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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe need to incorporate programming content into introductory physics is widely appreciated by the academic community \\cite{Fuller2006}. By some estimates, \\emph{at least 70\\%} of new STEM jobs in the US will require computer programming skills \\cite{labor2014} and in the sciences computer programming skills have become an essential part of many disciplines. In response to these shifts, groups like code.org and the ``hour of code\" have brought coding tutorials to wider and younger audiences \\cite{code}. These groups also influenced federal education legislation in the US. In particular, the Every Student Succeeds Act (ESSA), which was signed into law in December 2015, designates computer science as a ``core subject\". This is a significant change that places computer science on the same level as english and mathematics \\cite{coresubject}. The 2017-2018 school year was the first school year that this legislation was fully implemented.  Yet, for physics instruction, and perhaps even more generally, the task of re-imagining STEM courses with computer science as a crucial element is still far from complete. Although there are a number of universities that use coding activities in physics with vpython \\cite{Chabay_Sherwood2008}, and there exists significant research into using these activities in calculus-based introductory physics \\cite{Caballero_etal2012}, vpython exercises (or coding using some other software framework) are much less often used in algebra-based physics and at the high school level. \n\nWe were able to find two studies that reflect the difficulty of using coding activities in algebra-based physics at the high school level\\footnote{The open source ebook by \\citet{Titus_Esquembre2016} is also notable but there seems to be no published studies examining its appropriateness for various grade levels.}.  \\citet{Aiken_etal2013} describes a masters degree project by a high-school physics teacher who worked for two years to develop a vpython curriculum for a 9th grade high school physics class and found that only one third of the class successfully completed the exercises. \\citet{Aho2014} describes coding activities developed for a high school classroom that use the R programming language. Although they are not very specific in stating precisely what fraction of the students struggled with the exercises, they do indicate that a significant number of students needed extra time outside of class to complete the activities, and these students frequently needed extra practice to learn the R syntax. To mitigate this in future work, \\cite{Aho2014} proposes to set aside a week-long R programming tutorial for the students at the beginning of the year, which is a luxury most high school teachers do not have. The indications from \\citet{Aiken_etal2013} and \\citet{Aho2014} underscore the need to develop a curriculum that adds programming into \\emph{algebra-based} physics with a higher success rate.\n\n\\begin{figure*}\n\\includegraphics[width=2.4in]{Traditional_approach.pdf}\\includegraphics[width=2.4in]{Hybrid_approach.pdf}\\includegraphics[width=2.4in]{PhetPhyslet_approach_v2.pdf}\n    \\caption{Illustrations of different approaches to computationally-enriched physics content. The left panel illustrates the typical structure of a code in a traditional intermediate-to-advanced level physics-major computational physics course, emphasizing that the student has control over essentially the entire code. The right panel shows the typical structure of a web interactive in which students interact with a visualization but do not see or have any control of the underlying code. The center panel shows a hybrid approach with a high degree of interactivity but the students do see and potentially modify the parts of the code that advance system variables, even though code related to visualization remains fixed and invisible to the student.}\n    \\label{fig:approach}\n\\end{figure*}\n\nAs will be discussed, we use a javascript-based language called p5.js, which was designed to be a text-based (as opposed to block-based) language with a gentle learning curve for absolute beginner programmers. In principle, the exercises we describe here could be reproduced in vpython (or some other language) and used in a similar way. While there are clearly advantages and disadvantages to both vpython and javascript, the comparison of the two is not the subject of this paper. Instead we wish to emphasize the need for coding activities that would be appropriate for an algebra-based physics classroom. As discussed later, an important way to judge the appropriateness of these activities is the perceived difficulty of students who complete coding activities.\n\nThe left hand panel in Figure~\\ref{fig:approach} illustrates what we describe as the ``traditional\" computational physics approach that appears in an intermediate-to-advanced physics-major computational physics course, or in a physics-major course that has been re-tooled to include significant computational content. In this setting, the student is given complete control of the computer program, including the advance of variables (which may involve specifying forces and advancing positions and velocities, or it may involve the evolution of abstract quantities like wave functions). If visualization is needed, the student is typically given full control of a plotting program. Although there may be some template that the student is given and other advice may be provided, overall, the student has a high level of control. The drawback for this approach is that significant class time is often required for students to familiarize themselves with coding practices. Given the time constraints of a typical algebra-based college physics course, or high school physics course, this approach is in-feasible for most instructors.\n\nThe right hand panel in Figure~\\ref{fig:approach} describes interactive physics simulations in which the students do not see the code. This approach is extensively used by the PhET collaboration \\cite{PhET} and by the ``Physlet\" physics community \\cite{physlet}, and many studies have shown its utility for teaching scientific concepts \\cite[e.g.][]{Perkins_etal2006,Podolefsky_etal2010}. Largely for this reason, PhET and similar activities have been put into widespread use.\n\nThe central panel in Figure~\\ref{fig:approach} outlines the ``hybrid\" approach that we adopt in this paper in which the student does see and potentially modifies the code that evolves system variables (which is similar to the traditional approach), and there is some kind of interactive visualization that is produced in which the simulation responds to user input (which is similar to the PhET\/Physlet approach). However there are still aspects of the code, particularly related to visualization, that the student does not see in order to substantially reduce the cognitive load \\cite{Jong_2010,cogload} by shortening the length and minimizing the complexity of the program. The intention is to remove ``extraneous cognitive load\" associated with the graphical user interface among other things, and focus on the aspects of the code that directly determine or update physical quantities. Our assumption is that the ``intrinsic cognitive load\" of setting and updating the physical quantities using the target concepts and relationships is within students' abilities. As will be illustrated in this paper, the portion of the code with which the students interact can be concise both textually and conceptually, and still produce interesting game-like interactives that emphasize kinematic and diagrammatic concepts like force, acceleration, velocity, and their vector representations.\n\nTo provide some comparison to other works in the literature, there may also be some overlap with our approach and that of \\cite{kordakai2010}, who describes a graphically enriched coding environment for teaching computer science and outlines how their activities align with the educational theories of various authors. The Netlogo project \\cite{netlogo} is another comparable effort which borrows from earlier efforts to incorporate programming into schools, but we are not familiar enough with how Netlogo activities are used in introductory physics to say more than this.\n\nAlthough there are exceptions \\cite{Taub_etal2015,Weintrop_etal2016,Titus_Esquembre2016,netlogo}, interactive activities where students key-in commands and ``play\" their code like a video game, are typically not a part of programming exercises at the introductory level. In the Matter \\& Interactions curriculum that integrates vpython into calculus-based physics \\cite{Chabay_Sherwood2015}, many of these programs, such as the three-body gravitational simulation or the 3D pendulum \\cite{Chabay_Sherwood2008}, are designed for the student to perform coding tasks and then passively watch the execution of the program (except perhaps for changing the perspective). And while there are a large number of exercises currently available on the AAPT's Partnership for the Integration of Computation into Undergraduate Physics (\\href{http:\/\/compadre.org\/PICUP}{compadre.org\/PICUP}), only a few of them involve a high level of interactivity as the program is running.\nOur hypothesis is that this interactive, game-like approach with a concisely-written code will create a fun and approachable experience for students who might otherwise find a programming task to be intimidating, making it an ideal choice for engaging students in introductory courses\n\nThis paper is only the beginning of a research effort to validate this hypothesis. We will describe a set of computer programming activities designed for absolute beginner programmers in first-semester introductory physics (mechanics) classes, that were used during four semesters at Ohio State's Marion campus. Survey results will be presented that examine student perceptions from completing the first exercise, and probe the percentage of weak or absolute beginner programmers in the classroom.\n\nAlthough there is good work in the literature describing how numerical exercises can be connected with laboratory activities \\cite[e.g.][]{Serbanescu_etal2011}, we consider this out of scope for the present work. The javascript-based language p5.js does have capabilities to interact with Arduino circuit boards, making this an interesting possibility for future work.\n\n\\section{Overview of Programming Activities for Mechanics}\n\nIn a semester course of introductory physics at Ohio State University (OSU) at the regional campus in Marion, we include six required programming activities and a seventh activity that is optional or extra credit. In most other ways, the course is identical to the same course on the Columbus campus. The official description of this course is calculus-based physics I, but on all OSU campuses students only need to be concurrently enrolled in calculus in order to take the course, and as a result the calculus content in the course is rather limited. Moreover, the students at OSU's regional campuses are less prepared than their peers on OSU's Columbus campus. During the data gathering, incoming OSU Marion students had an average ACT score near 22 (in 2014 \\cite{osumarion2014} and 2015 \\cite{osumarion2015}) or 22.5 (in 2016 \\cite{osumarion2016}) whereas students admitted directly to the Columbus campus over the same time span had an average ACT score close to 29 \\cite{osucolumbus}. The limited calculus in the course and the comparably poor ACT performance of the students make interesting venue for integrating programming exercises into introductory physics with the end goal of creating a curriculum that might succeed in the high school physics classroom.\n\nEach activity is designed to take about an hour to complete. Students are not explicitly assigned to groups or pairs, but the classroom setup involves six tables of four, so students will tend to collaborate on the activities and this is not discouraged. To date, about 125 students from OSU's regional campus in Marion have completed the exercises mentioned below. The activities are graded on the completion of required steps.\n\nAll of the exercises illustrate the velocity, acceleration and force vectors. The first exercise gives the student much of the code that they will need, only asking them to make small, guided modifications, which we will describe in the next section. All of the exercises build off of each other in a way that would make it hard for a student to start in the middle of the sequence.  Additionally, all of the exercises contain ``challenges\" that encourage the student to develop some functionality that often adds an interesting element to the game. The list of exercises is as follows:\n\\begin{enumerate}\n    \\item Planetoids (similar to the classic game ``Asteroids\")\n    \\item Lunar descent (similar to the classic game ``Lunar lander\")\n    \\item Bellicose birds (similar to the popular game ``angry birds\")\n    \\item Planetoids with momentum\n    \\item Planetoids with torque\n    \\item Planetoids with a spring (harmonic motion)\n    \\item Extra credit: Bellicose birds with energy\n\\end{enumerate}\n\nThis sequence is designed to accompany a typical physics course on classical mechanics where momentum is not introduced until mid-way through the course, followed by concepts of torque and, later, harmonic motion. The ``Bellicose birds with energy\" exercise is made available to students in the middle of the course when energy is introduced, but this exercise is more difficult than the others because it is the only exercise that deals explicitly with the integration scheme. For simplicity, all the exercises adopt Euler-Cromer integration \\cite{Cromer1981} except for ``Bellicose birds with energy\", which describes the trapezoidal method in terms that an algebra-based physics student should be able to understand.\n\nIn this paper we provide a rather extensive description of the first exercise (``Planetoids\") including student survey responses. This section provides a context to this exercise since essentially all the exercises listed above are derived from this ``Planetoids\" exercise.  These activities will be described in detail in later work. We will only add here that some of these additional activities use a graphing system to plot various relevant physical quantities (such as velocity) over time in the bottom right corner of the screen.\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=5in]{planetoids_fig_improved.png}\n    \\caption{The code (left) and corresponding interactive (right) that the student sees at the beginning of the first exercise. This code is written with the processing javascript library p5.js. As a result, the code has a C\/C++ like syntax except that draw() replaces main() and draw() is run 60 times per second until the user stops the program. The interactive (right) shows a ship traveling towards the right with constant velocity indicated by a red velocity vector. On the left panel the student sees about 50 lines of code, but about half of these lines are spaces or comments.}\n    \\label{fig:planetoids}\n\\end{figure*}\n\n\\section{The Planetoids Game}\n\nThe choice of an ``Asteroids\"-like game for the first activity is intentional. A natural environment for illustrating Newton's laws is free space, away from any sources of gravity. We are not unique in using this situation as a starting point. \\cite{white1984} found learning gains from students interacting with a video game with a similar premise, and no doubt other authors adopt a similar approach. The advantage of this environment is that objects in motion will continue with the same velocity, moving in a straight line, unless a force is acts upon them. The classic game ``Asteroids\" illustrates this well with a ship that drifts through free space, except when its rockets fire to avoid asteroids that are also drifting through free space. The net force is either zero, or constant in the direction the ship is pointing.\n\n\\subsection{Learning Goals}\n\\label{sec:learning}\n\nThe learning goals for this exercise are as follows:\n\n\\begin{enumerate}\n    \\item Understand how to convert a simple 1D code into a 2D code\n    \\item Understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space\n\\end{enumerate}\n\nThe list above is intentionally short because we do not expect that during this 1-2 hour activity that the student will be able to absorb the subtleties of computational thinking \\cite{Weintrop_etal2016}\nor become proficient with the javascript-based coding framework to the point where they can comfortably make numerous modifications to the code. In the following subsections we discuss how the activity is structured to reinforce the two learning goals mentioned above, and we point out various difficulties that students often have.  We discuss additional learning goals and extensions to the activity in later sections.\n\n\\subsection{Structure of the Program and Design Choices}\n\nFigure ~\\ref{fig:planetoids} shows what the student sees at the beginning of this exercise. Initially, the ship can only move in the $x$ direction and the first task is to allow the ship to rotate when the user presses the left and right arrow buttons by changing the value of $\\theta$. It is worth commenting on Fig.~\\ref{fig:planetoids} in detail because even at this stage there are a number of choices that have been made that could affect student learning. One important choice made to simplify the cognitive load for the student is to ``hide\" a significant amount of code in the \\texttt{display()} function. In this example, there are about three times more lines of code defining the \\texttt{display()} function than the $\\approx 50$ lines of code that the student sees and modifies\\footnote{We attempted to re-create this exercise in vpython using glowscript.org but found that (at least currently) there is no way of setting up a second page of code where subroutines can be defined without being in plain view by the student, which is a barrier to implementing this ``hybrid\" approach. This may or may not be a limitation with other browser-based python development environments like trinket.io or jupyter.org}.\n\nAnother important choice is to hide the variable types. There are no \\texttt{float}, \\texttt{int}, \\texttt{double} or \\texttt{var} declarations used to initialize the variables. Instead, variables are implicitly declared to be floating point decimals and the number of characters that the student sees is minimized. This syntax is essentially the same as used in Matlab, which is a popular language for absolute beginner programmers. We use the processing javascript library p5.js for these exercises and as a result the code shown in Fig.~\\ref{fig:planetoids} is javascript which does not produce an error for missing variable types. A possible drawback of postponing the discussion of variable types is that the difference between global and local variables is not explained at this stage. Students may not realize that \\texttt{accelx}, which is only used and defined inside of an \\texttt{if} statement, is a local variable while \\texttt{deltaVx} is a global variable, but this is unlikely to cause a problem at this stage. Our philosophy is to explain subtleties like these in the step-by-step tutorial only if absolutely necessary for completing a particular exercise.\n\nThe structure of the program in Fig.~\\ref{fig:planetoids} is an important choice that may affect student learning. The sections of the code are as follows:\n\\begin{enumerate}\n\\item Variable initializations \n\\item the \\texttt{draw()} function -- velocity and position advance\n\\item the \\texttt{draw()} function -- keyboard inputs\n\\item the \\texttt{draw()} function -- \\texttt{display()} function followed by other user-defined graphics\n\\end{enumerate}\nIt is understood that the \\texttt{draw()} function is run many times per second so that after the \\texttt{display()} function is executed the program will go back to the beginning of \\texttt{draw()} and advance the velocity and position again and go through the whole sequence again until the user presses stop\\footnote{An optional ``Hello world\" activity demonstrates that adding code to write a simple message to the browser console while inside of the \\texttt{draw()} function will result in that message being written many times over because the \\texttt{draw()} function is being run many times per second.}. Because \\texttt{draw()} is being run again and again, one could easily change the sequence so that, for example, the \\texttt{display()} function would be first and the velocity and position advance would be last. The drawback of this approach is that when the student parses the code for the first time they would see the physics content of the code \\emph{last}. In a physics course, our primary interest lies in directing the students' attention to how the physics content, such as $d=vt$ for example, is implemented in the code, with discussions of the programming concepts such as syntax, variable types, and the structure of the algorithm being supplementary to that.\n\nFollowing the physics section there is a line of code \\texttt{deltaVx = 0} ($\\Delta v_x = 0$) which is accompanied by a comment ``velocity is unchanged if there are no forces\". This is just a restatement of Newton's first law in a form that a computer can understand. Following this, the program checks if the user is pressing certain buttons on the keyboard.\n\nThe drawback to this physics-first, keyboard commands later approach is that the student may not fully appreciate that the program holds on to the global variable \\texttt{deltaVx}, which is determined from the keyboard command section, only using it again at the beginning of the \\emph{next} iteration of \\texttt{draw()}.\n\nIn the written step-by-step directions, the user is asked to put non-zero values in the section of the keyboard input section that changes the angle of the ship. Then the student is asked to enable motion in the $y$ direction by imitating the code for advancing the velocity and position in the $x$ direction. Finally, the student is asked to determine the correct change in velocity due to a constant force (thrust) in the $y$ direction. This involves realizing that while $\\cos \\theta$ gives the component of the force oriented in the $x$ direction, one must use $\\sin \\theta$ to obtain the component of the force in the $y$ direction.  Students are given a hint that it is either a cosine, sine, or tangent function that gives the correct behavior.\n\nAt each step in the tutorial, the student can click links to see and interact with how the program should work at a particular stage, but without seeing the source code for the completed step. This is an important capability that gives the student instant guidance on whether they have completed a particular programming task correctly, leaving the instructor more time to spend on subtle issues.\n\nCommon mistakes that students make include forgetting to set \\texttt{deltaVy = 0}, in which case the ship accelerates uncontrollably in the $y$ direction. Students rarely self-diagnose this issue because the ship appears to behave correctly if the thrusters are repeatedly fired and it is only when the student stops firing the thrusters that the uncontrollable acceleration becomes obvious. When students interact with the correct version of the program (as described in the previous paragraph) they should notice this difference in behavior but the problem is subtle enough that this problem is easy to miss.\n\nAnother frequent mistake is that students tend to do a quick copy paste of the acceleration code for the $x$ direction to the $y$ direction without changing the trigonometric function from cosine to sine. This causes $\\Delta v_y = \\Delta v_x$ and as a result the ship only travels on a diagonal line regardless of the angle $\\theta$. Students have an easier time self-diagnosing this issue because the problem is easy to see and they are told that the trigonometric function in the line of code that determines $\\Delta v_y$ should be either a cosine, sine, or tangent.