Tensor renormalization group methods for spin and gauge models

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 2014 Tensor renormalization group methods for spin and gauge models Haiyuan Zou University of Iowa Copyright 2014 Haiyuan Zou This dissertation is available at Iowa Research Online: Recommended Citation Zou, Haiyuan. "Tensor renormalization group methods for spin and gauge models." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 TENSOR RENORMALIZATION GROUP METHODS FOR SPIN AND GAUGE MODELS by Haiyuan Zou A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa August 2014 Thesis Supervisor: Professor Yannick Meurice

3 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Haiyuan Zou has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the August 2014 graduation. Thesis Committee: Yannick Meurice, Thesis Supervisor Michael Ogilvie Vincent Rodgers Wayne Polyzou Maxim Khodas

4 To my parents. ii

5 ACKNOWLEDGEMENTS First of all, I would like to sincerely thank my PhD advisor, Professor Yannick Meurice, for his guidance, encouragement, and tremendous support and help throughout my graduate career. During the last six years, Yannick has been encouraging and helping me in my study and research. His great kindness and patience helped me to pass over the times of discouragement and frustration. I learned a lot from his problem solving skills and profound knowledge. Yannick encouraged me to try different projects to help me get a broad view of physics. In all, This work would never have been accomplished without Yannick s meticulous guidance and continuous support. I would also like to gratefully acknowledge Professor Tao Xiang and Dr. Zhiyuan Xie for teaching me tensor networks algorithm. The extensive discussion with them greatly broadened my knowledge of both methods and models. I also want to thank Professor Vincent Rodgers, Professor Wayne Polyzou, Professor Maxim Khodas, Professor Michael Ogilvie, Professor John Schweitzer, and Professor Mary Hall Reno for their help and valuable suggestions. Many thanks to my colleagues Dr. Daping Du, Dr. Yuzhi Liu, Dr. Alan Denbleyker, Dr. Alexei Bazavov, Judah Unmuth-Yockey, Zechariah Gelzer and other friends who always gave me support. Finally, I would like to give my special thanks to my fiancee Li Teng and my parents. Li has been supportive of me ever since we met with each other. Without her effort in it, I could never make it finished. I m grateful to my parents for their encouragement and support. iii

6 ABSTRACT In this thesis, we study the physics and critical behavior of several statistical models, e.g. O(N) model and lattice gauge models by using different approach, which including the conventional perturbative expansion and later and mostly, tensor renormalization group methods. We first start from the exact solvable model, 1-d O(2) model with open boundary conditions (OBC) and periodic boundary conditions (PBC) on a finite lattice and take the perturbative approach. We discuss the error of perturbative series by comparing it to the exact solution of the partition function and the average energy. The error (nonperturbative part) for both boundary conditions can be parametrized as Aβ B e Cβ but with different coefficients. For PBC, the error comes from the vortices solutions. We calculate the weak coupling expansion for finite PBC systems up to term 12 by using a new method and modify the ordinary perturbative series of the 1-link model. We also compare the small E expansion of the density of states with numerical values. We search for the Fisher s zeros for a system with PBC and construct Migdal-Kadanoff flows for the OBC system. We conclude that the Fisher s zeros control the geometrical properties of the flows. Motivated by recent attempts to find nontrivial infrared fixed points in 4- dimensional lattice gauge theories, we discuss 2D nonlinear O(N) sigma models on a finite lattice, in the large-n limit. We explain the Riemann sheet structure and singular points of the finite L mappings between the mass gap and the t Hooft iv

7 coupling. We calculated the complex zeros of the partition function and constructed two types of renormalization group (RG) flow in the complex coupling plane. The results also provide evidence to a general statement that the complex conjugated pair of zeros closest to the real axis can be seen as a gate controlling the complex flows. Both the perturbative approach and the conventional analytical method to understand the phase transition properties are difficult to be extended to higher dimensional models. We apply tensor renormalization group method to different models. Numerically, for models with complex coupling or with chemical potential, a computational issue called sign problem preventing direct MC simulations appears. However, the Tensor Renormalization Group (TRG) method is insensitive to the problem. By using one tensor renormalization group method named Higher-Order tensor renormalization group, the partition function can be calculated accurately even at large imaginary coupling. We then search for the zeros of the partition function for both the 2D Ising model (as a comparison) and 2D classical XY model. From the finite size scaling of the lowest zeros (with smallest imaginary part) for XY model, we did different fits to show that the zeros pinch the real axis convexly as the volume increases. The result agrees with the Kesterlitz-Thouless transition assumption and the locations of zeros for larger volume can be predicted. We use character expansion to construct exact blocking formulas for some spin and gauge models in the tensor language, e.g. O(N) models, Z 2 gauge model, and U(1) lattice gauge models. But using gauge fixing and applying a trick to symmetrize the RG procedure, we obtain reasonable numerical result for gauge models. We work on the 2D classical XY model v

8 with chemical potential, another model with sign problem. By using TRG method, sign problem can be solved and the phase diagram of the model can be obtained. By a transformation of coordinates the phase diagram looks very similar to the phase diagram of the Bose-Hubbard model. By the analysis of the structure of the local tensor, we have a conjecture connecting the behavior of the tensor with the phase transition properties of spin and Abelian gauge models. vi

