar masses are then\n$\\sqrt{2} m_{1,2}$.  Without loss of generality, we can label the\nscalar fields such that $\\Phi_1$ has the larger vacuum expectation\nvalue, and is responsible for the first of the symmetry-breakings.\n\nAfter the first symmetry-breaking, the theory has 't Hooft-Polyakov\nmonopole solutions with mass \\cite{Forgacs:2005vx}\n\\begin{equation}\n\\label{eq:monopolemass}\nM_\\text{m} = \\frac{4\\pi v_1}{g}\nf_\\mathrm{m}\\left(\\frac{2\\lambda}{g^2}\\right); \\qquad f_\\mathrm{m}(1)\n\\approx 1.238.\n\\end{equation}\nAfter the second symmetry-breaking, the theory has string solutions, with mass per unit length\n\\begin{equation}\n\\label{eq:stringtension}\n\\mu = \\pi v_2^2 f_\\mathrm{s}\\left(\\frac{2\\lambda}{g^2}\\right), \n\\end{equation}\nwhere $f_\\mathrm{s}(1) = 1$.\n\n\nAs described in \\cite{Hindmarsh:2016lhy}, in the generic case $m_1^2 >\nm_2^2$ this system has a discrete global $Z_2\\times Z_2$ symmetry\n$\\Phi_{1} \\to \\pm \\Phi_{1}$ and $\\Phi_{2} \\to \\pm \\Phi_{2}$.  The\nstring solutions break it down to $Z_2$. The resulting kinks\ninterpolating between the two string solutions, called beads\n\\cite{Hindmarsh:1985xc}, can be interpreted as 't Hooft-Polyakov\nmonopoles with their flux confined to two tubes.  When $m_1^2 =\nm_2^2$, the global symmetry is enlarged by the transformation $\\Phi_1\n\\to \\Phi_2$ to $D_4$, the square symmetry group, which is broken to\n$Z_2$ by strings. The resulting kinks are labelled by a $Z_4$\ntopological charge. A pair of these kinks has the same charge as a\nmonopole on a string, hence the name semipole.\n\nFinally, when $m_1^2 = m_2^2$ and $\\kappa=\\lambda$, there is a global O(2) symmetry \n\\begin{equation}\n\\label{e:U1Sym}\n\\Phi \\to e^{i\\al} \\Phi \\quad \\text{and} \\quad \\Phi \\to \\Phi^*,\n\\end{equation}\nwhere $\\Phi = \\Phi_1 + i \\Phi_2$.  The phase of the complexified\nadjoint scalar $\\theta$, defined by $\\tan \\theta = |\\Phi_2|\/|\\Phi_1|$,\nchanges smoothly along the string.  In this case the string supports\npersistent supercurrents, proportional to the gradient of the phase\nalong the string.\n\nIn order to achieve greater dynamic range, it is common practice in\ncosmic string simulations to scale the couplings and mass parameters\nwith factors $a^{1-s}$, where $a$ is the cosmological scale factor and\n$0 \\le s \\le 1$.  This is done in such a way as to keep the scalar\nexpectation value fixed As a result, the physical string width grows\nfor $s < 1$, but the string tension depends only on the ratio of the\nscalar self coupling to the square of the gauge coupling, and so stays\nconstant.  The dynamics of a string network at $s=0$ are very similar\nto those at $s=1$~\\cite{Daverio:2015nva}.\n\nBy contrast, the monopole mass $M_\\text{m}$ is inversely proportional to\nits radius, and so $M_\\text{m}$ and the dynamical quantity $d_\\text{BV}$ both grow\nthroughout simulations with $s < 1$.  It is therefore not clear how\nthe necklaces should behave in this case: the growing mass might lead\none to expect that the monopole RMS velocity should decrease, and the\nmonopole density increase.  We will see however that necklaces behave\nsimilarly with $s=0$ as they do with $s=1$.\n\n\n\n\\section{Lattice implementation}\n\\label{s:LatImp}\n\n\\subsection{Discretisation and initial conditions}\n\nWe simulate the system by setting temporal gauge $A_0 = 0$ and then\ndiscretising the system on a comoving 3D spatial lattice.  The\nHamiltonian of this model in the cosmological background takes the\nform\n\\begin{multline}\n\\label{e:ModHam}\nH(t) = \\frac{1}{2g^2a^{2(s-1)}} \\sum_{x,i,a} \\epsilon_i^a(x,t)^2 + \\frac{1}{2} a^2 \\sum_{x; \\; n,a} \\; \\pi_n^a(x,t)^2 \\\\\n + \\frac{4}{g^2a^{2(s-1)}} \\sum_{x; \\; i<j} \\left( 1- \\frac{1}{2} \\text{Tr}\n\\; U_{i j} (x,t) \\right) \\\\ \n- a^2 \\sum_{x; \\; i, n} 2 \\; \\mathrm{Tr}\\;\n\\Phi_n(x)U_i(x)\\Phi_n(x+\\hat{\\imath}) U_i^\\dag (x)  \\\\\n+ a^2 \\sum_{x,n} 6 \\, \\mathrm{Tr} \\; \\Phi_n^2 + a^{4} \\sum_{x} V(\\Phi_1, \\Phi_2)\n\\end{multline}\nwhere the link matrices are $U_\\mu = u^0 + i \\sigma^a u^a$ with\n$(u^0)^2 + u^a u^a = 1$ and\n\\begin{equation}\n\\epsilon_i^a = -(i\/2) \\tr (\\sigma^a \\dot U_i U_i^\\dag).\n\\end{equation}\nWith the time-varying constants, the potential becomes\n\\begin{multline}\nV(\\Phi_1, \\Phi_2) = \\frac{1}{a^{2(1-s)}} \\left[ m_1^2 \\mathrm{Tr} \\; \\Phi_1^2 + \\lambda (\\mathrm{Tr} \\;\n\\Phi_1^2)^2 + m_2^2 \\mathrm{Tr} \\; \\Phi_2^2  \\right. \\\\ \n\\left. + \\lambda (\\mathrm{Tr} \\;\n\\Phi_2^2)^2  + \\kappa (\\mathrm{Tr} \\; \\Phi_1 \\Phi_2)^2 \\right].\n\\end{multline}\n\n\nThe parameter $s$ can be chosen to be smaller than its physical value\n1, in order that the comoving width of the monopoles and strings\n$\\Mwid \\sim (a^sm_1)^{-1}$, $\\Swid \\sim (a^sm_2)^{-1}$, does not\nshrink below the lattice spacing during the simulation\n\\cite{Bevis:2006mj}. This extends the time range over which a\nsimulation can be run.\n\n\nWe evolve our lattice equations of motion with a standard Leapfrog\nmethod, and the damping term is handled using the Crank-Nicolson\nmethod.  