\n\n\\subsection{Challenges}\n\nStudents must also implement 1-2 ``challenges\". The challenges in this exercise include creating ``planetoids\" (a word play on the astronomical term planetesimals) that drift across the screen using the \\texttt{drawEllipse()} function and adding reverse thrusters when the down arrow is pressed (which can be done by copying the code from the up arrow and adding minus signs to change the direction of the force). Students can also allow the ship to shoot a projectile using the \\texttt{drawPoint()} function and the code includes an \\texttt{if} statement that detects if spacebar is pressed for this purpose. This latter task is more difficult than the others because the projectile must be launched in the same direction as the ship whereas the planetoids can be given a random velocity using the \\texttt{random()} function. One should also include the velocity of the ship when determining the velocity of the projectile as a fun illustration of Galilean invariance. Most students will just implement the reverse thrusters challenge.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=3.3in]{notorque_lab_mod_posy.png}\n    \\caption{A screenshot from an activity where the student explores how changing the force of the rocket's thrust and the mass of the ship affects one's ability to avoid randomly drifting ``planetoids\". This follows code modification tasks that enable the ship to move in two dimensions (instead of one dimension as in Fig.~\\ref{fig:planetoids}).}\n    \\label{fig:withplanetoids}\n\\end{figure}\n\n\nA more recent modification to this activity that was added after the study is to give the student a code that includes a number of drifting planetoids that will cause the game to end if the ship runs into one of them (Fig.~\\ref{fig:withplanetoids}). The student is asked to explore the effect of changing the force of the ship's thrust and the mass of the ship on surviving in the game. This task helps foster a discussion of how it is only the ratio of the force to the mass that matters to the acceleration of a rocket in free space.\n\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=3.4in]{experience.pdf}\\includegraphics[width=3.4in]{level_of_difficulty.pdf}\n    \\includegraphics[width=3.4in]{vectors.pdf}\\includegraphics[width=3.4in]{fun2.pdf}\n    \\end{center}\n    \\vspace{-0.6cm}\n    \\caption{Survey results from Ohio State Marion students who completed the first programming exercise (rocket in free space). Results are cumulative from four semesters of students (Spring 2015 - Fall 2016).} \n    \\label{fig:survey}\n\\end{figure*}\n\n\\section{Student data}\n\\label{sec:survey}\n\n\nAfter the student completes the Planetoids exercise, there is a detailed online survey that probes their experience in completing the activity. While the questions in this survey are qualitative and involve student self-reporting, the results can offer insight on whether the level of difficulty of the first exercise is appropriate and whether students find the exercises to be enjoyable to complete.\nFigure~\\ref{fig:survey} summarizes the results of the survey from four semesters of students (Spring 2015 -- Fall 2016). The upper left plot in Fig.~\\ref{fig:survey} shows that there are a significant number of absolute beginner programmers and weak programmers in the class. There were also a significant number of students who reported ``some experience\" which may have meant that they were currently enrolled in a required C++ course, but had not had significant experience with coding prior to this.\n\nThe upper right plot in Fig.~\\ref{fig:survey} shows that the difficulty level seems to be appropriate for the population of students, with a significant number of students selecting ``Easy!\". The lower right plot in Fig.~\\ref{fig:survey} indicates that many of the students found the programming activities to be enjoyable or fun. Students also have many positive things to say about the programming exercises in written evaluations at the end of the course after all of the exercises have been completed.\n\nThe bottom left plot in Fig.~\\ref{fig:survey} summarizes student responses to the question ``Did the programming lab help you understand vectors better?\" Although students can only provide a subjective estimation for how much they have learned, studies have shown that information of this kind can be valuable and even predictive other measures of student success \\cite{Sawtelle_etal2012}.\n\n\\section{Success Relative to Learning Goals}\n\nIn an earlier section (\\ref{sec:learning}) we outlined two learning goals for the exercise. The student survey data in \\ref{sec:survey} can provide some qualitative or indirect insight on whether these goals were met. In particular, the ``Level of Difficulty\" question, which is asked after the completion of the code, relates to the learning goal of ``Understand how to convert a simple 1D code into a 2D code\" since this is the main activity of the exercise. Unfortunately we do not have precise data to pinpoint the perceived difficulty for the subset of students who reported the least prior programming experience. But with only 1 student reporting ``Extremely Difficult!\" and 8 students reporting ``Difficult!\" compared to the 39 students who reported either ``No\" or ``a little bit\" of prior programming experience, the data supports the idea that students were able to complete the 1D to 2D conversion of the code without severe difficulty. Whether they fully understand the changes that were made is another important question that we can probe in future work.\n\nThe other learning goal was ``Understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space\". Although we do not have a direct probe of this learning goal, the question ``Did the programming lab help you understand vectors better?\" relates to this learning objective in an indirect way.  Many of the students found the exercise to be at least ``somewhat\" helpful in understanding vectors. As mentioned in the last section, student self-reporting can be useful and even predictive of student learning \\cite{Sawtelle_etal2012}. In retrospect, one wonders if even more students would have reported understanding vectors better if there had been a part of the exercise where the student gives the ship an initial velocity and interacts with the program from that starting point, or if we had included the activity described earlier where students change the force (thrust) and mass of the ship (Fig.~\\ref{fig:withplanetoids}) to see the effect on the motion in avoiding asteroids (an activity which was only added later).\nIt is also key to note that learning gains can only be achieved if students do actually engage with the activity. The question most closely related to this was ``Was the programming lab fun?\" An overwhelming majority of the students found the exercise to be ``enjoyable\" or ``fun\" which suggests that they did significantly play around with the simulation (which demonstrates the relationship between force, velocity and acceleration vectors in an interactive way, making it very relevant to the goal of better understanding these vectors). It is therefore reasonable that there may be sufficiently high student \"buy-in\" to warrant further study, and further optimization of the user interface to maximize learning outcomes as described above.\nNevertheless, the questions discussed here are still oblique, self-reported measures of student learning on these learning goals and we do not wish to overstate the results we obtained. \n\nIn future work we can directly probe the second learning goal using, for example, the rocket questions from either the Animated Force Concept Inventory by \\citet{Dancy2006} or the conventional Force Concept Inventory \\cite{FCI}, and other questions that ask students to identify the correct force, velocity and acceleration vectors in different situations. Importantly, we can compare results for these questions from students who complete a coding activity, and a ``control group\" of students who only play around with the interactive for that coding activity for some period of time but without actually seeing or modifying the code. This will probe whether coding activities of the kind we discuss here, which involves multiple steps where students modify the code and check the behavior of the program, cause students to look more critically at the interactives they produce than they would if they did not have to perform coding tasks.\n\n\n\n\n\n\n\n\\section{Summary and Conclusion}\n\nIn this paper we illustrate a ``hybrid\" approach to incorporating computer programming activities into introductory physics courses by describing a coding activity that resembles the classic asteroids game. The approach is so named because activities like the one described here produce interactives that bear some resemblance to web interactives that groups like PhET and Physlet have produced, but unlike PhET and Physlet, the student works with and modifies the code that evolves the system. In a ``traditional\" computational physics course the student would have a great deal of control over producing visualizations. To reduce the cognitive load for weak or absolute beginner programmers in our study, the parts of the code that are unrelated to physics are hidden away in a \\texttt{display()} function so that the student sees and works with only about 50 lines of code. In this sense our approach is a kind of ``hybrid\" between canned interactives and mature computational physics exercises that are typically used in physics-major courses.\n\nThe first exercise in our suite of activities is an interactive simulation that resembles the classic game ``asteroids\". The learning goals of this activity are to (1) understand how to convert a simple 1D code into a 2D code and (2) to understand how force, velocity and acceleration vectors relate to the motion of a ship traveling in free space. The activity includes scaffolding and hints to make the task of modifying the 1D code into 2D more manageable.\n\nIn an introductory class at OSU Marion where a substantial fraction of the students are weak or absolute beginner programmers, student survey data ($N \\approx 80-85$) confirms that most students, including those with weak or absolute beginner programming experience, are able to complete the activity without severe difficulties. We interpret this as evidence that the first learning goal is being met. \n\nWe are still only just beginning to investigate the effectiveness of the second learning goal. We discuss survey results that provide some insight into student experiences with the exercises, which in an indirect way addresses the second learning goal. However, this is no substitute for directly probing student learning with carefully chosen questions. In future work we will use the Animated Force Concept Inventory \\cite{Dancy2006}, and other assessments to probe whether students understand the relationship between velocity and acceleration vectors. Of particular importance is whether the task of making modifications to the code and checking for the effect of these modifications on the interactive program will cause students to think more critically about the physics concepts than they would by playing around with a ``canned\" interactive. This may be the real value of integrating coding at this level.   \n\n\n\nWe welcome inquiries from educators who may wish to use this suite of coding activities in their courses. Individual exercises and solution sets (including the planetoids game described here) are available at \\url{http:\/\/compadre.org\/PICUP}\n\n\\acknowledgements\n\nThe authors thank Chris Britt and Michael Hardesty for their collaboration on a p5.js learning management system. Chris Orban thanks Kathy Harper, Gregory Ngirmang, and Kelly Roos for discussions. This project was made possible through a Connect and Collaborate Grant, a program supporting innovative and scholarly engagement programs that leverage academic excellence of The Ohio State University in mutually beneficial ways with external partners. Support also comes from the American Institute of Physics Meggers Award.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{} Introduction}\n\nThe complex interplay between spin, charge, and lattice degrees of freedom in the quasi-two dimensional copper-oxide \nhigh temperature superconductors have been the subject of intense interest since the discovery of\nsuperconductivity in the La$_{2-x}$Ba$_x$CuO$_4$\\ system some 21 years ago\\cite{Bednorz:1986}.  Both La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ display a fascinating series of structural, magnetic and superconducting phase transitions as a function of temperature\\cite{Kastner:1998}.  While La$_{2-x}$Ba$_x$CuO$_4$\\ was the first layered cuprate high T$_{c}$ superconductor to be discovered, difficulties associated with the \ngrowth of high quality single crystals have significantly limited its study.  As a result the La$_{2-x}$Ba$_x$CuO$_4$\\ family is much less studied than the La$_{2-x}$Sr$_x$CuO$_4$\\ family and other high temperature superconductors which have an extended history of being grown and characterized in single crystal form\\cite{Kastner:1998}, such as the YBa$_{2}$Cu$_3$O$_{7-\\delta }$\\ and Bi$_2$Sr$_2$CaCu$_2$O$_8$\\ families\\cite{Birgeneau:2006,Eschrig:2006,Fong:1999,Castellan:2006}.  \n\nRecently, significant progress has been made in growing the La$_{2-x}$Ba$_x$CuO$_4$\\ family of materials in single crystal form, and  this has enabled several important new studies of this and related systems\\cite{Fujita:2004,Reznik:2006,Tranquada:2004,Kimura:2005}.  It is therefore timely to perform high resolution structural studies of these new single crystals, and to compare to previous studies on La$_{2-x}$Ba$_x$CuO$_4$\\ in polycrystalline form\\cite{Suzuki:1989,Suzuki:1989a}.\n\nOne of the many interesting properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ family is the sequence of structural phase transitions which this material displays on cooling \nbelow room temperature for underdoped Ba concentrations (x$\\lesssim$ 0.18). Previous studies on polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ shows three different structures, which proceed from High Temperature Tetragonal (HTT, $I4\/mmm$), to Middle Temperature Orthorhombic (MTO, $Cmca$) and finally to Low Temperature Tetragonal (LTT, $P4_{2}\/ncm$)\\cite{Axe:1989,Axe:1989a,Suzuki:1989,Suzuki:1989a,Adachi:2001}. The HTT$\\to$MTO and the MTO$\\to$LTT phase transition temperatures are referred to as $T_{d1}$ and $T_{d2}$, respectively.   The HTT$\\to$MTO transition is \ncontinuous, while the MTO$\\to$LTT transition is known to be strongly discontinuous.  These structures are closely \nrelated to the magnetic and electronic properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ families.  The phase diagram of the La$_{2-x}$Ba$_x$CuO$_4$\\ system contains a dome\nof LTT phase, which is centred around x=0.125.  This Ba-concentration corresponds to a steep depression of the superconducting $T_C$ as a function of concentration, known as the 1\/8 anomaly\\cite{Axe:1989a,Moodenbaugh:1988}.  The La$_{2-x}$Sr$_x$CuO$_4$\\ system shows a much smaller $\\sim$ 10$\\%$ dip in $T_C$ at x=0.125 and the absence of the LTT phase at low temperatures\\cite{Nagano:1993,Radaelli:1994}.  The 1\/8 anomaly within the LTT phase also corresponds to strong\nincommensurate magnetic long range order at temperatures just below the completion of the MTO-LTT phase transition\\cite{Fujita:2004,Tranquada:2004}.  Clearly, the structural,\nmagnetic, and superconducting properties of the La$_{2-x}$Ba$_x$CuO$_4$\\ and La$_{2-x}$Sr$_x$CuO$_4$\\ systems are strongly coupled.\n\nThe critical phenomena associated with the HTT-MTO transition has been previously studied in pure La$_{2}$CuO$_{4}$ as well as in La$_{2-x}$Sr$_x$CuO$_4$\\ in single crystal and polycrystal form\\cite{Birgeneau:1987,Vaknin:1987,Boni:1988,Braden:1994,Ting:1993,Thurston:1989}, as single crystals of these materials have existed for some time.  These\nstudies show the HTT$\\to$MTO phase transition to be characterized with an order parameter critical exponent $\\beta$ varying from 0.28 to 0.37\\cite{Birgeneau:1987,Vaknin:1987,Boni:1988,Braden:1994,Ting:1993,Thurston:1989}. Studies on polycrystalline samples of La$_{2-x}$Ba$_x$CuO$_4$\\ by Susuki et al\nproduced estimates for $\\beta$ $\\sim$ 0.33\\cite{Suzuki:1989,Suzuki:1989a}, and which are consistent with expectations for 3D universality\\cite{Collins:1989}.\n\nIn this paper, we report the successful growth of large La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals with x=0.095 and 0.08, and a high resolution x-ray diffraction study on the x=0.125, 0.095 and  0.08 compounds in this family. This study focusses on a comparison between the structural and superconducting phase diagrams in polycrystalline and single crystal materials, critical phenomena associated with the HTT$\\to$MTO phase transition, and the nature of the LTT phase in x=0.125 and 0.095 samples at low temperatures.\n\n\n\\section{\\label{} Experiment details}\n\\subsection{\\label{} Crystal Growth}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{map125_arxiv.eps}\n\\caption {(a), High resolution longitudinal scans of the (3, 3, 0)$_{HTT}$ Bragg peak in single crystal La$_{2-x}$Ba$_x$CuO$_4$, x=0.125 are shown as a function of temperature. (b) Representative longitudinal scans at T=290 K, 120 K, and 20 K from which the color contour map in (a) was made.}  \n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{map095_arxiv.eps}\n\\caption {(a), High resolution longitudinal scans of the (3, 3, 0)$_{HTT}$ Bragg peak in single crystal La$_{2-x}$Ba$_x$CuO$_4$, x=0.095 are shown as a function of temperature. (b) Representative longitudinal scans at T=290 K, 120 K, and 20 K from which the color contour map in (a) was made.}  \n\\end{figure}\n\n\n\nWe studied three high quality La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals with x=0.125, 0.095 and 0.08.  All crystals were grown by using traveling solvent, floating zone image furnace techniques. The x=0.125 sample was grown separately, and the details of this growth have been previously discussed\\cite{Fujita:2004,Tranquada:2004}.\n\nThe x=0.095 and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ single crystal growths followed similar processess, and employed polycrystalline La$_{2}$O$_{3}$, BaCO$_{3}$ and CuO as starting \nmaterials to make the initial, polycrystalline feed rod and solvent. For the production of the feed rods, the starting materials were mixed to\ngive an initial ratio of La:Ba:Cu=1.875:0.125:1. These materials were mixed, ground, and annealed at 980$^{\\circ}$C for 12 hours \nin air. This process was repeated twice in order to ensure homogeneous feed rods.  To compensate for Cu evaporation during the crystal growth, the pre-annealed feed rods were mixed with extra CuO. A further 1\\% and 2\\% mol CuO was added to the starting polycrystalline materials and thoroughly mixed to prepare the two final feed rods, respectively. The final feed rods were heated to a temperature of 1190$^{\\circ}$C, at a rate of 100$^{\\circ}$C\/hour.  They were held at this temperature for 12 hours.  We also employed a solvent, formed from the original polycrystalline feed rod, with CuO added so as to reach a final ratio of constituent atoms (La$_{1.875}$Ba$_{0.125}$):Cu=3:7. After mixing and sintering, small disks weighing $\\sim$ 0.44 g were cut out and used as solvents in the subsequent single crystal growths.\n\nThe single crystal growths were carried out using a four-mirror image furnace (Crystal System Inc.). A small pure La$_{2}$CuO$_{4}$ single crystal was employed as the seed rod for both growths. The growths were carried out in an O$_2$ atmosphere at pressures of 165 kPa and 182 kPa for the two crystal growths.  The growth rate was 1mm\/h with a counter-rotation speed of 25 rpm for feed and seed rods for both growths.\n\n\n\n\nUpon completion of the growths, the as-grown single crystals were kept above 100$^{\\circ}$C in a furnace to prevent hydrolysis of the \nmaterial, which is known to be problematic for single crystal La$_{2-x}$Ba$_x$CuO$_4$. The two crystals, which are identified in this study as being at x=0.095 and 0.08, were of almost identical dimensions of 80 mm long by  5 mm in diameter as-grown. Within the first week following completion of the growths, the initial $\\sim$ 30 mm of the crystals turned to dust as a result of hydrolysis of the second phase. The undamaged part of both crystals was stable.  They had approximate dimensions of 50 mm long by 5 mm in diameter for x=0.095 and 55 mm long by 5 mm in diameter for x=0.08.\nThese volumes are sufficiently large for advanced characterization by neutron scattering techniques, and indeed a program of neutron measurements has been carried out on these samples\\cite{Dunsiger:2007}.  \n\nWe note that while the two crystal growths were initiated with similar starting materials, and the growths followed similar procedures, the Ba\/La ratio, as identified by T$_{d1}$ and T$_{d2}$, were different at the $\\sim$ 15$\\%$ level. This originates from Cu evaporation during the growth. All the phase transitions observed (structural, magnetic, and superconducting) are nevertheless very sharp in temperature, indicating excellent homogeneity of concentration within the individual single crystals.\n\n\\subsection{\\label{} X-ray diffraction}\n\nSingle crystal samples with approximate dimensions 8 mm$\\times$8 mm$\\times$1 mm for x=0.125, and 5 mm$\\times$5 mm$\\times$1 mm for x=0.095 and 0.08, were cut from large single crystals of La$_{2-x}$Ba$_x$CuO$_4$.  These were sequentially attached to the cold finger of a closed cycle refrigerator and mounted within a four circle x-ray \ndiffractometer.  Cu K$_{\\alpha 1}$ radiation from an 18kW rotating anode x-ray generator was selected using a perfect Germanium (111) single crystal monochrometer. A Bruker Hi-Star multi-wire area detector was placed on the detector arm, 76 cm from the sample allowing an angular resolution of approximately  0.01 degrees to be achieved. All measurements focused on (3, 3, 0)$_{HTT}$ Bragg peak of the samples, using notation appropriate to the high temperature tetragonal phase. As we were interested in critical phenomena, the sample was mounted in a Be can and in the presence of a helium exchange gas and the sample temperature was stabilized to $\\sim$ 0.005 K for all measurements. \n\n\\section{\\label{} Experimental Results}\n\n\\subsection{\\label{} Identification and Nature of Phases}\n\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{LattvsT_all.eps}\n\\caption {(a) The orthorhombic strain vs. temperature is plotted for La$_{2-x}$Ba$_x$CuO$_4$\\ x=0.125, 0.095 and 0.08 single crystal samples. The open and filled symbols represent warming and cooling cycles, respectively.  The orthorhombic strain is obtained by fitting longitudinal scans, shown in Figs. 1 and 2.  (b) The same orthorhombic strain vs. temperature as in (a) but now plotted vs T\/T$_{d1}$ and the strain has been scaled (for the x=0.125 sample, by a factor of 2.4) to emphasize universal behavior for T\/T$_{d1}$ greater than 0.8. }  \n\\end{figure}\n\nTwo dimensional maps of the scattering around the (3, 3, 0)$_{HTT}$ Bragg peaks of all three x=0.125, 0.095 and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ samples were acquired as a function of temperature.  Each data set consisted of a sample angle rock through the Bragg peak which was integrated in the vertical \ndirection and plotted as a function of scattering angle, 2$\\theta$.  A longitudinal cut through this two dimensional data set was performed, giving rise to the longitudinal scans shown in Fig. 1b for the x=0.125 sample, and Fig. 2b for the x=0.095 sample.  Similar data sets taken over a more restricted temperature regime for the x=0.08 sample are of similar quality, but are not shown.\n\nThese data sets can be put together to display the full temperature dependence of the longitudinal scans, and this is what is shown in Figs. 1a and 2a for the x=0.125 and x=0.095 samples, respectively.  These data sets clearly show the bifurcation of a single Bragg peak into two, and then back into one, as the temperature is decreased from room temperature to 20 K, signifying the sequence of phase transitions HTT$\\to$MTO$\\to$LTT.   The fact that two Bragg features can be seen in a single longitudinal scan within the MTO phase is indicative of twinning within the orthorhombic phase, although the two twin domains which are observed do not possess equal volume fraction within the crystal;  one Bragg feature is considerably stronger in intensity than the other.  A minority and majority twin domain is clearly present, but the relevant volume fraction can change from one thermal cycle to the next.  For example, the x=0.095 data set shown in Fig. 2(a) shows data from two independent thermal cycles, one ending with a lowest temperature of $\\sim$ 200 K, while the next beginning a new thermal cycle at 200 K.  