Topology and Phase Transitions: The Case of the Short Range Spherical Model

We characterize the topology of the phase space of the Berlin-Kac spherical model in the context of the so called Topological Hypothesis, for spins lying in hypercubic lattices of dimension d. For zero external field we are able to characterize the topology exactly, up to homology. We find that, even though there is a continuum of changes in the topology of the corresponding manifolds, for d ≥ 3 there are abrupt discontinuities in some topological functions that could be good candidates to associate with the phase transitions that occur at the thermodynamic level. We show however that these changes do not coincide with the phase transitions and conversely, that no topological discontinuity can be associated to the points where the phase transitions take place. At variance with what happens in the Mean Field version of this same model, we show that these abrupt topological changes are accessible thermodynamically. We conclude that, even in short range systems, the topological mechanism does not seem to be responsible for the triggering of a phase transition. We also analyze the case of spins connected to a macroscopic number of (but not all) neighbors, and find that, similar to the results found for the fully connected version, in this case the topological hypothesis seems to hold: the phase transition coincides with an accumulation point of the topological changes present in configuration space. The question of the ensemble equivalence in the short range spherical model is also considered.     

Discord under the influence of a quantum phase transition

This paper studies the discord of a bipartite two-level system coupling to an XY spin-chain environment in a transverse field and investigates the relationship between the discord property and the environment’s quantum phase transition.The results show that the quantum discord is also able to characterize the quantum phase transitions.We also discuss the difference between discord and entanglement,and show that quantum discord may reveal more general information than quantum entanglement for characterizing the environment’s quantum phase transition.     

Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.     

Quantum phase transitions in electronic systems

Quantum phase transitions occur at zero temperature when some non‐thermal control‐parameter like pressure or chemical composition is changed. They are driven by quantum rather than thermal fluctuations. In this review we first give a pedagogical introduction to quantum phase transitions and quantum critical behavior emphasizing similarities with and differences to classical thermal phase transitions. We then illustrate the general concepts by discussing a few examples of quantum phase transitions occurring in electronic systems. The ferromagnetic transition of itinerant electrons shows a very rich behavior since the magnetization couples to additional electronic soft modes which generates an effective long‐range interaction between the spin fluctuations. We then consider the influence of rare regions on quantum phase transitions in systems with quenched disorder, taking the antiferromagnetic transitions of itinerant electrons as a primary example. Finally we discuss some aspects of the metal‐insulator transition in the presence of quenched disorder and interactions.     

含有Dzyaloshinskii-Moriya相互作用的自旋1键交替海森伯模型的量子相变和拓扑序标度

利用张量网络表示的无限矩阵乘积态算法研究了含有Dzyaloshinskii-Moriya (DM)相互作用的键交替海森伯模型的量子相变和临界标度行为.基于矩阵乘积态的基态波函数计算了系统的量子纠缠熵及非局域拓扑序.数据表明,随着键交替强度变化,系统从拓扑有序的Haldane相转变为局域有序的二聚化相.同时DM相互作用抑制了系统的二聚化,并最终打破系统的完全二聚化.另外,通过对相变点附近二聚化序的一阶导数和长程弦序的数值拟合,分别得到了此模型相变的特征临界指数a和b的值.结果表明,随着DM相互作用强度的增强, a逐渐减小,同时b逐渐增大. DM相互作用强度影响着此模型的临界行为.针对此模型的临界性质的研究,揭示了量子自旋相互作用的彼此竞争机制,对今后研究含有DM相互作用的自旋多体系统中拓扑量子相变临界行为提供一定的借鉴与参考.     

Gauge theories of Josephson junction arrays

We show that the zero-temperature physics of planar Josephson junction arrays in the self-dual approximation is governed by an Abelian gauge theory with a periodic mixed Chern-Simons term describing the charge-vortex coupling. The periodicity requires the existence of (Euclidean) topological excitations which determine the quantum phase structure of the model. The electric-magnetic duality leads to a quantum phase transition between a superconductor and a superinsulator at the self-dual point. We also discuss in this framework the recently proposed quantum Hall phases for charges and vortices in presence of external offset charges and magnetic fluxes: we show how the periodicity of the charge-vortex coupling can lead to transitions to anyon superconductivity phases. We finally generalize our results to three dimensions, where the relevant gauge theory is the so-called BF system with an antisymmetric Kalb-Ramond gauge field.     

一维扩展量子罗盘模型的拓扑序和量子相变

应用矩阵乘积态表示的无限虚时间演化块算法,研究了扩展的量子罗盘模型.为了深入研究该模型的长程拓扑序和量子相变,基于奇数键和偶数键,引入了奇数弦关联和偶数弦关联,计算了保真度、奇数弦关联、偶数弦关联、奇数弦关联饱和性与序参量.弦关联表现出三种截然不同的行为:衰减为零、单调饱和与振荡饱和.基于弦关联的以上特征,给出了量子罗盘模型的基态序参量相图.在临界区,局域磁化强度和单调奇弦序参量的临界指数β=1/8表明:相变的普适类是Ising类型.此外,保真度探测到的相变点、连续性与非连续性和序参量的结果一致.     

