张量重正化群方法及其应用

张量重正化群方法是近年来发展起来的一种新的数值计算方法,它将经典配分函数和量子波函数的张量网络表示与重正化群方法相结合,在强关联系统的数值研究中,发挥着越来越重要的作用。文章以经典统计模型和量子格点模型为例,简要介绍了张量重正化群的一些基础知识和研究给定物理模型的一般性思路,并对张量重正化群未来可能的发展方向和亟待解决的问题进行了讨论。     

有限温度量子多体系统与热态张量网络

量子多体系统热力学性质的精确模拟在理论和实验方面都具有重要的价值。局域相互作用量子多体系统的热态满足互信息(mutual information)面积律,对于这样的系统,热态张量网络可以提供满足面积律的精确"波函数"拟设,提供了模拟有限温度系统的有力手段。文章介绍了关联格点模型在有限温度下的热态张量网络刻画及相关模拟方法。作者按照世界线热态张量网络和级数展开热态张量网络来分别介绍,并讨论了自由能极小变分原理与重正化群剪裁的优化原则。世界线框架内,人们发展了转移矩阵重正化群,基于纯化策略的有限温度密度矩阵重正化群,以及张量网络的线性重正化群等方法。在此基础上,介绍作者新近提出的级数展开热态张量网络方法,该方法受随机级数展开量子蒙特卡罗方法的启发,突破了世界线方法的局限,提高了有限温度计算重正化群模拟的精度标准,并且在计算阻挫量子自旋链模型时不会有负符号问题。此外,文章讨论了在两维格点系统上推广有限温度张量网络计算的进展和未来展望。     

对《二维六角形晶格伊辛模型的重正化群解》一文的进一步计算

针对《二维六角形晶格伊辛模型的重正化群解》一文中有关〈V〉0的计算进行了修正,给出了新的重正化群的变换、重正化群的线性化变换矩阵以及临界指数.     

O(N)模型破缺相的红外重正化群计算

利用一个与质量有关的重正化方案,研究了O(N)模型破缺相的红外重正化群性质.得到了与质量有关的一圈重正化群系数以及在红外极限下的二圈重正化群系数.利用标准的微扰场论计算方法,发现特殊的抵消效应,该效应使得二圈的重正化群系数与一圈重正化群系数在红外极限下相同.     

W(112)p(2×1)-O化学吸附系统有序-无序相变的重整化群理论分析

用位置空间的重整化群方法,讨论W(112)p(2×1)-O化学吸附系统的有序-无序相变。采用周期性集团的递推关系,进行了将格点组成单胞,再将单胞组成周期性集团的4×4重整化变换,考虑最近邻、次近邻氧原子之间的对相互作用以及三个氧原子之间和四个氧原子之间的相互作用,求出重整化群耦合参数K′_α和格点间相互作用参数K_α之间的关系,确定了固定点K_α^*求出不同覆盖度下的临界温度T_c,得到与实验结果相符合的W(112)p(2×1)-O相图。 关键词：     

格点规范理论(Ⅰ)

本文介绍了由Wilson等人发展起来的处理粒子间强相互作用的格点规范理论。由于这个理论是建立在点阵上的规范理论,故首先讨论了点阵上体系的场论性质和统计物理性质之间的联系,介绍了处理粒子禁闭问题的Wilson判据,点阵的哈密顿形式。然后讨论了各种具体模型的计算方法,如规范场的点阵模型、紧致QED模型、费米子模型、阿贝尔Higgs模型等。在此基础上,总结出Wilson定理。本文也讨论了格点规范理论中的实空间重正化群方法,介绍了Heisenberg平面模型的重正化群分析,一维的二维的复现关系及Migdal近似。最后评介了近年来对于Wilson回路算子的一些研究,内容包括’t Hooft代数和Wilson回路算子方程等。     

格点规范理论(Ⅰ)

本文介绍了由Wilson等人发展起来的处理粒子间强相互作用的格点规范理论。由于这个理论是建立在点阵上的规范理论,故首先讨论了点阵上体系的场论性质和统计物理性质之间的联系,介绍了处理粒子禁闭问题的Wilson判据,点阵的哈密顿形式。然后讨论了各种具体模型的计算方法,如规范场的点阵模型、紧致QED模型、费米子模型、阿贝尔Higgs模型等。在此基础上,总结出Wilson定理。本文也讨论了格点规范理论中的实空间重正化群方法,介绍了Heisenberg平面模型的重正化群分析,一维的二维的复现关系及Migdal近似。最后评介了近年来对于Wilson回路算子的一些研究,内容包括’t Hooft代数和Wilson回路算子方程等。     

分形格点中伊辛模型的临界行为

分形格点是一类特殊的格点,它具有非整数的维度,且打破了平移不变性.本文对分形格点中伊辛模型的临界行为进行了研究.在这个系统中存在从有序到无序的连续相变,本文利用张量网络重正化群算法计算了不同位置格点上的物理量,并据此在不同空间位置拟合出了相应的临界指数.由于平移对称性的缺失,发现临界指数的拟合结果对空间位置有依赖关系.另外,在分形格点中的不同位置检验了临界指数间的标度关系(hyperscaling relations),最终发现在某些格点上所有的标度关系全部成立,而在另外一些格点上则只有部分的标度关系成立.     