In the first of these, the high angle Bragg peak is the majority domain, while in the second cycle, the lower angle Bragg peak is the majority domain. \n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{Latt_Tzoom.eps}\n\\caption {The orthorhombic strain is plotted vs reduced temperature, (T-T$_{d1}$\/T$_{d1}$) for the x=0.125, 0.095, and 0.08 La$_{2-x}$Ba$_x$CuO$_4$\\ samples at small values of reduced temperature, near T$_{d1}$.  The open and filled symbols show data from warming and cooling cycles, respectively.  Fits of the data to the form of the order parameter squared vs reduced temperature, Eq. 1, \nused to extract values of $\\beta$ are shown as the solid lines.}  \n\\end{figure}\n\nThe fact that we observe both twin domains in the MTO phase means that the peak positions, the lattice parameters, and consequently the\northorhombic strain, 2(a-b)\/(a+b), can be determined as a function of temperature.  This is shown for all three samples in Fig. 3a.  The single (3, 3, 0)$_{HTT}$ Bragg peak breaks into (6, 0, 0)$_{MTO}$ and (0, 6, 0)$_{MTO}$ near T$_{d1}$=232 K, 272 K and 305 K in the x=0.125, 0.095, and 0.08 samples, respectively, before reforming into a single (3, 3, 0)$_{LTT}$ Bragg peak near T$_{d2}$=60 K, 45 K, and 35 K, respectively.\n\nExamination of Figs 1-3 shows two qualitative features of the evolving structures.  Note that for ease of comparison, the 2$\\theta$ range of the\nscattering in Figs. 1 and 2 is the same.  First, the orthorhomic strain decreases quite substantially with increasing Ba concentration.  The lowest\ntemperature strain, for example, in the  x=0.125 sample is roughly half that of the x=0.095 sample.  Secondly and more importantly, the longitudinal profile of the (3, 3, 0)$_{LTT}$ peak at the lowest temperatures measured, well within the LTT phase, is considerably broader than the\ncorresponding profile of (3, 3, 0)$_{HTT}$.  This is true for both the x=0.125 sample and the x=0.095 sample as can be seen by comparing the top and bottom panels of Fig. 1b (for x=0.125) and Fig. 2b (for x=0.095).  This shows that the LTT phase is either an admixture of a tetragonal and an orthorhombic phase, as was suggested by electron microscopy on \nan earlier generation of La$_{2-x}$Ba$_x$CuO$_4$\\ crystals\\cite{Zhu:1994}, or that it is itself othorhombic with a very small orthorhombic strain.  In either case it is not as ``tetragonal\" as the HTT phase, and is consistent with the ``less orthorhombic\" low temperature structures proposed previously for La$_{2-x}$Sr$_{x-y}$Ba$_y$CuO$_4$ single crystals\\cite{Fujita:2002b}. \n\n\n\\subsection{\\label{} Critical Phenomena at the HTT$\\to$MTO Phase Transition}\n\nLongitudinal scans of the form shown in Fig. 1b and 2b were fit for the purpose of extracting the peak positions in 2$\\theta$ and \ntherefore the d spacings associated with the MTO phase.  This is straightforward for data far removed from the HTT$\\to$MTO phase transition, as the two peaks are well defined and separated, as can be seen in the middle panels of Fig. 1b and 2b.  Closer to the phase transition, one peak may appear as a shoulder to the other, and it is more difficult to ascribe unique values to the two lattice parameters.  We fit these data in two different ways in order to attain robust values for the lattice parameters close to the transition.  One of these was to simply fit the longitudinal scans to sums of Lorentzians or Lorentzians raised to an adjustable exponent, while a second technique was to look for zeros in the derivatives of the intensity as a function of 2$\\theta$.  These gave consistent results for the lattice parameters, giving us confidence that the orthorhombic strain could be estimated accurately close to the transition. However, this technique also gives non-zero values for the orthorhombic strain, albeit relatively small ones, within the HTT phase.  \n\nPrevious work on the HTT$\\to$MTO phase transition in polycrystalline La$_{2-x}$Sr$_x$CuO$_4$\\ and La$_{2-x}$Ba$_x$CuO$_4$\\ samples show the \northorhombic strain to scale as the square of the order parameter\\cite{Birgeneau:1987,Boni:1988,Suzuki:1989,Suzuki:1989a}.  Consequently we examined \nthe critical behaviour of the orthorhombic strain in our La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals by fitting the measured strain as a function of temperature to: \n\\begin{equation}\n\\label{ }\n\\Delta=\\Delta_{0}\\times  (\\frac{T_{d1}-T} {T_{d1}})^{2\\beta}+Background\n\\end{equation}\nwhere the square of the order parameter, $\\Delta$, is the orthorhombic strain, 2(a-b)\/(a+b), and the background accounts for finite strain within the HTT phase introduced by the fitting process described above. The results of this fitting is shown in Fig. 4, which shows the orthorhombic strain as \na function of reduced temperature, (T-T$_{d1}$)\/T$_{d1}$, in the region of small reduced temperature close to T$_{d1}$.  Clearly this description of the data is very good.  It results in accurate estimates for both $\\beta$ and T$_{d1}$.  These are T$_{d1}$=232.3 $\\pm$ 0.7 K, 271.7 $\\pm$ 1 K, and 305.4 K $\\pm$ 1 K for the x=0.125, 0.095, and 0.08 samples, respectively.  The extracted values for $\\beta$ are 0.35 $\\pm$ 0.03, 0.34 $\\pm$ 0.04 and 0.28 $\\pm$ 0.06, respectively. \n\nUsing these values of T$_{d1}$ for each of the three samples, we can scale the plot of orthorhombic strain vs temperature, Fig. 3a, so as to give scaled orthorhombic strain vs T\/T$_{d1}$, which is shown in Fig. 3b.  We see that above T\/T$_{d1}$ $\\sim$ 0.8 the orthorhombic strains for all three samples collapse to a single curve.  We therefore expect universal behaviour in this regime, which is borne out by the similarity in the extracted values for the critical exponent $\\beta$ at all three Ba concentrations.\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{BetaChi.eps}\n\\caption {The dependence of critical exponent $\\beta$ and goodness-of-fit parameter $\\chi^2$ are shown as a function  of the assumed value of T$_{d1}$ for x=0.125 (upper panel), x=0.095 (middle panel) and x=0.08 (lower panel) La$_{2-x}$Ba$_x$CuO$_4$\\ single crystal samples.  The uncertainty in $\\beta$ is largely determined by the uncertainty in critical temperature T$_{d1}$.}  \n\\end{figure}\n\n\nThe uncertainties associated with the critical exponent $\\beta$ are largely determined by the uncertainties in the critical temperature, T$_{d1}$, derived from the fits to the critical behaviour.  We performed fits to Eq. 1 using T$_{d1}$ set to a range of values around the approximate phase transition temperature, and then allowed the fit to adjust the other parameters in Eq. 1.  This gives a monotonically increasing estimate for $\\beta$ as a function of increasing T$_{d1}$.  Best estimates for $\\beta$ and T$_{d1}$ are given by the minimum in the goodness-of-fit parameter $\\chi^2$ which we define as:\n\\begin{equation}\n\\label{ }\n\\chi^{2}=\\frac{\\sum(\\Delta_{measured}-\\Delta_{calculated})^{2}} {N}\n\\end{equation}\nwhere N is the number of data points.\n\n$\\beta$ and $\\chi^2$ are shown as a function of T$_{d1}$ for the x=0.125 (top panel), x=0.095 (middle panel), and x=0.08 (bottom panel) samples in Fig. 5.  The uncertainty in $\\beta$ is determined by the corresponding uncertainty in T$_{d1}$, and it is roughly 10$\\%$ for the x=0.125 and 0.095 samples where we have an extended data set throughout the MTO phase, and roughly 20$\\%$ for the x=0.08 sample where the data set is restricted to temperatures close to T$_{d1}$.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{Latt_Tlog.eps}\n\\caption {The orthorhombic strain is plotted as a function of reduced temperature, (T$_{d1}$-T)\/T$_{d1}$, on a log-log scale for the x=0.125, 0.095 and 0.08 single crystal La$_{2-x}$Ba$_x$CuO$_4$\\ samples.  The open and filled symbols show results from warming and cooling cycles, respectively.  For comparison power law behavior showing $\\beta$=0.35, indicative of the theoretically expected 3D XY universality class, is indicated as the straight line on this log-log plot.}  \n\\end{figure}\n\nInvestigation of the critical properties at the HTT$\\to$MTO phase transition in polycrystaline La$_{2-x}$Sr$_x$CuO$_4$\\cite{Birgeneau:1987,Boni:1988}\\ and La$_{2-x}$Ba$_x$CuO$_4$\\cite{Suzuki:1989,Suzuki:1989a}\\ anticipated 3D XY universality on the basis of a Landau expansion appropriate to this ferroelastic system.  These early results on polycrystalline systems were consistent with the $\\beta$=0.35 expected from 3D XY universality\\cite{Le-Guillou:1977,Le-Guillou:1980}.  However, these earlier estimates for $\\beta$ spanned the range from 0.28 to 0.37, ignoring uncertainties associated with the estimates, which covers all standard 3D universality classes: Heisenberg ($\\sim$0.37), XY ($\\sim$0.35), Ising ($\\sim$0.32) and which begins to approach values consistent with tricritical phenomena (0.25)\\cite{Collins:1989}.\n\nFigure 6 shows the orthorhombic strain, 2(a-b)\/(a+b), plotted as a function of the reduced temperature, (T$_{d1}$-T)\/T$_{d1}$, on a log-log plot \nin order to identify the expected power law regime.  For comparison a straight line appropriate to $\\beta$=0.35 and 3D XY universality is also plotted.  For each sample, two data sets are plotted, one for a warming run and one for a cooling run.  We observe very similar power law behaviour in all three samples, and behaviour which is very much consistent with 3D XY universality as anticipated theoretically.  We also see, at least for the x=0.125 and 0.095 samples for which we have data over the entire MTO phase regime in temperature, that a single power law is a remarkably good descriptor of the data over a very large temperature regime.  There appears to be a slight increase in slope for reduced temperatures greater than $\\sim$ 0.2, but overall, power law-like growth of the orthorhombic strain is observed over almost two decades in reduced temperature.  This is in contrast to most critical phenomena, wherein asymptotic critical behaviour is expected to cross over to a mean field-like regime, as one moves away from the critical temperature.\n\nTaken together our orthorhombic strain measurements show critical behaviour at the HTT$\\to$MTO phase transition in single crystal La$_{2-x}$Ba$_x$CuO$_4$\\ over a broad range of concentration which is characterized by $\\beta$=0.34 $\\pm$ 0.04.  This result clearly demonstrates 3D universality, and is consistent with 3D XY universality which is expected based on Landau theory.  It is also largely consistent with previous experimental work on single crystal and polycrystal La$_{2-x}$Sr$_x$CuO$_4$\\ and polycrystalline La$_{2-x}$Ba$_x$CuO$_4$, much of which centred on measurements of superlattice Bragg peak intensities within the MTO structure, as opposed to measurements of the orthorhombic strains\\cite{Braden:1994,Thurston:1989}.  Superlattice Bragg peak intensities near continuous phase transitions can be difficult to interpret, as they can be influenced by extinction and by fluctuations above the phase transition.  This latter effect manifests itself in upwards curvature and difficulty identifying a precise phase transition temperature, which in turn can lead to uncertainty in critical exponents. \n\n\\begin{table}\n\\caption{\\label{tab:table1} Summary of structural and superconducting phase transition temperatures in single crystal La$_{2-x}$Ba$_x$CuO$_4$}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nx&$T_{d1}$(K)&$T_{d2}$(K)&$T_{c}$(K) & $\\beta$\\\\\n\\hline\n0.125 & 232.3 & 60 & 4\\footnotemark[1] & 0.35 $\\pm$ 0.03\\\\\n0.095 & 271.7& 45\\footnotemark[2] & 27\\footnotemark[2] & 0.34 $\\pm$ 0.04 \\\\\n0.08& 305.4 & 35\\footnotemark[2] & 29\\footnotemark[2] & 0.28 $\\pm$ 0.06 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{From Ref.~\\onlinecite{Fujita:2004}.}\n\\footnotetext[2]{From Ref.~\\onlinecite{Dunsiger:2007}.}\n\\end{table}\n\n\n\\subsection{\\label{} Phase Diagram and Comparison to Polycrystalline Materials}\n\nIt is of interest to compare the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram known to characterize pre-existing polycrystalline samples with that determined for the high quality single crystals\nin the present studies.  A rather detailed comparison can be carried out, as two structural and one superconducting transition temperature characterize La$_{2-x}$Ba$_x$CuO$_4$\\ samples in this underdoped concentration range.  The phase transitions measured for the single crystals in this study are summarized in Table 1.  The critical exponent $\\beta$\nrelevant to the HTT$\\to$MTO structural transition is also shown in the same table for reference.\n\nThe superconducting transition temperatures were determined from SQUID magnetometry as reported by Dunsiger et al.\\cite{Dunsiger:2007} for the x=0.095 and x=0.08 samples, and \nby Fujita et al.\\cite{Fujita:2004} for the x=0.125 sample.  The strongly first order  MTO$\\to$LTT transition is measured both by the abrupt change in the orthorhombic strain seen in Fig. 1 and 2, for the x=0.125 and x=0.095  samples, respectively, as well as by the appearance of the (0, 1, 0) superlattice Bragg peak intensity as again reported by Dunsiger et al.\\cite{Dunsiger:2007} for the x=0.095 and x=0.08 samples, and by Fujita et al. for the x=0.125 sample\\cite{Fujita:2004}. \n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{PhaseLBCO_1.eps}\n\\caption {Phase boundaries identifying structural and superconducting phases of La$_{2-x}$Ba$_x$CuO$_4$\\ single crystals are plotted on the phase diagram derived from \npreviously studied polycrystalline samples.  The structural transitions at T$_{d1}$ and T$_{d2}$ are indicated by filled squares, while superconducting T$_C$'s are indicated\nby open circles.  The first order transition at T$_{d2}$ is indicated by a bar $\\sim$ 10 K wide, showing the onset to completion of the phase transition.  Solid lines showing phase boundaries from polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ are taken from Adachi et al.\\cite{Adachi:2001} }\n\\end{figure}\n\n\nFigure 7 shows the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram with HTT, MTO, and LTT phases indicated. The HTT$\\to$MTO and MTO$\\to$LTT transitions are shown as filled squares for the three Ba concentrations measured.  The discontinuous transition at T$_{d2}$ is indicated as a bar, in order to show the onset to completion of the transition, which is $\\sim$ 10 K wide.  T$_{d2}$ in Table 1 is the midpoint of the transition.  The superconducting transitions are given by the open circles, and they indicate the onset of the superconductivity, which is also what is listed in Table 1.  Previous results for these same phase boundaries as determined for polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ samples are shown as the solid lines in Fig. 7.  These results were extracted from Adachi et al.\\cite{Adachi:2001}  and are reproduced here.\n\nAs can be seen on inspection of Fig. 7, the agreement between the structural and superconducting phase boundaries in polycrystalline La$_{2-x}$Ba$_x$CuO$_4$\\ and the new floating zone image furnace grown single crystals is remarkably good.  The absolute values for T$_{d2}$ are systematically high, at the 10$\\%$ level for the polycrystalline materials as compared to the single crystals, but overall the full level of agreement is excellent.  In particular we see that good agreement between the two for T$_{d1}$ means that this transition can be used as an accurate marker for the Ba concentration in single crystal La$_{2-x}$Ba$_x$CuO$_4$, as T$_{d1}$ has such strong Ba dependence.  The image furnace single crystals were grown without crucibles, and are expected to be of higher purity than the corresponding polycrystalline materials grown from a flux melt in a crucible.  The similarity between the overall phase diagrams in polycrystalline and image furnace grown single crystal La$_{2-x}$Ba$_x$CuO$_4$, implies an insensitivity of these phase boundaries to this level of imperfection.\n\n\\section{\\label{} Conclusions}\n\nWe have successfully grown large single crystals of La$_{2-x}$Ba$_x$CuO$_4$\\ with x=0.095 and 0.08 using floating zone image furnace techniques.  These single crystals are sufficiently large so as to enable neutron scattering studies, which will be reported separately\\cite{Dunsiger:2007}.  High resolution single crystal x-ray diffraction measurements were carried out on these samples, as well as on a high quality x=0.125 single crystal.  These measurements focus on the (3, 3, 0)$_{HTT}$ Bragg peak and show the HTT$\\to$MTO$\\to$LTT sequence of structural phase transitions known to be relevant to underdoped La$_{2-x}$Ba$_x$CuO$_4$.  The measurements also clearly show anomolous longitudinal broadening of the (3, 3, 0)$_{LTT}$ Bragg peaks in the x=0.095 and x=0.125 samples at low temperatures, indicating that the LTT phase is not a simple tetragonal phase, but rather an admixture of tetragonal and orthorhombic phases, or an orthorhombic phase with very small orthorhombic strain.  Critical\northorhombic strain measurements near the continuous HTT$\\to$MTO phase boundary show clear 3D universality, with universal behavior observed in the orthorhombic strain vs T\/T$_{d1}$ for the three x=0.125, 0.095 and 0.08 samples.  The best estimate for a common critical exponent $\\beta$ for these samples is $\\beta$=0.34 $\\pm$ 0.04, which is consistent with 3D XY universality expected theoertically for such ferroelastic transitions.  A detailed comparison of the La$_{2-x}$Ba$_x$CuO$_4$\\ phase diagram incorporating structural and superconding phase boundaries at this underdoped concentration regime indicates excellent agreement with pre-existing data based on polycrystalline samples.\n\nIt is a pleasure to acknowledge the contributions of Ms. Ann Kallin to the single crystal growth.  This work was supported by NSERC of Canada.  Gu was supported by the US Department of Energy under contract number DE-AC02-98CH10886.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA carbon-based materials have now become the most promising candidates for nanoelectronic applications. Applications of carbon nanotubes (CNTs) and graphenes have already emerged; for example, field effect transistors (FETs) \\cite{Javey2003,Misewich2003,Ouyang2007}, electrical interconnects \\cite{Avouris2007}, and sensors \\cite{Siwy2010,Prasongkit:2011}. However, linear carbon wires; namely, cumulene and polyyne, received poor attention owing to the difficulty of getting access to the pure wire \\cite{Haley:2010jc}. The cumulene, in particular, was proposed as the ideal molecular wire \\cite{Lang:1998}. However, it is still challenging to create such a metal-molecule junction in the experiment, becuse cumulene wire is generally unstable.\n\n\n\nThe appearance of the cumulene bridging CNTs or graphene was recently observed in the experiment \\cite{Jin:2009p592,Chuvilin2009,Troiani:2003p990,Marques:2004p465} by using electron irradiation inside a high resolution transmission electron microscope (TEM), thus demonstrating a mechanical way of producing the carbon wire. An axial strain on the graphene or CNT was induced by the high-energy electron beam of TEM resulting in the fracture, and led to the establishing carbon wire-CNT junction \\cite{Jin:2009p592,Troiani:2003p990}. The observed carbon wires were more stable than those produced with previous approaches; however, the long wire ($>$10 carbon atoms) appeared to be unstable \\cite{Troiani:2003p990}. Recently, the carbon wire bridged between graphene sheets was shown to perform as a bistable switch \\cite{Standley2008,Zhang:2012}, operating for many thousands of cycles without degradation, which can be interesting from an application point of view.\n\n\nThe theoretical investigations have been carried out on the structure of carbon wire-CNT junctions \\cite{Marques:2004p465,Enyashin:2005ve}. Before breaking of the junction, the carbon wire bridging between CNTs or graphene sheets can transform into either cumulene or polyyne \\cite{Ajayan:1998p1241, Jin:2009p592,Marques:2004p465}. By using the non-equilibrium Green's function (NEGF) approach, Khoo \\emph{et. al}\\cite{Khoo:2008p688} observed negative differential resistance in the carbon wire-capped CNT junction, bonded through sp$^{3}$ bond. Recently, the carbon wire connected to graphene sheets \\cite{Zhang:2010fm,Shen2010} (representing infinite radii of CNT) have been theoretically studied, revealing different electronic functions such as molecular switches \\cite{Erdogan:2011bt}, molecular rectifiers \\cite{Zeng:2011jo}, and molecular spintronics devices\\cite{Zanolli2010}. Carbon wires connected to gold \\cite{Prasongkit:2010p780,Crljen:2007p452}, lithium \\cite{Zhang13} and fullerene \\cite{Wang:2011jf} electrodes have also been investigated. \n\n\nIn this paper, we present the transport properties of the short carbon wire suspended between CNT electrodes by employing NEGF technique based on density functional theory. Varying the gap width between the electrodes changes the wire structure and contact geometries in the junction. The latter is specific for CNT electrodes, in which the bridging site can appropriately adjust to the change of junction lengths. We have focused on the variation in conductance of the carbon wire via repeated compression\/elongation of the junction, effectively leading to a prominent difference in conductance. We consider here the zigzag (4,0)CNT and armchair (4,4)CNT, due to a very small diameter of the CNTs observed in the experiment before fracture. \\cite{Troiani:2003p990}. The influence of CNT chiralities (zigzag and armchair) on conductance has been investigated. In addition, oscillating behavior of conductance, typical for cumulenes of different lengths \\cite{Lang:1998,Lang:2000tp,Emberly2009} is reproduced. Upon junction stretching, the current-voltage characteristics of the carbon wire-zigzag junctions show the high- and low-conduction states (referred as ON and OFF states) at elevated bias, corresponding to the cumulene and polyyne structures, respectively. It should be emphasized that we produced the ON and OFF states by changing the wire configuration without breaking wires, while Standley et al.\\cite{Standley2008} have demonstrated experimentally that the switch works by breaking the cumulene wire.\n\n\nExperimental studies of the graphene and CNT edge have revealed several complex rearrangements \\cite{Caglar,Huang2009,Liu:2009iu}. The migration of carbon wire along the graphene edge was observed in the experiment \\cite{Jin:2009p592}: the carbon wire could jump along the graphene or CNT edges with a change of bonding site at the junction due to a strain accumulating along the chain. We show the conductance of the C$_5$ wire connected to all possible bridge sites on the (4,0)CNT.  \nInteresting features in the electron transport properties of the carbon wires are revealed, including the possibility of current modulation via carbon wire shrinking and stretching without breaking wires. We employ a detailed analysis of transport channels in carbon wire-CNT junctions to explain our results.\n\n\n\\section{Computational Methods}\n\n\\begin{figure}[tp]\n\\begin{minipage}[ht]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[width = 8 cm]{cnt_unit.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Two-probe systems for measuring the conductance of carbon wires, connected to semi-infinite zigzag (4,0)CNT (left panel) and armchair (4,4)CNT (right panel).}\\label{grap_unit}\n\\end{figure}\n\nIn this section, we briefly describe details of the computational method and the geometrical setup procedure performed in the present work.\n\nThe carbon wire-CNT junction is referred as a two-probe system, divided into three regions: the left and right electrodes, and the central region (see Fig. \\ref{grap_unit}). To construct the carbon wire-CNT junction, an infinite cumulene and CNT were first optimized separately, and then the short wire was placed between electrodes. The central region includes the CNT on either side of the junction in order to ensure that the perturbation effect from the carbon wire edge is sufficiently screened. The carbon wire-CNT junction was optimized again, allowing all atoms in the central region to relax. Then, the gap width between electrodes was varied with a step of 0.2 \\AA.\n\n\nAll optimizations were carried out by using density functional theory (DFT) as implemented in the SIESTA code\\cite{Soler2002}. Our calculation was performed within the non spin-polarized generalized gradient approximation (GGA) \\cite{Perdew1996} method with a single-$\\zeta$ with polarization (SZP) basis, which has already been presented its validity for the short carbon wire-CNT junction \\cite{Khoo:2008p688}. The atomic core electrons were modeled with Troullier-Martins norm-conserving pseudo potential \\cite{Troullier1991}, and valence states are 2s2p for C. The real-space integrations were performed using 170 Ry cutoff, assuring the energies and forces were converged. All atoms in the central region were allowed to move until the forces were less than 0.01 eV\/\\AA.