9 TABLE OF CONTENTS LIST OF TABLES ix LIST OF FIGURES x CHAPTER 1 INTRODUCTION Motivation and Overview TRG D O(2) MODEL Classical Solutions perturvative method Density of States, Fisher s zeros, and RG flow Monte Carlo Simulation of Configurations D NONLIEAR O(N) SIGMA MODELS The model The Gap Equation Fisher s zeros Complex RG flows Rescaling Two-lattice matching TENSOR RENORMALIZATION GROUP Sign problems D complex Ising Model D complex O(2) Model O(2) with a chemical potential New phase diagram Comparison with the Worm algorithm The 3D case D O(3) model Gauge Models D Z 2 gauge U(1) lattice gauge vii

10 D = D = D = Symmetric method CONCLUSION APPENDIX A COMPARISON OF IMPROVED PERTURBATIVE METHODS A.1 Two different methods A.1.1 The Large Field Cutoff Method A.1.2 The PPS Method A.2 Comparison of Partition Functions A.3 Logarithms of Partition Functions A.4 Results for the Anharmonic Oscillator REFERENCES viii

11 LIST OF TABLES Table 2.1 coefficients a n L and 2q b solutions The lowest zeros with D s = 16, 20, and The lowest zeros with D s = 40 and Different fitting for L = 4, 8...L max by fixing ν = 1/ Solutions of N ix

12 LIST OF FIGURES Figure 1.1 One HOTRG step.(a) A HOTRG contraction of the state along the y axis. (b) Steps of contraction and recombination. Figure is from Ref. [83] Error graphs. OBC (Left): Errors of the average energy series with order 2, 4,..., 20 (Blue), PBC with L = 36 (Right): Errors of the average energy series with order 2, 4,..., 12 (Blue). Red lines correspond to the results in Eq. (2.17) Errors of different series to order 2, 4,..., 20. Black: Hadamard series; Blue: modified Hadamard series (n = 10); Red: modified Hadamard series (n = 20) Small E expansion with order 4, 8,..., 20 and the numerical data n(e) Zeros of the partition function (PBC) with different volumes and zeros of the partition function (OBC) MK complex flows for the system (OBC) and zeros from the two different boundary conditions Final configurations after sweeps. Top: Starting from a random configuration for PBC; Middle: Starting from an ordered configuration for PBC; Bottom: Startinf from a random configuration for OBC Relation between L and 2q for D = Zeros, poles and singular points of b(m 2 ) in the M 2 plane. Left: for 4 4; Right: 8 8 lattices The blending small crosses (x, blue online) are the b images of two lines of points located very close above and below the [ 8, 0] cut in infinite volume; the crosses (+) are the images of the singular points for L = 32. The images of the 4 closest singular points appear as boxes Singular points and cuts. Left: In the b plane for L = 4; Right: The mapped images in the M 2 plane x

13 3.5 Zeros of partition function for L = 6, N = 2 (boxes), and images of the singular points of b(m 2 ) (crosses). The images of the solutions f = 0 are given with the third symbol Number of zeros in a fixed region of the b-plane. Left: fixing N = 2, L is a variable. Right: fixing L = 2, N is a variable Density of state function for L = 4 and N = Singular points, zeros of partition function, and Re(f ) = 0 in E plane for L = 2,N = 4. The line (red online), corresponds to Re(f ) = 0 with Re(f ) > 0 above it RG flows. Top: RG flows by rescaling and image of the cut at infinite volume; Bottom: Flows by rescaling, singular points, CSPs, zeros of partition functions (N = 2), and f = 0 (N = 2) in b plane for 6 6 lattice system RG flows for the 2-lattice matching between 8 8 and 4 4 lattices. Circles and triangles are the singular points for L = 4 and L = b versus b. Left: results from rescaling; Right: results from 2-lattice matching Fisher curves and the zeros at finite volume for a L = 8 system. Complex β in region 1, 2, and 3 are displayed Left: The real part of the normalized partition function for region 2, result from the HOTRG with D s = 30, MC, and the exact solution. Right: relative error of the real part of free energy from HOTRG calculation, minimum at Imβ = π/ Left: The real part of the normalized partition function for region 3, result from HOTRG with D s = 30, MC, and the exact solution. Right: relative error of the real part of free energy from HOTRG calculation, minimum of the error curve at Imβ = π/4 approximately The real part of the normalized partition function for β near the Fisher zero i (the big filled circle on the real axis): result from the HOTRG with D s = 10, 20, and 40 (D s = 30 result is not shown as it is close to the D s = 40 case), MC, and exact solution The relative error of the real part of free energy for HOTRG calculation with D s = 10, 20, 30, and 40. Vertical line corresponds to the lowest zero. 70 xi

14 4.6 The distributions of normalized singular values λ/λ 1 for β 0 in region 1, 2, and 3, with D s = 40. There are 1600 singular values for each case Zeros of Real () and Imaginary ( ) part of the partition function of Ising model at the volume 8 8 from the HOTRG calculation with D s = 40 are on the exact solution lines. Gray dots: MC reweighting solution. Thick Black curve: the region of confidence for the MC reweighting result, above this line, the MC error is large Fisher s zeros of the XY model with length L = 4, 8, 16, 32, 64, and 128 (from up-left to down-right) at different D s. For L = 128, only D s = 40 and 50 are shown Zeros of the XY model with linear size L = 4, 8, 16, 32, 64, and 128 (from up-left to down-right) calculated from HOTRG with D s = 40, and 50 and zeros with L = 4, 8, 16, and 32 from MC. The curve is a model for trajectory of the lowest zeros The second normalized eigenvalues (λ 2 /λ 1 ) of the transfer matrix and the particle number density N at β = 0.06 from HOTRG calculation with the number of states D st = 15 are shown for the 1+1 D O(2) model. As a comparison, the particle number density N 3 and the second normalized eigenvalues (λ 2 /λ 1 ) 3 for the spin-1 projection (3-states) is also shown Phase diagrams. Top: The phase diagram for the 1+1 D O(2) model in the β-µ plane. Bottom: The phase diagram in the β-βe µ /2 plane. In both graphs, the lines labeled by 3s stand for the phase separation lines of the spin-1 (3-states) system Phase diagram for the 1+1 D O(2) model at β t = 10 in β e -µ e plane is shown. The lines labeled by 3s stand for the phase separation lines of the spin-1 (3-state) system Phase diagram for the 2+1 D O(2) model tensor for O(3) model Average energy by TRG, weak and strong expansion, and MC. The MC data are from Ref. [9] A tensor and B tensors A new basic cell. Left: Cell in an original cube. Right: The equivalent T 6 tensor, its center is (1/4, 3/4, 3/4) in the original cube xii