Dynamics of a quantum phase transition

We present two approaches to the dynamics of a quench-induced phase transition in the quantum Ising model. One follows the standard treatment of thermodynamic second order phase transitions but applies it to the quantum phase transitions. The other approach is quantum, and uses Landau-Zener formula for transition probabilities in avoided level crossings. We show that predictions of the two approaches of how the density of defects scales with the quench rate are compatible, and discuss the ensuing insights into the dynamics of quantum phase transitions.     

Phase Transitions and Topology Changes in Configuration Space

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly strong topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: (i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; (ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.     

Chemical memory erasure; a photochemical model

In this paper we propose an exactly solvable model of a topological insulator defined on a spin- \(\tfrac{1}{2}\) square decorated lattice. Itinerant fermions defined in the framework of the Haldane model interact via the Kitaev interaction with spin- \(\tfrac{1}{2}\) Kitaev sublattice. The presented model, whose ground state is a non-trivial topological phase, is solved exactly. We have found out that various phase transitions without gap closing at the topological phase transition point outline the separate states with different topological numbers. We provide a detailed analysis of the model’s ground-state phase diagram and demonstrate how quantum phase transitions between topological states arise. We have found that the states with both the same and different topological numbers are all separated by the quantum phase transition without gap closing. The transition between topological phases is accompanied by a rearrangement of the spin subsystem’s spectrum from band to flat-band states.     

Topology, phase transitions, and the spherical model

The topological hypothesis states that phase transitions should be related to changes in the topology of configuration space. The necessity of such changes has already been demonstrated. We characterize exactly the topology of the configuration space of the short range Berlin-Kac spherical model, for spins lying in hypercubic lattices of dimension d. We find a continuum of changes in the topology and also a finite number of discontinuities in some topological functions. We show, however, that these discontinuities do not coincide with the phase transitions which happen for d > or = 3, and conversely, that no topological discontinuity can be associated with them. This is the first short range, confining potential for which the existence of special topological changes are shown not to be sufficient to infer the occurrence of a phase transition.     

Geometric phase contribution to quantum nonequilibrium many-body dynamics

We study the influence of geometry of quantum systems underlying space of states on its quantum many-body dynamics. We observe an interplay between dynamical and topological ingredients of quantum nonequilibrium dynamics revealed by the geometrical structure of the quantum space of states. As a primary example we use the anisotropic XY ring in a transverse magnetic field with an additional time-dependent flux. In particular, if the flux insertion is slow, nonadiabatic transitions in the dynamics are dominated by the dynamical phase. In the opposite limit geometric phase strongly affects transition probabilities. This interplay can lead to a nonequilibrium phase transition between these two regimes. We also analyze the effect of geometric phase on defect generation during crossing a quantum-critical point.     

Chiral universality class of normal-superconducting and exciton condensation transitions on surface of topological insulator

New two-dimensional systems such as the surfaces of topological insulators (TIs) and graphene offer the possibility of experimentally investigating situations considered exotic just a decade ago. These situations include the quantum phase transition of the chiral type in electronic systems with a relativistic spectrum. Phonon-mediated (conventional) pairing in the Dirac semimetal appearing on the surface of a TI causes a transition into a chiral superconducting state, and exciton condensation in these gapless systems has long been envisioned in the physics of narrow-band semiconductors. Starting from the microscopic Dirac Hamiltonian with local attraction or repulsion, the Bardeen–Cooper–Schrieffer type of Gaussian approximation is developed in the framework of functional integrals. It is shown that owing to an ultrarelativistic dispersion relation, there is a quantum critical point governing the zero-temperature transition to a superconducting state or the exciton condensed state. Quantum transitions having critical exponents differ greatly from conventional ones and belong to the chiral universality class. We discuss the application of these results to recent experiments in which surface superconductivity was found in TIs and estimate the feasibility of phonon pairing.     

Quantum phase transitions and vortex dynamics in superconducting networks

Josephson-junction arrays are ideal model systems to study a variety of phenomena such as phase transitions, frustration effects, vortex dynamics and chaos. In this review, we focus on the quantum dynamical properties of low-capacitance Josephson-junction arrays. The two characteristic energy scales in these systems are the Josephson energy, associated with the tunneling of Cooper pairs between neighboring islands, and the charging energy, which is the energy needed to add an extra electron charge to a neutral island. The phenomena described in this review stem from the competition between single-electron effects with the Josephson effect. They give rise to (quantum) superconductor–insulator phase transitions that occur when the ratio between the coupling constants is varied or when the external fields are varied. We describe the dependence of the various control parameters on the phase diagram and the transport properties close to the quantum critical points. On the superconducting side of the transition, vortices are the topological excitations. In low-capacitance junction arrays these vortices behave as massive particles that exhibit quantum behavior. We review the various quantum–vortex experiments and theoretical treatments of their quantum dynamics.     

