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Chinese Physics B

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

Yan-Wei Dai, Sheng-Hao Li, Xihao Chen

Published Jan 17, 2022• 1 citation
Publisher

Abstract

We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.

DOI
10.1088/1674-1056/ac4bd1

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We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.
We numerically investigate the two-dimensional q-state quantum Potts model on the infinite square lattice by using the infinite projected entangled-pair state (iPEPS) algorithm. We show that the quantum fidelity, defined as an overlap measurement between an arbitrary reference state and the iPEPS ground state of the system, can detect q-fold degenerate ground states for the Z_{q} broken-symmetry phase. Accordingly, a multiple bifurcation of the quantum ground-state fidelity is shown to occur as the transverse magnetic field varies from the symmetry phase to the broken-symmetry phase, which means that a multiple-bifurcation point corresponds to a critical point. A (dis)continuous behavior of quantum fidelity at phase transition points characterizes a (dis)continuous phase transition. Similar to the characteristic behavior of the quantum fidelity, the magnetizations, as order parameters, obtained from the degenerate ground states exhibit multiple bifurcation at critical points. Each order parameter is also explicitly demonstrated to transform under the Z_{q} subgroup of the symmetry group of the Hamiltonian. We find that the q-state quantum Potts model on the square lattice undergoes a discontinuous (first-order) phase transition for q=3 and q=4 and a continuous phase transition for q=2 (the two-dimensional quantum transverse Ising model).
We establish an intriguing connection between quantum phase transitions and bifurcations in the ground-state fidelity per lattice site, and construct the universal order parameter for quantum Ising model in a transverse magnetic field on an infinite-size square lattice in two spatial dimensions, a prototypical model with symmetry breaking order. This is achieved by computing ground-state wave functions in the context of the tensor network algorithm based on the infinite projected entangled-pair state representation. Our finding is applicable to any systems with symmetry breaking order, as a result of the fact that, in the conventional Landau-Ginzburg-Wilson paradigm, a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking captured by a local order parameter. In addition, a bifurcation in the reduced fidelity between two different reduced density matrices is also discussed.
For any translation-invariant quantum lattice system with a symmetry group G, we propose a practical and universal construction of order parameters which identify quantum phase transitions with symmetry-breaking order. They are defined in terms of the fidelity between a ground state and its symmetry-transformed counterpart, and are computed through tensor network representations of the ground-state wave function. To illustrate our scheme, we consider three quantum systems on an infinite lattice in one spatial dimension, namely, the quantum Ising model in a transverse magnetic field, the quantum spin-1/2XYX model in an external magnetic field, and the quantum spin-1 XXZ model with single-ion anisotropy. All these models have symmetry group Z(2) and exhibit broken-symmetry phases. We also discuss the role of the order parameters in identifying factorized states.
Abstract This paper explores an interesting relationship between the ground-state fidelity quantum phase transitions and bifurcations per lattice site, and proposes a universal order parameter for the quantum Ising model for a square lattice of infinite-size in two spatial dimensions as a prototype model with symmetry breaking order. The approach is based on computing ground-state wave functions using a tensor network algorithm utilizing a representation with infinite projected entangled-pair states. The results can be applied to any systems with symmetry breaking order, because in the conventional Landau-Ginzburg-Wilson paradigm a quantum system subjected to a phase transition exhibits spontaneous symmetry breaking quantified by a local order parameter. A reduced fidelity bifurcation between two different reduced density matrices is also explored.
We have investigated an intriguing connection between bifurcations, reduced fidelity per lattice site, local order parameter, universal order parameter, entropy and quantum phase transitions in the ground state for quantum three-state Potts model with two coupled infinite-size ladder system, in the context of the tensor network algorithm. The tensor network algorithm produces degenerate symmetry-breaking ground-state wave functions arising from the Z3 symmetry breaking, each of results from a randomly chosen initial state. We expect that our approach might provide further insights into critical phenomena in quantum many-body infinite lattice systems in condensed matter physics.
Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the qqq-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 XXXXXXXXX and anisotropic XYXXYXXYX models in an external magnetic field, and the antiferromagnetic spin-1 XXZXXZXXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.