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arXiv:1202.6147 · Journal of the Physical Society of Japan

Non-local Correlations in the Haldane Phase for an XXZ Spin-1 Chain: A Perspective from Infinite Matrix Product State Representation

Yao Heng Su, Sam Young Cho, Bo Li, Hong-Lei Wang, Huan-Qiang Zhou

Published Jun 20, 2012• 36 citations
arXivHTMLPDF

Abstract

String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results: $Δ_{c2} = 0$ for the XY-Haldane phase transition and $Δ_{c3} = 1.185$ for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be $c \simeq 1.0$ at $Δ_{c2} = 0$ and $c \simeq 0.5$ at $Δ_{c3} = 1.185$, respectively, which shows that the XY-Haldane phase transition at $Δ_{c2} = 0$ belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at $Δ_{c2} = 1.185$ belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.

DOI
10.1143/jpsj.81.074003

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String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results: Δc2=0Δ_{c2} = 0Δc2​=0 for the XY-Haldane phase transition and Δc3=1.185Δ_{c3} = 1.185Δc3​=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be c≃1.0c \simeq 1.0c≃1.0 at Δc2=0Δ_{c2} = 0Δc2​=0 and c≃0.5c \simeq 0.5c≃0.5 at Δc3=1.185Δ_{c3} = 1.185Δc3​=1.185, respectively, which shows that the XY-Haldane phase transition at Δc2=0Δ_{c2} = 0Δc2​=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at Δc2=1.185Δ_{c2} = 1.185Δc2​=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.
In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β = 1/8 and the cental charges c = 1/2 at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the Von Neumann entropy and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in previous studies.
The characterization of the quantum phase transition in a lowdimensional system has attracted a considerable amount of attention in quantum manybody systems. As one of the fundamental models in quantum magnetism, spin-1 models have richer phase diagrams and show more complex physical phenomena. In the spin-1 antiferromagnetic XXZ model, the Haldane phase and the Nel phase are the gapped topologic phases which cannot be characterized by the local order parameters. To characterize the nature in such phases, one has to calculate the non-local long range order parameters. Normally, the non-local order parameter in the topological phase is obtained from the extrapolation of finite-sized system in numerical study. However, it is difficult to extract the critical exponents with such an extrapolated non-local order parameter due to the numerical accuracy. In a recently developed tensor network representation, i.e., the infinite matrix product state (iMPS) algorithm, it was shown that the non-local order can be directly calculated from a very large lattice distance in an infinite-sized system rather than an extrapolated order parameter in a finite-sized system. Therefore, it is worthwhile using this convenient technique to study the non-local orders in the topological phases and characterize the quantum criticalities in the topological quantum phase transitions. In this paper, by utilizing the infinite matrix product state algorithm based on the tensor network representation and infinite time evolving block decimation method, the quantum entanglement, fidelity, and critical exponents of the topological phase transition are investigated in the one-dimensional infinite spin-1 bond-alternating XXZ Heisenberg model. It is found that there is always a local dimerization order existing in the whole parameter range when the bond-alternative strength parameter changes from 0 to 1. Also, due to the effect of the bond-alternating, there appears a quantum phase transition from the long-rang ordering topological Nel phase to the local ordering dimerization phase. The von Neumann entropy, fidelity per lattice site, and order parameters all give the same phase transition point at c = 0.638. To identify the type of quantum phase transition, the central charge c 0.5 is extracted from the ground state von Neumann entropy and the finite correlation length, which indicates that the phase transition belongs to the two-dimensional Ising universality class. Furthermore, it is found that the Nel order and the susceptibility of Nel order have power-law relations to |-c|. From the numerical fitting of the Nel order and its susceptibility, we obtain the characteristic critical exponents ' = 0.236 and ' = 0.838. It indicates that such critical exponents from our method characterize the nature of the quantum phase transition. Our critical exponents from the iMPS method can provide guidance for studying the properties of the phase transition in quantum spin systems.
We investigate string correlations in an infinite-size spin-1/2 bond-alternating Heisenberg chain. By employing the infinite matrix product state representation with the infinite time evolving block decimation method, a finite string correlation for extremely large lattice distances is directly observed, in contrast to an extrapolated extreme value for finite size chains. We find that a topological quantum phase transition occurs between two different phases separated and characterized by two different long-range string orders in the space of bond-alternating interactions. Also, the critical exponent $\ensuremath{\beta}$ from the long-range string orders is obtained as $\ensuremath{\beta}=1/12$ and the central charge at the critical point is obtained as $c\ensuremath{\simeq}1$, which shows that the topological phase transition belongs to the Gaussian universality class. In addition, it is shown that, for the topological quantum phase transition, the phase boundary can be captured by the singular behavior of the von Neumann entropy and the pinch point of the fidelity per site.
Quantum entanglement and non-local string order are investigated in the one-dimensional spin-1 bond-alternating Heisenberg model for quantum phase transitions. Employing the infinite matrix product state representation, ground state wavefunctions are numerically obtained by using the infinite time evolving block decimation approach in the infinite lattice system. From a bipartite entanglement measure, i.e., von Neumann entropy, a quantum phase transition point corresponding to the singular point of the von Neumann entropy between the Haldane phase and the dimerised phase is found to be δc=0.26. In addition, based on the infinite matrix product state algorithm, the non-local string order is calculated. The result shows that the non-local string order is able to characterize the topological phase transitions.
We have developed a numerical procedure to clarify the critical behavior near a quantum phase transition by analyzing a multipoint correlation function characterizing the ground state. This work presents a successful application of this procedure to the string order parameter of the S=1S=1S=1 XXZXXZXXZ chain with uniaxial single-ion anisotropy. The finite-size string correlation function is estimated by the density-matrix renormalization-group method. We focus on the gradient of the inverse-system-size dependence of the correlation function on a logarithmic plot. This quantity shows that the finite-size scaling sensitively changes at the critical point. The behavior of the gradient with increasing system size is divergent, is stable at a finite value, or rapidly decreases to zero when the system is in the disordered phase, at the critical point, or in the ordered phase, respectively. The analysis of the finite-size string correlation functions allows precise determination of the boundary of the Haldane phase and estimation of the critical exponent of the correlation length. Our estimates of the transition point and the critical exponents, which are determined only by the ground-state quantities, are consistent with results obtained from the analysis of the energy-level structure. Our analysis requires only the correlation functions of several finite sizes under the same condition as a candidate for the long-range order. The quantity is treated in the same manner irrespective of the kind of elements which destroy the order concerned. This work will assist in the development of a method for directly observing quantum phase transitions.
We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as ∝r−α\propto r^{-α}∝r−α using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-1/21/21/2 degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large ααα (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small ααα (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at αc=2.49(1)α_c = 2.49(1)αc​=2.49(1) and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamical exponent z≠1z \neq 1z=1. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.