15 4.18 Highlight of the basic cell with labels The new T 6 tensor with axial gauge T 4 tensor which contains two A tensors and one B tensor D U(1) tensor in a hypercube red (online) dots: A tensors. blue (online) dots: B tensors normal HOTRG method in 2D new HOTRG method in 1D D Z 2 gauge model tensor D Z 2 gauge model results. Left: Small β expansion. Right: The second eigenvalue D Z 2 gauge model tensor A.1 Number of significant digits for the simple integral obtained from field cutoffs with φ max = 2 (blue), φ max = 3 (red), φ max = 4 (green) A.2 Error graphs. Left: Number of significant digits obtained from the PPS modification for the simple integral with regulator N = 2 (blue), N = 3 (red), N = 4 (green). Right: Asymptotic value (black) for the PPS method at large λ: N = 2 (blue), N = 3 (red), N = 4 (green) A.3 Number of significant digits for the simple integral obtained with a field cutoff with φ max = 2 at order 7, 11, 15, as a function of λ (red lines with the blue asymptotic line increasing as λ increases), from the PPS method with N = 2 at the same orders (red lines with the blue asymptotic line decreasing as λ increases), and from regular perturbative method at the same orders (black) A.4 Error graphs for the anharmonic oscillator. Black: regular perturbation theory at order 15. Blue line (decreasing when λ increases): the PPS method at regular N = 4. Green: the PPS method at order 15. Blue line (increasing when λ increases): cutoff method for a cut = 2. Red: cutoff method at order 9,13,17,21, xiii

16 1 CHAPTER 1 INTRODUCTION 1.1 Motivation and Overview From the last century, Physicists are getting closer to the success in understanding and discovering the big pictures that how nature behaves from different energy and size scales. For example, on the high energy and fundamental particle scales, particles are described by the standard model, which has been checked by benchmark experiments [16, 17, 14, 15] in the last 30 years. On the lower energy and condensed matter scales, Landau s symmetry-breaking and Fermi liquid theory can describe lots of phenomena in many-body systems. However, many novel and exciting phenomena beyond Landau s theories of strongly-correlated systems have been discovered, like the fractional quantum Hall effect [77], high-t c superconductivity [7]. Theoretically, the analogy between statistical mechanics and quantum field theory makes the developments in condensed matter theory and high-energy physics influence upon each other. Practically, to understand the new phenomena, people study simplified models which mimic the real physics. Many of the models are on a lattice. However, not all of the models can be exactly solved. One needs to find reliable numerical methods, For example, Quantum Monte Carlo (QMC) [76], Density Matrix Renormalization Group (DMRG) [81], Tensor Renormalization Group (TRG) [51] and so on. In this thesis, we focus on the specially chosen models with the most fundamental symmetries listed by Polyakov [73] or these models with additional terms. For the model

17 2 cannot be exactly solved, we discuss the application of one tensor network (or TRG) method, named Higher Order Tensor Renormalization Group (HOTRG) method [83] to these models in Chapter 4. Discrete Global Symmetries: The simplest example is Ising models with Z 2 symmetry. In 2D, the exact solution was provided by Onsager [68] in the thermodynamic limit and by Kaufman [42] on finite lattices. The Ising model becomes the test-bed for all numerical methods in the first step. With the local order parameter σ, the Ising model is the simplest example of Landau s phase transition theory, With the property of self-duality, the high and low temperature expansions of the model can be related with each other [49]. In 3D, although there is no exact solution so far, people have accurates result from different numerical methods. For example, right now Tensor networks method provides very accurate result for the critical temperature so far [83]. In section 4.1.1, To test the TRG method to the case with sign problem, we apply the TRG method to 2D ising model on a finite lattice with complex coupling and find that the numerical results agree with Kaufman s exact solution [20]. In general, models with more complicated discrete groups, like Z n, are also interesting because of their rich phase transition properties [24]. For n 4, there is only one critical temperature of second order phase transition, which is Ising-like. For n 5, there is one phase in between the known ferromagnetic phase and paramagnetic phase. As n goes to the infinite limit, the intermediate