More details of our numerical methods can be found in\nAppendix~\\ref{app:lattice}.\n\nWe perform simulations with both $s=1$ and $s=0$, with two different\nexpansion rate parameters, defined as\n\\begin{equation}\n\\label{e:ExpRatPar}\n\\nu = d\\ln a \/ d\\ln t.\n\\end{equation}\nWe will see that the quantities of most interest described in the next\nsection behave in similar ways, justifying the use of $s=0$.\n\nOur initial conditions for $\\Phi_{1,2}$ are uniformly distributed\nrandom values in the range $[-0.5,0.5]$ for each component\n$\\phi_{1,2}^a$, while for the SU(2) gauge field on the lattice we\ngenerate a random SU(2) matrix from four Gaussian random numbers\n$\\{u^0, u^a\\}$ which we then normalise to obtain a unitary matrix of\ndeterminant 1.\n\nWe first run for a period of time with relatively strong damping\n($\\sigma = 0.25$, see Appendix~\\ref{app:lattice}) before switching to\nstandard Hubble damping at $t_{0,\\text{H}}$ (see\nTable~\\ref{tab:s1runs}).  The momenta at the end of the damping phase\nare about a thousandth the size of those arising initially from the\nrandom initial conditions.\n\nThe procedure of seeding random fields at each site followed by a\nperiod of over-damped evolution is standard for modelling initial\nconditions for topological defects.  The important feature is that the\ncorrelations vanish beyond a certain length scale, which is bounded\nabove by the causal horizon \\cite{Kibble:1976sj}.  A finite\ncorrelation length is a sufficient condition for defects to form.  In\nall numerical experiments to date, the fields subsequently evolve\ntowards a self-similar or scaling configuration which at large\ndistances is independent of the initial conditions. An explicit check\nof the scaling in Abelian Higgs string simulations with two different\nsets of initial conditions was made in \\cite{Bevis:2010gj}, although\nsee also \\cite{Rajantie:2008bc} for a discussion of possible scaling\nviolation by super-horizon correlations in truly thermal initial\nconditions).\n\n\nWe then run with $s=-1$ for a period until time $t_\\text{cg}$, during which\nthe comoving string width grows linearly.  After $t_\\text{cg}$, $s$ is set to\nits physical value $s=1$.  The reason for this period of core growth\nis to accelerate the preparation of the string network: the time taken\nfor the fields to settle to their vacua is of order $\\Mwid$ and\n$\\Swid$, so it is helpful to arrange for them to be small while the\nfields are relaxing.  The graphs of $\\Mwid$ and $\\Swid$ are shown in\nAppendix~\\ref{app:expanding} in Fig.~\\ref{f:ComCorWid}.\n\nWhen $\\nu=0$ or when $s=0$, there is no need for the period of core\ngrowth, and data taking can begin at $t_{0,\\text{H}}$.\n\nOur initial conditions are designed as a compromise between removing\nunwanted short-distance fluctuations and allowing the strings to form\nin a reasonable time.\n\nDue to the initial cooling period, the lattice ultraviolet modes\nremain strongly suppressed during the evolution of the string network.\nThis is justified physically, because in the early universe the local\nenergy density within the strings is much larger than the energy\ndensity of the thermal background (although, in a given volume, the\ntotal energy of the thermal background can be larger than the energy\nof the string network). Thus, the thermal modes are expected to have\nlittle influence on the string evolution.  This also helps us to avoid\nthe problems associated with the thermal ultraviolet modes in\nreal-time lattice equations of motion~\\cite{Arnold:1997yb}.\n\n\\subsection{Numerical tests}\n\nIn simulations where expansion and the Hubble damping were turned off\n($\\nu=0$), energy conservation was better than $0.1\\%$ over the period\nfrom $t=t_{0,\\mathrm{H}} = 42.5$ to $t=720$.  The root mean square\nper-site relative Gauss law violation $\\overline G \/ \\overline \\rho$\nnever exceeded $3\\times 10^{-15}$ during our simulations, approaching\nthis value only in the initial heavy damping phase. For more details\nsee Appendix~\\ref{app:lattice}, and in particular\nEq.~(\\ref{eq:gausslaw}).\n\nIn the expanding case with $s=1$, comoving energy conservation was\nobeyed to 0.1\\% for simulations with $m^2_1 = 0.25$, $m^2_2 = 0.1$,\nwhile the relative Gauss law violation $\\overline G \/ \\overline \\rho$\nwas at most $8 \\times 10^{-4}$, a value reached at the start of the\ncore growth phase.\n\nWe also tested whether the lattice spacing was acceptable: if the\nlattice is too coarse, velocities tend to be reduced as the kinetic\nenergy of a defect can be converted into radiation.\\footnote{For\n  methods of mitigating this energy loss see\n  Refs.~\\cite{Moore:1996wn,Hindmarsh:2014rka}.}  For these tests we\ncompared $s=0$ and $s=1$ simulations at $m_1^2 = m_2^2 = 0.25$ with\nthose at $m_1^2 = m_2^2 = 0.1$, both with $\\kappa=2$.  The string RMS\nvelocities at $s=1$ differed by about $1\\%$ between simulations with\ndifferent masses, suggesting that any effect of the lattice spacing on\nthe dynamics of the strings is minor.  