\n\n\\begin{figure}[tp]\n\\begin{minipage}[ht]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[width = 8.5 cm]{sum.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Various structures of the short carbon wires (C$_{4}$ and C$_{5}$) bridged between zigzag (4,0)CNTs and armchair (4,4)CNTs, in which the gap width is varied. Different geometrical structures and bridging sites of the carbon wires are represented by I, II, III, etc. The linear carbon wire can transform into either cumulene ``C\" or polyyne ``P\". }\\label{sum}\n\\end{figure}\n\nThe transport calculation was performed with the SMEAGOL code \\cite{Rocha:2005,Rocha:2006}, based on the combination of the NEGF and DFT. The basis set and the real-space integrations used in the electron transport calculation were the same as that of geometrical optimization part. We have performed test calculations by using a double-$\\zeta$ with polarization (DZP) basis set, resulting in only minor change of the transmission function.\n\n\nThe current through the junction was calculated using the Landauer-Buttiker formula \\cite{Datta1995,Brandbyge2002}:\n\n\\begin{equation}\nI=\\frac{2e}{h}\\int^{\\infty}_{-\\infty} dE [f(E,\\mu_L)-f(E,\\mu_R)]T(E,V),\n\\end{equation}\nwhere $\\mu_L$ and $\\mu_R$ are the electrochemical potentials of the left and right electrodes, respectively. $T(E,V)$ is the transmission coefficient at energy $E$ and bias voltage $V$, which is evaluated as\n\\begin{equation}\nT(E,V)=\\mbox{Tr}[G(E)\\Gamma_{L}G^{\\dag}(E)\\Gamma_{R}],\n\\end{equation}\nwhere $G(E)$ and $G^{\\dag}(E)$ are retarded and advanced Green's function of the central region.\n\n\n\\section{Atomic structures}\\label{sec3}\n\nWe categorize our systems into two groups: the carbon wire connected to the zigzag and armchair CNTs. We focus on the interval of 1D carbon wire existence. Pulling the wire beyond this interval results in the wire breakage, while compressing the wire more leads into folding structures. Fig.~\\ref{sum} demonstrates the various wire structures and contact geometries of the  C$_{4,5}$ wire connected to the zigzag (4,0)CNT and to the armchair (4,4)CNT, varying the gap width. The zigzag and armchair junctions are labeled as (4,0)@C$_{n}$ and (4,4)@C$_{n}$ respectively, where $n$ is the number of carbon atoms in the wire. Different geometrical structures of the wires are labeled by I, II, III, etc. The linear carbon wire can take either cumulene ``C\" or polyyne ``P\" structures.\n\n\nLet us first discuss geometrical structure of the carbon wire-zigzag(4,0)CNT junction. As illustrated in Fig.~\\ref{sum}, the C$_{4}$ wire is slightly bent due to compression of the junction, labelled as (4,0)@C$_{4}$-I(C), and then becomes the straight-wire when the junction is gradually pulled apart, labelled as (4,0)@C$_{4}$-II(C). These C$_{4}$ wires, forming two bonds to each side of CNTs, are the cumulene structures with double bonds between neighbouring atoms. The central bond length of C$_{4}$ wire can vary in the range of $\\sim$ 1.23 \\AA \\ - 1.36 \\AA \\, depending on the variation of gap width. Note that the bond length alternation of the C$_4$ wire cannot be observed, since there is only one central bond in the wires. However, we have tested for C$_{6,8}$ wires, showing the bond length alternation of $\\sim$ 0.03-0.05 \\AA \\ upto the gap width. Before wire rupture, the C$_{4}$ wire forms one bond to each side of electrodes, labelled as (4,0)@C$_{4}$-III(P). At this step, the average bond lengths of the C$_{4}$ wires are 1.27 \\AA \\ and 1.40 \\AA, which is a polyyne structure due to the triple and single bond alternation.\n\n\nFor the C$_{5}$ wire, as seen in Fig.~\\ref{sum}, the bent wire is observed in the compressed junction, labelled as (4,0)@C$_{5}$-I(C). Then it becomes straight while pulling the junction, labelled as (4,0)@C$_{5}$-II(C). These C$_5$ wires, forming two bonds to each side of CNTs, show the cumulene structure. During compressing and pulling of the (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C), the bond lengths of the cumulene wires vary in the range of $\\sim$ 1.26 \\AA \\ - 1.34 \\AA. There is no bond length alternation for the odd cumulene wire. \n\n\nIf we continue to stretch the junction, the C$_{5}$ wire forms one bond to one side of CNTs and two bonds to the other side, labelled as (4,0)@C$_{5}$-III(P). Note that this configuration is not even metastable for the C$_{4}$ wire due to broken symmetry. The average bond lengths of the C$_{5}$  wire are $\\sim$ 1.28 \\AA \\ and 1.42 \\AA, revialing the polyyne structure. In particular, the bonding geometry of the (4,0)@C$_{5}$-III(P) is similar to a transition state in the possible migration pathway for the migration of a 3\\% strained carbon chain along the graphene sheet with zigzag edge reported in Ref. \\citenum{Jin:2009p592}. We would like to emphasize that a curvature of the CNT plays an important role in the structure of the wire at this stage of elongation: the tube edge becomes capped instead of being an open-ended one. This results in a long bond along a chord across the CNT towards the next hexagon, thus defining a single bond to the wire and yielding shorter triple bonds in polyyne. The far more symmetric cumulene structure is less affected by the CNT curvature, but the bonding geometry is changed as compared to graphene, namely the wire is bonded to the site between two adjacent hexagons. Before breaking of the contact, the C$_{5}$ wire forms one bond to each side of CNTs, and becomes the cumulene structure again, (4,0)@C$_{5}$-IV(C). Typically, bond lengths of single, double, and triple bonds are 1.54 \\AA, 1.34 \\AA \\ and 1.2 \\AA, respectively \\cite{Hino2003}. Before the wire fractures, the central bond lengths of (4,0)@C$_{5}$-IV(C) are stretched so that those bond lengths can be $\\sim$ 1.40-1.43 \\AA, which are intermediate between single bond and double bond.\n\n\nIn the case of armchair junction, the wire structures and contact geometries at each step of varying the gap width are similar to that of the zigzag junction (Fig.~\\ref{sum}); thus, both zigzag and armchair junctions are labeled in the same way. Furthermore, we find that the geometrical structures of the carbon wire-graphene junctions are also similar to those of the carbon wire-CNT junction discussed above.\n\n\nIn conclusion, the gap width variation leads to a change in the wire structures and bonding geometries of the carbon wire connected to the zigzag (4,0)CNTs or armchair (4,4)CNTs. Moreover, the effect of odd-even numbered wires has played an important role for a difference in the wire structure and bonding at the junction for both cases. The observed bond lengths of the wires agree well with previous works \\cite{Senapati:2005p2333,Molder:2004dw,Khoo:2008p688}. In particular, the geometrical structure of the carbon wire bridging CNTs are similar to the experimentally observed structure of the cumulenes connected to the graphene, \\cite{Jin:2009p592} showing pathways for the migration of the carbon wire along the graphene edge. Differences in the geometry of the junctions as compared to the reported for the graphene are attributed to the finite curvature of the CNT electrodes.\n\\section{Electron transport properties}\n\\subsection{Zigzag vs. armchair junction}\n\n\n\\begin{figure*}[tp]\n\\begin{center}\n\\includegraphics[scale = 0.75]{binding-tran-cnt.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Binding energies vs gap width of the C$_4$ and C$_5$ wires connected to the (4,0)CNT (left panel) and (4,4)CNT (right panel) electrodes, and corresponding zero-bias transmission on the energy-length plane $T(E, L)$ (bottom). The location of a transition state of structures is marked by red dots (shown in the upper panel). The length-dependent transmission is correlated with the binding energies and wire structures.}\\label{binding}\n\\end{figure*}\n\nFig.~\\ref{binding} presents the binding energies and the zero-bias transmission $T(E, V=0)$ projected on the energy-length plane of the C$_{4}$ and C$_{5}$ wires connected to the zigzag (4,0)CNTs and armchair (4,4)CNTs. The length-dependent transmission is correlated with the binding energies and wire structures. We show that varyation of the gap width results in a substantial change in the wire structure and contact geometries which in turn affects transport properties of the junction.\n\nLet us first discuss the zero-bias transmission $T(E,V=0)$ of the zigzag junctions in some details. We note that the effect of (4,0)CNT curvature plays an important role in the transport properties; therefore, the metallic behavior of the (4,0)CNT electrode has been observed. For the C$_5$ wire-zigzag junction, shown in Fig.~\\ref{binding}a, the cumulene wires (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C), show a broad resonance peak at the Fermi level, resulting in the high conductance. With stretching the junction, the wire changes structure to the polyyne, (4,0)@C$_{5}$-III(P), in which the intensity of the transmission peak drops to almost zero around the Fermi level. Before breakage, the wire changes structure back to the cumulene one, in which the transmission peak around the Fermi level appears again. The average conductance values of each structure: (4,0)@C$_{5}$-I(C), (4,0)@C$_{5}$-II(C), (4,0)@C$_{5}$-III(P) and (4,0)@C$_{5}$-IV(C) are 0.9G$_{0}$, 1G$_{0}$, 0.06G$_{0}$ and 0.37G$_{0}$, respectively. From the $T(E,V=0)$ discussed above, the electronic transport properties differ for the zigzag and armchair junctions. Obviously, charge transport for the polyyne wire is strongly suppressed in  the zigzag case, indicating that a switching behavior will be expected at low bias voltage. We emphasize that, due to the high curvature of the (4,0)CNT, the polyyne structure is defined with 1.28 \\AA \\ - 1.42 \\AA \\ bond alternation.\n\n\nFor the C$_4$ wire-zigzag junction, its transport properties are similar to those of the C$_5$ wire. We obtain the broadened resonance peak at the Fermi level for the cumulene; (4,0)@C${_4}$-I(C) and (4,0)CNT@C$_{4}$-II(C), but there is no resonance peak for the polyyne; (4,0)@C${_4}$-III(P). The average conductance of each structure; (4,0)@C$_{4}$-I(C), (4,0)@C$_{4}$-II(C) and (4,0)@C$_{4}$-III(P), is 0.9G$_{0}$, 0.4G$_{0}$ and 0.01G$_{0}$, respectively. Note that the conductance of (4,0)@C$_{4}$-I(C) (bent C$_4$ wire) is very high, because of a short separation between the leads of the C$_4$ compressed wire.\n\n\nNext, we discuss the zero-bias transmission function $T(E,V=0)$ of the armchair junction, as shown in Fig.~\\ref{binding}b.  We find that the structural change in the wire through varying the gap width affects the value of $T(E,V=0)$. Both C$_4$ and C$_5$ wires show the transmission peaks around the Fermi energy. Unlike zigzag junction, the resonance peaks, lying below the Fermi level ($E< E_F$) of the armchair junction, do not disappear from changing the wire structure from cumulene to polyyne, but the transmission drops by a factor of 2. In the armchair case, there is not enough difference between the conductance of cumulene and polyyne wires to show the switching characteristics useful for electronic applications.\n\n\\begin{figure}[bp]\n\\begin{minipage}[tp]{1.0\\linewidth}\n\\begin{center}\n\\includegraphics[scale = 0.2]{tdc.pdf}\n\\end{center}\n\\end{minipage}\n\\caption{Zero-bias transmission on the energy-length plane $T(E, L)$ and corresponding projected density of $p_{m=1}$ and $p_{m=-1}$ states for the carbon wire stretched between armchair (4,4)CNT electrodes at the two lengths corresponding to the II(C) and III(P) configurations as indicated in the Fig. \\ref{binding}.}\\label{tdc}\n\\end{figure}\n\nThe difference of the detailed transport behavior for both junctions is due to their entirely different electronic structures and bonding geometries. The bonding geometries are different for armchair and zigzag tubes (see Fig. \\ref{sum}). It is seen that, for example, a tetragon is formed at the junction for the cumulene-zigzag junction; (4,0)@C$_{4,5}$-II(C), whereas the cumulene-armchair junction; (4,4)@C$_{4,5}$-II(C), has a pentagon. Also, the bonding geometries for the polyyne structure of both tubes are different.\n\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width = 8 cm]{oscillate.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Zero-bias conductance of (4,0)@C$_{n}$-II, n= 4-9.}\\label{oscillate}\n\\end{figure}\n\n\nTo elucidate the transport properties of the armchair junction, we show the $T(V=0,L)$ and corresponding projected density of the carbon wire stretched between armchair (4,4)CNTs (Fig.~\\ref{tdc}). For the two different widths of the gap between the electrodes, the density of $p_{m=1}$ and $p_{m=-1}$ states along the molecule calculated with DFT method. The density of states correlates well with the $T(E)$ map since the carbon wire is directly coupled to the CNT. Molecular states with $p_{m=1}$ and $p_{m=-1}$ symmetry correspond to the two conducting channels in the shorter junction. In the longer junction, one of the C-C bonds breaks, and the density distribution changes for $p_{m=-1}$ states, effectively opening a wide gap in this channel and charging the molecule with $\\approx$0.1e. We note that slight mismatch of the energies, the width of states and transmission resonances are due to the further adjusted charge of the system in NEGF calculation with open boundary conditions as compared to the neutral state of the system in DFT calculation of projected DOS.\n\n\n\n\n\nThe large difference in conductance between the cumulene and polyyne can be understood from the electronic structure of the junction: electrons tunnel through extended $p_x$ and $p_y$ orbitals delocalized both in the wire and in the CNT. In the case of armchair junction, the cumulene configuration shows the states contributing to both channels lying close to the Fermi level of the system. In the polyyne configuration, a gap over 2 eV opens in one of the channels with the $p$ orbitals lying along the CNT surface tangent plane, which reduces the number of conduction channels available close the Fermi level from two to one. In the case of the zigzag junction, the states remaining close to the Fermi level in polyyne configuration become additionally localized. Localization occurs when the defined single bond is created from last atom in the chain towards CNT. As we have discussed earlier, because of the CNT curvature, the wire connected to the tube becomes effectively capped. This leads to localization of the previously delocalized $p$-bonding along the CNT-carbon wire-CNT structure. \n\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[width = 8 cm]{iv.pdf}\n\\end{center}\n\\vspace{-20pt}\n\\caption{The $I-V$ characteristics of (a) (4,0)@C$_{4}$ (bottom panel) (b) (4,0)@C$_{5}$ (top panel) varying the gap width.}\\label{iv}\n\\end{figure}\n \nFig.\\ref{oscillate} exhibits the calculated zero-bias conductance of (4,0)@C$_{n}$-II(C), n= 4-9. Note that we selected (4,0)@C$_{n}$-II(C) in studying the oscillating conductance of the cumulene wire because that structure is the straight cumulene wire providing the highest conductance. Similar to the cumulene wire connected to metal leads\\cite{Prasongkit:2010p780, Zhang13}, we observe the oscillatory characteristics in conductance of the cumulene wire-CNT junction, resulting from the difference in the electronic properties between even- and odd-cumulene wires \\cite{Lang:1998,Prasongkit:2010p780}. We find that the conductance of the odd-wire is higher than that of even-wire by $\\sim$ 60 \\%. The conductance values do not show any pronounced dependence on the wire length when its length increased from four to nine atoms, resulting from the ballistic transport character of electrons through the short wire\\cite{Prasongkit:2010p780}.\n\n\n\n \\begin{figure}[tp]\n \\begin{center}\n \\includegraphics[width = 6 cm]{pdos_b.pdf}\n \\end{center}\n \\caption{The illustration of the C$_5$ wire bridging between (4,0)CNTs. By twisting the CNT, we can change bridging sites and bonding at the junction (top panel). The bridge sites of the left and right (4,0)CNT leads and the corresponding conductance of each structure (bottom panel).}\\label{pdosb}\n \\end{figure}\n \n \nTo compare our findings for small radii CNTs to the large ones, we note that the latter have low curvatures and are well represented by flat graphene electrodes. Graphene electrodes are also used in the recent TEM observations and electronic structure studies by Ref. \\citenum{Jin:2009p592, Erdogan:2011bt} and others. We represent infinite radii of CNTs by studying the transport properties of the carbon wire connected to zigzag- and armchair-edge graphenes. For the zigzag graphene junction, we find that no zero-bias conductance is observed due to $\\sim$ 2 eV band gap of the zigzag edge-graphene; thus, no current is expected at $V_b > 1$V. For the armchair graphene junction, the structural change of the carbon wires connected to the armchair edge graphene electrodes make the transmission resonance peak around the Fermi level shift in energy but with little change in its magnitude.\n\n\nIn the following, we will concentrate on a potential ON\/OFF switch applications of the carbon wire-CNT junction. The electron transport properties of the carbon wire-(4,0)CNT junction will be discussed in the next section.\n\n\n\\subsection{(4,0)CNT-Carbon wire-(4,0)CNT}\n\nTo confirm the possibility of electrical switching behavior, as illustrated in Fig.\\ref{iv}, we present the calculated $I-V$ characteristics (IVCs) of (a) (4,0)@C$_{5}$ (top panel) (b) (4,0)@C$_{4}$ (bottom panel), varying the gap width. We find that the ON\/OFF current ratio of the (4,0)@C$_{5}$ is higher than that of the (4,0)@C$_{4}$. At $V_b$=0.2V, the ON\/OFF ratio of (4,0)@C$_{4}$ and (4,0)@C$_{5}$ takes the value of $\\sim$ 7 and 13, respectively, indicating electrical switching characteristics at low bias, coupled to the mechanical stretching of the CNT. We note that a very small diameter of the (4,0)CNTs plays an important role in their transport properties; thus, the metallic behavior of the (4,0)CNT electrode has been observed.\n\n\nFrom the IVCs (Fig.\\ref{iv}), the charge transport of (4,0)@C$_{4}$-III(P), (4,0)@C$_{5}$-III(P) and (4,0)@C$_{5}$-IV(C) is suppressed, whereas the (4,0)@C$_{4}$-II(C), (4,0)@C$_{5}$-I(C) and (4,0)@C$_{5}$-II(C) show a high-current state. It is important to notice at this point that the cumulene wires do not always give rise to the high conductance: a variation in conductance of the cumulene wire depends on the bonding geometries.\n\n\n  \\begin{figure}[tp]\n   \\begin{center}\n   \\includegraphics[width = 9 cm]{pdos_a.pdf}\n   \\end{center}\n   \\caption{The zero-bias transmission function and the PDOS of {$\\mathrm{p_x}$} and {$\\mathrm{p_y}$} orbitals of the cumulene C$_5$ wire, corresponding to the wire configuration in Fig. \\ref{pdosb}.}\\label{pdosa}\n   \\end{figure}\n\n\nThere is a probability that, in the experiment, the wire can jump to other sites when the CNT is effectively twisted. Consequently, we have investigated the geometrical structures and its corresponding conductance of the C$_5$ wire connected to all possible bridge sites on the (4,0)CNT, as presented in Fig. \\ref{pdosb}. The wire structures have changed into either cumulenes or polyynes depending on the bridge site and gap width. Independently of the bridge site, we find that the polyyne wire is always a low conductance state. For the cumulene wire, interestingly, the results show a possibility to modulate the conductance of the zigzag junction by changing bridge sites, and bonding at the interface. The range of variation in conductance of the cumulene wires is $\\simeq$ 0.1G$_0$-1G$_0$.\n\n\nTo explain the variation in conductance of the cumulene wire, we analyzed the transport channels via PDOS of the wire. We observe that the alignment of PDOS is determined by bridging sites and bonding at the junction. As we have already known that the cumulene wire has two $\\pi$  orbitals around the Fermi level resulting from $\\mathrm{p_x}$ and {$\\mathrm{p_y}$} orbitals \\cite{Shen2010,Prasongkit:2010p780}, the PDOS of the wire is decomposed into $\\mathrm{p_x}$ and $\\mathrm{p_y}$ components, as demonstrated in Fig. \\ref{pdosa}. Our results can be classified into three cases according to bonding geometries at the interface. First, for the cumulene wire forming two bonds to each side of CNTs with two planes parallel to each other ((4,0)@C$_ 5$-I(C),II(C),VIII(C)), there are two states; $\\mathrm{p_x}$ and  {$\\mathrm{p_y}$}, with the same energy position at Fermi level. Consequently, electrons can propagate through $\\mathrm{p_x}$ and {$\\mathrm{p_y}$} eigenchannels of the carbon wire, showing a high conductance state. Apparently, the PDOS peak positions, consisting of the two transport channels, show the transmission function close to 2$G_0$. Second, for the cumulene forming one bond to the CNT leads; ((4,0)@C$_ 5$-IV(C),V(C)), there is the only {$\\mathrm{p_y}$} state lying at the Fermi level, whereas the $\\mathrm{p_x}$ state exists above the Fermi energy. This results in a decrease of conductance owing to only one transport channel at the Fermi level. Third, for the cumulene wire forming two bonds to the CNT lead with two planes perpendicular to each other; ((4,0)@C$_ 5$-VI(C)), the PDOS projected to $\\mathrm{p_x}$ and  {$\\mathrm{p_y}$} around the Fermi level is very low, causing a drop in conductance. We can therefore conclude that the variation in conductance of the cumulene wire is determined by the transport channels at the Fermi level, depending on the bridge sites and its bonding to the CNT leads.\n\n\n\\section{Summary}\nWe have performed first principles calculations to investigate the transport properties of the carbon wire between zigzag (4,0)CNTs and armchair (4,4)CNT electrodes. The gap width between the electrodes is varied and corresponding conductance variation upon the compression\/elongation of the junction is calculated. Varying the gap width make the carbon wire change the structures (cumulene or polyyne) and contact geometries. We have observed the migration pathways of the carbon wire along the edge, which agrees well with the experimental results.\n\nWe find that the transport properties of the junction are significantly affected by the choice of chirality (zigzag or armchair). For the zigzag junction, the distinguishable ON- and OFF-state is observed, corresponding to the cumulene and polyyne structure, respectively. The zigzag CNT junction can be reversibly switched between ON- and OFF-state through varying the gap width between electrodes. In contrast, for the armchair junction, there is not enough difference in conductance to perform switching. The difference of the detailed transport behavior of both junctions is due to their entirely different electronic structures and bonding geometries, in which the high and low conductance states correspond to the cumulene and polyyne structures, respectively. In the studied configuration, the cumulene usually yields higher conductance than the polyyne wire. However, the cumulene wire can show the variation in conductance in the range from 0.1G$_0$ to 1G$_0$ upon the bridging geometries and sites. Their qualitative difference in transport mechanisms, resulting in the difference in the conductance state, was discussed by means of the transport channel at the Fermi level.\n\n\nOscillatory behavior in conductance of the cumulene wires with different lengths is demonstrated. We find that odd-cumulene wires yield higher conductance than the even-cumulene wires and ballistic transport behavior. The calculated ON\/OFF ratios of the odd-wire are larger than that of the even-wire, resulting from the difference in the electronic properties between the odd- and even-wires.\n\nGraphene leads, representing infinite radii of CNTs, have also been considered. However, graphene lead with zigzag edges has a band gap, and no zero-bias conductance is observed. Due to this semiconducting character of the zigzag electrode, switching behavior will be observed at a high bias voltage. In other aspects, the results for the graphene leads (infinite radii of CNT) are similar with that of CNT leads. \n\n\nIn contrast to metal electrodes, the CNTs can act as true nanoscale electrodes and there is a possibility that the carbon wire can jump to any site of the edge. We therefore investigate how the conductance is affected by the bonding site at the junction for the C$_5$ wire connected to (4,0)CNT electrodes. The range of variation in conductance of the cumulene wires is $\\sim$ 0.1G$_0$--1G$_0$, revealing a potential to tune the conductance. The observed variation is explained by the PDOS analysis of p-orbital components of the cumulene wire. \n\nFinally, the carbon wire-zigzag CNT junctions demonstrate the prominent difference in conductance between ON and OFF states via compression\/elongation junctions without breaking the wire and may be a promising candidate for mechano-switching device in molecular and nanoelectronics.