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arXiv:1506.05027

Network topology transition at criticality

Chung-Pin Chou, Yihua Wang, Ming-Chiang Chung

Published Jun 16, 2015• 0 citations
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Abstract

Many-body systems when continuous phase transition occurs are mainly built in the interrelationship between particles, implemented through many-body correlations. Some of them may exhibit so-called topological order hardly measured by experiments. Therefore we need, beyond mean-field theory, the complex-systems approach that stresses the systemic complexity of many-body network at criticality. According to our previous study, network space experiences the homogeneous-heterogeneous transition invisible in traditional phase transitions. The network robustness can be a useful indicator to capture the critical phenomena of phase transitions with/without symmetry breaking. In this work, we demonstrate the idea of the change of robust networks is successfully applied to the well-known 1D quantum and 2D classical XY models.

DOI
10.48550/arxiv.1506.05027

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Many-body systems when continuous phase transition occurs are mainly built in the interrelationship between particles, implemented through many-body correlations. Some of them may exhibit so-called topological order hardly measured by experiments. Therefore we need, beyond mean-field theory, the complex-systems approach that stresses the systemic complexity of many-body network at criticality. According to our previous study, network space experiences the homogeneous-heterogeneous transition invisible in traditional phase transitions. The network robustness can be a useful indicator to capture the critical phenomena of phase transitions with/without symmetry breaking. In this work, we demonstrate the idea of the change of robust networks is successfully applied to the well-known 1D quantum and 2D classical XY models.
Can the topology of a network that consists of many particles interacting with each other change in complexity when a phase transition occurs? The answer to this question is particularly interesting to understand the nature of the phase transitions if the distinct phases do not break any symmetry, such as topological phase transitions. Here we present a novel theoretical framework established by complex network analysis for demonstrating that across a transition point of the topological superconductors, the network space experiences a homogeneous-heterogeneous transition invisible in real space. This transition is nothing but related to the robustness of a network to random failures. We suggest that the idea of the network robustness can be applied to characterizing various phase transitions whether or not the symmetry is broken.
The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
Determination and characterization of criticality in two-dimensional (2D) quantum many-body systems is one of the most important challenges and problems in modern physics. Based on the scaling theory in one-dimensional (1D) systems and tensor networks, we propose an efficient scheme to access the criticality of 2D quantum states. We demonstrate that the boundary state of a critical infinite projected entangled pair state can be described by 1D conformal field theory, which provides a solid indicator that allows one to identify and characterize the criticality of 2D states. Our scheme is verified on the resonating valence bond (RVB) states on kagome and square lattices, where the boundary state of the honeycomb RVB is found to be described by a c=1c=1c=1 conformal field theory. We apply our scheme also to the ground state of the spin-1/2 XXZ model on honeycomb lattice, illustrating the difficulties of standard variational tensor network approaches to study the critical ground states. Our scheme is of high versatility and flexibility, and can be further applied to investigate the quantum criticality in many other phenomena, such as finite-temperature and topological phase transitions.
We establish an intriguing connection between quantum phase transitions and bifurcations in the reduced fidelity between two different reduced density matrices for quantum lattice many-body systems with symmetry-breaking order. Our finding is based on the observation that in the conventional Landau–Ginzburg–Wilson paradigm a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking that is captured by a local-order parameter, which in turn results in an essential change of the reduced density matrix in the symmetry-broken phase. Two quantum systems on an infinite lattice in one spatial dimension, i.e. a quantum Ising model in a transverse magnetic field and a quantum spin-1/2 XYX model in an external magnetic field, are considered in the context of the tensor network algorithm based on the matrix product state representation.
The physics that emerges when large numbers of particles interact can be complex and exotic. The collective behaviour may not reflect the underlying constituents, for example fermionic quasiparticles can emerge from models of interacting bosons. Due to this emergent complexity, manybody phenomena can be very challenging to study, but also very useful. A theoretical understanding of such systems is important for robust quantum information storage and processing. The emergent, macroscopic physics can be classified using the idea of a quantum phase. All models within a given phase exhibit similar low-energy emergent physics, which is distinct from that displayed by models in different phases. In this thesis, we utilise tensor networks to study many-body systems in a range of quantum phases. These include topologically ordered phases, gapless symmetry-protected phases, and symmetry-enriched topological phases.
We discuss the boundary critical behaviors of two-dimensional (2D) quantum phase transitions with fractionalized degrees of freedom in the bulk, motivated by the fact that usually it is the one-dimensional boundary that is exposed and can be conveniently probed in many experimental platforms. In particular, we mainly discuss boundary criticality of two examples: (i) the quantum phase transition between a 2D Z2{Z}_{2}Z2​ topological order and an ordered phase with spontaneous symmetry breaking; (ii) the continuous quantum phase transition between metal and a particular type of Mott insulator [U(1)\mathrm{U}(1)U(1) spin liquid]. In particular, we obtain the critical exponents and scaling laws of these exotic quantum phase transitions when the systems are probed from the boundary in proposed experimental setup. These critical exponents obtained are significantly different from those one would see through bulk measurements. This theoretical study could be relevant to many purely 2D systems, where recent experiments have found correlated insulator, superconductor, and metal in the same phase diagram.