However, with $s=0$, monopole\nand semipole RMS velocities were as much as 10 \\% higher at the lower\nmass, indicating that there is some lattice friction at $m_1^2 = m_2^2\n= 0.25$.  Our $s=0$ runs are therefore carried out at $m_1^2 = m_2^2 =\n0.1$.\n  \n\\section{Measurements}\n\\label{s:Mea}\n\n\\subsection{Network length scale}\n\nWe measure the number of monopoles $N$, the string length $L$, and\nstudy length scales derived from them.  We obtain the number of\nmonopoles $N$ by computing the residual unbroken $\\mathrm{U}(1)$ gauge\nfield using projectors derived from $\\Phi_1$, the Higgs field which\nforms the monopoles. From this, we can compute the divergence of the\neffective magnetic field and hence the magnetic charge. We give fuller\ndetails of the U(1) projection in Appendix~\\ref{app:projectors}, based\non Ref.~\\cite{Davis:2000kv}.\n\nWe compute the length of string by counting the plaquettes with a\ngauge-invariant ``winding'' in the U(1) subgroups formed by projection\nwith the scalar field $\\Phi_1$, which is the heavier one in the\nnon-degenerate case.\n\nThe comoving string length $L$ is then defined to be the number of\nplaquettes with winding.  It is possible to include a geometric\ncorrection to $\\xi_\\mathrm{s}$ to account for the fact that counting\nthe winding number gives the Manhattan distance along the string\nrather than the true string length~(see Ref.~\\cite{Scherrer:1997sq}).\nWe choose to omit it, which should be borne in mind when comparing to\nother field theory simulations~\\cite{Bevis:2006mj,Daverio:2015nva}.\n\n\\subsection{Monopole density}\n\nSeveral further quantities can be derived from $N$ and $L$.  First,\nthe average comoving string and monopole separations,\n\\begin{equation}\n\\label{e:xismDef}\n\\xi_\\mathrm{s} = (V\/L)^{1\/2} \\quad \\text{and} \\quad \\xi_\\mathrm{m} = (V\/N)^{1\/3}.\n\\end{equation}\n\nWe define the average comoving monopole separation along the string\n\\begin{equation}\nd=L\/N = \\frac{\\xi_\\text{m}^3}{\\xi_\\text{s}^2}.\n\\end{equation}\nand the average number of monopoles per comoving length of string\n\\begin{equation}\nn = N\/L = 1\/d.\n\\end{equation}\nThe quantity $r$ defined in (\\ref{e:rDef}) can be thought of as the\nnumber of monopoles per unit physical length relative to the length\nscale $d_\\text{BV}$. The string and monopole separations can be combined into\none network length scale $\\xi_\\text{n}$, defined as\n\\begin{equation}\n\\label{e:xinDef}\n\\frac{1}{\\xi_\\text{n}^2} = \\frac{1}{\\xi_\\text{s}^2}\\left( 1 + r \\right).\n\\end{equation}\nThe energy density of the necklace is proportional to $\\xi_\\text{n}^{-2}$, and\nwhen $r > 1$ the majority of the energy in the network is due to the\nmonopoles.\n\nNote that in the degenerate cases $m_2^2\/m_1^2 = 1$ with $\\kappa=1$, the\npoints where $\\Phi_1$ vanishes recorded by our monopole search\nalgorithm are not special: there is no local maximum in the energy\ndensity.  However, they can be used as convenient markers of the phase\n$\\theta$, defined after Eq.~(\\ref{e:U1Sym}).\n\n\n\n\\subsection{Monopole and string velocities}\n\nWe use the positions of the strings and monopoles to compute the\nstring root-mean-square (RMS) velocity $\\bar v$, and the monopole RMS\nvelocity $\\bar{v}_\\text{m}$.\n\nUsing the projection methods discussed in\nAppendix~\\ref{app:projectors}, we record a list of the lattice cells\nthat contain magnetic charge every few timesteps. We then take these\nlists for two timesteps and form a distance matrix for every pair of\nmonopoles in the system. If the time interval $\\delta t$ is much\nsmaller than $\\xi_\\mathrm{m}$, we can assume that pairing each\nmonopole at the later timestep with the closest one at the earlier\ntimestep captures the same monopole at two different times.  On the\nother hand, the time interval between measurements has to be large\nenough that lattice-scale discretisation ambiguities do not induce\nnoise~\\cite{Hindmarsh:2014rka}. We will therefore compare results for\nseveral different $\\delta t$.\n\nThere are a number of standard algorithms to find the choice of\npairings in a distance matrix that minimises the total distance.  We\nused a simple `greedy' algorithm that found the smallest entry in the\nentire distance matrix, then removed that monopole pair, repeating\nuntil all monopoles at the later time were paired up. This algorithm\nhas the advantage of being easy to code, on the other hand it scales\nas the square of the number of monopoles.\n\nThe system has periodic boundary conditions, and so a `halo' region is\nincluded from the other side of the lattice to ensure that all\npossible subluminal monopole separations will be found. Once we have\ndetermined all the pairings, we remove spurious superluminal pairings\n(typically $\\lesssim 1\\%$ of measurements) and use the results to\ndetermine $\\bar{v}_\\text{m}$. We considered $\\delta t = 5$, $10$ and $15$ and\nfound convergence in the resulting curves. We used $\\delta t= 15$ for\nour results.  The difference from $\\delta t=10$ can be considered as a\nsystematic uncertainty, but in practice it is comparable to or smaller\nthan the random error.