18 3 phase takes over the ferromagnetic phase at large β (O(2) limit). Continuum Abelian Global Symmetries: An interesting example in this category is the classical O(2) models. for the 1D case, it s exactly solvable. In Chapter 2, we present a new method to obtain the higher order weak coupling expansion. By looking the difference between the expansion and the exact solution, we can relate the missing part with classical solutions. In 2D, it has a nontrivial infinite order phase transition, the Kosterlitz-Thouless (KT) transition, governed by the combination and separation of vortices and anti-vortices [48, 47]. The property of the transition can be understood by renormalization group analysis [48]. In Section 4.1.2, We calculate the Fisher s zero by using the HOTRG method. The scaling of the zeros in finite volume agrees well with the KT transition assumption [20]. The same model with a chemical potential µ is also interesting because it has a sign problem and it can be mapped into a Bose-Hubbard model [87]. In Section 4.2, By using the HOTRG method, we get the phase diagram of the model in both 2D and 3D. Non-Abelian Global Symmetries: In 2D O(N) models, N > 2, due to the strong interactions of the Goldstone bosons, there is no phase transition. In the large N limit, the model shares the same properties as the QCD model with fermions. In Chapter 3, we try to solve the large-n limit case in finite volume [62], we calculate the Fisher s zero and construct the RG flows. From the structure of Fisher s zeros and flow, we find

19 4 that Fisher s zeros form the boundary of RG flows [19]. We also have the tensor formulation in section 4.3 and the numerical solution is comparable with the Monte Carlo (MC) calculation for N = 3. Discrete and Continuum Gauge Symmetries: The simplest case of discrete gauge symmetry is Z 2 gauge model. In 2D, it is trivial like 1D Ising model. In 3D, the model is dual to 3D Ising model. It has a phase transition but without local order parameter. For the Continuum gauge symmetry case, one interesting example is the U(1) lattice gauge model. In 3D, it has no phase transition. In 4D, it presumably has a first order phase transition. In section 4.4, we formulate the gauge models in different ways and discuss the numerical results. 1.2 TRG Recently the tensor network states idea has become increasingly powerful in the analysis of strongly correlated quantum systems. The development of the techniques also provide a bridge for physicists in different areas. Historically, the development of the tensor network states is starting from White s DMRG [81] (which is 1D matrix-product state (MPS) in the tensor language [69]) for 1D quantum system. Then, Projected entangled pair states (PEPS) provide a generalization of MPS to higher dimensions [78]. After that, different or generalized wave-function ansatz are developed, for example, Multiscale Entanglement Renormalization Ansatz (MERA) [79], Projected Entangled Simplex States (PESS) [82] and so on. The classical version

20 5 of tensor network states from transfer matrices method [65], tensor renormalization group (TRG) [51], second renormalization group (SRG) [84] till recent TRG method based on the higher-order singular value decomposition (HOTRG) [83] can be effectively applied to classical statistical models. In this thesis, we will apply the HOTRG method to the models we mentioned above. Statistical models and field theories with local interactions can be represented as tensor-network states. the partition function and expectation value of a local operator can be obtained by the contraction of the product of local tensors.

21 6 Figure 1.1: One HOTRG step.(a) A HOTRG contraction of the state along the y axis. (b) Steps of contraction and recombination. Figure is from Ref. [83]. Z = T r i T i (1.1) O = T r[o j i j T i ]/Z (1.2) Take the 2D case for example, starting from the initial local tensor T (0), 2N HOTRG steps are applied in the two directions alternately to get a coarse-grained tensor corresponding to a system with volume 2 N 2 N. At the nth step, firstly, a

22 7 contracted tensor, M (n), is defined by connecting two local tensors T (n 1) with the number of states D s (Fig.1): M (n) xx yy = i T (n 1) (n 1) x 1 x 1yiT x 2 x, (1.3) 2 iy where x = x 1 x 2 and x = x 1 x 2. In both directions, the number of states becomes Ds 2 for the M tensor if we don t do other approximations. Secondly, the new local tensor, T (n), is formed by applying an unitary transformation followed by a truncated U (n) to the two sides of M (n) with product states x and x, T (n) xx yy = ij U (n) ix M (n) ijyy U (n) jx. (1.4) In practical, we need to truncate the number of states of the new T (n) tensor to the order of D s. Thus, determining an optimal truncate unitary matrix U is a crucial step. Mathematically, we are trying to find the minimal error for T (n) M (n), the difference between the truncated case T (n) and the original one without truncation M (n). By EckartYoung theorem [22], we can see that for each step, the U matrix can be determined by taking the singular value decomposition (SVD) for a matrix or higher-order SVD (HOSVD) for a tensor. In this case we need to take the HOSVD of the M tensor, M xx yy = ijkl S ijkl U L xiu R x ju U yku D y l (1.5) where S is the core tensor of M. Defining a Q matrix as, Q M M = UΛU, (1.6)

23 8 where the matrix M x,x yy is converted from the tensor M xx yy by regrouping its indices x yy into a single one. It is easy to find that S i,:,:,: 2 = Λ i (1.7) Thus, by taking the SVD of matrix Q, we obtain the same U getting from taking the HOSVD of M. By keeping the number of states to D s, we mean keeping the eigenvectors corresponding to the first D s largest singular values of Q (first D s columns of U). The T tensor is projected into D s new states in each direction without losing too much information. By iterating many times (N step) of HOTRG, we can get expectation values in either finite volume (small N) or infinite volume limit(results converging for large number of iterations).