\n\n\\begin{acknowledgments}\nJ.P. has partially been supported by the Nanotech- nology Center (NANOTEC), NSTDA, Ministry of Sci- ence and Technology, Thailand, through its program of Center of Excellence Network. A.G. and R.A. gratefully acknowledge financial support from Carl Tryggers Stiftelse f\\\"or Vetenskaplig Forskning and U3MEC, Uppsala. The calculations were performed at the high performance computing centers UPPMAX within the Swedish National Infrastructure for Computing.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\al$ be an algebraic number of degree $d \\geq 2$ over $\\Q$ with conjugates $\\al_1=\\al,\\al_2,\\dots,\\al_d$. An additive linear relation\n\\begin{equation}\\label{rysys}\nk_1\\al_1+k_2\\al_2+\\dots+k_d\\al_d=0\n\\end{equation}\nwith some $k_1,k_2,\\dots,k_d \\in \\Q$ is called {\\it nontrivial} if \n$k_i \\ne k_j$ for some $1 \\leq i<j \\leq d$.\nThus, the relation\n$${\\rm Trace}(\\al):=\\al_1+\\al_2+\\dots+\\al_{d} =0$$\nand its rational multiples $\\sum_{j=1}^d r \\al_j=0$, where $r \\in \\Q$, are {\\it trivial} linear relations. (These hold for conjugates of any algebraic number whose trace is zero.)\nNote that if \\eqref{rysys} is a nontrivial linear relation with some $k_1,k_2,\\dots,k_d \\in \\Q$ then, by multiplying all the $k_i$ by their common denominator, we can assume that $k_1,k_2,\\dots,k_d \\in \\Z$. Accordingly, we call the sum $$|k_1|+|k_2|+\\dots+|k_d| \\in \\N$$ the {\\it length}\nof the relation \\eqref{rysys}. \n\nThe investigation of nontrivial linear relations \\eqref{rysys} in conjugates of algebraic numbers \nhas begun with the papers of Kurbatov \\cite{kurb1, kurb2, kurb3}. In \\cite{smy2}, Smyth obtained some useful results and also formulated several natural conjectures on the possibility of \\eqref{rysys} which are still wide open; see also his previous paper \\cite{smy1}. Further results on this subject have been obtained by several authors in \\cite{bds,dix,ds1,ds2,dub,dub2,girst,fl1,fl2,vall}. \n\nRecently, in \\cite{DHJ}\nit was shown that there is a unique Pisot number $\\al=(1+\\sqrt{3+2\\sqrt{5}})\/2$ with minimal polynomial $x^4-2x^3+x-1$ satisfying the nontrivial linear relation $$\\al_1+\\al_2-\\al_3-\\al_4=0$$ of length $4$. \nRecall that\nan algebraic integer $\\al>1$ is called a {\\it Pisot number} if its other conjugates over $\\Q$ (if any) all lie in the open unit disc $|z| < 1$. \nThis answers two questions raised earlier in \\cite{smy0}. For instance, this implies that no two conjugates of a Pisot number can have the same imaginary part. See also a subsequent paper \\cite{DJ} for some further analysis of some simple linear relations of small length.\n\nIn the present paper, we investigate additive linear relations in conjugates of a Salem number. Recall that\nan algebraic integer $\\al>1$ is called a {\\it Salem number} if  its other conjugates over $\\Q$ all lie in the closed unit disc $|z| \\le 1$ with at least one conjugate lying on the circle $|z|=1$. \n\nThroughout, if $\\al>1$ is a Salem number \nof degree $d=2s \\geq 4$ we label its conjugates\nas in the theorem below.\n\n\n\\begin{theorem}\\label{pirmoji}\nLet $\\al_1=\\al>1$ be a Salem number of degree $d=2s \\geq 4$ with conjugates $\\al_1,\\dots,\\al_d$ satisfying\n$\\al_2=1\/\\al_1$ and \n$\\al_{2j}=1\/\\al_{2j-1}=\\overline{\\al_{2j-1}}$ for $j=2,\\dots,s$. \nIf for some rational numbers $k_i$, $i=1,\\dots,d$, and for some totally real algebraic number $\\gamma$ we have \n\\begin{equation}\\label{duu}\nk_1\\al_1+k_2\\al_2+\\dots+k_d\\al_d=\\gamma,\n\\end{equation}\nthen \n$k_{2j-1}=k_{2j}$ for each $j=1,\\dots,s$.\n\\end{theorem}\n\nIn particular, the theorem obviously holds for $\\gamma=0$. So, every linear relation \\eqref{rysys} in the conjugates $\\al_i$, $i=1,\\dots,d$, of a Salem number $\\al$ is induced by the linear relation \n\\begin{equation}\\label{rysys1}\nk_1\\be_1+k_3\\be_2+\\dots+k_{2s-1}\\be_s=0\n\\end{equation}\nin conjugates of the respective totally real algebraic integer $\\be_1=\\be:=\\al+1\/\\al>2$ whose other conjugates\nare\n$$\\be_j=\\al_{2j-1}+\\al_{2j}=\\al_{2j-1}+1\/\\al_{2j-1}=\\al_{2j-1}+\\overline{\\al_{2j-1}}=2\\Re \\al_{2j-1} \\in (-2,2)$$ for $j=2,\\dots,s$.  If $f$ is the minimal polynomial of a Salem number $\\al$ of degree $d=2s$ and $g$ is the minimal polynomial of \n$\\be=\\al+1\/\\al$ of degree $s$ then they are related by the identity\n$\nf(x)=x^{s} g(x+1\/x).\n$\nThen, as in \\cite{chr}, we call $g$ the {\\it trace polynomial} of $f$. \nNote that $f$ is irreducible if and only if $g$ is irreducible. Also,\n${\\rm Trace}(\\al)=\\sum_{j=1}^d \\al_j= \\sum_{i=1}^s \\be_i= {\\rm Trace}(\\be)$. \n\nBy \\cite{kurb3} (or \\cite{ds1}), the only relation with conjugates $\\be_1=\\be,\\dots,\\be_p$ \nof an irreducible polynomial of prime degree $p$ can be of the form \n$$r\\be_1+r\\be_2+\\dots+r\\be_p=0,$$ where $r \\in \\Q$. Hence, \nthe only possible linear relation with conjugates of a Salem number $\\al$\nwith degree $2p$\nis $r {\\rm Trace}(\\al)=0$, where $r \\in \\Q$. This relation is trivial.\n\nSo, in particular Theorem~\\ref{pirmoji} implies that \n\n\\begin{corollary}\\label{keturi}\nIf $p$ is a prime number then there are no nontrivial linear relations in conjugates of a Salem number of degree $d=2p$.  \n\\end{corollary}\n\nBy \\cite{mcsm}, it is known that there are Salem numbers of every integral trace. The degree of a Salem number with negative trace \n$-t$ is quite large if $t \\in \\N$ is large. Earlier, in \\cite{mcsm0} \nSmyth has shown that there are Salem numbers with trace $-1$ of every even degree\n$d \\geq 8$. \n\nHere, by a similar argument, we show that\n\n\\begin{theorem}\\label{keturi-1}\nFor any even $d \\geq 6$ there is a Salem number of degree $d$\nwith trace $0$.    \n\\end{theorem}\n\nIn Corollary~\\ref{hbhbhb} below, we list of all $4$ possible Salem numbers of degree $6$ and trace $0$. Note that there are no Salem numbers \nof degree $4$ and trace $0$. Indeed, otherwise the minimal polynomial of\nsuch a Salem number would be $x^4+ax^2+1$, with $a \\in \\Z$, which is impossible. \n\nOur next theorem describes the minimal length  of nontrivial linear relations between conjugates of a Salem number and the minimal degree\nof a Salem number for which a nontrivial linear relation may occur. \n\n\\begin{theorem}\\label{antroji}\nSuppose $\\al>1$ is a Salem number with conjugates $\\al_1=\\al,\\al_2,\\dots, \\al_d$ over $\\Q$ labelled as in Theorem~\\ref{pirmoji}. \n\\begin{itemize}\n\\item[$(i)$]\nIf for some\nintegers  $k_1,k_2,\\dots,k_d$, not all zero, the nontrivial linear relation \\eqref{rysys} holds then its length \nmust be at least $6$.\nFurthermore, there exist Salem numbers $\\al$ of degree $12$ whose six conjugates satisfy the following nontrivial linear relation of length $6$:\n$$\n\\al_1+\\al_2+\\al_3+\\al_4+\\al_5+\\al_6=0.\n$$\n\n\\item[$(ii)$] The smallest degree of a Salem number with a nontrivial linear relation between its conjugates is $8$.  Furthermore, there exist Salem numbers $\\al$ of degree $8$ whose conjugates satisfy the following nontrivial linear relation:\n$$\n\\al_1+\\al_2+\\al_3+\\al_4-\\al_5-\\al_6-\\al_7-\\al_8=0.\n$$\n\\end{itemize}\n\\end{theorem}\n\n\n\n\\section{Auxiliary results}\n\nWe begin with two simple lemmas. \n\n\\begin{lemma}\\label{trecioji}\nThe cubic polynomial $x^3-ax+b \\in \\R[x]$ has three distinct roots in the interval $(-2,2)$ iff $0<a<4$ and\n\\begin{equation}\\label{pima}\n \\max\\Big(2a-8, -\\frac{2a\\sqrt{a}}{3\\sqrt{3}}\\Big)<b < \\min\\Big(8-2a, \\frac{2a\\sqrt{a}}{3\\sqrt{3}}\\Big),\n\\end{equation}\nand two distinct roots in $(-2,2)$ and one root in $(2,+\\infty)$ iff $3<a<12$\nand\n\\begin{equation}\\label{pima2}\n-\\frac{2a\\sqrt{a}}{3\\sqrt{3}}< b < -|2a-8|.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{proof}\nSet $$h(x):=x^3-ax+b.$$Since\n$h'(x)=3x^2-a$, the polynomial $h$ has only one real root \nif $a \\leq 0$. \n\nSuppose $a>0$.  Set $x_0:=\\sqrt{a\/3}$. \nThen, the polynomial $h$\nhas three distinct roots in $(-2,+\\infty)$ iff $-2<-x_0$ (i.e., $0<a<12$), \n\\begin{equation}\\label{mm1}\nh(-2)=-8+2a+b<0, \n\\end{equation}\n\\begin{equation}\\label{mm2}\nh(-x_0)=\\frac{2a\\sqrt{a}}{3\\sqrt{3}}+b>0\n\\end{equation}\nand\n\\begin{equation}\\label{mm3}\nh(x_0)=-\\frac{2a\\sqrt{a}}{3\\sqrt{3}}+b<0.\n\\end{equation}\n\nClearly, all three roots belong to $(-2,2)$ if, in addition, we have $h(2)=8-2a+b>0$. Combined with \\eqref{mm1}, \\eqref{mm2} and \\eqref{mm3}\nthis proves \\eqref{pima}. Evidently, \\eqref{pima} is only possible for some $b$ when its left hand side does not exceed its right hand side, that is, when $0<a<4$. \n\n \nSimilarly, two roots of $h$ are in $(-2,2)$ and one root in $(2,+\\infty)$ when one has \\eqref{mm1}, \\eqref{mm2}, and\n$h(2)=8-2a+b<0$. (As $h$ is increasing in the interval $(x_0,2)$, the inequality \\eqref{mm3} automatically holds.) Evidently, all these inequalities combine into \\eqref{pima2}. Here, as $0<a<12$, it is easy to see that the inequality $$|2a-8|<\\frac{2a\\sqrt{a}}{3\\sqrt{3}}$$ holds only for $a$ in the range $3<a<12$, so only for such $a$ one can find some $b$ satisfying\n\\eqref{pima2}.\n\\end{proof}\n\nObserve that there are only $7$ pairs of integers $(a,b)$ satisfying the conditions $3<a<12$ and \\eqref{pima2}, namely,\n$(4,-1)$, $(4,-2)$, $(4,-3)$, $(5,-3)$, $(5,-4)$, $(6,-5)$ and $(7,-7)$. \nHowever, the polynomials $x^3-4x-3$, $x^3-5x-4$ and \n$x^3-6x-5$ are reducible. Other four polynomials $x^3-4x-1$, $x^3-4x-2$, $x^3-5x-3$ and \n$x^3-7x-7$ are irreducible. So, Lemma~\\ref{trecioji} implies that \n\n\\begin{corollary}\\label{hbhbhb}\nThere are exactly four Salem numbers of degree $6$ with trace $0$. \nTheir minimal polynomials are: $$x^6-x^4-x^3-x^2+1, \\quad\nx^6-x^4-2x^3-x^2+1,$$ \n$$x^6-2x^4-3x^3-2x^2+1, \\quad x^6-4x^4-7x^3-4x^2+1.$$ \n\\end{corollary}\n\n\\begin{lemma}\\label{trecioji-1}\nLet $h(x) \\in \\Z[x]$ be a monic polynomial of degree $k \\geq 2$ with \n$k-1$ roots in the interval $(-2,1\/4)$ and one root in $(-6,-2)$.  Then,\n$$f(x):=(-1)^k x^{2k} h\\big((x+1\/x)(1-x-1\/x)\\big) \\in \\Z[x]$$ is a monic reciprocal polynomial of degree $4k$ which defines a Salem number of degree $d=4k$ in case it is irreducible over $\\Q$. Moreover, the conjugates of this Salem number $\\al$ labelled as in Theorem~\\ref{pirmoji}\nsatisfy \n\\begin{equation}\\label{aaa}\n\\al_1+\\al_2+\\al_3+\\al_4=\\dots=\\al_{4k-3}+\\al_{4k-2}\n+\\al_{4k-1}+\\al_{4k}=1.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{proof}\nLet $\\ga_1 \\in (-6,-2)$ and $\\ga_2<\\dots<\\ga_{k} \\in (-2,1\/4)$ be the roots of $h$. Consider the monic polynomial $g(x):=(-1)^k h(x(1-x))$. Then, its roots\n\\begin{equation}\\label{bbbb}\n\\be_{2j-1}:=\\frac{1 + \\sqrt{1-4\\ga_j}}{2} \\quad \n\\text{and} \\quad\n\\be_{2j}:=\\frac{1 - \\sqrt{1-4\\ga_j}}{2},\n\\end{equation}\nwhere $j=1,\\dots,k$,\nsatisfy $\\be_1=(1+\\sqrt{1-4\\ga_1})\/2>2$, $\\be_2=(1-\\sqrt{1-4\\ga_1})\/2 \\in (-2,-1)$ and $\\be_3,\\dots,\\be_{2k} \\in (-1,2)$. So, $g$ has $2k-1$\nroots in $(-2,2)$ and one root greater than $2$. Clearly, by \\eqref{bbbb}, we have\n\\begin{equation}\\label{bee}\n\\be_{1}+\\be_2=\\dots=\\be_{2k-1}+\\be_{2k}=1.\n\\end{equation}\n\nNow, as the roots $\\al_1=\\al>1,\\al_2=1\/\\al,\\dots,\\al_{4k-1},\\al_{4k}=1\/\\al_{4k-1}$ of $$f(x)=x^{2k}g(x+1\/x)=(-1)^k x^{2k}h\\big((x+1\/x)(1-x-1\/x)\\big)$$ satisfy  $\\be_j=\\al_{2j-1}+\\al_{2j}=\\al_{2j-1}+1\/\\al_{2j-1}$ for each $j=1,\\dots,2k$, we see that \\eqref{bee} implies \\eqref{aaa}. Furthermore,   $\\al$ is a Salem number of degree $4k$ provided that $f$ is irreducible\nover $\\Q$. \n\\end{proof}\n\n\nWe made some calculations related to Lemma~\\ref{trecioji-1}. It turns out that there exactly $15$ quadratic polynomials $h$ satisfying the conditions of the lemma and thus producing $15$ Salem numbers \nof degree $8$ satisfying \\eqref{aaa} with $k=2$. For instance, $x^2+4x+1$ is such a quadratic polynomial $h$. Also, there are exactly \n$30$ cubic, $20$ quartic and $4$ quintic polynomials $h$ producing\n$30$ Salem numbers of degree $12$ (satisfying \\eqref{aaa} with $k=3$), $20$ Salem numbers of degree $16$ (satisfying \\eqref{aaa} with $k=4$) and $4$ Salem numbers of degree $20$ (satisfying \\eqref{aaa} with $k=5$), respectively.  In the case $k=5$, the example of $h$ is  $$x^5+9x^4+22x^3+16x^2-x-1.$$ This gives a Salem number $\\al$ of degree $20$ with minimal polynomial \n$$x^{20}-5x^{19}+11x^{18}-19x^{17}+26x^{16}-29x^{15}+\n27x^{14}-19x^{13}+8x^{12}+x^{11}$$ $$-5x^{10}+\nx^9+8x^8-19x^7+27x^6-29x^5+\n26x^4-19x^3+11x^2-5x+1$$\nwhose conjugates satisfy \\eqref{aaa} with $k=5$. \n\nThe first part of the next lemma was inspired by Lemma 1 of Beukers and Smyth in \\cite{beusmy}. Essentially, it is  a version of their algorithm \\cite{beusmy} to locate cyclotomic points on curves, specialized to the case of sequences of polynomials that produce Salem numbers from Pisot numbers. Also, the second part of Lemma \\ref{Lemma_cyclotomic_factors} is loosely related to the work on irreducibility of polynomials of the type $x^nf(x)+g(x) \\in \\Z[x]$ and on the sequences  and covering systems of integers by Schinzel \\cite{schi}, Filaseta et al. \\cite{fifoko, fima}, although these irreducibility results are not of direct relevance here. Throughout, $f^*(x) = x^{\\deg{f}}f(1\/x)$ stands for the \\emph{reciprocal polynomial} of $f(x)$.\n\n\\begin{lemma}\\label{Lemma_cyclotomic_factors}\nFor $n \\in \\N$, consider the sequence of polynomials\n\\[\ng_n(x) := x^nf(x) + \\eps f^{*}(x),\n\\] where $\\eps \\in \\{-1, 1\\}$ and $f(x) \\in \\Z[x]$ satisfies $f^*(x) \\ne \\pm f(x)$.  \nSuppose that a root of unity $\\zeta \\in \\C$ is also a root of some polynomial $g_n(x)$. Then, $\\zeta$ must appear among the zeros of at least one of the following polynomials:\n\\[\nf(x^2)f^*(x)^2 + \\eps f(x)^2f^*(x^2), \\qquad f(x)^2f^*(-x^2) \\pm  f(-x^2)f^*(x)^2,\n\\]\n\\[\nf(x)f^*(-x) \\pm  f(-x)f^*(x).\n\\]\nIn particular, if none of these polynomials is identically zero, then the set of all such possible roots of unity $\\zeta$ is finite.\n\nIn addition to this, if $f(\\zeta) \\ne 0$ then the root of unity $\\zeta$ is a zero of $g_n(z)$ if and only if $n$ belongs to the arithmetic progression $\\ell k+r$, $k=0, 1, 2, \\dots$, where $r$ is a fixed integer in the range $0 \\leq r < \\ell$ and $\\ell={\\rm ord}(\\zeta)$ denotes the multiplicative order of $\\zeta$. \n\\end{lemma}\n\n\\begin{proof}\nAs $\\zeta$ is the root of unity, by Lemma 1  of \\cite{beusmy} (or Lemma 2.1 of \\cite{mcsm0}, at least one of the three numbers $\\zeta^2$, $-\\zeta^2$, $-\\zeta$ must be an algebraic conjugate of $\\zeta$ over $\\Q$.\nMultiplying $g_n(x)=x^n f(x)+\\eps f^*(x)$ by $x^n f(x)-\\eps f^*(x)$ we see that the polynomial $h(x)=x^{2n}f(x)^2 - f^*(x)^2$ has a zero at $x=\\zeta$.\n\nIf $\\zeta^2$ is conjugate of $\\zeta$, then one also has $g_n(\\zeta^2)=0$. Combining this with $h(\\zeta)=0$ yields\n\\[\n\\left\\{\n\\begin{split}\n\\zeta^{2n}f(\\zeta)^2 \t&-  f^*(\\zeta)^2\t\t&= 0,\\\\\n\\zeta^{2n}f(\\zeta^2)\t&+ \\eps f^*(\\zeta^2)\t&= 0.\n\\end{split}\n\\right.\n\\]\nHence,\n\\[\n\\begin{vmatrix}\nf(\\zeta)^2 \t &- f^*(\\zeta)^2\\\\\nf(\\zeta^2) &\\eps f^*(\\zeta^2)\\\\\n\\end{vmatrix} = \\eps f(\\zeta)^2f^*(\\zeta^2)  + f(\\zeta^2)f^*(\\zeta)^2 = 0.\n\\]\nThus, $\\zeta$ is the root of $f(x^2)f^*(x)^2 + \\eps f(x)^2f^*(x^2)$.\n\nSuppose next that $-\\zeta^2$ is a conjugate to $\\zeta$. Then, using $g_n(-\\zeta^2)=0$ and $h(\\zeta)=0$, one concludes that $\\zeta$ is the root of the polynomial\n $f(x)^2f^*(-x^2) +\\eps (-1)^n f(-x^2)f^*(x)^2$. \n\nIn the case when $-\\zeta$ is conjugate to $\\zeta$, from $g_n(\\zeta)=g_n(-\\zeta)=0$ one obtains $\\zeta^n f(\\zeta)+\\eps f^*(\\zeta)=0$ and $(-\\zeta)^n f(-\\zeta)+\\eps f^*(-\\zeta)=0$, which yields that $\\zeta$ is a root of $f(x)f^*(-x) +(-1)^{n+1} f(-x)f^*(x)$.\n \nFinally, if a root of unity $\\zeta$ of order $\\ell$ satisfies $g_n(\\zeta)=0$, then $g_{n+\\ell}(\\zeta)=0$. Furthermore, if $\\zeta$ is a common root of $x^{n_1}f(x) +\\eps f^*(x)$ and $x^{n_2}f(x)+ \\eps f^*(x)$, then $(\\zeta^{n_2}-\\zeta^{n_1})f(\\zeta)=(\\zeta^{n_2-n_1}-1)\\zeta^{n_1}f(\\zeta)=0$. By $f(\\zeta) \\ne 0$, it follows that $\\zeta^{n_2 - n_1}=1$. Thus, $\\ell \\mid (n_2 - n_1)$ and so all such $n$ form an arithmetic progression with difference $\\ell$, as claimed. \n\\end{proof}\n\n\\section{Proofs of the theorems}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{pirmoji}]\nAssume that $k_{2i} \\ne k_{2i-1}$ for some $i$ in the range $1 \\le i \\le s$. Let $G$ be the Galois group of the normal extension of $\\Q(\\al,\\ga)$ over $\\Q$, and let $\\sigma$ be an automorphism of $G$ which maps $\\al_{2i-1}$ to $\\al_1=\\al$.  Then, $\\sigma(\\al_{2i})=\\sigma(1\/\\al_{2i-1})=1\/\\al$, so that \\eqref{duu} maps into\n\\begin{equation}\\label{kjh}\n\\sigma(\\gamma)=k_{2i-1}\\al+k_{2i}\/\\al+ t_3\\al_3+\\dots+t_d\\al_d,\n\\end{equation}\nwhere $t_3,\\dots,t_d \\in \\Q$ is a permutation of the list obtained from the initial list \n$k_1,\\dots,k_d$ by excluding the elements  $k_{2i-1}$ and $k_{2i}$. \n\nConsider the following equality which is complex conjugate to \\eqref{kjh}:\n\\begin{equation}\\label{kjh1}\n\\overline{\\sigma(\\gamma)}=k_{2i-1}\\al+k_{2i}\/\\al+ t_3\\overline{\\al_3}+\\dots+t_d\\overline{\\al_d}.\n\\end{equation}\nSince $\\overline{\\sigma(\\ga)}=\\sigma(\\ga)$ and $\\al_{2j}=\\overline{\\al_{2j-1}}$ for $j=2,\\dots,s$, by adding \\eqref{kjh} and \\eqref{kjh1}, we obtain\n$$\n2\\sigma(\\gamma) = 2k_{2i-1}\\al+2k_{2i}\/\\al+ w_2(\\al_3+\\al_4)+\n\\dots+w_{s}(\\al_{d-1}+\\al_d),\n$$\nwhere $w_{j}=t_{2j-1}+t_{2j}$ for $j=2,3,\\dots,s$. \nAdding $2(k_{2i}-k_{2i-1})\\al$ to both sides we deduce that\n$$2\\sigma(\\gamma)+ 2(k_{2i}-k_{2i-1})\\al=w_1(\\al_1+\\al_2)+w_2(\\al_3+\\al_4)+\\dots+w_{s}(\\al_{d-1}+\\al_d),$$\nwhere $w_1=2k_{2i}$.\n\nAs we already observed above, the number $\\be_1=\\be=\\al+1\/\\al=\\al_1+\\al_2$ is totally real with conjugates $\\be_2=\\al_3+\\al_4$, \\dots, $\\be_s=\\al_{d-1}+\\al_d$. Hence,\nthe number \n$$2(k_{2i}-k_{2i-1})\\al=w_1\\be_1+w_2\\be_2+\\dots+w_{s}\\be_s-2\\sigma(\\gamma)$$\nis a linear form (with rational coefficients $w_1,\\dots,w_s,-2$) in totally real algebraic numbers $\\be_1,\\dots,\\be_s,\\sigma(\\gamma)$.\nThus, it must be totally real itself. However, the number $2(k_{2i}-k_{2i-1})\\al \\ne 0$ is\nnot totally real, since it has a non-real conjugate $2(k_{2i}-k_{2i-1})\\al_3$. \nThis is a contradiction which completes the proof of the theorem. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{keturi-1}] Assume that there exists a smallest even degree $d$ (where $d \\geq 8$ by Corollary~\\ref{keturi-1}), such that there are no Salem numbers of that degree $d$ with trace $0$. We will track down and ultimately eliminate all such possible $d$ by considering 3 sequences of polynomials, given explicitly by Salem's original construction \\cite{salem1, salem2}.\n\nWe start with a Salem sequence \n\\[\ng_n(x)=x^n(x^3-x-1) + (-x^3-x^2+1), \\quad n \\geq 2.\n\\] Then $g_n(x)$ either posseses cyclotomic factors or it is a minimal polynomial of a Salem number of trace $0$; see \\cite{boyd, salem1, salem2}. Since we have assumed that no Salem number of degree $d$ and trace $0$ exists, the polynomial $g_n(x)$ of degree $d=\\deg{g_n}=n+3$ must be reducible, that is, it must be divisible by a cyclotomic polynomial $\\Phi_{\\ell}(x)$, where $\\ell \\in\\N$.\n\nTo find cyclotomic factors of $g_n(x)$, we apply\nLemma~\\ref{Lemma_cyclotomic_factors} with $f(x)=x^3-x-1$ and $\\eps=1$. The following candidates appear as factors of auxiliary polynomials described in\nLemma~\\ref{Lemma_cyclotomic_factors} (with $\\eps=1$):\n\\[\n\\Phi_1(x) = x-1, \\qquad \\Phi_2(x)=x+1, \\qquad \\Phi_8(x)=x^4+1\\]\n\\[\n\\Phi_{12}(x)=x^4-x^2+1, \\qquad \\Phi_{18}(x)=x^6-x^3+1,\n\\]\n\\[\n\\Phi_{30}(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1.\n\\]  Since none of the five auxiliary polynomials is zero identically, this list is complete. \n\nTo see which of these candidates actually show up, one can apply the periodicity property stated in the second part of Lemma~\\ref{Lemma_cyclotomic_factors}. After computation of ${\\rm gcd}(g_n(x), \\Phi_{\\ell}(x))$, $0 \\leq n \\leq \\ell-1$ for $\\ell=1, 2, 8, 12, 18, 30$ it turns out that $g_n(x)$ has cyclotomic factors precisely for the degrees $d=n+3$ in one of the arithmetic progressions:\n\\[\nd \\in \\{2k+1, 8k+2, 12k+1, 18k+17, 30k+24\\},\n\\]\nwhere $k=0, 1, 2, \\dots $. As $d$ must be even, we restrict all such possible $d$ to two arithmetic progressions: $d \\in \\{8k+2, 30k+24\\}$.\n\nNext, we take the second sequence\n\\[\nh_n(x) = \\frac{x^n(x^2-x-1) - (-x^2-x+1)}{x-1},  \\quad n \\geq 2.\n\\]\nAlthough now $f(x)=x^2-x-1$ contributes the coefficient $-1$ of $x^{n+1}$ to $g_n(x)$, one regains trace $0$ after division by $x-1$. \nLet us apply \nLemma~\\ref{Lemma_cyclotomic_factors} to the polynomial $g_n(x)=(x-1)h_n(x)$ with this new choice of $f(x)$ and $\\eps=-1$. The candidate cyclotomic factors are:\n\\[\n\\Phi_1(x) = x - 1, \\qquad \\Phi_2(x) = x + 1,\\qquad \\Phi_3(x)= x^2 + x + 1,\n\\]\n\\[\n\\Phi_6(x)= x^2 - x + 1,\\qquad \\Phi_{12}(x)=x^4 - x^2 + 1.\n\\] As above, the computation of gcd's with first $12$ polynomials of the sequence yields the list of possible bad degrees $d=n+1$:\n\\[\nd \\in \\{ 2k+1, 3k+ 2, 6k+3, 12k +4\\}.\n\\]\nThis list also accounts for the single occurrence of the multiple factors, namely, $(x-1)^2$ in $g_4(x)$. Bad degrees must be even, so we are left with $d \\in \\{6k+2, 12k+4\\}$. \n\nLet us combine this with the arithmetic progressions obtained from the first sequence:\n\\[\nd \\in \\{8k+2, 30k+24\\} \\cap \\{6k+2, 12k+4\\}.\n\\]\nNotice that all integers $30k+24$ are divisible by $6$, while none of $6k+2$ or $12k+4$ are. Therefore, $d \\notin \\{30k+24\\}$, and hence $d \\in \\{8k+2\\}$. Next, notice that $12k+4$ is divisible by $4$ while $8k+2$ is not. Consequently, $d \\notin \\{12k+4\\}$. It follows that\n\\[\nd \\in \\{8k+2\\} \\cap \\{6k+2\\} = \\{24k + 2\\}.\n\\]\n\nTo eliminate this possibility,  let us consider the third sequence, constructed with $f(x)=x^3-x^2-1$ and $\\eps=-1$:\n\\[\nh_n(x) = \\frac{x^n(x^3-x^2-1) - (-x^3-x+1)}{x-1}, \\quad n \\geq 2.\n\\]\nThis time, by Lemma~\\ref{Lemma_cyclotomic_factors} the candidates for cyclotomic divisors are\n\\[\n\\Phi_1(x)= x - 1, \\quad\n \\Phi_2(x)= x + 1, \\quad\n \\Phi_3(x)= x^2 + x + 1, \\quad\n \\Phi_4(x)= x^2 + 1, \\quad\n \\]\n \\[\n \\Phi_6(x)= x^2 - x + 1, \\quad\n \\Phi_{10}(x)= x^4 - x^3 + x^2 - x + 1, \\quad\n \\Phi_{18}(x)= x^6 - x^3 + 1.\n\\]\nNow, bad degrees $d=n+2$ for this sequence $h_n(x)$ are\n\\[\nd \\in \\{2k+1, 3k+1, 4k+3, 6k+4, 10k+5, 18k+ 6\\}.\n\\] This last list accounts for the factor $(x-1)^2$ of $g_5(x)$ for a single value $n=5$. Since $d$ is even, $d \\notin \\{2k+1, 4k+3, 10k+5\\}$. Since $d \\in \\{24k+2\\}$ would have remainder $2 \\pmod{3}$, we have $d \\notin \\{3k+1, 6k+4\\}$. Finally, $d \\notin \\{18k+6\\}$, since $24k+2$ is not divisible by $6$. This exhausts the list of possibilities, so no such bad degrees can exist. Hence, for each even $d \\geq 6$, we can find a Salem number of degree $d$ and trace $0$ in one of the three Salem sequences that were considered above.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{antroji}]\nSuppose that the relation \\eqref{rysys} holds with some $k_j \\in \\Z$, not all zero, and conjugates $\\al_j$ of a Salem number $\\al$ labelled as in Theorem~\\ref{pirmoji}.  Then, by Theorem~\\ref{pirmoji}, we \nhave $k_{2j}=k_{2j-1}$ for $j=1,\\dots,s$. Setting $\\beta_j=\\al_{2j-1}+1\/\\al_{2j-1}$ for $j=1,\\dots,s$ we find that\n\\eqref{rysys1} holds, namely,\n$k_{1}\\beta_1+k_3\\beta_2+\\dots+k_{2s-1}\\beta_s=0$.\n\nIn order to prove the first part of the theorem we need to show that $|k_1|+|k_3|+\\dots+|k_{2s-1}| \\geq 3$. \nFor a contradiction, assume that $$|k_1|+|k_3|+\\dots+|k_{2s-1}| \\leq 2.