\n\nFor the string velocities, a very similar approach was adopted, using\nthe positions of the plaquettes threaded by string.  As many\nplaquettes can be threaded by the strings in the system, the above\npairing and distance finding algorithms were parallelised. Even so,\ndetermining the string velocity for a few hundred thousand plaquettes\nbetween a pair of timesteps took about five minutes on 120 processors.\nFor this reason, string velocities are not computed at early times,\nwhen the number of plaquettes becomes too large.  The corresponding\nmonopole measurement takes about a second, and can be performed\nthroughout the simulations.\n\n\n\\section{Results}\n\nWe run over several different parameter choices for both $s=1$ and\n$s=0$. \n\nThe parameters cover both the degenerate ($m_1^2 = m_2^2$) and\nnon-degenerate cases, and allow us to explore the three possible\nglobal symmetries of the string solutions, namely $\\mathrm{O}(2)$,\n$D_4$, and $Z_2\\times Z_2$. In the degenerate case three\ncross-couplings $\\kappa$ are considered: the special case $\\kappa =\n2\\lambda$ having $\\mathrm{O}(2)$ symmetry, and both $\\kappa >\n2\\lambda$ and $\\kappa < 2\\lambda$.  For the non-degenerate case,\nhaving $Z_2 \\times Z_2$ symmetry, we explore various ratios of $m_1^2$\nto $m_2^2$.\n\nTwo different expansion rate parameters $\\nu = 0.5, 1$ were chosen,\nwhere $\\nu$ is defined in Eq.~(\\ref{e:ExpRatPar}).  The choice $\\nu =\n1$ represents a radiation-dominated universe.  While $\\nu = 0.5$ does\nnot correspond to any realistic cosmology, it is useful to explore the\nimpact of different expansion rates.  Simulating in a matter dominated\nbackground ($\\nu=2$) does not give enough dynamic range for reliable\nresults.\n\n\nAll runs are carried out with $m_1^2 = 0.25$ ($s=1$) and $m_1^2 = 0.1$\n($s=0$).  The parameter choices are listed in Tables \\ref{tab:s1runs}\nand \\ref{tab:runs}.  The scale factor is normalised so that $a=1$ at\nthe end of the simulation.\n\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\t\\begin{tabular}{lllll|lll|lll}\n\t\t\t$m_1^2$ & $m_2^2$ & $g$ & $\\lambda$ & $\\kappa$ & $M_\\text{m}$ & $\\mu$ & $d_\\text{BV}$ & $\\nu$ & $t_{0,\\text{H}}$ & $t_\\text{cg}$ \\\\\n\t\t\t\\hline\n\t\t\t0.25 & 0.25 & 1 & 0.5 & 2 & 11 & 1.6 & 7 & 1 & 30 & 230 \\\\\n\t\t\t0.25 & 0.25 & 1 & 0.5 & 1 & 11 & 1.6 & 7 & 1 & 30 & 230 \\\\\n\t\t\t\\hline\n\t\t\t0.25 & 0.1 & 1 & 0.5 & 1 & 11 & 0.63 & 17.5 & 0.5 & 42.5 & 242.5 \\\\\n\t\t\t0.25 & 0.1 & 1 & 0.5 & 1 & 11 & 0.63 & 17.5 & 1 & 42.5 & 242.5 \\\\\n\t\t\t\\hline\n\t\t\t0.25 & 0.05 & 1 & 0.5 & 1 & 11 & 0.31 & 35 & 0.5 & 60 & 260 \\\\\n\t\t\t0.25 & 0.05 & 1 & 0.5 & 1 & 11 & 0.31 & 35 & 1 & 60 & 260 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:s1runs} List of parameters for $s=1$\n          (physical) runs, with dimensionful parameters given in units\n          of the lattice spacing $a$.  Potential parameters\n          (\\ref{e:ScaPot}) are shown along with the isolated monopole\n          mass $M_\\text{m}$ and the isolated string tension $\\mu$ computed\n          using\n          Eqs.~(\\ref{eq:monopolemass})~and~(\\ref{eq:stringtension}).\n          The length scale $d_\\text{BV}$ as computed using\n          Eq.~(\\ref{eq:dbvdefn}) is also shown.  Finally, we quote the\n          expansion rate parameter $\\nu = d\\ln a\/d \\ln t$, the time at\n          which we change to Hubble damping during our simulations,\n          $t_{0,\\text{H}}$, and the time at which core growth ends and\n          strings and monopoles reach their true physical width\n          $t_\\text{cg}$.  All these simulations have lattice size 720 and\n          duration 720. }\n\\end{table}\n\n\n\\begin{table}[t]\n\t\\begin{center}\n\t\t\\begin{tabular}{lllll|lll|l}\n\t\t\t$m_1^2$ & $m_2^2$ & $g$ & $\\lambda$ & $\\kappa$ & $M_\\text{m}$ & $\\mu$ & $d_\\text{BV}$ & $t_{0,\\text{H}}$ \\\\\n\t\t\t\\hline\n\t\t\t0.1 & 0.1 & 1 & 0.5 & 2 & 6.96 & 0.628 & 11.1 & 30 \\\\\n\t\t\t0.1 & 0.1 & 1 & 0.5 & 1 & 6.96 & 0.628 & 11.1 & 30 \\\\\n\t\t\t0.1 & 0.1 & 1 & 0.5 & 0.5 & 6.96 & 0.628 & 11.1 & 30 \\\\\n\t\t\t\\hline\n\t\t\t0.1 & 0.04 & 1 & 0.5 & 1 & 6.96 & 0.251 & 27.7 & 67.1 \\\\\n\t\t\t0.1 & 0.02 & 1 & 0.5 & 1 & 6.96 & 0.126 & 55.4 & 94.9 \\\\\n\t\t\t0.1 & 0.01 & 1 & 0.5 & 1 & 6.96 & 0.0628 & 111 & 134 \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{\\label{tab:runs} List of simulation parameters for\n          runs with $s=0$, as for Table \\ref{tab:s1runs}.  The\n          expansion rate parameter is $\\nu=1$ (radiation era) for all\n          simulations.  At $s=0$ the physical size of the monopole and\n          string cores grows in proportion to the scale factor.  All\n          these simulations have lattice size 720 and duration 720.}\n\\end{table}\n\nThe units are defined such that the lattice spacing $\\Delta x$ is 1.\nAll simulations are carried out on a $720^3$ lattice, with timestep\n$\\Delta t = 0.25$ after the initial heavy damping period ends at\n$t_{0,\\text{H}}$, for a total time $720$, or one light-crossing time\nof the box.  