24 9 CHAPTER 2 1D O(2) MODEL We choose the 1-d O(2) model on a finite lattice with both periodic boundary condition (PBC) and open boundary condition (OBC) as the starting point and obtain the leading nonperturbative part of the partition function or the average energy by certain nontrivial solutions of the classical equations of motion. These solutions correspond to the topological charges of the system, and can be shown in different ways. After introducing the exact solutions of the partition function of the model, we solve the equations of motion in Sec. 2.1 by using the saddle point approximation for a system with PBC and get the nonperturbative part of the partition function which has the form e Evβ β B. E v is the energy of the configuration with integer winding numbers (topological charges). We also get the nonperturbative part of an OBC system with a form e 2β β B. By using a new method based on asymptotic behavior and the continuous limit of a summation, we can get the higher order weak coupling expansion of the partition function. Looking at the difference between the numerical values and the truncated series, we have a signal for the topological charge solutions. We also modify the perturbative series of the simplest 1-link model by the Hadamard expansion. We calculate the density of states n(e) in Sec By the comparison between the small E expansion of n(e) and the numerical solution, we find that modification is needed when E > E v, which implies that the nonperturbative part is related to the topological charge solutions. We calculate Fisher s zeros in different volumes for a PBC system and get the Migdal-Kadanoff flows for an OBC system. We

25 10 find that the zeros separate the flows with different properties into different regions. In Sec. 2.4, we use Monte Carlo methods to obtain a stable configuration for a particular volume system with a fixed large coupling, β. If we start from random spins, the configuration stops at the state with winding number equal to 1, which corresponds to the topological charge solution again. In order to get the normal result with small fluctuations at large β, besides starting from an ordered configuration, we can bypass this nontrivial solution by using an OBC system with the same volume and coupling. In a D-dimensional lattice with N sites, we introduce a unit-length vector S i = (cos θ i, sin θ i ) per site for the O(2) model (the classical XY model). The partition function reads as follows: Z[β] = π dθ i π 2π e β <ij> i N (1 S i S j ) π dθ i = [1 cos(θ i θ j )] π 2π e β <ij> i N (2.1) in which i and j are nearest-neighbor sites. We consider the O(2) model in a L-link, one-dimensional lattice with different boundary conditions. For an OBC system, there are L + 1 sites for L links lattice. The partition function can be factorized using the change of variables θ i θ i+1 = θ i where 1 i L. Which would give Z[β] OBC = [e β I 0 (β)] L (2.2) where we used the expansion e β cos(θ θ ) = I n (β)e in(θ θ ), n Z(Integers) (2.3) n=

26 11 (I n are the modified Bessel function). From Eq. (2.2) we obtain that the partition function for a L-link chain is just the product of L independent one-link partition functions. For a PBC system, there are L sites for a L-link chain. The partition function cannot be factorized because of the constraint L (θ i θ i+1 ) = 0, and θ L+1 θ 1 is needed. Using Eq. (2.3), the partition function obtained is: i=1 Z[β] PBC = Z[β] OBC V [β] (2.4) where V [β] = n= I L n (β), n Z(Integers). (2.5) I L 0 (β) From Eq. (2.2) and (2.4), we can construct the PBC partition function with L links as the product of the L-link OBC partition function and the factor V [β]. 2.1 Classical Solutions For a model of statistical mechanics, one can obtain many local minima of the action, S({θ i }), by solving the classical equations of motion. Ordinary perturbation theory can be developed from the trivial minimum: θ i = θ 0 (all the θ i are the same). And there is great interest in the nontrivial solutions. Polyakov named the nontrivial minima as pseudo-particle solutions with a density which is proportional to exp( E/g 2 ) [72]. The energy E depends on the number of pseudo-particles. In Ref. [8], the authors proved instantons with winding number 1 exist for compact non-abelian gauge groups (BPST instanton). In this section, the nontrivial solution for 1-d O(2) model with different bound-

27 12 ary conditions is discussed in detail. We find the nonperturbative part of the partition function or average energy corresponds to the nontrivial solution (instanton solution for PBC particularly) with the weight exp( Eβ), in which E is the corresponding energy of the nontrivial solution. We first consider a L-site PBC system because the saddle point approximation can be used to obtain the leading order of the nonperturbative part. Staring from the action S({θ i }) βe({θ i }) = β L [1 cos(θ i θ i+1 )] (2.6) i=1 the minima can be solved by S({θ i }) θ i = 0. (2.7) {θi } For PBC, there are L 1 independent θ i. So we can fix one variable θ L = θ 0 and solve L 1 equations. Elements of the (L 1) (L 1) matrix A can be solved by A ij = 2 S({θ i }) θ i θ j {θi } (2.8) in which the matrix A is symmetric. By considering all the solutions of Eq. (2.7) and using the saddle point approximation, we have the leading order partition function with all the minima Z[β] = {θi} e βe({θ i}) (2π) L 1 det A. (2.9) To solve Eq. (2.7), we change variables θ i θ i+1 = θ i. Because of the compactness, there are L 1 equations of motion

28 13 L 1 sin θ i + sin( θ i) = 0, i = 1, 2,..., L 1. (2.10) i=1 Using sum-to-product formula, Eq. (2.10) has two types of solutions type I: sin(θ i + 1 θ 2 j) = 0, type II: cos( 1 θ 2 j) = 0. (2.11) j i For the set of equations of motion, type II corresponds to nearly π-jump solutions. The number of negative eigenvalues equals the number of type II equations. So the only positive definite solution is from the set with all type I equations. In this j i case, θ i = 2kπ L (2.12) and A = β cos( 2kπ L ) , cos( 2kπ ) > 0. (2.13) L Then, the leading order of the partition function from all the saddle points (Eq. (2.9)) has the solution Z[β] k [0,[L/4]) 2 exp{ βl[1 cos( 2kπ L )]} 1. (2.14) L[2πβ cos( 2kπ )]L 1 L For the OBC system, we only need to consider the simplest case with 1-link because any large volume system can be treated as the combination of independent 1-link systems. Unfortunately, the saddle point approximation fails to get the nonperturbative term because of the negative eigenvalues in the nontrivial solution.