$$\nThe case when $|k_{2j-1}|=2$ for some $j$ (and so other $k_{2i-1}$ are all zeros) is clearly impossible, since \n$\\pm 2 \\be_j \\ne 0$. Therefore, we must have $|k_{2i-1}|=|k_{2l-1}|=1$, where $i<l$, and \n$k_{2j-1}=0$ for each $j \\ne i,l$. Dividing both sides of the relation $k_{2i-1}\\be_{i}+k_{2l-1}\\be_l=0$ by $k_{2i-1}$, we find that $\\be_{i}=-k_{2l-1}\\be_{l}\/k_{2i-1}=\\pm \\be_{l}$. Since $\\be_{i} \\ne \\be_l$, the only possibility is $\\beta_{i}=-\\be_{l}$. Applying to it any automorphism $\\sigma$ that maps $\\be_{i}$ to $\\be_1>2$ one obtains $\\be_1=-\\sigma(\\be_l)$. Here, the left hand side is a real number greater than $2$, whereas the right hand side belongs to the interval $(-2,2)$, which is a contradiction. \n\nIn order to prove the existence of a Salem number of degree $12$ with \nrequired linear relation among its conjugates we can\ntake, for instance, the following two pairs of real numbers $(a,b)$: \n$$(a_1,b_1)=(5-\\sqrt{2}, -3+2\\sqrt{2}) \\quad \\text{and} \\quad  (a_2,b_2)=(5+\\sqrt{2},-3-2\\sqrt{2}).$$ \n\nHere, the first pair $(a_1,b_1)$ satisfies $0<a_1<4$ and \\eqref{pima}, since $b_1=-0.171572\\dots$ and the left and right hand sides of \\eqref{pima}\nare\n$-0.828427\\dots$ and $0.828427\\dots$, respectively.\nThus, by Lemma~\\ref{trecioji}, $x^3-a_1x+b_1$ has three roots in $(-2,2)$. \n\nThe second pair $(a_2,b_2)$ satisfies $3<a_2<12$ and\n\\eqref{pima2}, because $b_2=-5.828427\\dots$ and the left and right hand sides of \\eqref{pima2}\nare\n$-6.252637\\dots$ and $-4.8284427\\dots$, respectively.\nHence, by Lemma~\\ref{trecioji}, $x^3-a_2x+b_2$ has two roots in $(-2,2)$ and one greater than $2$.  \n\nConsequently, their product\n\\begin{align*}\ng(x) :&=(x^3-a_1x+b_1)(x^3-a_2x+b_2) \\\\&= (x^3-5x-3+\\sqrt{2}(x+2))(x^3-5x-3-\\sqrt{2}(x+2)) \\\\&=\nx^6-10x^4-6x^3+23x^2+22x+1\n\\end{align*}\nhas $5$ roots in $(-2,2)$ and one greater than $2$. Now,\n$$f(x):=x^6g(x+1\/x)$$\nequals to\n$$x^{12}-4x^{10}-6x^9-2x^8+4x^7+7x^6+4x^5-2x^4-\n6x^3-4x^2+1.$$\nThis polynomial defines a Salem number $\\al=2.502568\\dots$, since $f$ is irreducible over $\\Q$. \n\nWe remark than none of the choices with $\\sqrt{2}$ replaced by\n$\\sqrt{3}$ or $\\sqrt{5}$ works.  The pairs $(a_1,b_1)=(5-\\sqrt{3},-3+2\\sqrt{3})$ and $(a_2,b_2)=(5+\\sqrt{3},-3-2\\sqrt{3})$ satisfy the requirements of Lemma~\\ref{trecioji}. However, the polynomial $g$ (and so $f$) is reducible:\n\\begin{align*}\ng(x):&= (x^3-5x-3+\\sqrt{3}(x+2))(x^3-5x-3-\\sqrt{3}(x+2)) \\\\&\n=x^6-10x^4-6x^3+22x^2+18x-3 \\\\&\n=(x^2-3)(x^4-7x^2-6x+1).\n\\end{align*}\nSimilarly, with the pairs $(a_1,b_1)=(5-\\sqrt{5},-3+2\\sqrt{5})$ and $(a_2,b_2)=(5+\\sqrt{5},-3-2\\sqrt{5})$ one also obtains $g$ with $5$ roots in $(-2,2)$ and one in $(2,+\\infty)$, but $g$ (and so $f$) is reducible:\n\\begin{align*}\ng(x):&= (x^3-5x-3+\\sqrt{5}(x+2))(x^3-5x-3-\\sqrt{5}(x+2)) \\\\&\n=x^6-10x^4-6x^3+20x^2+10x-11 \\\\&\n=(x^2+x-1)(x^4-x^3-8x^2+x+11).\n\\end{align*}\n\n\nBy Corollary~\\ref{keturi}, there no Salem numbers of degree $4$ or $6$\nwith a nontrivial linear relation among its conjugates. \nTo give the example of a Salem number of degree $8$ with nontrivial\nlinear relation among its conjugates \nwe can take, for instance,\n$h(x):=x^2+4x+1$ with roots $\\ga_1=-2-\\sqrt{3}$ and $\\ga_2=-2+\\sqrt{3}$ satisfying the conditions of \nLemma~\\ref{trecioji-1}.  \nThen, \n\\begin{align*}\nf(x):&=x^4 h\\big((x+1\/x)(1-x-1\/x)\\big) \\\\&=\nx^8-2x^7+x^6-2x^5+x^4-2x^3+x^2-2x+1\n\\end{align*}\nis irreducible. Hence, by Lemma~\\ref{trecioji-1}, $f$ defines a Salem number \n$\\al=1.994004\\dots$ of degree $8$ whose conjugates satisfy \\eqref{aaa}\nwith $k=2$.\n\nAs above, not every choice of an irreducible $h$ produces the irreducible polynomial $f$.\nFor example, selecting\n$h(x):=x^2+4x+2$ whose roots $\\ga_1=-2-\\sqrt{2}$ and $\\ga_2=-2+\\sqrt{2}$ satisfy the conditions of \nLemma~\\ref{trecioji-1}, we get\nthe polynomial\n\\begin{align*}\nf(x):&=x^4 h\\big((x+1\/x)(1-x-1\/x)\\big) \\\\&=\nx^8-2x^7+x^6-2x^5+2x^4-2x^3+x^2-2x+1 \\\\&=\n(x^4+1)(x^4-2x^3+x^2-2x+1)\n\\end{align*}\nwhich is reducible.\n\\end{proof}\n\n{\\bf Acknowledgements.} \nThe reserach of the first named author was funded by the European Social Fund according to the activity 'Improvement of researchers' quali\\-fication by implementing \nworld-class R\\&D projects' of Measure  No. 09.3.3-LMT-K-712-01-0037. \nThe post-doctoral position of the second named author is supported by the Austrian Science Fund (FWF) project M2259 Digit Systems, Spectra and Rational Tiles under the Lise Meitner Program.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nPhotosynthesis in plants, bacteria and algae involves light harvesting\ncomplexes. Solar energy creates excitons in these so-called ``antenna\ncomplexes,'' which are subsequently transported to the reaction center.\nUnderstanding this transport process involves the study of the couplings of the\nmolecular vibrations and the protein scaffold that holds the complex together\nand their impact on the dynamics. Approximate simulations of these excitation\nenergy transfer (EET) processes are often performed using the\nRedfield~\\cite{ishizakiAdequacyRedfieldEquation2009} and F\\\"orster resonance\nenergy transfer (FRET)~\\cite{novoderezhkinTheTheoryofForster1996}. While decent\nin certain parameter regimes, their accuracy cannot be implicitly assumed. The\ncomplexity of simulating quantum dynamics accurately grows exponentially with\nthe number of dimensions. For EET systems, the Hilbert space has a large\ndimensionality, which consequently makes simulating these systems\ncomputationally challenging.\n\nTypically, rigorous wave function-based methods like the density matrix\nrenormalization group (DMRG)~\\cite{whiteDensityMatrixFormulation1992,\nwhiteRealTimeEvolutionUsing2004,\nschollwockDensitymatrixRenormalizationGroup2005,\nschollwockDensitymatrixRenormalizationGroup2011a,\njiangFiniteTemperatureDynamical2020}, the multiconfiguration time-dependent\nHartree (MCTDH)~\\cite{beckMulticonfigurationTimedependentHartree2000} and its\nmulti-layer extension\n(ML-MCTDH)~\\cite{wangMultilayerFormulationMulticonfiguration2003} have often\nbeen used to simulate the dynamics of extended systems. These methods decompose\nthe system using various tensor networks to provide a compressed\nrepresentation. Though they have been used to study the systems in presence of\nvibrational manifolds~\\cite{renTimeDependentDensityMatrix2018}, the cost\nincreases with the number of such vibrational modes and the temperature of the\nsimulation. This is because the wave function-based approaches proceed by\ntruncating the basis set corresponding to the bath. In such a framework,\nincorporating a continuum of such states at a finite temperature becomes\ncomputationally very challenging~\\cite{tanimuraNumericallyExactApproach2020}.\n\nQuantum systems coupled to vibrational dissipative manifolds are most often\nsimulated using reduced density matrix methods. Foremost among these are the\nhierarchical equations of motion (HEOM)~\\cite{tanimuraTimeEvolutionQuantum1989,\ntanimuraNumericallyExactApproach2020,\nyanEfficientPropagationoftheHierarchical2021} and quasi-adiabatic propagator\npath integral (QuAPI)~\\cite{makriTensorPropagatorIterative1995,\nmakriTensorPropagatorIterative1995a}. Though historically, HEOM has been\nexceptionally popular for simulating large quantum\nsystems~\\cite{strumpferLightHarvestingComplexII2009,\nstrumpferEffectCorrelatedBath2011, strumpferExcitedStateDynamics2012}, recent\nwork on path integrals~\\cite{makriModularPathIntegral2018,\nkunduRealTimePathIntegral2020, kunduOriginVibrationalFeatures2021,\nboseMultisiteDecompositionTensor2022} has made it possible to study these\nsystems as well. Notably, the modular path integral\n(MPI)~\\cite{makriModularPathIntegral2018} has been used to study the exciton\ntransfer in bacteriochlorophyll aggregates~\\cite{kunduRealTimePathIntegral2020}.\nThe semiclassical partially linearized density matrix path integral approach\nalong with accurate spectral density have also been used by\n\\citeauthor{leeSemiclassicalPathIntegral2016}~\\cite{leeSemiclassicalPathIntegral2016}\nto study the Fenna-Matthew-Olson complex.\n\nWe have recently developed a multisite tensor network path integral method\n(MS-TNPI)~\\cite{boseMultisiteDecompositionTensor2022} using the framework of\ntensor network path\nintegral~\\cite{boseTensorNetworkRepresentation2021,\nbosePairwiseConnectedTensor2022, strathearnEfficientNonMarkovianQuantum2018,\njorgensenExploitingCausalTensor2019}. MS-TNPI starts with a tensor network\ndecomposition of the system, similar to what is commonly used in\nDMRG~\\cite{schollwockDensitymatrixRenormalizationGroup2005,\nschollwockDensitymatrixRenormalizationGroup2011,\nschollwockDensitymatrixRenormalizationGroup2011a,\npaeckelTimeevolutionMethodsMatrixproduct2019} and extends it to incorporate the\nFeynman-Vernon influence functional~\\cite{feynmanTheoryGeneralQuantum1963}. In\norder to achieve this, the decomposition along the system spatial dimension is\nextended to the temporal dimension creating a 2D tensor network. It is along\nthis temporal dimension that the influence functional is applied in the form of\na matrix product operator (MPO). MS-TNPI, being based on the Feynman-Vernon\ninfluence functional, can handle arbitrary spectral densities describing the\ndissipative environment. The resultant 2D MS-TNPI network can be efficiently\ncontracted to yield the time-dependent reduced density matrix corresponding to the\nextended quantum system represented in the form of a matrix product state\n(MPS). This representation contains the full Hilbert space of the system. Thus,\nthe method is not limited to problems that can only be formulated in the first\nexcitation Frenkel subspace. It allows for simulations of higher order spectra\nand many-body observables while ensuring that the dissipative medium is still\ntreated in a numerically exact manner.\n\nTo accurately simulate the excitonic dynamics in B850, high quality\nparameterizations of the environment are essential. Much effort has gone into\nsuch studies. Starting from simulations of photosynthetic complexes using model\nspectral densities~\\cite{ishizakiTheoreticalExaminationQuantum2009,\nishizakiAdequacyRedfieldEquation2009}, studies have incorporated descriptions\nusing experiments like fluorescence line narrowing\nspectra~\\cite{ratsepElectronPhononVibronic2007,\nboseAllModeQuantumClassical2020} and fully theoretically simulated spectral\ndensities~\\cite{olbrichTimeDependentAtomisticView2010,\nolbrichTheorySimulationEnvironmental2011, maityDFTBMMMolecular2020}. The\nbenefit of using a theoretically simulated spectral density is the internally\nconsistent of treatment the high frequency ``quantum'' region comprising of\nrigid vibrations and low frequency ``classical'' region primarily made of\nro-translational modes. Both regions need to be accounted for to obtain\naccurate dynamics. In this paper, we study the dynamics and\nabsorption spectrum of the exciton transport in the B850 ring of LH2. Molecular\ndynamics-based descriptions of the dissipative medium coupled to the\nchlorophyll ring are available in the form of spectral\ndensities~\\cite{olbrichTimeDependentAtomisticView2010}. This spectral density\ncaptures the effect of the high frequency rigid molecular vibrations as well as\nthe ro-translational modes primarily coming from the protein scaffold using\nmolecular dynamics.\n\nThis paper is organized as follows. Section~\\ref{sec:method} summarizes the\nMS-TNPI method and network that is used for the simulations. We describe how the\nring conformations can be included after making minor modifications to the\npropagator. We also show how the absorption spectrum can be calculated using\nMS-TNPI. This tensor network formulation, utilizing the many-body reduced\ndensity matrix corresponding to the extended system, allows a single simulation\nto give the entire spectrum. Then we discuss the B850 system under study and\nthe simulation results in Sec.~\\ref{sec:results}. We analyse the symmetries\npresent in the dynamics and demonstrate the spectra corresponding to different\napproximations. We also explore the temperature effects on the absorption\nspectrum under the assumption that the solvent spectral density invariant over\nthe temperature range and show how the behavior of the approximate spectra is\nqualitatively different. The simulations are computationally quite cheap,\nprobably owing to the structure of the Frenkel model. We also discuss the\ndifferences between this model and the well-known Ising model in presence of a\ndissipative medium. Finally, we end the paper with some concluding remarks in\nSec.~\\ref{sec:conclusions}.\n\n\\section{Method}\\label{sec:method}\n\n\\subsection{Multisite Tensor Network Path Integral}\\label{sec:PI_Exp}\n\nConsider an extended quantum system consisting of $P$ sites coupled with\nvibrational modes described by the following Hamiltonian:\n\\begin{align}\n    \\hat{H} & = \\hat{H}_0 + \\sum_{i=1}^P \\hat{V}_i\n\\end{align}\nwhere $\\hat{H}_0$ is the Hamiltonian describing the quantum system and\n$\\hat{V}_i$ captures the interaction of the $i$\\textsuperscript{th} site with\nits local vibrational modes.\n\nFor an EET process, the individual system sites are chromophores. The\n$i$\\textsuperscript{th} site can be represented by the two states, a ground\nstate, $\\ket{\\phi_i^g}$, and an excited state, $\\ket{\\phi_i^e}$. The Hamiltonian\ncorresponding to the quantum system consequently can be expressed as a Frenkel\nmodel:\n\\begin{align}\n    \\hat{H}_0 = &\\sum_{i=1}^P E_i \\dyad{e_i} + \\sum_{i=1}^{P-1} J_i \\left(\\dyad{e_{i+1}}{e_i} + \\dyad{e_i}{e_{i+1}} \\right)\\label{eq:frenkel}\\\\\n    \\text{where }\\nonumber \\\\\n    &\\ket{e_i} = \\ket{\\phi_i^e}\\otimes\\prod^\\otimes_{j\\ne i}\\ket{\\phi_j^g}.\\label{eq:1exciton_state}\n\\end{align}\nHere, $\\ket{e_i}$, as expressed by the direct product of the site-local basis\nin Eq.~\\ref{eq:1exciton_state}, is the single-exciton state with the excitation\nlocalized on the $i$\\textsuperscript{th} site. The electronic excitation energy\nof the $i$\\textsuperscript{th} system is $E_i$. Here we have assumed that only\nthe nearest neighbor units are coupled through the coupling $J_i$. This\ntypically represents a chain. However, B850 is a ring, so there is an extra\ninteraction term between the $i=1$ and $i=P$ sites, $H_{1,P} =\nJ_P\\left(\\dyad{e_P}{e_1} + \\dyad{e_1}{e_P}\\right)$.\n\nUnder Gaussian response theory, the effect of the dissipative medium can be\nmapped onto a bath of harmonic oscillators:\n\n\\begin{align}\n    \\hat{V}_i & = \\sum_{l} \\frac{p^2_{i,l}}{2m_{i,l}} + \\frac{1}{2}m_{i,l}\\omega_{i,l}^2 \\left(x_{i,l} - \\frac{c_{i,l} \\hat{s}_i}{m_{i,l} \\omega_{i,l}^2}\\right)^2\\label{eq:harmonicbath},\n\\end{align}\nwhere $\\omega_{i,l}$ and $c_{i,l}$ are the frequency and coupling of the\n$l$\\textsuperscript{th} mode of the $i$\\textsuperscript{th} site, respectively.\nThe system operator, $\\hat{s}_i$ associated with the $i$\\textsuperscript{th}\nsite, couples the site with its local vibrations. For EETs, the $\\hat{s}_i$\noperators are typically characterized by $\\hat{s}_i\\ket{\\phi^g_i} = 0$ and\n$\\hat{s}_i\\ket{\\phi^e_i} = 1$. The site-vibration interaction is described by a\nspectral density~\\cite{caldeiraPathIntegralApproach1983,\nmakriLinearResponseApproximation1999}\n\\begin{align}\n    J_i(\\omega) & = \\frac{\\pi}{2}\\sum_l\\frac{c_{i,l}^2}{m_{i,l}\\omega_{i,l}}\\delta(\\omega-\\omega_{i,l}),\n\\end{align}\nwhich is related to the energy-gap autocorrelation function obtained using\nclassical trajectory-based methods.\n\nFor the case of the B850 ring, \\citet{olbrichTimeDependentAtomisticView2010}\nhave simulated the correlation function using classical molecular dynamics for\nthe trajectories and ZINDO\/S-CIS for the excitation energy. This correlation\nfunction on the $i$\\textsuperscript{th} site was subsequently fit as a sum of\nexponentials and damped oscillations:\n\\begin{align}\n    C_i(t) &= \\sum_{j=1}^{N_\\text{exp}} \\eta_j e^{-\\gamma_j t} + \\sum_{j=1}^{N_\\text{osc}} \\tilde\\eta_j\\cos(\\omega_j t) e^{-\\tilde\\gamma_j t}\\label{eq:corrfunc}\n\\end{align}\nwhere $N_\\text{exp}$ is the number of exponentials and $N_\\text{osc}$ is the\nnumber of damped oscillations needed to fit the correlation function. Here,\n$\\eta_j$ is the strength of the $j$\\textsuperscript{th} exponential with a\ndecay rate $\\gamma_j$. The tildes correspond to the damped oscillations.\nThe spectral density is given as a cosine transform of this correlation function and has the following\nform~\\cite{olbrichTimeDependentAtomisticView2010,olbrichTheorySimulationEnvironmental2011}:\n\\begin{align}\n    J_i(\\omega) & = \\frac{2}{\\hbar} \\tanh\\left(\\frac{\\hbar\\omega\\beta}{2}\\right) \\int_0^\\infty C_i(t) \\cos(\\omega t) \\dd{t}\\label{eq:spect_base}\\\\\n                & = \\frac{2}{\\hbar}\\tanh\\left(\\frac{\\hbar\\omega\\beta}{2}\\right)\\left[\\sum_{j=1}^{N_\\text{exp}} \\frac{\\eta_j\\gamma_j}{\\gamma_j^2 + \\omega^2} \\right.\\nonumber \\\\\n              & \\left.+ \\sum_{j=1}^{N_\\text{osc}} \\frac{\\tilde{\\eta}_j\\tilde{\\gamma}_j}{2\\left(\\tilde\\gamma_j^2 + (\\omega - \\omega_j)^2\\right)}\\right]\n\\end{align}\nThe correlation functions and thus the spectral densities can, in general, be\ndifferent for the various sites. However, for B850, due to cylindrical symmetry\nof the complex, all the spectral densities (and correlation functions) are\nidentical according to~\\citet{olbrichTimeDependentAtomisticView2010}.\nExperimentally, these baths are often reported by their Huang-Rhys factors.\nMany simulations have been previously done with HEOM using a Drude-Lorentz\nspectral density~\\cite{strumpferEffectCorrelatedBath2011,\nstrumpferLightHarvestingComplexII2009}, which is a specialization of the above\nform with $N_\\text{exp}=1$ and $N_\\text{osc}=0$. Of course, such a form is far\nless flexible in accounting for the full physics of the problem. In particular,\nit misses out on the contributions from the rigid molecular vibrations.\n\nIt should be noted that there are a variety of ways of evaluating quantum\ncorrelation functions from purely classical data as summarized\nin Refs.~\\cite{kimEvaluationQuantumCorrelation2002,\nkimEvaluationQuantumCorrelation2006}. It has been recently shown that the\nso-called harmonic approximation, where the hyperbolic tangent is replace by its\nhigh temperature limit, yields better agreement with the quantum\ncorrelation function. It has also been shown to better maintain the temperature independence\nof the spectral density~\\cite{valleauOnTheAlternatives2012}. Additionally, the\napproach employed by~\\citet{olbrichTimeDependentAtomisticView2010} suffers from\nthe \"geometry mismatch\" problem, i.e., the potential surface used for the\nclassical molecular mechanics (MM) part of the simulation does no have the same normal modes and\nfrequencies as the corresponding quantum potential. In particular, it has been\nshown that for a gas phase BChl molecule the frequencies predicted by the MM\nsurface are at substantially higher then those corresponding to the quantum surface~\\cite{leeModelingElectronicNuclearInteractions2016}.   \nNewer methods of calculating spectral densities designed to remedy these\nissues are available~\\cite{leeModelingElectronicNuclearInteractions2016,\nmaityDFTBMMMolecular2020}. It is not entirely clear to what extent these deficiencies in the\nspectral density effect observables like the population dynamics and spectra. As\nMS-TNPI is derived completely independently from the spectral density used, it\noffers an excellent means to investigate the effects of these improved spectral\ndensities on observables of interest. While not done here, this will be the\ntopic of future research.\n\nThe reduced density matrix of the extended quantum system can be\nrepresented as a path integral expression:\n\\begin{align}\n    \\tilde\\rho(S^\\pm_N, N\\Delta t) & = \\Tr_\\text{bath} \\mel{S^+_N}{\\rho(N\\Delta t)}{S^-_N} \\\\ \\label{eq:RDM_Simple}\n                                   & = \\sum_{S_0^\\pm}\\sum_{S^\\pm_1}\\cdots\\sum_{S^\\pm_N-1} \\tilde\\rho(S^\\pm_0, 0) P_{S_0^\\pm\\cdots S^\\pm_N} \\\\\n                                   & = \\sum_{S_0^\\pm}\\sum_{S^\\pm_1}\\cdots\\sum_{S^\\pm_N-1} \\tilde\\rho(S^\\pm_0, 0) P^{(0)}_{S_0^\\pm\\cdots S^\\pm_N} F\\left[\\left\\{S^\\pm_n\\right\\}\\right]\\label{eq:RDM_Full}\n\\end{align}\nwhere $\\tilde\\rho$ is the reduced density matrix at an arbitrary time,\n$P_{S^\\pm_0\\cdots S^\\pm_N}$ is the path amplitude tensor,\n$P^{(0)}_{S^\\pm_0\\cdots S^\\pm_N}$ is the bare path amplitude tensor and $F$ is\nthe Feynman-Vernon influence functional~\\cite{feynmanTheoryGeneralQuantum1963}.\nThe system states at the $n$\\textsuperscript{th} time point are collectively\ndenoted by $S_n^\\pm$. (The superscript of ``$\\pm$'' denotes the combined\nforward-backward state, with the ``$+$'' and ``$-$'' coordinates defining the\nbra and ket sides of the reduced density matrix respectively. The state of the\n$i$\\textsuperscript{th} site at the $n$\\textsuperscript{th} time point will be\ndenoted by $s^\\pm_{i,n}$. In these notations, the first index will be the\nspatial index and the second index will be the temporal one. The\nforward-backward state of the $i$\\textsuperscript{th} site, $s^\\pm_{i,n}$, can\ntake values corresponding to $\\dyad{\\phi^g_i}$, $\\dyad{\\phi^g_i}{\\phi^e_i}$,\n$\\dyad{\\phi^e_i}{\\phi^g_i}$, or $\\dyad{\\phi^e_i}$.) The key terms appearing\nhere are the bare path amplitude tensor, the influence functional and the path\namplitude tensor. The bare path amplitude tensor, $P^{(0)}_{S^\\pm_0\\cdots\nS^\\pm_N}$, contains the full information of the system independent of the\nsolvent. It is given by:\n\\begin{align}\n    P^{(0)}_{S^\\pm_0\\cdots S^\\pm_N} & = K(S^\\pm_0, S^\\pm_1, \\Delta t) K(S^\\pm_1, S^\\pm_2, \\Delta t)\\nonumber \\\\\n                                    & \\times\\cdots K(S^\\pm_{N-1}, S^\\pm_N, \\Delta t),\n\\end{align}\nwhere $K$ is the so-called ``forward-backward propagator'' obtained from a\ndirect product of the forward, $U$, and backward, $U^\\dag$, system propagators,\n\\begin{align}\n    K(S^\\pm_n, S^\\pm_{n+1}, \\Delta t) &= U(S^+_n, S^+_{n+1}, \\Delta t)\\,U^\\dag(S^-_n, S^-_{n+1}, \\Delta t).\n\\end{align}\nThe influence functional, $F$, encodes the interaction of the system with the\nsolvent. Since the vibrational modes are site local, it can be expressed as a\nproduct of site-specific influence functionals:\n\\begin{align}\n    F\\left[\\left\\{S^\\pm_n\\right\\}\\right] &= \\prod_{i=1}^P F_i\\left[\\left\\{s^{\\pm}_{i,n}\\right\\}\\right],\n\\end{align}\n\nwhere\n\n\\begin{align}\n    F_i\\left[\\left\\{s^{\\pm}_{i,n}\\right\\}\\right] & = \\exp\\left(-\\frac{1}{\\hbar}\\sum_{k} \\Delta s_{i,k} \\sum_{k'\\le k}\\left(\\Re\\left(\\eta^i_{k,k'}\\right)\\Delta s_{i,k'} \\right.\\right.\\nonumber \\\\\n                                            & \\left. \\vphantom{\\sum_{k\\le k'}}\\left. + 2i\\Im\\left(\\eta^i_{k,k'}\\right) \\bar{s}_{i,k'}\\right)\\right).\n\\end{align}\nThe bath response function, $C_i(t)$ in Eq.~\\ref{eq:corrfunc}, discretized\nalong the QuAPI system\npath~\\cite{makriTensorPropagatorIterative1995,makriTensorPropagatorIterative1995a}\nfor the $i$\\textsuperscript{th} site is given by $\\eta_{kk'}^i$. Additionally, $\\Delta\ns_{i,k} = s^{+}_{i,k} - s^{-}_{i,k}$ and $\\bar{s}_{i,k} =\n\\tfrac{1}{2}\\left(s^{+}_{i,k} + s^{-}_{i,k}\\right)$. (The method is, of course,\nnot tied down to any specific form of the spectral density like the Drude or\nthe Ohmic forms. The $\\eta$-coefficients can be expressed in terms of integrals\nover the spectral density.) Because the B850 ring necessitates identical\nspectral densities, the $\\eta$-coefficients are the same for every site. The\npath amplitude tensor, $P_{S^\\pm_0\\cdots S^\\pm_N}$ is effectively the product\nof the bare path amplitude tensor and the influence functional. It contains the\nfull information of the system embedded in the solvent.\n\nTo simplify the discussion, let us briefly neglect the effects of the solvent.\nUnder this condition, the summations in Eq.~\\ref{eq:RDM_Full} can be performed\nindependently, since the influence functional, which couples the system at\ndifferent time points, is omitted when there is no system-solvent interactions.\nIn this case, the density matrix can be evaluated iteratively, \n\\begin{align}\n    \\tilde\\rho(S^\\pm_n, n \\Delta t)  & = \\sum_{S_{n-1}^\\pm}\\tilde\\rho(S^\\pm_{n-1}, (n-1)\\Delta t) K(S^\\pm_{n-1}, S^\\pm_{n}, \\Delta t). \\label{eq:RDM_Iter}\n\\end{align}\nThis has the same form as matrix-vector multiplications. The storage and\ncomputational complexity of these expressions grow exponentially with the\nnumber of system sites making direct simulations of extended systems\npractically impossible.  However, for many extended systems, the correlations\nbetween system sites decrease rapidly with the distance between them.