In principle, correlations can start to be established\nafter half a light-crossing time. However, the only massless\nexcitations are waves on the string, and the strings are much longer\nthan the box size even at the end of the simulations.  The network\nlength scale does not show any evidence for finite-size effects,\nalthough it is possible that the slight increase in $d$ for semipoles\nand supercurrents at $t \\gtrsim 360$ in Fig.~\\ref{f:n_both} is a sign\nof the limited simulation volume.\n\n\n\nEach set of parameter choices is run for 3 different realisations of\nthe initial conditions, and our results are statistical averages over\nthese runs.\n\nWe investigate the monopole density with the two different measures\nintroduced in Section \\ref{s:Mea}, the monopole-to-string density\nratio $r$ and the number of monopoles per unit comoving length of\nstring $n$.\n\n\\subsection{Network length scale}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[clip=true,width=0.5\\textwidth]{xi_s1-all.pdf}\n\\includegraphics[clip=true,width=0.5\\textwidth]{xin-all.pdf}\n\\end{center}\n\\caption{\\label{fig:xin} Plot of the network length scale $\\xi_\\text{n}$,\n  defined in Eq.~(\\ref{e:xinDef}), with core growth parameter $s=1$\n  (top) and $s=0$ (bottom).  Fits to linear growth are also shown,\n  within the range indicated by the vertical dashed lines.  The\n  gradients of the fit are given in Tables \\ref{tab:fits_s1} and\n  \\ref{tab:fits_s0}.}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:xin} we plot the comoving necklace network length\nscale $\\xi_\\text{n}$, defined in Eq.~(\\ref{e:xinDef}), for $s=1$ (top) and\n$s=0$ (bottom).\n\nAll cases show linear growth with time, which means that the network\nis scaling.  We perform fits in the range $360 < t < 480$, which while\nin excess of the half light crossing time for the system, allows time\nfor the scaling behaviour to develop.  There are small differences in\nthe slope between simulations with different mass ratios, although\nthere is not enough dynamic range to ensure that they are not\ninherited from differences in the initial conditions.  There is also\nevidence that the lower expansion rate $\\nu = 1\/2$ the slope is lower,\ni.e. that the average necklace density is higher.\n\n\n\n\\begin{table}[h!]\n\\begin{center}\n\n\\begin{tabular}{lll|l|l}\n$m_1^2$ & $m_2^2$ & $\\kappa$ & $\\nu$  & $\\xi_\\mathrm{n}$ gradient \\\\\n\\hline\n0.25 & 0.25     & 2   & 1 & $0.171 \\pm 0.002$ \\\\\n0.25 & 0.25     & 1  & 1 & $0.168 \\pm 0.004$ \\\\\n\\hline\n0.25 & 0.1   & 1 & 0.5 & $0.154 \\pm 0.001$ \\\\\n0.25 & 0.1   & 1 & 1 & $0.171 \\pm 0.002$ \\\\\n\\hline\n0.25 & 0.05    & 1  & 0.5 & $0.158 \\pm 0.002$ \\\\\n0.25 & 0.05    & 1  & 1 & $0.165 \\pm 0.004$ \\\\\n\\hline\n\\end{tabular} \n  \n\\end{center}\n\\caption{\\label{tab:fits_s1} Gradients for the network comoving length\n  scale $\\xi_\\mathrm{n}$, from the fits shown in the graphs of $\\xi_n$\n  against conformal time $t$ for $s=1$ in Fig.~\\ref{fig:xin} (top). }\n\n\\bigskip\n\n\\begin{tabular}{lll|l}\n$m_1^2$ & $m_2^2$ & $\\kappa$  & $\\xi_\\mathrm{n}$ gradient \\\\\n\\hline\n0.1 & 0.1     & 2   & $0.154 \\pm 0.005$ \\\\\n0.1 & 0.1     & 1  & $0.150 \\pm 0.003$ \\\\\n0.1 & 0.1     & 0.5  & $0.163 \\pm 0.008$ \\\\\n\\hline\n0.1 & 0.04   & 1 & $0.141 \\pm 0.004$ \\\\\n0.1 & 0.02    & 1  & $0.143 \\pm 0.001$ \\\\\n0.1 & 0.01     & 1  & $0.126 \\pm 0.001$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:fits_s0} Gradients for the network comoving length\n  scale $\\xi_\\mathrm{n}$, from the fits shown in the graphs of $\\xi_n$\n  against conformal time $t$ for $s=0$ in Fig.~\\ref{fig:xin}\n  (bottom). }\n\n\\end{table}\n\n\n\n\\subsection{Monopole density}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[clip=true,width=0.5\\textwidth]{r_s1-all.pdf}\n  \\end{center}\n  \\caption{\\label{f:r_s1} The ratio of monopole to string energy\n    density (\\ref{e:rDef}) in simulations with $s=1$.  The legend\n    gives the expansion rate parameter $\\nu = d \\log a\/d \\log t$, the\n    mass ratio of the fields $m_2\/m_1$, and in the degenerate case the\n    value of the cross-coupling $\\kappa$, which is otherwise $\\kappa=1$.\n    The mass parameter $m_1^2 = 0.25$. }\n\\end{figure}\n\n\nIn Fig.~\\ref{f:r_s1} we plot the ratio of monopole to string energy\ndensity $r$, defined in (\\ref{e:rDef}), against time in units of\n$m_1^{-1}$, for all parameters given in Table \\ref{tab:s1runs}.  Note\nthat $m_1^{-1}$ is approximately the monopole size.\n\n\nWe see that $r$ decreases after the formation of the string network,\nwith what appears to be a power law after the core growth period has\nfinished.\n\nThe significance of the power law is clearer if we plot the comoving\nlinear monopole density on the string $n$, again in units of\n$m_1^{-1}$ (Fig.~\\ref{f:n_both}).  We can see from the figure that,\nwith the possible exception of the mass-degenerate cases ($m_2^2\/m_1^2\n= 1$) at $s=1$, $n$ appears to tend to a constant at large time.\nHence the comoving separation of the monopoles remains the same order\nof magnitude as its value at the formation of the strings.