29 14 However, we can use the minimal error estimate method in Ref. [59] to get the leading order of the nonperturbative part Z Npert [β] 1 2π β 1 e 2β. (2.15) From the partition function for PBC (Eq. (2.14)) and OBC (Eq. (2.15)), we can calculate the average energy per link: E(β) = Z (β) + 1. (2.16) Z(β)L We define the leading, nonperturbative part of the average energy as the difference between numerical average energy, E, and the energy calculated from the trivial saddle point, E 0. These nonperturbative parts are very different for open and periodic boundary conditions, (E E 0 )/E e 2β 2 πβ (OBC.) e βev 2E v L cos(2π/l) (L 1)/2 (PBC.) (2.17) where E v = L[1 cos( 2π )] is the energy that corresponds to the next saddle point L from the classical equations of motion. In the next section, numerical results confirm these relations. 2.2 perturvative method Dyson s instability shows that perturbative expansions are divergent for QED [21], and it was established for many other models [31]. People are interested in new methods of modified perturbation theories. in Appendix A, we consider two

30 15 improved perturbative methods. People believe that instantons play an important role in the leading nonperturbative term of a field theory in the weak-coupling region [13]. Another effective way to understand physical properties of a model is finding the connection between the weak and strong coupling expansion.a good example of this that the self-duality of the 2-d Ising model gives the right critical temperature [49]. In QED, there is a conjecture called Montonen-Olive duality in which Noether charge and topological charge exchange roles as weak or strong coupling [64]. In non-abelian cases, like 2-d O(N) (N 3) σ model, people are interested in the topological contribution. The O(N) mode has a nontrivial topological charge in (N 1) dimensions. In Ref. [11], authors investigated the topological actions in 1-d O(2) and 2-d O(3) models. In Section 2.1, we obtained the leading order of the nonperturbative part of the partition function and the average energy. In this section, we will calculate the higher order, ordinary perturbative series (weak coupling expansion), and obtain the difference between the truncated series and the numerical value of the quantity to check if the main nonperturbative part is right or not. The usual way to get the weak coupling expansion is by using Feynman rules. By introducing collective coordinates in treating the zero modes and adding a source j µ (n) for θ (n) instead of θ(n), we obtain the weak coupling expansion of the PBC partition function up to order 1/β 3 (or the average energy to order 1/β 4 ), which is very similar to Horsley and Wolff s treatment of the compact U(1) model [38]. For the OBC case, the Feynman rules are the same except that the zero modes are kept. We first present the detailed

31 16 calculation for both cases. We use Feynman rules to calculate the first 4 weak-coupling coefficients of the partition function. Firstly, we consider a PBC. system. In general, we use d dimensions, and set d = 1 at the end. Then, we define the field θ (n, µ) = θ(n+µ) θ(n) in the µ direction for site n. Here, θ (n, µ) is not a vector because of the case θ(n + µ) = θ(n + ν). We use {θ (n)} to represent the new independent variable. In 1-d, it is just θ i. θ (n, µ) is invariant under the global transformation θ (n, µ) θ (n, µ) + b. Then, we can introduce a term to exclude these zero modes, ( ) 1 = dbδ θ (n, µ) b (2.18) n,µ Because of the compactness of θ (n, µ), the change θ (n, µ) θ (n, µ) + L d b gives a trivial b integration. So only a factor δ[ θ (n, µ)] is left. Inserting Eq. (2.18) n,µ into the partition function and extending the range of integration to (, ): Z[β] ( ) ( d{θ (n)}δ θ (n, µ) exp β ) [1 cos(θ (n, µ))]. (2.19) n,µ n,µ Changing θ(n) β 1/2 θ(n) and introducing a source j µ (n) for θ (n, µ) gives the parturbation series: ( Z[β] β (Ld 1)/2 exp β ) {1 cos(β 1/2 / j µ (n)) [β 1/2 / j µ (n)] 2 /2} Z 0 [j] n,µ Z 0 [j] exp ( 1 2 ) j µ (n)d µ,ν (n m)j ν (m) n,µ j=0 (2.20)

32 17 The propagtor d is: d µ,ν = µ ν F (n) (2.21) with F (n) the solution of µ µ F (n) = δ n,0 1/L d, µ F (n) = 0. (2.22) n In a Fourier representation F is: F (n) = 1 2L d k 0 exp(ik n) (1 cos k µ ) µ (2.23) where k µ ranges over the set {0, 2π/L,, [(L 1)/L]2π}. We can calculate the series of ln Z by using diagrammatic notation. There are only connected diagrams for ln Z. For example, Then, = d µ,ν (m n) 4, m,n µ,ν = d µ,ν (k m) 2 d ν,κ (m n) 2 d κ,µ (n k) 2. k,m,n µ,ν,κ E F ln Z[β] = const + b 0 ln β + b 1 /β + b 2 /β 2 + b 3 /β 3 + O(1/β 4 ), (2.24) with b 0 = 1 2 (Ld 1), b 1 = 1 8, b 3 = b2 = , 48