\nThus, the large tensors (e.g., $\\tilde\\rho$ and $K$) representing every\nparticle in the system can be efficiently factored into a network of smaller\ntensors corresponding to a single particle each. This fact is widely utilized\nby methods like time-dependent\nDMRG~\\cite{daleyTimedependentDensitymatrixRenormalizationgroup2004} and\ntime-dependent variational principal\n(TDVP)~\\cite{haegemanTimeDependentVariationalPrinciple2011,yangTimedependentVariationalPrinciple2020}.\nIn this representation, the reduced density matrix becomes an MPS, \n\n\\begin{align}\n    \\tilde{\\rho}\\left(S_n^\\pm, n\\Delta t\\right) &= \\sum_{\\left\\{\\alpha_{(i,n)}\\right\\}} A_{\\alpha_{(1,n)}}^{s^\\pm_{1,n}} A_{\\alpha_{(1,n)},\\alpha_{(2,n)}}^{s^\\pm_{2,n}}\\cdots A_{\\alpha_{(P-1,n)}}^{s^\\pm_{P,n}},\n\\end{align}\nthe forward-backward propagator an MPO,\n\\begin{align}\n    K\\left(S_{n}^\\pm, S_{n+1}^\\pm, \\Delta t\\right)  = & \\sum_{\\left\\{\\alpha_{(i,n)}\\right\\}}\n    W^{s_{1,n}^\\pm, s_{1,n+1}^\\pm}_{\\alpha_{(1,n)}}  W^{s_{2,n}^\\pm,\n    s_{2,n+1}^\\pm}_{\\alpha_{(1,n)},\\alpha_{(2,n)} }\\nonumber                      \n                                 \\\\\n                                                      & \\cdots W^{s_{P-1,n}^\\pm,\n                                     s_{P-1,n+1}^\\pm}_{\\alpha_{(P-2,n)},\\alpha_{(P-1,n)}}W^{s_{P,n}^\\pm,\n                             s_{P,n+1}^\\pm}_{\\alpha_{(P-1,n)}},\\label{eq:prop_MPO}\n\\end{align}\nand the matrix-vector multiplication becomes an MPO-MPS application. Here,\n$\\alpha_{i,n}$ is the ``bond'' index that connects the $i$\\textsuperscript{th}\nsite at time-step $n$ to the $(i+1)$\\textsuperscript{th} site at the same time\nstep. It is called a ``spatial'' bond index because it connects points that are\nspatially separated. The structures of the MPS and MPO are shown in\nFig.~\\ref{fig:rho_MPS}. The maximum and average bond dimension associated with\nthe $n$\\textsuperscript{th} time step is  $m(n)=\\max_{i}\n\\left(\\dim(\\alpha_{(i,n)})\\right)$ and $\\bar{m}(n)=\\tfrac{1}{P}\\sum_i\n\\dim(\\alpha_{(i,n)})$, respectively. The efficiency of these factorizations can\noften be characterized by the maximum bond dimension. Roughly speaking, the\nsmaller the resulting bond dimension the more efficient the MPO\/MPS\nfactorization. \n\n\\begin{figure}\n    \\subfloat[MPS representation of the density matrix]{\\includegraphics[scale=0.2]{fig01.pdf}}\n    \\hspace{2cm}\\subfloat[MPO representation of the propagator]{\\includegraphics[scale=0.35]{mpo.png}}\n    \\caption{Matrix product representations of the density matrix and the propagator.}\\label{fig:rho_MPS}\n\\end{figure}\n\nFor most problems, there are many ways to construct the propagator MPO that\noften involve a trade off between the maximum time-step and bond dimension. The\ndevelopment of optimal propagator MPOs has been an object of intense research\nover the years~\\cite{paeckelTimeevolutionMethodsMatrixproduct2019,\nZaletel2015a, haegemanTimeDependentVariationalPrinciple2011,\nyangTimedependentVariationalPrinciple2020}. In this work, we use a modified\nsecond-order Suzuki-Trotter split propagator MPO that is commonly used with the\n(second-order) time-evolved block decimation method\n(TEBD)~\\cite{whiteRealTimeEvolutionUsing2004,\ndaleyTimedependentDensitymatrixRenormalizationgroup2004,\nvidalEfficientSimulationOneDimensional2004}. These MPOs are generally used\nto simulate systems with nearest neighbor couplings. For many photosynthetic\nsystems, the ring topology is biologically relevant. As discussed earlier, the\nring Hamiltonian has an extra coupling, $H_{1P}$ between the\n$1$\\textsuperscript{st} and the $P$\\textsuperscript{th} sites. Now, the\npropagator element between two points $S^+_n$ and $S^+_{n+1}$ for the ring can\nbe written as\n\\begin{align}\n    U_{\\text{ring}} & (S^+_n,S^+_{n+1},\\Delta t) \\approx\\nonumber\n    \\\\ & \\mel{S^+_{n+1}}{e^{-i H_{1P} \\Delta t\/{2\\hbar}} U_{\\text{chain}}\n    (\\Delta t) e^{-i H_{1P} \\Delta t\/{2\\hbar}}}{S^+_{n}},\n\\end{align}\nwhere $U_{\\text{chain}}$ is the standard second-order TEBD propagator for the\nchain. The resulting propagator MPO for the ring is obtained by multiplying the\nMPOs corresponding to the ``long bond,'' $e^{-iH_{1P}\\Delta t\/2\\hbar}$, together\nwith that of the chain. It is feasible to construct the propagator in a\ncylindrical form, reflecting the true ring symmetry of the system. However,\nthere may be other performance concerns. This would be evaluated in a future\nwork.\n\n\nThe approach discussed till now is a density matrix version of methods like\nTEBD or time-dependent DMRG. The next step is to incorporate the effects of the\nsolvent by accounting for the time non-locality of the influence functional.\nSince the different time points can no longer be uncoupled, traditional time\nstep iteration is impossible. In principle, this would cause the computational\ncomplexity to increase exponentially with the number of time steps; however,\nthis cost can be avoided by performing an additional DMRG-like tensor\ndecomposition along the time axis~\\cite{boseTensorNetworkRepresentation2021,\nboseMultisiteDecompositionTensor2022}. The resulting 2D tensor network, factored\nin both space and time, forms the foundation of MS-TNPI.\n\nTo construct the MS-TNPI network, one starts with the MPO representations of the\nforward-backward propagator between each of the time-points and uses them to\nconstruct a fully factorized tensor network description of the bare path\namplitude tensor $P^{(0)}$:\n\\begin{align}\n    P^{(0)}_{S_{0}^{\\pm}\\cdots S_{N}^{\\pm}} & = \\sum_{\\left\\{\\beta_{n}\\right\\}} \\mathbb{T}^{S_{0}^{\\pm}}_{\\beta_{0}}\n    \\cdots \\mathbb{T}^{S_{n}^{\\pm}}_{\\beta_{n-1}, \\beta_{n}}\n    \\cdots\\mathbb{T}_{\\beta_{N-1}}^{S_{N}^{\\pm}}.\\label{eq:MS-PA_MPS}\n\\end{align}\nHere, each $\\mathbb{T}$ is in a matrix product representation decomposed along\nthe site axis, and $\\beta_n$ is the bond dimension connecting the tensors at\ntime-point $n$ to the one at $n+1$. These indices are called ``temporal'' bonds\nbecause they connect points on the same site but different times. The 2D\nstructure of the MS-TNPI network is demonstrated in Fig.~\\ref{fig:ms-tnpi}.\n(For convenience, a more detailed derivation of the tensors that make up\n$\\mathbb{T}$ is provided in Appendix~\\ref{app:MS-TNPI-Network}.) Though\nEq.~\\ref{eq:MS-PA_MPS} has been written for the bare path amplitude tensor, the\nfull tensor also would have a practically identical form. The main difference\nbeing whether the influence functional has been incorporated.\n\n\\begin{figure}\n    \\includegraphics[scale=0.2]{fig02.pdf}\n    \\caption{2D MS-TNPI tensor network.}\\label{fig:ms-tnpi}\n\\end{figure}\n\nSince the influence functional has not yet been applied, contracting the 2D\nMS-TNPI network, as it stands right now, yields the time-evolved density matrix\nfor the isolated extended system expressed in the form of an MPS. Using the\ntensor network path integral~\\cite{boseTensorNetworkRepresentation2021}, one can\ndefine the influence functional MPO for each site or monomer unit and apply it\nto the corresponding site as illustrated in Fig.~\\ref{fig:MS-TNPI-influence}.\nThe relevant equations for the influence functional MPO have been summarized in\nAppendix~\\ref{app:IF-MPO}. Now, upon contraction, the network gives the\nresulting time-evolved reduced density matrix. (Though, we have presumed a lack\nof correlation between the baths on different sites, it is possible to extend\nthe structure to take correlation effects into account as well by applying\noperators that connect the ``rows'' corresponding to the different sites. In\nabsence of those correlation effects, the influence functional MPOs directly\naffect only the temporal bond dimension and not the spatial bond dimension.)\n\n\\begin{figure}\n    \\includegraphics[scale=0.2]{fig03.pdf}\n    \\caption{Schematic depiction of the application of the influence functional\n    MPOs to selected sites of the extended\nsystem.}\\label{fig:MS-TNPI-influence}\n\\end{figure}\n\nLet us examine the 2D MS-TNPI network corresponding to the path amplitude\ntensor in more depth. Along the time axis, it consists of the local path\namplitude tensor for each of the sites, effectively generalizing the TNPI\nstructure to multiple sites. On the other hand, along the space axis, each of the ``columns''\nrepresents the full state of the system at that time point and is effectively a\ngeneralization of the reduced density matrix as propagated by time-dependent\nDMRG methods. The network allows the possibility of many different algorithms\nfor contracting it. While the present work uses a contraction scheme that\npreserves the columns, and consequently obtains the entire time propagated\ndensity matrix as an MPS, future explorations could yield other interesting and\nperformant schemes.\n\n\nNa\\\"ively speaking, the network should have one column for each time point of propagation. This is due to the\nnon-Markovian memory induced by the bath. This represents the growth of\ncomputational cost with the propagation time. Much of this exponential\ncomplexity is already controlled through the tensor network decomposition and\nthe accompanying truncated singular value decomposition filtration schemes.\nHowever, this is not enough by itself. It is well-known that in condensed\nphase environments, the memory dies away with the temporal distance between two\npoints. It is, therefore possible to truncate the memory length to say $L$\ntime-steps and use $L$ as a convergence parameter. This is achieved through a\nprocedure for iterative propagation of the density matrix for the extended\nsystem.\n\nWhen iteration starts, there are $L$ time-steps and, consequently, $L+1$ columns in\nthe MS-TNPI network. These are labeled as $C_n$ for $1\\le n\\le L+1$. For the\ninitial step of iteration, let $C_0 = \\tilde\\rho(0)$ written as an MPS. (If, as\nin Sec.~\\ref{sec:spectra},  we are simulating a correlation function $C(t) =\n\\Tr[U(t)\\,\\rho(0)\\,A\\,U^\\dag(t)\\,B]$, $C_0 = \\tilde\\rho(0)\\,A$.) The iteration\nmethod can be summarized in the following series of steps. (An MPO-MPS\nmultiplication is written as $\\otimes$ in the following steps.)\n\\begin{enumerate}\n    \\item Update $C_0$ by multiplying it by the MPO $C_1$. $C_0 \\leftarrow C_1\n     \\otimes C_0$. (Note that the first column, $C_1$, is an MPO and the\n     resulting $C_0$ an MPS.)\n    \\item Slide all the columns back by one step. $C_j \\leftarrow C_{j+1}$ for\n     $j<L$. (Now, the new first column is no longer an MPO and has two temporal\n     bond indices.)\n    \\item Update $C_L$ and insert the $C_{L+1}$ to account for the propagator\n        between the penultimate and the last time steps. (The working equations\n        are in Appendix~\\ref{app:MS-TNPI-Network}.) \\item Apply the influence\n        functional MPO to each row.\n    \\item Trace over the site indices of $C_1$ to turn it into an MPO again.\n\\end{enumerate}\nSteps (1) -- (5) are repeated as many times as required to get the full\ndynamics. This procedure is schematically represented in Fig.~\\ref{fig:iter}.\n\n\\begin{figure}\n    \\includegraphics[scale=0.2]{fig04.pdf}\n    \\caption{Schematic illustrating the iteration scheme in the MS-TNPI framework.}\\label{fig:iter}\n\\end{figure}\n\nThese calculations involve manipulation of high-dimensioned tensors factorized\nin different forms. The cost of applying the IF MPO is\n$\\mathcal{O}\\left(m_{t}^3w_{p}^2w_{I}^2d^2\\right)$, and the cost of the\ncontraction is $\\mathcal{O}\\left(m^3w_{p}^2m_{t}\\right)$. Here, $m_{t}$ is the\nmaximum temporal bond dimension, $m$ is the maximum bond dimension of the\ncontracting MPS, $w_{I}$ is the maximum bond dimension of the IF MPO. The\nmaximum bond dimension of the forward-backward propagator of the bare system is\ndenoted by $w_{p}$ and $d=2$ is the dimensionality of a typical system site.\nThe computational cost is linear in the number of sites or system size, which\nappears as a prefactor in the formal scaling expressions. Though the magnitude\nof $m_t$ might be dependent on the memory length, $L$, the exponential growth\nof complexity within memory is effectively curtailed. The cost propagation\nbeyond the memory span of $\\tau = L\\Delta t$ is strictly linearly proportional\nto the number of steps of dynamics beyond the memory length simulated. The\ndominant cost is determined by the particular parameters under consideration.\nFor more details about the contractions involved, please consult\nRef.~\\cite{boseMultisiteDecompositionTensor2022}.\n\n\\subsection{Absorption Spectrum}\\label{sec:spectra}\n\nAbsorption spectra are calculated as the Fourier transform of dipole-dipole time\ncorrelation function,\n\\begin{align}\n    \\sigma(\\omega) & \\propto \\Re \\int_0^t e^{i\\omega t} C(t)\\dd{t}                       \\\\\n    C(t)           & = \\Tr\\left(\\hat\\mu(t)\\hat\\mu(0)\\rho(0)\\right).\\label{eq:mu_mu_corr}\n\\end{align}\nHere, $\\hat\\mu(t) = \\sum_{i=1}^P \\hat U^\\dag(t)\\hat\\mu \\hat U(t)$ is the\ntime-evolved total dipole operator, $\\hat\\mu = \\sum_{i=1}^P \\hat\\mu_i$. Generally\nspeaking, the local dipole operators do not point in the same direction.\nTherefore, using the site-local basis, $\\hat\\mu_i = \\vec{d}_i\n(\\dyad{\\phi^g_i}{\\phi^e_i} + \\dyad{\\phi^e_i}{\\phi^g_i})$, where $\\vec{d}_i$ is\nthe dipole moment vector corresponding to the $i$\\textsuperscript{th} unit. For\ncalculating the absorption spectrum,\n\\begin{align}\n    \\rho(0)       & = \\tilde\\rho(0)\\otimes\\prod_i \\frac{\\exp(-\\beta V_i)}{Z_i} \\\\\n    \\tilde\\rho(0) & = \\prod^\\otimes_i \\dyad{\\phi^g_i},\n\\end{align}\nwhere the vibrational manifold associated with the $i$\\textsuperscript{th} unit\nis distributed thermally at an inverse temperature, $\\beta=\\tfrac{1}{k_B T}$, on\nthe ground Born-Oppenheimer surface. The partition function for this\ndistribution is given by $Z_i$.\n\nIn the path integral notation of Sec.~\\ref{sec:PI_Exp}, the correlation\nfunction, Eq.~\\ref{eq:mu_mu_corr}, can be written as\n\\begin{align}\n&C(N \\Delta t) = \\sum_{S_{0}^{\\pm}\\ldots S_{N}^{\\pm}}\\,\\sum_{S_{0}^{'\\pm}}\\,\\sum_{S_{N}^{'\\pm}}\\,\\nonumber\\\\ \n&\\delta_{S_{N}^{'+}, S_{N}^{'-}}\\,\\tilde{\\rho}\\left(S_0^\\pm,0\\right)\\,\\hat{\\mu}\\left(S_0^{\\pm},S_0^{'\\pm}\\right)\\,P_{S_{0}^{'\\pm}\\cdots S_{N}^{\\pm}}\\,\\hat{\\mu}\\left(S_N^{\\pm},S_N^{'\\pm}\\right),\\label{eq:mu_mu_corr_PI}\n\\end{align}\nwhere $\\hat{\\mu}\\left(S_n^{\\pm},S_n^{'\\pm}\\right) =\n\\mel{S^{'+}_n}{\\hat{\\mu}}{S^{+}_n} \\mel{S^{-}_n}{\\mathbb{I}_n}{S^{'-}_n}$ and\n$\\mathbb{I}_n$ represents the identity operator of the full forward-backward\nspace at $n$\\textsuperscript{th} time point. It is worth noting that in this\ncase, the total dipole operator only acts on the forward space; furthermore, it\nis possible to analytically represent it as an MPO, which is given in\nAppendix~\\ref{app:Dipole_MPO}. Interestingly, in this form, the MPO is extremely\ncompact, having a bond dimension of just two. Since we have an MPO expression\nfor the total dipole operator, the correlation function can be efficiently\ncomputed by applying this MPO to the initial density MPS as well as the final\ntime-propagated MPS before taking the trace, as per Eq.~\\ref{eq:mu_mu_corr_PI}.\nBy applying the total dipole moment MPO at multiple intermediate points, it\nshould also be possible to calculate higher order response functions at minimal\nextra computational cost.\n\nFrom here, we could proceed directly to computing the correlation function.\nHowever, the high frequency nature of the electronic absorption necessitates the\nuse of very short time steps, thereby increasing the memory length.\nAdditionally, these correlation functions decay slowly which means the total\nnumber of simulation steps would be fairly large. Together, these two factors\nserve to increase the computational complexity of these simulations. Such a\ndirect approach is, therefore, rather inefficient. Fortunately, though, in\nthis case, it is possible to transform the entire problem into a numerically\nsimpler problem.\n\nThis transformation begins by shifting each site by a constant energy term,\n$\\bar{E}$, that has the same order of magnitude as the electronic excitation\nenergy ($\\bar{E}\\approx \\SI{12000}{\\per\\cm}$ for B850). The system Hamiltonian,\nEq.~\\ref{eq:frenkel}, can then be rewritten as\n$\\hat{H}_{0}=\\hat{\\bar{H}}_{0}+\\hat{\\mathcal{H}}_{0}$, where\n\\begin{align}\n    &\\hat{\\bar{H}}_{0}   = \\bar{E}\\sum_{i=1}^{P}\\dyad{e_{i}}\\quad\\text{and}\\\\\n    &\\hat{\\mathcal{H}}_{0}  = \\sum_{i=1}^{P}\\epsilon_{i}\\dyad{e_{i}} + \\sum_{i=1}^{P-1} J_i \\left(\\dyad{e_{i+1}}{e_i} + \\dyad{e_i}{e_{i+1}} \\right),\n\\end{align}\nwith $\\epsilon_{i} = E_{i} - \\bar{E}$. $\\hat{\\bar{H}}_{0}$ commutes with\n$\\hat{\\mathcal{H}}_{0}$, so we can factor\n$\\hat{\\bar{U}}=\\exp(-\\tfrac{i}{\\hbar}\\hat{\\bar{H}}_{0} t)$ out of the propagator\nwithout incurring any Trotter error. Next, we identify the ``bra'' side of\n$\\hat\\mu(0)\\tilde\\rho(0)$ (corresponding to the backward path) with the\nelectronic ground state. On the ``ket'' side, which corresponds to the forward\npath, it is in a state with a single excitation. Due to the block-diagonal\nstructure of the Frenkel Hamiltonian, the bra of the time-evolving operator\nremains in the ground state and the ket remains in the manifold of singly\nexcited states. In other words, the backward path remains in the ground state,\nand forward path only populates the first excited subspace. In this subspace,\n$\\hat{\\bar{H}}_{0}=\\bar{E}$ and $\\hat{\\bar{U}}=e^{-i \\bar{E}t\/\\hbar}$ Thus,\nwe have:\n\\begin{align}\n    C(t) & = e^{-i \\bar{E} t\/\\hbar} \\mathcal{C}(t),\\label{eq:factor_corr}\n\\end{align}\nwhere $\\mathcal{C}(t)$ is the dipole moment autocorrelation function obtained\nusing the propagator $\\hat{\\mathcal{U}}$ corresponding to\n$\\hat{\\mathcal{H}}_{0}$. Because the extremely high frequency oscillations have\nbeen factorized out, the time-steps can now be larger. The multiplication by\nthe fast rotating phase is done as a post-processing step and is equivalent to\na shift of the absorption lineshape to account for the redefining of the zero\nof energy. It is worth noting that since the backward path remains in the\nground state, we could have derived a more compressed representation of the\ninfluence functional MPO. While this optimization was not needed here, it may\nbe required in the future.  \n\nBefore concluding this subsection, it is instructive to explore the form\nof the ``initial state''. We note that though $\\tilde\\rho(0)$ is a separable\nstate, the initial state, defined as the product with the direct sum of the site\nlocal $\\hat\\mu_j$ operators, is surely not separable.\n\\begin{align}\n    \\hat\\mu(0)\\tilde\\rho(0) & = \\sum_j \\hat\\mu_j\\tilde\\rho(0)                  \n                         \\\\ & = \\sum_j \\vec{d}_j \\prod_{k\\ne j}\n              I_k\\otimes \\left(\\dyad{\\phi^g_j}{\\phi^e_j}+\\dyad{\\phi^e_j}{\\phi^g_j}\\right) \\prod_i \\dyad{\\phi^g_i} \\\\\n                            & = \\sum_j\\vec{d}_j \\prod_{k\\ne j} \\dyad{\\phi^g_k} \\otimes \\dyad{\\phi^e_j}{\\phi^g_j}\\label{eq:sumMPS}\n\\end{align}\nwhere $I_k$ is the identity operator on the $k$\\textsuperscript{th} site. Each\nof the operators in the summand of Eq.~\\ref{eq:sumMPS} is in a direct product\nform. This sum over multiple such operators causes the sites to be entangled and\nthe effective initial state to be non-separable. While we have written out the\nequation explicitly for the absorption spectrum, this issue of non-separability\nand entanglement of the initial condition is a consequence of the operators\ninvolved in the correlation function. This feature is common to most spectra of\ninterest. In fact, for the emission spectrum, the initial condition has even\ngreater entanglement. To our knowledge, MPI is the only other method that is\nable to use influence functionals for general extended quantum systems; however,\nsince it treats the system sites sequentially, it is not designed to handle\nnon-separable initial states.  This means that the simulation would require\nseparate runs, each corresponding to a different term in the sum. The fact that\nthe final result comes from a trace over a non-direct product operator further\nincreases the number of runs that would be be required. Thus, an MPI calculation\nof the correlation function, given by Eq.~\\ref{eq:mu_mu_corr}, is likely many\ntimes more costly than a simple calculation of the population dynamics. On the\nother hand, since MS-TNPI is compatible with the MPS\/MPO framework, and the\ntotal dipole operator can be expressed as an extremely compact MPO; it can\ncalculate the correlation function at practically the same cost as the\npopulation dynamics.\n\n\\subsection{A Note About Convergence}\\label{sec:converge}\n\nAs with any numerical approach, the simulations included in this work involve a\nvariety of different convergence parameters. Here, we give a brief description\nof the key parameters as well as a quick outline of the procedure used. Loosely\nspeaking, these parameters can be grouped into two categories: those arising\nfrom the path integral (i.e., time-step and memory length) and those coming\nfrom the SVD compression of the network.  Under our compression scheme, the\nsingular values, $\\lambda_{n}$, are discarded such that\n\\begin{align}\n  \\frac{\\sum_{n\\in\\text{discarded}}\\lambda_{n}^{2}}{\\sum_{n}\\lambda_{n}^{2}}<\\chi.\n\\end{align}\nThe particular value of truncation threshold, $\\chi$, used depends on the part\nof the network being compressed.  We used two different truncation thresholds,\nhere, $\\chi_t$ (used for compressing the bonds along the temporal axis) and\n$\\chi_s$ (for the spatial axis). Conceptually, the value of $\\chi_t$ changes\nwith the memory length, $L$, while the value of $\\chi_s$ changes with the number\nof sites and the strength of the couplings between them. Of course, while doing\nthe calculation, these clean conceptual divisions do not hold and the two\ndimensions start affecting each other.\n\nTypically, one starts by choosing a particular value of time-step and $L$, and\niterating through different cutoffs to achieve convergence with respect to\nthem. Subsequently, the time-step and $L$ are changed, repeating the process of\nconverging the cutoffs at each step, to find the largest converged time-step\nand the smallest $L$. Unlike typical system-solvent decomposed methods, here,\nthe Trotter error caused by the time-step stems from both the system-solvent\nsplit as well as the system-system split. The memory length, $L$, however is\nonly caused by the local baths. \n\n\\section{Results}\\label{sec:results}\n\n\\begin{figure}\n    \\includegraphics[scale=0.3]{B850.pdf}\n    \\caption{Side and top views of the B850 ring of LH2. The molecular vibrations and the protein scaffolding affect the dynamics.}\\label{fig:b850}\n\\end{figure}\n\nThe B850 ring of LH2, shown in Fig.~\\ref{fig:b850}, is an important component of\nphotosynthetic complexes. It has been previously studied with approximate\nspectral densities. Here, we use the accurate spectral densities derived\nby~\\citet{olbrichTimeDependentAtomisticView2010} to model the interaction of the\nsystem with the rigid molecular vibrations and the impact of the protein\nscaffolding. The resultant spectral density obtained along MD trajectories with\nZINDO\/S-CIS calculations for the energy gap is shown in\nFig.~\\ref{fig:spect_dens}. It is well-known that the B850 ring can be decomposed\ninto constituent dimers with high intra-dimer electronic couplings. The\ncouplings between the different dimers is considerably smaller. The electronic\ncouplings between the nearest neighbors, calculated using the method of\ntransition charges from electrostatic potentials\n(TrEsp),~\\cite{rengerTheoryExcitationEnergyTransfer2009,\nmadjetIntermolecularCoulombCouplings2006} alternate between\n\\SIlist{173;140}{\\per\\centi\\meter}~\\cite{olbrichTimeDependentAtomisticView2010}.\nNotably, these values are significantly less than ones derived from experiments\nbecause they take environmental screening effects into account. As a point of\ncomparison and to understand the system better, we also consider the\nexperimentally derived electronic couplings of \\SIlist{315;245}{\\per\\cm} as\nreported by~\\citet{freibergExcitedStateDynamics2009} in their experimentally fit\nmodels. These parameters have been used\nby~\\citet{strumpferEffectCorrelatedBath2011} in their study of the dynamics of\nB850 ring coupled with a Drude-Lorentz spectral density. Other experimentally\nderived numbers~\\cite{tretiakBacteriochlorophyllCarotenoid2000} are also of\nsimilar magnitude.