\n\nThere is some evidence for a slow increase in $n$ for the degenerate\ncases $m_2^2\/m_1^2 = 1$ at $s=1$, which may be due to semipole\nannihilations being less probable than monopole-antimonopole\nannihilations -- some pairings of semipoles cannot\nannihilate~\\cite{Hindmarsh:2016lhy}.  However, the increase occurs\nafter a half-light crossing time for the simulation box, so this may\nbe a finite volume effect.\n\nWe illustrate the ability of semipoles to avoid annihilation in\nFig.~\\ref{f:semipole_end}, which depicts two strings winding around\nthe periodic lattice when the total length of string and the semipole\nnumber has stabilised.  One can see that on one of the strings, the\nsemipole density is much higher, and examination of multiple snapshots\nprior to this one shows that semipoles have repelled each other.\nHowever, the high semipole density may be an artefact of the periodic\nboundary conditions, which have prevented the strings from shrinking\nin length any further.  Without this shrinking, semipoles are not\nforced together, so there is less likelihood of overcoming the\nrepulsion and annihilating.\n\nIn the degenerate cases $m_2^2\/m_1^2 = 1$ with $\\kappa=1$, we recall that\nthe recorded monopole positions are just places where the phase of the\ncomplexified scalar has the value $\\theta=\\pm\\pi\/2$.  The fact that the\ncomoving distance between these points remains approximately constant\nindicates that the comoving RMS current is constant, and so the\nphysical RMS current decreases in inverse proportion to the scale\nfactor.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[clip=true,width=0.5\\textwidth]{n_s1-all.pdf}\n\\includegraphics[clip=true,width=0.5\\textwidth]{n-all.pdf}\n\\end{center}\n\\caption{\\label{f:n_both} The number of monopoles per comoving string\n  length in simulations with $s=1$ (top) and $s=0$ (bottom). The\n  legend gives the expansion rate parameter $\\nu = d \\log a\/d \\log t$,\n  the mass ratio of the fields $m_2\/m_1$, and in the degenerate case\n  the value of the cross-coupling $\\kappa$, which is otherwise $\\kappa=1$.\n  The mass parameter $m_1^2 = 0.25$ ($s=1$) and $m_1^2 = 0.1$\n  ($s=0$).}\n\\end{figure}\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.3\\textwidth]{{visit-kappa2-late-nocredit}.jpeg}\n  \\end{center}\n\\caption{\\label{f:semipole_end} A small $360^3$ box at $t = 1080$,\n  simulated at $m_2^2\/m_1^2 = 1$ and $\\kappa = 2$.  The high density\n  of semipoles on one of the strings shows that semipoles can avoid\n  annihilation in some cases.  }\n\\end{figure}\n\n\nIn the $s=0$ case, the increased dynamic range means we can attempt a\nmeaningful fit to investigate the relaxation to the constant $n$\nevolution.  In Fig.~\\ref{fig:nfit}, we show a graph of $n - n_\\infty$,\nwhere the asymptotic value of the linear monopole density $n_\\infty$\nis taken from a fit to the functional form\n\\begin{equation}\n\\label{e:nFit}\nn = n_\\infty + A\\exp(-B m_1 t).\n\\end{equation}\nFits are shown with dashed lines, and fit parameters are given in\nTable \\ref{t:nFitPar}.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{ll|lll}\n$m_1^2$ & $m_2^2$ & $\\frac{n_\\infty}{m_1}$ & $A$ & $B$  \\\\\n\\hline\n0.1 & 0.04 & 0.036 & 0.031 & 0.0072 \\\\\n0.1 & 0.02 & 0.023 & 0.060 & 0.0104 \\\\\n0.1 & 0.01 & 0.025 & 0.075 & 0.0134 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{t:nFitPar} Parameters for the fit of the linear\n  monopole density data in Fig.~\\ref{fig:nfit} to the function\n  (\\ref{e:nFit}).  All simulations are radiation era, with $s=0$.  }\n\\end{table}\n\nThe fits confirm the visual impression that the linear monopole\ndensity is asymptoting to a constant non-zero value, and also support\nthe exponential ansatz for the relaxation.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[clip=true,width=0.5\\textwidth]{nfit.pdf}\n\\end{center}\n\\caption{\\label{fig:nfit} The difference of the linear monopole\n  density $n$ from its asymptotic value $n_\\infty$.  The parameter\n  $n_\\infty$ is extracted from a fit of $n$ to a constant to\n  exponential decay [see Eq.~(\\ref{e:nFit})]; the fits are shown as\n  dashed lines.  Both $n$ and the time are scaled by $m_1$ to make\n  dimensionless quantities.  Only those values of $m_2\/m_1$ where a\n  reliable fit is possible are shown; for other values, the change in\n  $n$ is too small.  }\n\\end{figure}\n\n\n\n\\subsection{Monopole velocities}\n\n\\begin{figure*}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[clip=true,width=0.45\\textwidth]{monopole-velocities-s1-scaled-dbv.pdf}\n\t\t\\includegraphics[clip=true,width=0.45\\textwidth]{string-velocities-s1-scaled-dbv.pdf}\n\t\t\\includegraphics[clip=true,width=0.45\\textwidth]{monopolesvel-all_dbv.pdf}\n\t\t\\includegraphics[clip=true,width=0.45\\textwidth]{stringsvel-all_dbv.pdf}\n\t\\end{center}\n\t\\caption{\\label{fig:rmsvels1} Plot of $\\bar v$ and $\\bar{v}_\\text{m}$,\n          the root mean square string and monopole\/semipole\n          velocities, for $s=1$ (top) and $s=0$ (bottom). The time\n          axis is scaled by $d_\\text{BV}$, defined in Eq.