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ひねり境界条件を使った多重臨界点付近の相転移線の計算手法

Shunji Moriya

Published Mar 23, 2022• 0 citations
OpenAlex

Abstract

A point where several critical lines intersect is called a multicritical point.Near such a point, multiple critical phenomena interfere and a finite-size correction becomes large.Therefore, conventional methods cannot be applied near the multicritical point.We propose a new method to numerically calculate a transition point for a quantum spin chain near the multicritical point.We treat a bond-alternating (BA) XXZ model for S = 1/2, 1, 3/2.In a ground-state phase diagram of this model, a 2D Gaussian universality transition line bifurcates into two 2D Ising universality transition lines, which make a multicritical point.At this point, Berezinskii-Kosterlitz-Thouless transition occurs, where the correlation length diverges singularly.To deal with a 2D Ising universality, we review a transverse field 1D quantum Ising (TFI) model, corresponding to the classical 2D Ising model with transfer matrix.A relation between a disorder phase and an order phase is known as the Kramers-Wannier duality transformation.We find a proper duality transformation that is exact in the finite-size TFI model with a periodic and anti-periodic boundary condition.This shows that the energies for the two boundary conditions are crossing at the transition point.A free fermion field theory is derived from the critical Ising model by taking a continuum limit.From a conformal field theory, we verify that the energy-crossing is realized in the 2D Ising universality class, not only in the TFI model.We apply our method to the BA XXZ model.The energy-crossing happens in boundary conditions that are twisted around a z-axis and y-axis.In an anisotropic limit, two energies in a finite-size are crossing at the transition point since the BA XXZ model is identical to the TFI model.on the other hand, at the multicritical point, the finite-size correction vanishes by an isotropy and a twist translation symmetry.Therefore, near the multicritical point, our method has a smaller finite-size correction.In this paper, we also review the 2D Gaussian universality.By a bosonization, the BA XXZ model can be transformed to a phase Hamiltonian composed of boson operators.The degenerate energies for a twisted boundary condition are split by a perturbation around a Gaussian fixed point.By our method, we numerically calculate a transition point of the BA XXZ model for S = 1/2, 1, 3/2.As expected, the finite-size correction of numerical results becomes very small near the multicritical point.i