由流方程求解非零温条件下t-J模型能谱

t_J模型是研究电子强关联作用和高T_c超导理论的重要模型之一. 将重正化群流 方程方法应用于t_J模型，分别解出了t_J模型在非零温条件下费米子和玻色子元激发能谱的 解析式. 并将结果与采用格林函数方法求解的结果作了比较，两者基本一致.重正化群流方 程方法解出的结果是解析解，而格林函数方法求解的结果是数值解. 关键词： 重正化群 t－J模型 流方程     

重正化群与渗流理论

本文由一系列讲演组成,内容包括:临界现象与渗流,标度理论,位置空间重正化群与渗流,位置空间重正化群用于热力学相变,动量空间重正化群与高斯模型和动量空间重正化群用于S~4模型。     

Quantum spins and quasiperiodicity: a real space renormalization group approach

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.     

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

The fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.     

Some rigorous results on majority rule renormalization group transformations near the critical point

We consider the majority rule renormalization group transformation with two-by-two blocks for the Ising model on a two-dimensional square lattice. For three particular choices of the block spin configuration we prove that the model conditioned on the block spin configuration remains in the high-temperature phase even when the temperature is slightly below the critical temperature of the ordinary Ising model with no conditioning. We take as the definition of the infinite-volume limit an equation introduced in earlier work by the author. We use a computer to find an approximate solution of this equation and verify a condition which implies the existence of an exact solution.     

The Fisher-Hartwig Formula and Entanglement Entropy

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in spin chains. In the paper we review these recent developments. We use the Fisher-Hartwig formula, as well as the recent results concerning the asymptotics of the block Toeplitz determinants, to calculate entanglement entropy of large block of spins in the ground state of XY spin chain.     

Adatom diffusion on a square lattice. Theoretical and numerical studies

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to . Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions. Received 1st September 1998 and Received in final form 8 March 2000     

Nonequilibrium photo dynamics of low-dimensional strongly correlated electron systems

One of the outstanding contemporary challenges in condensed matter physics is to understand the dynamics of interacting quantum systems exposed to an external perturbation. We theoretically examine nonequilibrium photo dynamics and its interplay of charge, spin, and lattice degrees of freedom on a Hubbard-Holstein chain in one dimension and a t-J-Holstein square lattice in two dimensions. In the chain, performing dynamical density-matrix renormalization group calculations, we find that many phonons generated dynamically after photo irradiation in Mott insulators cause initial relaxation process. On the other hand, in the square lattice with model parameters as relevant for cuprates, a Lanczos-type exact diagonalization calculation shows that the majority of absorbed energy flows into spin subsystem rather than phonon subsystem.     

Evolution by damage spreading in Kauffman model

In Kauffman's Boolean automata model on the square lattice, the Darwinian fitness of survival can be defined as the fraction of elements which do not change from one iteration to the next. Biological mutations are simulated by filpping one bit in the rule of one site. Selection of the fitter mutant then optimizes the whole lattice completely. This optimization is particularly effective near the critical point of the transition to chaos, but is in itself not a critical phenomenon. Also a two-dimensional spin glass can be optimized in this way.     

Critical behavior of XY spin chain with Dzyaloshinsky-Moriya interaction studied in terms of Loschmidt echo

In this paper, the critical behavior of the general XY spin chain with the Dzyaloshinsky-Moriya (DM) interaction is studied by means of a Loschmidt Echo (LE) calculation. LE presents a Gauss decay in the region of magnetic field intensity |λ|<1 and an exponential decay in the region of |λ|>1. There exists a critical spin chain size N_C. When spin chain size is larger than N_C, the value of λ corresponding to the minimum value of LE (λ_m) is independent of the spin chain size and keeps a stable value. In the region of λ<0, the stable value is same for different DM interactions. In the region of λ>0, the stable value varies with changing DM interaction.     

Topological estimator of block entanglement for Heisenberg antiferromagnets

We introduce a computable estimator of block entanglement entropy for many-body spin systems admitting total singlet ground states. Building on a simple geometrical interpretation of entanglement entropy of the so-called valence bond states, this estimator is defined as the average number of common singlets to two subsystems of spins. We show that our estimator possesses the characteristic scaling properties of the block entanglement entropy in critical and noncritical one-dimensional Heisenberg systems. We invoke this new measure to examine entanglement scaling in the two-dimensional Heisenberg model on a square lattice revealing an "area law" for the gapped phase and a logarithmic correction to this law in the gapless phase.     

Spin dynamics and magnetic correlation length in two-dimensional quantum heisenberg antiferromagnets

The correlated spin dynamics and temperature dependence of the correlation length xi(T) in two-dimensional quantum (S = 1/2) Heisenberg antiferromagnets (2DQHAF) on a square lattice are discussed in light of experimental results of proton spin lattice relaxation in copper formiate tetradeuterate. In this compound the exchange constant is much smaller than the one in recently studied 2DQHAF, such as La2CuO4 and Sr2CuO2Cl2. Thus the spin dynamics can be probed in detail over a wider temperature range. The NMR relaxation rates turn out to be in excellent agreement with a theoretical mode-coupling calculation. The deduced temperature behavior of xi(T) is in agreement with high-temperature expansions, quantum Monte Carlo simulations, and the pure quantum self-consistent harmonic approximation. Contrary to the predictions of the theories based on the nonlinear sigma model, no evidence of crossover between different quantum regimes is observed.