~(\\ref{eq:dbvdefn}).\n        }\n\\end{figure*}\n\nFig.~\\ref{fig:rmsvels1} shows the RMS velocities of the strings,\nmonopoles and semipoles for different masses, cross-couplings $\\kappa$,\nand expansion rate parameters $\\nu$.  The RMS velocities all appear to\nasymptote at the same rate $d_\\text{BV}^{-1}$ to a constant value.\n\nWe see that the RMS string velocities are all around $0.5$.  When the\nfield mass parameters $m_1$ and $m_2$ are different, the RMS monopole\nvelocities are also all about 0.5, independent of the mass ratio and\nthe expansion rate.  If the mass parameters are the same, the RMS\nmonopole velocity at about $0.63$ is a little higher than the RMS\nstring velocity.  RMS velocities are consistent between $s=1$ and\n$s=0$, with the exception of the semipoles at $s=0$, which appear to\nmove a little slower ($\\bar{v}_\\text{m} \\simeq 0.6$) than at $s=1$ ($\\bar{v}_\\text{m}\n\\simeq 0.68$).\n\n\nThe higher velocities of the semipoles should make collisions more\nfrequent than those between monopoles and antimonopoles.  However, as\nobserved in the Introduction, semipole collisions need not result in\nannihilation, and so the higher velocities do not necessarily result\nin a lower monopole density.\n\nWe interpret the difference $\\bar v_\\text{rel}^2 = \\bar{v}_\\text{m}^2 - \\bar v^2$ as the mean\nsquare relative velocity of the monopoles and semipoles along the\nstring.  One can estimate that, for semipoles, $\\bar v_\\text{rel} \\simeq 0.3$,\nwhile there is little evidence for relative motion of monopoles.\n\n\n\\section{Conclusions}\n\nWe have carried out simulations of non-Abelian cosmic strings, formed\nby the symmetry-breaking scheme SU(2)$\\to Z_2$ by two adjoint scalar\nfields.  This theory has classical solutions which can be interpreted\nas 't Hooft-Polyakov monopoles or semipoles~\\cite{Hindmarsh:2016lhy}\nthreaded by non-Abelian strings.  We observe the formation of cosmic\nnecklaces, consisting of networks of strings and monopoles or\nsemipoles.\n\nOur simulations were carried out in a cosmological background\ncorresponding to a radiation dominated era, and also one with half the\nexpansion rate of a radiation-dominated universe, testing the effect\nof the expansion rate.  We performed simulations both with the true\nexpanding universe equations of motion, and allowing the cores of the\ntopological defects to grow with the expansion of the universe.  Core\ngrowth has been shown not to significantly affect the dynamics of\nstrings \\cite{Bevis:2006mj,Bevis:2010gj,Daverio:2015nva}, but its\neffect on the dynamics of necklaces is important to check.\n\nIn all cases, our numerical results are consistent with the evolution\ntowards a scaling network of necklaces, with both the density of\nstrings and the density of monopoles proportional to $t^{-2}$.  We\nobtain scaling with or without core growth, giving confidence that\nscaling is a robust feature of a necklace network. A necklace network\nshould therefore contribute a constant fraction to the energy density\nof the universe.\n\n\nWe observe that the number of monopoles per unit comoving length of\nstring $n$ changes little from its value at the formation of the\nstring network: monopole annihilation on the string is therefore not\nas efficient as envisaged in Ref.~\\cite{BlancoPillado:2007zr}, and the\naverage comoving separation of monopoles along the string $d = 1\/n$\nremains approximately constant.  The monopole to string density ratio\n$r$ therefore decreases in inverse proportion to the scale factor, and\ndoes not increase as proposed in Ref.~\\cite{Berezinsky:1997td}.  The\nRMS monopole velocity is close to the RMS string velocity, implying\nthat the monopoles have no significant motion along the string.  In\nparticular, the suggestion that the monopole RMS velocity should be\n50\\% larger than the string RMS velocity \\cite{BlancoPillado:2007zr},\ndue to the extra degree of freedom or motion, is not supported.\n\n\nThe number per unit comoving length of semipoles is also approximately\nconstant in the simulations with core growth, but grows slightly in\nthe simulations using the true equations of motion.  We do not have\nlarge enough dynamic range to establish whether this is a finite\nvolume effect.  The semipole RMS velocity is higher than the string\nRMS velocity, indicating some relative motion of the semipoles along\nthe string.  Annihilation is still inefficient despite the relative\nmotion, indicating that repulsion between semipoles is an important\nfactor in the dynamics.\n\nIn the special case where the strings carry a supercurrent, the\ncomoving distance between points where the $\\Phi_1$ field vanishes $d$\nalso stays approximately constant.  The supercurrent along the string\ncan be estimated as $j \\sim 1\/ad$, where $a$ is the scale factor, and\nshould therefore decrease.  This suggests that current is lost from\nshrinking loops of string, which would tend to prevent the formation\nof cosmologically disastrous stable string loops\n\\cite{Ostriker:1986xc,Copeland:1987th,Davis:1988ij}.