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A point where several critical lines intersect is called a multicritical point.Near such a point, multiple critical phenomena interfere and a finite-size correction becomes large.Therefore, conventional methods cannot be applied near the multicritical point.We propose a new method to numerically calculate a transition point for a quantum spin chain near the multicritical point.We treat a bond-alternating (BA) XXZ model for S = 1/2, 1, 3/2.In a ground-state phase diagram of this model, a 2D Gaussian universality transition line bifurcates into two 2D Ising universality transition lines, which make a multicritical point.At this point, Berezinskii-Kosterlitz-Thouless transition occurs, where the correlation length diverges singularly.To deal with a 2D Ising universality, we review a transverse field 1D quantum Ising (TFI) model, corresponding to the classical 2D Ising model with transfer matrix.A relation between a disorder phase and an order phase is known as the Kramers-Wannier duality transformation.We find a proper duality transformation that is exact in the finite-size TFI model with a periodic and anti-periodic boundary condition.This shows that the energies for the two boundary conditions are crossing at the transition point.A free fermion field theory is derived from the critical Ising model by taking a continuum limit.From a conformal field theory, we verify that the energy-crossing is realized in the 2D Ising universality class, not only in the TFI model.We apply our method to the BA XXZ model.The energy-crossing happens in boundary conditions that are twisted around a z-axis and y-axis.In an anisotropic limit, two energies in a finite-size are crossing at the transition point since the BA XXZ model is identical to the TFI model.on the other hand, at the multicritical point, the finite-size correction vanishes by an isotropy and a twist translation symmetry.Therefore, near the multicritical point, our method has a smaller finite-size correction.In this paper, we also review the 2D Gaussian universality.By a bosonization, the BA XXZ model can be transformed to a phase Hamiltonian composed of boson operators.The degenerate energies for a twisted boundary condition are split by a perturbation around a Gaussian fixed point.By our method, we numerically calculate a transition point of the BA XXZ model for S = 1/2, 1, 3/2.As expected, the finite-size correction of numerical results becomes very small near the multicritical point.i
We propose a new method to numerically calculate transition points that belongs to 2D Ising universality class for quantum spin models.Generally, near the multicritical point, in conventional methods, a finite size correction becomes very large.To suppress the effect of the multicritical point, we use a z-axis twisted boundary condition and a y-axis twisted boundary condition.We apply our method to an S = 1 2 bond-alternating XXZ model.The multicritical point of this model has a BKT transition, where the correlation length diverges singularly.However, with our method, the convergence of calculation is highly improved, thus we can calculate the transition point even near the multicritical point.
We study boundary criticality at the Nishimori multicritical point of the two-dimensional (2D) random-bond Ising model. Using tensor-network methods, we construct a family of microscopic boundary conditions that incorporates both boundary-spin rotation and boundary disorder. We identify three conformal boundary fixed points, corresponding to free, fixed, and random boundary conditions, and map out the boundary renormalization group flows among them. We extract the corresponding boundary conformal data, including the boundary entropies and the scaling dimensions of boundary primary operators, which characterize the boundary universality class. At the free boundary fixed point, we uncover the multifractal scaling of boundary-spin fields. We further complement the numerical results with a controlled renormalization group analysis. Finally, we connect the boundary conformal data to quantum error-correcting codes, establishing a bridge between boundary universality class and boundary decoding threshold.
We consider a classical random-bond Ising model (RBIM) with binary distribution of $\ifmmode\pm\else\textpm\fi{}K$ bonds on the square lattice at finite temperature. In the phase diagram of this model there is the so-called Nishimori line which intersects the phase boundary at a multicritical point. It is known that the correlation functions obey many exact identities on this line. We use a supersymmetry method to treat the disorder. In this approach the transfer matrices of the model on the Nishimori line have an enhanced supersymmetry osp(2n+1∣2n),\mathrm{osp}(2n+1|2n),osp(2n+1∣2n), in contrast to the rest of the phase diagram, where the symmetry is osp(2n∣2n)\mathrm{osp}(2n|2n)osp(2n∣2n) (where n is an arbitrary positive integer). An anisotropic limit of the model leads to a one-dimensional quantum Hamiltonian describing a chain of interacting superspins, which are irreducible representations of the osp(2n+1∣2n)\mathrm{osp}(2n+1|2n)osp(2n+1∣2n) superalgebra. By generalizing this superspin chain, we embed it into a wider class of models. These include other models that have been studied previously in one and two dimensions. We suggest that the multicritical behavior in two dimensions of a class of these generalized models (possibly not including the multicritical point in the RBIM itself) may be governed by a single fixed point, at which the supersymmetry is enhanced still further to osp(2n+2∣2n).\mathrm{osp}(2n+2|2n).osp(2n+2∣2n). This suggestion is supported by a calculation of the renormalization-group flows for the corresponding nonlinear sigma models at weak coupling.
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
Abstract. The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function. PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.Hf Recently, there has been a lot of interest in generalised twisted partition functions of conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundary conditions in solvable lattice models of statistical mechanics [5]. Examples of such boundary conditions leading to the appearance of “exotic ” spinor operators in the
We investigate two-dimensional spin models and begin with an introduction to critical phenomena with emphasis on classical systems with boundaries and conformal invariance. Then we introduce the antiferromagnetic quantum Heisenberg model with frustrating interactions. Conformal invariance of a statistical mechanical system at criticality has powerful implications in two dimensions. With the use of conformal field theory, we investigate finite-size and boundary effects for a class of N-critical models that describe the simplest type of multicritical behaviour. In Papers I and II, correlation functions for the order parameter and other operators of these multicritical Ising models are derived for restricted domains (semi-infinite plane, infinite strip, disc and rectangle) with free or fixed boundary conditions. These yield the corresponding universal surface exponents and allow for a detailed analysis of correlations lengths, structure factors and response functions. The interplay between duality and boundary conditions is also made explicit. In Paper III, we analyse structure factors at criticality and show how apparent power- law scaling at intermediate momenta - induced by finite size and boundaries - depends on the boundary conditions applied. The predictions are confirmed by using correlation functions calculated in Papers I and II. Two-dimensional Heisenberg antiferromagnets have attracted much attention because of their relation to high-Tc superconductors. The second part of the thesis (Papers IV and V) treats spin-1/2 antiferromagnets defined on square- and honeycomb lattices with frustrating interactions. By using a Schwinger-boson mean-field theory, we calculate low-temperature quantum corrections to thermodynamic parameters, such as spin-wave velocity and transverse susceptibility. Their dependence on frustration is explicated and used to estimate the stability of the Néel state via a mapping to the nonlinear .sigma. model that describes the low-energy limit of the antiferromagnet.