Quantum phase transitions in a two-dimensional quantumXYXmodel: Ground-state fidelity and entanglement

Abstract

A systematic analysis is performed for quantum phase transitions in a two-dimensional anisotropic spin 1/2 anti-ferromagnetic XYX model in an external magnetic field. With the help of an innovative tensor network algorithm, we compute the fidelity per lattice site to demonstrate that the field-induced quantum phase transition is unambiguously characterized by a pinch point on the fidelity surface, marking a continuous phase transition. We also compute an entanglement estimator, defined as a ratio between the one-tangle and the sum of squared concurrences, to identify both the factorizing field and the critical point, resulting in a quantitative agreement with quantum Monte Carlo simulation. In addition, the local order parameter is "derived" from the tensor network representation of the system's ground state wave functions. Quantum critical phenomena are crucial in our understanding of the underlying physics in quantum manybody systems, especially in condensed matter systems, due to their relevance to high-T c superconductors, fractional quantum Hall liquids, and quantum magnets [1,2]. The latest advances in this area arise from quantum information science. Indeed, various entanglement measures have been widely applied to study condensed matter systems. Remarkably, for one-dimensional (1D) quantum systems, von-Neumann entropy, as a bipartite entanglement measure, turns out to be a good criterion to judge whether or not a system is at criticality [3,4,5,6,7,8,9]. On the other hand, fidelity, another basic notion in quantum information science, has demonstrated to be fundamental in characterizing phase transitions in quantum many-body systems [10,11,12,13]. This adds a new routine to explore quantum criticality in condensed matter physics from a quantum information perspective.However, with only a few notable exceptions [14,15], not much work has been done for two-dimensional (2D) quantum systems, due to great computational challenges. In fact, despite the existence of well-established numerical algorithms, such as exact diagonalization, quantum Monte Carlo (QMC), the density matrix renormalization group (DMRG) and series expansions, drawbacks become obvious when one deals with frustrated spin systems. A typical example is the QMC, which suffers from the notorious sign problem. However, a promising progress, inspired by new concepts from quantum information science, has been made in classical simulations of quantum many-body systems. The algorithms are based on an efficient representation of the system's wave functions through a tensor network. In particular, matrix product states (MPS) [16,17,18], a tensor network already present in DMRG, are used in the time-evolving block decimation (TEBD) algorithm to simulate time evolution in 1D quantum lattice systems [19,20], whereas projected entangled-pair states (PEPS) constitute the basis to simulate 2D quantum lattice systems [21,22].The aim of this paper is to show that the fidelity per lattice site, first introduced in Ref. [11], is able to unveil quantum criticalit...

Coherent and dissipative dynamics at quantum phase transitions

“…In quantum many-body systems, the establishment of a non-analytical behavior has been exploited to evidence CQTs in several contexts, which have been deeply scrutinized both analytically and numerically. We quote, for example, free-fermion models [422][423][424][425][426][427], interacting spin [428][429][430][431][432][433][434] and particle models [435][436][437][438][439][440][441], as well as systems presenting peculiar topological [442][443][444] and non-equilibrium steady-state transitions [445,446]. Recently, this issue has been also investigated at FOQTs, showing that the fidelity and the quantum Fisher information develop even sharper behaviors [381].…”

Universal Order Parameters and Quantum Phase Transitions: A Finite-Size Approach

“…The fidelity F (| φ 1 〉, | φ 2 〉) = |〈 φ 1 | φ 2 〉| between two states | φ 1 〉 and | φ 2 〉 scales as F (| φ 1 〉, | φ 2 〉) ~ d (| φ 1 〉, | φ 2 〉) L , with L the number of lattice sites. The fidelity per lattice site 4 5 6 7 d is the scaling parameter…”

“…Another approach makes use of the ground-state fidelity of a quantum many-body system 4 5 6 7 8 9 . For a quantum phase transition arising from SSB, a bifurcation appears in the ground-state fidelity per lattice site, with a critical point identified as a bifurcation point 10 11 12 .…”

Fidelity Mechanics: Analogues of the Four Thermodynamic Laws and Landauer’s Principle

“…The ground-state phase diagram may be mapped out by evaluating the ground-state fidelity per lattice site. As demonstrated in [ 26 , 27 , 28 , 29 , 30 , 49 , 50 , 73 , 74 , 75 ], the ground-state fidelity per lattice site is able to signal QPTs arising from symmetry-breaking order and/or topological order. Here, we restrict ourselves to briefly recall the definition of the ground-state fidelity per lattice site (also cf.…”