33 18 For d = 1, the propagator d reads d(n) = δ n,0 1/L, then, all the diagrams can be evaluated exactly. From Eq. (2.24), we can calculate the 1-d partition function: Z P BC [β] e E F = const β (L 1)/2 (a 0 + a 1 /β + a 2 /β 2 + a 3 /β 3 + O(1/β 4 )) (2.25) in which a 0 = 1, a 1 = L 8 a 3 = L ( 1 1 ) 2 (, a 2 = L L 128 L 62 ( L + 87 L L L 4 L 5 L 6 3L L L 4 ), ). (2.26) For OBC, there is no constraint δ[ n,µ θ (n, µ)], so we keep the zero modes everywhere and the propagator d is just d OBC (n) = δ n,0. Using the same method, we obtain the OBC partition function: Z OBC [β] = const β L/2 (c 0 + c 1 /β + c 2 /β 2 + c 3 /β 3 + O(1/β 4 )) (2.27) in which c 0 = 1, c 1 = L ( 8, c 2 = L ) (, c 3 = L L 3072 L ). (2.28) L 2 From Eq. (2.26) and Eq. (2.28), we find the asymptotic behavior of the OBC and PBC coefficients at L are the same a n (c n ) 1 n! ( L 8 )n. To get higher order terms by using Feynman rules, one needs to count diagrams increasing at a factorial rate with the order. We do need other methods to get larger orders. For the OBC case, one can easily get the expansion by taking the series of Eq. (2.2) to large orders, which cannot be done for the PBC case. For 1-d PBC O(2) model with L sites, we introduce a new and effective method to calculate the weak

34 19 coupling expansion. We check the result from the partition function with L = 2 up to order 20 (1/β 39/2 term). The basic idea of the new method is the following: we calculate the 1/β expansion of V [β] (Eq. (2.5)) and then take the product of the series of V [β] and Z[β] OBC to get the final expansion. Firstly, considering the ingredient of V [β], In(β) I 0 (β). One can replace this by its asymptotic behavior at large β: I n (β) I 0 (β) n2 exp( 2β ){1 + f[n, O( 1 )]} (2.29) β2 in which f[n, O( 1 n2 )] = β2 4β + 13n2 + 2n n2 + 5n 4 + (2.30) 2 48β 3 32β 4 Then, V [β] = n= exp( Ln2 2β ){1 + f[n, O( 1 β 2 )]}L (2.31) the leading term of which is the large β result of the Villian approximation [80]. in which By changing variables from n to n as n = n β, one has, V [β] = n β= exp( Ln 2 2 ){1 + g[n, O( 1 β )]}L (2.32) g[n, O( 1 β )] = 6n 2 + n 4 24β n n 4 132n 6 + 5n β 2 + (2.33) For large β, the variable n can be treated as a continuous variable. So we can change the summation of terms with n into an integral. Eq. (2.32) becomes a 1-d Gaussian integral: V [β] = β dn exp( Ln 2 2 ){1 + g[n, O( 1 β )]}L. (2.34)

35 20 By increasing the order of g[n, O( 1 )], we can calculate higher orders of 1/β, β and finally we can get the right weak coupling expansion of Z[β] PBC. For example, we get the coefficients for the system with L = 36 up to order 12 Z[β] L=36 = β 35/2 π 35/2 a n β n. (2.35) n=0 The coefficients a n are listed in Table 2.1. n 0 1 a n Table 2.1: coefficients a n From the 1/β expansion of Z[β], we can calculate the corresponding expansion of the average energy per link by using Eq. (2.16). To get the behavior of the nonperturbative part we can use error graphs of the significant digits vs. β, in which the significant digits are defined as, ( ) series exact S.D = log 10. (2.36) exact

36 21 From Fig. 2.1 we find that the leading nonperturbative part we got from Eq. (2.17) is the right behavior compared to the curves of the error graphs for both PBC and OBC. -log 10 [ (series-exact)/exact ] log 10 (e -2 )+c 1 errors (o.b.c) -log 10 [ (series-exact)/exact ] log 10 (e -E v )+c 2 errors (p.b.c) Figure 2.1: Error graphs. OBC (Left): Errors of the average energy series with order 2, 4,..., 20 (Blue), PBC with L = 36 (Right): Errors of the average energy series with order 2, 4,..., 12 (Blue). Red lines correspond to the results in Eq. (2.17).

37 22 It is difficult to fully modify the ordinary perturbative series by adding the nonperturbative terms. However, we can use the (modified) Hadamard expansion to get better results for a 1-link model. The Hadamard expansion can make the asymptotic series converge, and the modified Hadamard expansion makes the series converge faster [70]. Z(x) = e x I 0 (x): in which We consider the usual asymptotic expansion of the function Z(x) = 1 a k (2.37) 2πx (2x) k a k = k=0 ( Γ(k + 1/2) π ) 2 1 k!. The Hadamard Expansions of the function is Z(x) H = 1 a k 2πx (2x) P (k + 1, 2x) (2.38) k 2 k=0 in which P is a normalized incomplete gamma function. The Modified Hadamard Expansions is Z(x) M.H = 1 M 1 a k { 2πx (2x) P (k + 1 k 2, 2x) + T M,n(x)} (2.39) k=0 where T M,n is a modified n-term series of the tail: T M,n (x) = k=m a k (2x) k P (k + 1 2, 2x) = (2x) 1/2 e 2x k=m ( n = e 2x (2x) r+1/2 r=0 a k Γ(k + 3/2) 1 F 1 (1; k + 3/2; 2x) M Γ(r + 1/2) r!γ(r + 1) k=0 a k Γ(k + r + 3/2) If one plots the error graphs of the original Z(x) to different orders, one can get the same type of graph as we did in Fig We draw the error graph of the )

38 23 Hadamard series and the Modified Hadamard series to different order n, the result is shown in Fig From the graph, we conclude that the Hadamard series avoids the zigzag behavior in Fig. 2.1 and improves the results slightly by increasing the order. The Modified Hadamard series can improve small β result dramatically. -log 10 [ (series-exact)/exact ] Hadamard(H) series Modified H series (n=10) Modified H series (n=20) Figure 2.2: Errors of different series to order 2, 4,..., 20. Black: Hadamard series; Blue: modified Hadamard series (n = 10); Red: modified Hadamard series (n = 20).