\n\n\\begin{figure}\n     \\includegraphics{b850_kleinekathoefer}\n     \\caption{Spectral density corresponding to the B850 ring as calculated in\n Ref.~\\cite{olbrichTimeDependentAtomisticView2010}.}\\label{fig:spect_dens}\n\\end{figure}\n\nThe dynamics of exciton transport in the B850 ring corresponding to the initial\nexcitation of the 8\\textsuperscript{th} BChl unit ($\\tilde\\rho(0) =\n\\dyad{e_8}$) with the TrEsp couplings is shown in\nFig.~\\ref{fig:popln_kleinekathoefer}. For the calculations shown in here,\ntypically $\\Delta t = \\SI{4.84}{\\fs}$ yields converged results with a memory\nspan of $L\\Delta t = \\SI{24.19}{\\fs}$. The cutoffs for this problem are very\ndifferent along the temporal and spatial axes. Along the time-axis, $\\chi_t$\nconverged around $10^{-10}$ whereas the spatial cutoff, $\\chi_s$ was converged\naround $10^{-5}$ -- $10^{-7}$. A representative converged simulation of the\nfull dynamics takes around 6 hours on an Intel\\textregistered{}\nXeon\\textregistered{} Gold CPU. The runtime is, of course, extremely dependent\non the exact parameters of the system under study and the levels of singular\nvalue decomposition truncation that is being done. Because of the low\ncouplings, the first peak of the initially excited site happens at $\\approx\n\\SI{75}{\\femto\\second}$, which is significantly later than what is expected\nfrom experimentally derived couplings. As a comparison, we demonstrate the\ncorresponding dynamics of B850 ring parameterized by experimentally derived\nelectronic couplings in Fig.~\\ref{fig:popln_experiment}. Notice that in this\ncase, the prominent hump in the excited state population of the initially\nexcited monomer happens around $\\SI{37}{\\femto\\second}$. \n\n\\begin{figure}\n    \\subfloat[Excited state populations of each of the BChl sites.]{\\includegraphics{TrEsp_popln.pdf}}\n\n    \\subfloat[2D plot of the transfer of excitations from one BChl site to\n    another.]{\\includegraphics{TrEsp_popln_contour.pdf}}\n\n    \\caption{Excited states of the bacteriochlorophyll corresponding to the\n    B850 ring with the TrEsp couplings.}\\label{fig:popln_kleinekathoefer}\n\\end{figure}\n\\begin{figure}\n    \\subfloat[Excited state populations of each of the BChl sites.]{\\includegraphics{Exp_popln.pdf}}\n\n    \\subfloat[2D plot of the transfer of excitations from one site to\n    another.]{\\includegraphics{Exp_popln_contour.pdf}}\n\n    \\caption{Excited states of the bacteriochlorophyll corresponding to the B850\n    ring with the experimentally derived couplings.}\\label{fig:popln_experiment}\n\\end{figure}\n\nThe dynamics of the dimerized ring of identical chlorophyll molecules displays\nan interesting symmetry. Consider all pairs of BChl units equidistant from the\ninitially excited one, which in this case is the 8\\textsuperscript{th} unit.\nIf the electronic couplings between all nearest neighbor pairs were equal, the\ndynamics of the monomers of any pair would have been identical. However,\nbecause of the alternating nature of these couplings, such a symmetry would be\nabsent. Interestingly, this situation leads to a different symmetry. Now it is\nevery alternate pair that has identical dynamics and the other pairs have\ndifferent dynamics. Of course, because the number of units in this case is\neven, the unit diametrically opposite to the initially excited unit, the\n8\\textsuperscript{th} BChl unit in this case, is unique.\n\nNow, let us compare the dynamics of the dimerized\nBChl ring with the TrEsp couplings with a BChl ring with all the couplings set\nat the average value of the TrEsp couplings. Figure~\\ref{fig:popln_comp} shows\nthe dynamics corresponding to the non-dimerized (with average couplings,\nFig.~\\ref{fig:popln_comp}~(a)) and dimerized (alternating couplings,\nFig.~\\ref{fig:popln_comp}~(b)) ring. Note that in\nFig.~\\ref{fig:popln_comp}~(a), all the lines but the ones corresponding to 8\nand 16 are paired. For example, the dynamics corresponding to the\n7\\textsuperscript{th} and the 9\\textsuperscript{th} sites are identical (they\nhave the same colors in the figure, but different line styles, so it seems\nlike there is only one single line). The same applies to the dynamics of the\n6\\textsuperscript{th} and the 10\\textsuperscript{th} sites. However, as\ndiscussed, this is not the case in Fig.~\\ref{fig:popln_comp}~(b). For this\ncase, the dynamics of site 7 and site 9 are different, as is the dynamics of\nsite 5 and 11. However the dynamics of site 6 is same as that of site 10, as\nis the dynamics of sites 4 and 12.\n\n\\begin{figure}\n    \\subfloat[BChl ring with average intermonomer electronic couplings.\n    Excited state population of site 7 is same as that of site 9, site 6 is\n    equal to site 10, so on.]{\\includegraphics{TrEsp_short_time_avg.pdf}}\n\n    \\subfloat[Dimerized BChl ring with electronic couplings obtained using\n    TrEsp. Data is the same as Fig.~\\ref{fig:popln_kleinekathoefer} but shown\n    for a shorter time duration.]{\\includegraphics{TrEsp_short_time.pdf}}\n\n    \\caption{Dynamics for BChl rings with identical and alternating electronic\n        couplings.}\\label{fig:popln_comp}\n\\end{figure}\n\nTo further our understanding, consider the dynamics corresponding to a more\ninvolved initial state. Till now, we have discussed the dynamics following an\nexcitation of only a single site, the 8\\textsuperscript{th} site in our case.\nLet us assume that the initial density is defined by $\\tilde\\rho(0) =\n0.5\\dyad{e_8} + 0.25\\dyad{e_7} + 0.25\\dyad{e_9}$. The system is, therefore,\ninitially in a statistical ensemble with the 7\\textsuperscript{th},\n8\\textsuperscript{th} and 9\\textsuperscript{th} sites getting excited with\ndifferent probabilities. The dynamics is shown in Fig.~\\ref{fig:mixed_state}.\nThe coupling between the 7\\textsuperscript{th} and 8\\textsuperscript{th} BChl\nunits is higher than that between the 8\\textsuperscript{th} and\n9\\textsuperscript{th}. This leads to a transient build-up of excitonic\npopulation in the 7\\textsuperscript{th} site at around $\\SI{30}{\\fs}$, while\nthe population of the 9\\textsuperscript{th} site shows a more or less monotonic\ndecay. Not only does the 9\\textsuperscript{th} BChl unit receive population\nfrom the 8\\textsuperscript{th} unit slowly, it also quickly leaks population\ninto the 10\\textsuperscript{th} unit which is completely in the ground state\nbecause of a high electronic coupling.\n\n\\begin{figure}\n    \\includegraphics{TrEsp_mult_exc.pdf}\n    \\caption{Dynamics starting from $\\tilde\\rho(0) = 0.5\\dyad{e_8} + 0.25\\dyad{e_7} + 0.25\\dyad{e_9}$ using the TrEsp parameters.}\\label{fig:mixed_state}\n\\end{figure}\n\nA major consideration in multisite systems is the entanglement between the\nindividual sites. Here, we use the average bond dimension of the reduced\ndensity MPS as a measure of the entanglement. It is intuitively quite clear\nthat the presence of the bath should change the growth of this bond dimension\nand consequently the entanglement between the sites. We have\nshown~\\cite{boseMultisiteDecompositionTensor2022} that in the case of the Ising\nmodel, the coupling to the local baths severely restricts the growth of the\naverage bond dimension. In Fig.~\\ref{fig:entanglement}, we show the growth of\nthe average bond dimension for the B850 system both with and without the\npresence of the vibrational baths. It is surprising that in this case, the\naverage bond dimension, and consequently the intersite entanglement, of the\nbare B850 system does not really grow and is very small. Additionally, it is\nthe incorporation of the vibrational bath that leads to an increase in the bond\ndimension. Though it must be noted that the bond dimension despite being\ngreater in presence of the bath, is still quite small. This reversal of\npatterns vis-\\`a-vis the Ising model is probably unique to the Frenkel model\nand might be because of the block diagonal structure of the Hamiltonian. It\nmight also arise as a consequence of the nature of the quantum transport\nprocess.\n\n\\begin{figure}\n    \\includegraphics{bonddims.pdf}\n    \\caption{Average bond dimensions of the reduced density MPS for the B850\n    ring as a measure of average intersite entanglement in presence and absence\nof the vibrational bath.}\\label{fig:entanglement}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics{correlation_function}\n    \\caption{Correlation function after factorizing out the average excitation energy according to Eq.~\\ref{eq:factor_corr}.}\\label{fig:corr}\n\\end{figure}\n\nPopulations do not give a full account of the dynamics as explored through\nexperiments. We consider the absorption spectra corresponding to the two\ndifferent couplings. As a zeroth order approximation, we first consider the\ndipole moment vectors to point in the opposite directions for neighboring\nmonomers. The convergence parameters for these simulations are mostly the same as\nthose used for the populations. The most notable difference being the memory\nspan which is $L\\Delta t = \\SI{33.86}{\\fs}$ here. The correct dipole moment\nvectors are tangential to the B850 ring but are oriented in opposite\nmanners~\\cite{Hu1997a, strumpferLightHarvestingComplexII2009}. The mean optical\nexcitation energy is taken to be\n\\SI{12098}{\\per\\cm}~\\cite{olbrichTimeDependentAtomisticView2010}. First, we\nconsider the spectrum corresponding to the TrEsp couplings. The dipole moment\nautocorrelation function without the high-frequency oscillations is\ndemonstrated in Fig.~\\ref{fig:corr}. The spectra with the antiparallel dipole\nmoments and the correct dipole moments are shown in\nFig.~\\ref{fig:const_exc_spectrum}. (A representative converged spectrum\ncalculation takes roughly 2 hours on an Intel\\textregistered{}\nXeon\\textregistered{} Gold CPU.) Along with the numerically exact MS-TNPI\nresults, we report an approximate spectrum calculated within the second-order\ncumulant approximation~\\cite{damjanoviciExcitonsinPhotosynthetic2002}.\n\n\\begin{figure}\n    \\includegraphics{const_exc_spectrum.pdf}\n    \\caption{Absorption spectrum corresponding to the TrEsp couplings and constant excitation energies. Black dashed line: antiparallel dipole moments. Black solid line: correct dipole moments. Red solid line: second-order cumulant approximation.}\\label{fig:const_exc_spectrum}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics[scale=0.5]{exciton_eigenstates.png}\n    \\caption{Structure and spacings of the energies of the excitons for a ring with alternating couplings.}\\label{fig:exciton}\n\\end{figure}  \n\nThe analysis of the absorption spectrum in terms of the Frenkel excitons is\nwell-understood. In presence of a system with cylindrical symmetry, there are\ntwo bands of excitons as schematically demonstrated in\nFig.~\\ref{fig:exciton}~\\cite{Hu1997a}. The exact energies of the excitonic\neigenstates corresponding to the different system parameters is given in\nAppendix~\\ref{app:exciton}. The degenerate states are labeled as $\\ket{\\pm k}$.\nThe gap between the two bands is approximately $2|V_1 - V_2|$, where $V_1$ and\n$V_2$ are the two electronic couplings. Therefore, if there is a constant\ncoupling, the states $\\ket{4}$ and $\\ket{5}$ would be degenerate as\nwell~\\cite{strumpferLightHarvestingComplexII2009}. For the case where the dipole\nmoments are antiparallel, the lowest energy exciton, $\\ket{0}$, gets excited.\nHowever, it is well-known that the correct dipole moment actually excites into\nthe degenerate states of $\\ket{\\pm\n1}$~\\cite{Hu1997a,strumpferLightHarvestingComplexII2009}. Therefore, as shown in\nFig.~\\ref{fig:const_exc_spectrum}, we expect to see a small blue-shift of the\ncentral frequency of the peak corresponding to the correct dipole moments\nvis-\\`a-vis the antiparallel ones. The excitons are coupled to each other\nthrough interactions with the site-local baths. Thus, the peak is slightly\nshifted and significantly broadened in the presence of the dissipative medium.\nThe second-order cumulant approximation spectrum is quite red-shifted with\nrespect to the correct dipole moment MS-TNPI spectrum.\n\nNow, it is well-known that the electrostatics of the photosynthetic complex\noften induces a change in the excitation energies of the monomers. Typically,\nthe excitation energy alternates with a difference of around\n\\SI{197}{\\per\\cm}~\\cite{strumpferLightHarvestingComplexII2009}. We have also\nsimulated and plotted the absorption spectra corresponding to this case in\nFig.~\\ref{fig:diff_exc_spectrum}. The incorporation of this asymmetry in the\nexcitation energy gives rise to a smaller peak close to \\SI{12300}{\\per\\cm}\nwhen using the antiparallel dipole moments. This secondary peak is caused by\nexcitations into the highest energy exciton, $\\ket{9}$, which is now permitted\nby the symmetry. For both the correct and antiparallel dipole moments, the main\npeak in the spectrum is red-shifted in comparison to the case where the\nmonomers have the same excitation energy, due to a change in the eigenvalue\nspectrum. Also, it is interesting that the agreement with the approximate\nspectrum is much better when the varying excitation energies are incorporated.\n\n\\begin{figure}\n    \\includegraphics{diff_exc_spectrum.pdf}\n    \\caption{Absorption spectrum corresponding to the TrEsp couplings and varying excitation energies. Black dashed line: antiparallel dipole moments. Black solid line: correct dipole moments. Red solid line: second-order cumulant approximation.}\\label{fig:diff_exc_spectrum}\n\\end{figure}\n\n\\begin{figure}\n    \\includegraphics{spectrum_diff_params.pdf}\n    \\caption{Absorption spectra for TrEsp (red) and experimentally derived parameters (black) with correct dipole moments and asymmetry present. Dashed line: second-order cumulant approximation. Solid line: MS-TNPI spectra.}\\label{fig:spectrum_exp}\n\\end{figure}\n\nAs we had mentioned earlier, the electronic couplings obtained via TrEsp are\nsignificantly smaller than the ones derived from experiments. The same holds\nfor the mean optical excitation energy. We calculate the absorption spectrum\nfor the B850 ring with a mean optical excitation energy of\n$\\SI{12390}{\\per\\cm}$ and couplings of \\SIlist{315;245}{\\per\\cm} as reported\nby~\\citet{freibergExcitedStateDynamics2009}. The difference between the\nexcitation energies of consecutive chlorophyll units is once again taken to be\n$\\SI{197}{\\per\\cm}$. The comparison of the spectrum corresponding to these\nexperimental parameters with the one corresponding to the TrEsp parameters is\npresented in Fig.~\\ref{fig:spectrum_exp}. Clearly, the spectrum corresponding\nto the experimentally derived parameters is significantly blue-shifted with\nrespect to the TrEsp parameters. The peak at $\\SI{11763}{\\per\\cm}$ corresponds\nto $\\SI{850.12}{\\nm}$. This is in comparison to the TrEsp parameter peak at\n$\\SI{11666}{\\per\\cm}$ or $\\SI{857.19}{\\nm}$. Because of the higher electronic\ncouplings, the damping effect of the bath is less pronounced leading to a\nsignificantly sharper peak. It is interesting that in contrast to the MS-TNPI\nspectra, the peak widths of the two approximate spectra are quite similar to\neach other. The central frequencies of the approximate peaks have a difference\nof roughly $\\SI{78}{\\per\\cm}$ between them, which is smaller than the\ndifference between the MS-TNPI peaks. The peak of the true experimental\nspectrum~\\cite{olbrichTimeDependentAtomisticView2010} corresponding B850 region\nis red-shifted with respect to the TrEsp peak. This means that the agreement of\nthe peak corresponding to the higher excitation energy and couplings with the\nexperiment is worse than the agreement of the TrEsp peaks. Thus the effect of\nshielding, that leads to the smaller coupling values, are quite important.\n\n\\begin{figure}\n    \\includegraphics{spect_temp_comp.pdf}\n    \\caption{Comparison of spectra at \\SIlist{300;150;77}{\\kelvin} for the TrEsp parameters with varying excitation energies. Effects of temperature on ``solvent'' spectral density has been ignored for simplicity. Solid lines: MS-TNPI spectra. Dashed lines: second-order cumulant approximation.}\\label{fig:spect_temp_comp}\n\\end{figure}\n\nFinally, we explore how the approximate and exact absorption spectra change\nwith temperature. It needs to be noted that this exploration has a caveat. The\nspectral density coming from the protein and the rigid molecular vibrations is\nnot necessarily independent of the temperature. The validity of the spectral\ndensity across various temperature regimes would need to be verified on a\ncase-by-case basis. However, for simplicity, we would keep using the same\nspectral density across the temperatures. The spectra are calculated for the\nTrEsp parameters with alternating excitation energies at\n\\SIlist{300;150;77}{\\kelvin}. The comparison is shown in\nFig.~\\ref{fig:spect_temp_comp}. We see that as the temperature decreases, the\npeak of the exact spectrum gets sharper and shows a blue shift. This is\nconsistent with observations reported by~\\citet{chenOpticalLineShapes2009},\nthough, the magnitude of the blue shift, which is dependent on the exact\nparameters and spectral density, is much smaller their case. The approximate\nspectrum peaks, on the other hand, shows a red shift while getting sharper as\ntemperature decreases. This suggests that further investigation into various\napproximations for the spectra can give interesting insights into both the\nsystems under study and the nature of the approximations.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nUnderstanding electronic energy transfer processes is important. However,\nstudying the dynamics of extended systems with dissipative media is an\nextraordinarily challenging problem. System-solvent decomposition is a commonly\nused technique to accurately simulate open quantum systems. It handles the\nexponential scaling of quantum mechanics by limiting it to a small dimensional\nsubspace. However, with such extended systems, the exponential scaling of\nquantum mechanics is not sufficiently curbed to allow for efficient numerical\nsimulations. We have recently introduced MS-TNPI to address this problem using\na density matrix renormalization group-like decomposition along with\nFeynman-Vernon influence functional. Here, we use it to study EET in a B850\nring of LH2 with vibrational spectral densities obtained using molecular\ndynamics. Previous numerically exact studies of the dynamics of such systems\nhave typically been done with the Drude-Lorentz model spectral density.\n\nIn this paper, we have shown how MS-TNPI can be simply extended to account for\nthe ring structure that is almost ubiquitous in photosynthetic complexes in\npurple bacteria. MS-TNPI can efficiently simulate these systems as well. While\nwe use a ``flat'' 2D structure for simulating the ring system, it is\nconceivable that having the 2D structure turned into cylindrical form,\nreflecting the actual topology of the system, might bring additional\ncomputational benefits. Such ideas would be explored in the future. We have\nalso analyzed and massaged the expressions for the absorption spectrum to make\nit fit for MS-TNPI. Taking advantage of the availability of the full many-body\nreduced density matrix for the extended system, MS-TNPI can efficiently\nsimulate the required correlation functions and higher order response\nfunctions.\n\nWe have shown the impact of the different parameters on the direct EET\ndynamics in the B850 ring. The TrEsp couplings with the ZINDO\/S-CIS excitation\nenergies are generally much smaller than typical experimentally derived values.\nThe dynamics corresponding to both cases have been simulated. The bath has similar\neffects on both the parameters. However, owing to the faster oscillations\ncorresponding to the experimentally derived values, the oscillatory nature\npropagates even to the most distant BChl units before getting washed away.\nAdditionally, subtle effects stemming from the unequal electronic couplings get\namplified when using a more complex initial condition where multiple BChl units\nare statistically excited. Future work would focus on studying the impact of\nlight on B850, taking into consideration effects stemming from the varying\nalignments of the site-local dipole vectors and spatial inhomogeneity of the\nlight-BChl interaction.\n\nAdditionally, we have simulated the absorption spectrum, incorporating the full\nspectral density, for the ring using various approximations culminating in a\nsimulation with the most appropriate parameters. The B850 ring is characterized\nby non-parallel transition dipole moments and unequal monomer excitation\nenergies. To better understand the impact of these transition dipole moments,\nwe started with a very simple zeroth-order approximation where the transition\ndipole moments are anti-parallel. We show that consistent with excitonic wave\nfunction-based analysis, if the electronic excitation energies are identical,\nthere is only a single peak. However, the inhomogeneities induced by the local\nelectrostatic environment lead to a secondary peak that comes from excitation\ninto the highest energy excitonic level. Subsequently, we analyse the effect of\nusing the transition dipole moment with the correct form. Incorporation of the\ncorrect dipole moment operator along with the varying excitation energies still\nproduces a spectrum that is blue-shifted with respect to experimental\nspectra~\\cite{olbrichTimeDependentAtomisticView2010}. This is probably due to\ninaccurate excitation energies, electronic couplings, and limitations of the\nmodel. These calculations are compared with second order cumulant\napproximation~\\cite{damjanoviciExcitonsinPhotosynthetic2002}. The approximate\nspectra are consistently broader than the MS-TNPI calculations, and generally\nslightly red-shifted. It is interesting to note that the effect of the\ntemperature on the approximate spectra is qualitatively different from that on\nthe exact spectra. While on decreasing the temperature, the exact spectrum\nshifts to higher frequencies, the approximate spectrum shifts to lower\nfrequencies. Both the types of simulations show the sharpening of the\nabsorption spectrum at lower temperatures. Consequences of adding static\ndisorder to the Hamiltonian can be trivially incorporated in the MS-TNPI\nprocedure through an external Monte Carlo averaging of separate MS-TNPI runs. A\ndetailed exploration of such effects would be the topic of a future\nexploration.\n\nMS-TNPI does not restrict the simulation to the first excitation\nsubspace as many other methods do. These ``full space'' simulations, however,\nstill remain quite simple. The singular value decompositions involved seem to\nbe able to filter out the unnecessary information and lead to very compact\nrepresentations. We simulate the dynamics corresponding to a local excitation,\nand show that the entanglement between the sites as calculated by the average\nbond dimension does not grow exponentially with time, even in the absence of\nthe bath. This is unlike what happens in say the Ising\nmodel~\\cite{boseMultisiteDecompositionTensor2022}, and is probably due to the\nsparsity and the block-diagonal structure of the Frenkel Hamiltonian. It is\nalso interesting that unlike the case of the Ising model where the presence of\nthe bath controls the entanglement between different\nsites~\\cite{boseMultisiteDecompositionTensor2022}, in the case of the\nFrenkel-Holstein model, the presence of the bath actually serves to slightly\nincrease the entanglement. This deserves further study.\n\nWhile here we have explored the dynamics and the absorption spectra, other\nexperimentally realizable observables can be also be simulated with similar\nconceptual simplicity. Further investigation of other observables, especially\nmulti-time correlation functions and longer ranged interactions will be the\nfocus of future work. Incorporation of long ranged interactions, through more\nadvanced propagators (e.g., $W$\\textsuperscript{I,II} or TDVP based propagators), would be\nimportant in capturing dipole-dipole interactions between distant monomers in\nthe Hamiltonian. MS-TNPI provides a flexible scheme\nfor incorporation of increasingly complex Hamiltonians and effects of baths in\na unified framework, making it a lucrative method for studying quantum\ntransport in extended quantum systems.\n\n\\section*{Acknowledgments}\nA.\\,B. acknowledges the support of the Computational Chemical Science Center:\nChemistry in Solution and at Interfaces funded by the US Department of Energy\nunder Award No. DE-SC0019394. P.\\,W. acknowledges the Miller Institute for\nBasic Research in Science for funding.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}