Notes on Black Hole Phase Transitions

In these notes we present a summary of existing ideas about phase transitions of black hole spacetimes in semiclassical gravity and offer some thoughts on three possible scenarios or mechanisms by which these transitions could take place. We begin with a review of the thermodynamics of a black hole system and emphasize that the phase transition is driven by the large entropy of the black hole horizon. Our first theme is illustrated by a quantum atomic black hole system, generalizing to finite-temperature a model originally offered by Bekenstein. In this equilibrium atomic model, the black hole phase transition is realized as the abrupt excitation of a high energy state, suggesting analogies with the study of two-level atoms. Our second theme argues that the black hole system shares similarities with the defect-mediated Kosterlitz–Thouless transition in condensed matter. These similarities suggest that the black hole phase transition may be more fully understood by focusing upon the dynamics of black holes and white holes, the spacetime analogy of vortex and antivortex topological defects. Finally, we compare the black hole phase transition to another transition driven by an (exponentially) increasing density of states, the Hagedorn transition first found in hadron physics in the context of dual models or the old string theory. In modern string theory the Hagedorn transition is linked by the Maldacena conjecture to the Hawking–Page black hole phase transition in Anti-de Sitter (AdS) space, as observed by Witten. Thus, the dynamics of the Hagedorn transition may yield insight into the dynamics of the black hole phase transition. We argue that characteristics of the Hagedorn transition are already contained within the dynamics of classical string systems. Our third theme points to carrying out a full nonperturbative and nonequilibrium analysis of the large N behavior of classical SU(N) gauge theories to understand its Hagadorn transition. By invoking the Maldacena conjecture we can then gain valuable insight into black hole phase transitions in AdS space.     

Vortices and other topological defects in non-equilibrium phase transitions of superfluid He

Rapidly evolving non-equilibrium phenomena, associated with phase transitions from meta-stable states, provide examples of most complex dynamics. Condensed matter many-particle quantum systems, which are described in terms of a coherent quantum field, are a particularly useful environment for such studies. Among them, the ³He superfluids display the largest variety of topological defects of different dimensionality and structure, owing to the large degree of freedom in the rearrangement of their multi-component p-wave order parameter field. A few measurements exist which monitor the non-equilibrium phase transition dynamics and display defect formation.     

Fidelity in topological superconductors with end Majorana fermions

Fidelity and fidelity susceptibility are introduced to investigate the topological superconductors with end Majorana fermions. A general formalism is established to calculate the fidelity and fidelity susceptibility by solving Bogoliubov–de Gennes equations. Both clean and disordered systems are studied within this formalism, and the results show that the fidelity susceptibility serves as a valid indicator for the topological quantum phase transition which signals the appearance of Majorana fermions. Our study provides a useful tool to investigate the topological quantum phase transition in superconductors, which is helpful to find topological phases in various systems.     

Quantum Probing Topological Phase Transitions by Non‐Markovianity

Understanding the physical significance and probing the global invariants characterizing quantum topological phases in extended systems is a main challenge in modern physics with major impact in different areas of science. Here, a quantum‐information‐inspired probing method is proposed where topological phase transitions are revealed by a non‐Markovianity quantifier. The idea is illustrated by considering the decoherence dynamics of an external read‐out qubit that probes a Su–Schrieffer–Heeger (SSH) chain with either pure dephasing or dissipative coupling. Qubit decoherence features and non‐Markovianity measure clearly signal the topological phase transition of the SSH chain.     

Topological phase transition in anisotropic square-octagon lattice with spin–orbit coupling and exchange field

We investigate the topological phase transitions in an anisotropic square-octagon lattice in the presence of spin–orbit coupling and exchange field. On the basis of the Chern number and spin Chern number, we find a number of topologically distinct phases with tuning the exchange field, including time-reversal-symmetry-broken quantum spin Hall phases, quantum anomalous Hall phases and a topologically trivial phase. Particularly, we observe a coexistent state of both the quantum spin Hall effect and quantum anomalous Hall effect. Besides, by adjusting the exchange filed, we find the phase transition from time-reversal-symmetry-broken quantum spin Hall phase to spin-imbalanced and spin-polarized quantum anomalous Hall phases, providing an opportunity for quantum spin manipulation. The bulk band gap closes when topological phase transitions occur between different topological phases. Furthermore, the energy and spin spectra of the edge states corresponding to different topological phases are consistent with the topological characterization based on the Chern and spin Chern numbers.