39 Density of States, Fisher s zeros, and RG flow The density of states n(e) is the inverse Laplace transform of the partition function. n(e) = 1 2πi K+i K i dβe βe Z[β]. (2.40) The contour of integration is a vertical line in the complex β plane with K larger than the real part of all singularities of Z[β]. In the 1-d O(2) model, K can be an arbitrary real number. The definition itself gives us a way to obtain the numerical density of states for small volumes. Using Eq. (2.4), we can rewrite the partition function as Z[β] = m Z m[β], in which Z m [β] = exp( Lβ)I L m (β). We calculate n m (E) = 1 2πi K+i K i dβ exp(βe)z m [β] (2.41) by adding as many terms n m (E) as possible we can find n(e) m<m max n m (E). For small β, a finite number of terms can give us high precision results of n(e), which correspond to the region near E = L in the density of states. For large positive β (small E part), this method is not reliable. However we can get high precision results by using the weak coupling expansion of Z[β] in this case since the inverse Laplace transform of the 1/β expansion of Z[β] is just the small E expansion of n(e). We combine the results from two different methods for the system with L = 6 in Fig We find that the small E expansion fails for E much larger than the energy E v, which shows the importance of the nonperturbative contribution. This failure corresponds to the winding number one solution of the classical equation of motion.

40 data n(e) order 4 order 8 order 12 order 16 order 20 n(e) E Figure 2.3: Small E expansion with order 4, 8,..., 20 and the numerical data n(e). In this section, we discuss the Fisher zeros of the partition function at finite volume with PBC and OBC. For OBC, the zeros are just the zeros of the modified Bessel function I 0 (β), which are on the imaginary axis. For PBC, we use the contour

41 26 plot method to get the contours that correspond to the real and imaginary part of the partition function that equal zero. The intersections of two different kinds of curves are the approximate positions of the Fisher zeros. Starting from these approximate positions, we use Newton s method to get high precision results. Zeros found using different volumes are shown in Fig The zeros are very different for open and periodic boundary conditions. For PBC, zeros form layers in the direction of imaginary β which are independent of the volume; while in the direction of real β, the density increases with the volume. The real part of the first zero is proportional to 1/L. We also construct complex renormalization group (RG) flows by matching the correlation function from OBC systems with different lattice spacing S 0 S x L,β = x x S 0 S x L,β. (2.42) Under a particular RG transformation, one doubles the lattice spacing without changing the physical sizes, resulting in the number of sites being halved; in this case L = L/2. We compare an L-link lattice system with coupling β with an (L/2)-link system with β. Then we have the Migdal-Kadanoff recursion relation: ( ) 2 I1 (β) = I 1(β ) I 0 (β) I 0 (β ) (2.43)

42 Im zeros L=4 zeros L=8 zeros L=16 zeros L=32 zeros obc Re Figure 2.4: Zeros of the partition function (PBC) with different volumes and zeros of the partition function (OBC) In order to visualize this RG transformation, we construct a complex RG map as follows: from the initial βs near the unstable fixed points (solutions of I 1 (β) =

43 28 I 0 (β)) and a line with the real part equal to two, we solve the modified recursion relation I 2 1 (β)i 0 (β ) = I 1 (β )I 2 0 (β) to get β. The reason we use modified relation is to avoid the flow stopping near the singular regions due to a divide by zero issue. β is then treated as the starting point in the modified recursion relation to solve the new β (β β). This procedure is iterated several times until β reaches the stable fixed points (zeros of I 1 (β)) or singular points (zeros of I 0 (β)). The result is shown in Fig. 2.5 with the zeros of the PBC partition function. We have not found the RG flows for PBC system yet, but the behavior should be similar with what we obtained for the OBC system. In Fig. 2.5, zeros in the first layer separate two flows into different destinations. So we conjecture that in a PBC system with large volume, the lines of zeros separate different regions in the β plane where in each region β follows the flow to a particular fixed point or singular point.

44 Im 2 1 RG flows zeros L=4 zeros L=8 zeros L=16 zeros L=32 zeros obc Re Figure 2.5: MK complex flows for the system (OBC) and zeros from the two different boundary conditions 2.4 Monte Carlo Simulation of Configurations In this section, we simulate the model by using the Metropolis Monte Carlo technique. The program is based on Ref. [10] which is for PBC models. Firstly, we consider the PBC case. The volume is fixed at L = 40 and the coupling corresponds

45 30 to β = 20. Starting from a random configuration we use equilibration sweeps of the simulation, we then plot the final configuration of the system as shown in Fig. 2.6 (Top). This result corresponds to a metastable state with winding number one, which is similar to the domain wall configurations in 2-d XY model [10]. We conjecture that this state represents the first nontrivial solution of the classical equation of motion due to the compactness of the PBC. One can avoid this nontrivial minimum state by using an ordered start configuration. Fig. 2.6 (Middle) shows the configuration without winding after sweeps. We then try different volume and β to get the difference of the nearest spins θ i and find that θ i << π for large β, which shows the π jump is forbidden. Recently, Luscher and Schaefer proposed to avoid the topology barriers of Lattice QCD by using open boundary conditions [55, 56]. Following this idea, we simulate the 1-d O(2) model with OBC starting from random configuration. The result after sweeps is plotted in Fig. 2.6 (Bottom), which is a reasonable configuration with winding number zero.