\n\nWe are restricted to simulating necklace configurations with $r \\sim\n1$, so we are not able to fully test the robustness of the of the\nconstant comoving $d$ scaling regime.  Nonetheless, we find it\ninteresting to explore the consequences as it was not anticipated in\nprevious dynamical modelling, which envisaged that $d$ would either\nshrink to the string width ~\\cite{Berezinsky:1997td}, or grow with the\nhorizon size \\cite{BlancoPillado:2007zr}.  The absence of an\nsignificant relative velocity between monopoles and strings indicates\nthat monopoles are dragged around by the strings, independent of the\nratio of the energy scales.  The average string separation is of order\nthe conformal time $t$, which means that loops of string shrink and\nannihilate on that timescale. We infer that the main monopole\nannihilation channel is though collisions on shrinking loops of\nstring.\n\nAs argued in \\cite{Hindmarsh:2016lhy}, semipoles and monopoles are\ngeneric on strings in GUT models.  It is interesting to consider their\nobservational implications.  As usual with strings, one must\nextrapolate the results of numerical simulations to a much larger\nratio of the horizon size to the string width, and it is possible that\nsubtle effects change the scaling of the network.  It is clear in our\nsimulations that, just as with Abelian Higgs strings, our SU(2)\nstrings lose energy efficiently into Higgs and gauge radiation.\nHowever, the process that causes the strings to emit radiation of\nmassive Higgs and gauge fields is not well understood, and it may not\nbe efficient over the huge range of scales between today's horizon\nsize and the width of a GUT string. In this case, a necklace would end\nup behaving like ideal Nambu-Goto strings connecting massive\nparticles, as assumed in \\cite{Berezinsky:1997td} and\n\\cite{BlancoPillado:2007zr}.\n\nIn the case where field radiation is efficient, there is little\ndifference between a network of GUT strings with monopoles or\nsemipoles and an Abelian Higgs string network.  The network length\nscale grows in proportion to the horizon, and its energy density\nremains a constant fraction of the total. The energy is lost to\nmassive particles, which (if coupled to the Standard Model) will show\nup in the diffuse $\\gamma$-ray background.  Current observations from\nFermi-LAT indicate that the mass per unit length in Planck units\n$G\\mu$ is bounded above by $3 \\times 10^{-11} f^{-1}_\\text{SM}$, where\n$f_\\text{SM}$ is the fraction of the strings energy ending up in\n$\\gamma$-rays~\\cite{Mota:2014uka}. This fraction is likely to be close to\nunity in a GUT theory, and so such strings are essentially ruled out,\nas observed some time ago \\cite{Vincent:1997cx}. However, strings in a\nhidden sector are subject only to constraints from the Cosmic\nMicrowave\nBackground~\\cite{Moss:2014cra,Charnock:2016nzm,Lizarraga:2016onn},\nwhich are $G\\mu \\lesssim 10^{-7}$.\n\nIn the case where the string dynamics eventually changes over to\nNambu-Goto, the difference between a necklace network and an ordinary\ncosmic string network is more dramatic with our new picture that the\ncomoving distance between monopoles remains approximately constant\nfrom the time the strings formed.  For GUT scale strings forming along\nwith the monopoles, this is bounded above by the horizon distance at\nthe GUT temperature, or a few metres today.  Even if the scale of the\nU(1) symmetry-breaking is as low as a TeV, this distance is\nO($10^{12}$) m today, a factor $10^{-14}$ smaller than the horizon\nsize.  When horizon-size string loops are chopped off the long string\nnetwork, they will therefore have a large number of monopoles on them.\nNumerical investigations indicate \\cite{Siemens:2000ty} that such\nstring loops do not have periodic non-self-intersecting solutions.  We\ncan therefore expect them to quickly chop themselves up into smaller\nand smaller loops, some of which will be free of monopoles and find\nstable periodic non-self-intersecting trajectories.  In this case, the\ntypical loop size for a GUT scale string would be a few metres rather\nthan the horizon size.  Hence, the tight bounds on the Nambu-Goto\nstring tension from msec pulsar timing obtained by the European Pulsar\nTiming Array \\cite{Lentati:2015qwp} and NANOGrav\n\\cite{Arzoumanian:2015liz} would be avoided, as the gravitational\nwaves would be at frequencies inaccessible to direct observation.\n\n\n\\begin{acknowledgments}\nWe acknowledge fruitful discussions with Jarkko J\\\"arvel\\\"a during the\ninitial stages of this project. Our simulations made use of the COSMOS\nConsortium supercomputer (within the DiRAC Facility jointly funded by\nSTFC and the Large Facilities Capital Fund of BIS). DJW was supported\nby the People Programme (Marie Sk{\\l}odowska-Curie actions) of the\nEuropean Union Seventh Framework Programme (FP7\/2007-2013) under grant\nagreement number PIEF-GA-2013-629425.  MH acknowledges support from\nthe Science and Technology Facilities Council (grant number\nST\/L000504\/1).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}