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Study on quantum phase transition in a 3-leg ladder model based on tensor network algorithm

Shenghao Li, Yuanyuan Wu, Yuancheng Luo

Published Mar 18, 2025• 0 citations
Publisher

Abstract

Tensor network (TN) algorithms for strongly correlated electron lattice systems have become one of the most important numerical methods for studying such systems. Based on the tensor network algorithm for spin ladder systems, the study of quantum phase transitions (QPTs) in spin ladder systems through string order parameters has shown that quantum information theory provides a universal method for studying quantum phase transitions: without knowing the local or non-local order parameters of the system, or whether local or non-local order parameters exist, the ground-state wave function of the system can be obtained. If the ground-state lacks the traditional Landau long-range order and thus has no local order parameters, or if several non-local order parameters are found through other means, some topological quantum phases can be completely described, thereby greatly enriching the traditional Landau symmetry-breaking phase transition theory.

DOI
10.1117/12.3065168

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The Tensor Network (TN) algorithm for strongly related electronic quantum lattice systems has become the uppermost algorithm for studying strongly related electronic quantum lattice systems in numerical computation. Focusing on the TN algorithm based on 2-D spin systems, this paper discusses the theoretical principles of the Projected Entangled Pair States (PEPS) algorithm and its implementation process. It also explores the infinite PEPS (iPEPS) algorithm for simulating the spin-1/2 infinite 2-D square lattice quantum XYX model. The critical phenomena of the two-dimensional XYX quantum system are analyzed from the perspectives of ground-state fidelity and order parameters, and the quantum critical point is identified. It is anticipated that with the further development of TN algorithms, the single-site fidelity theory of quantum phase transitions (QPTs) can be applied to various quantum many-body systems in condensed matter physics, thereby deepening our understanding of the extremely rich states of matter in the quantum world. Research indicates that quantum information theory provides a universal method for the study of QPTs, which can identify several ground-state fidelities and order parameters through the ground-state wave function of the system, several ground state fidelity and local order parameters can be identified to fully describe some quantum phase transition points, greatly enriching the traditional Landau theory of symmetry breaking QPTs.
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Groundstate fidelity phase diagram of the fully anisotropic two-leg spin-12\tfrac{1}{2}21​XXZladder
The fully anisotropic two-leg spin- XXZ ladder model is studied in terms of an algorithm based on the tensor network (TN) representation of quantum many-body states as an adaptation of projected entangled pair states to the geometry of translationally invariant infinite-size quantum spin ladders. The TN algorithm provides an effective method to generate the groundstate wave function, which allows computation of the groundstate fidelity per lattice site, a universal marker to detect phase transitions in quantum many-body systems. The groundstate fidelity is used in conjunction with local order and string order parameters to systematically map out the groundstate phase diagram of the ladder model. The phase diagram exhibits a rich diversity of quantum phases. These are the ferromagnetic, stripe ferromagnetic, rung singlet, rung triplet, Néel, stripe Néel and Haldane phases, along with the two XY phases XY 1 and XY 2.
A large class of quantum phase transitions for quantum lattice systems are characterized by local order parameters. It is shown that local order parameters may be systematically constructed from tensor network representations of quantum many-body ground state wave functions by investigating the reduced density matrices for local areas on an infinite-size lattice. Depending on whether or not the system symmetries are spontaneously broken, and whether or not the ground state fidelity per lattice site is continuous, there are four categories of quantum phase transitions for systems with local order parameters. Quantum phase transitions characterized by nonlocal order parameters are discussed, aiming at better understanding quantum systems with topological order.
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Dimerized antiferromagnetic spin-1/2 chains and ladders demonstrate quantum critical phase transition, the existence or absence of which is dependent on the dimerization and the dimerization pattern of the chain and the ladder, respectively. The gapped phases can not be distinguished by the conventional Landau long-range order parameters. However, they possess non-local topological string order parameters which can be used to classify different phases. We utilize the self-consistent free fermionic approximation and some standard results for exactly solved models to analytically calculate the string order parameters of dimerized spin chains. As a complement parameter the gapped phases possess the topological number, called the winding number and they are characterized by different integer values of the winding number. In order to calculate the string order parameters and winding numbers in dimerized spin chains and two-leg ladders we use analytical methods such as the Jordan-Wigner transformation, mean-field approximation, duality transformations, and some standard results available for the exactly 1D solve models. It is shown that the winding number provides the complementary framework to the string order parameter to characterize the topological gapped phases.