Kibble-Zurek behavior in one-dimensional disordered topological insulators

Spontaneous symmetry breaking is a foundational concept in physics. In condensed matter, it characterizes conventional continuous phase transitions but is absent at topological phase transitions such as the Berezinskii–Kosterlitz–Thouless (BKT) transition—as in the BKT case the expected norm (i.e., the magnitude) of the U(1) order parameter vanishes in the thermodynamic limit at all nonzero temperatures. Phenomena consistent with low-temperature broken symmetry have been observed, however, in many different BKT experiments. Examples include recent experiments on superconducting films and the seminal work on two-dimensional arrays of Josephson junctions. While the inaccessibility of the above thermodynamic limit partially explains this paradox in finite systems, the full dynamical framework of symmetry breaking at the BKT transition remains unresolved. Here we provide this by introducing the broader concept of . This encompasses both spontaneous symmetry breaking and the BKT case by allowing the expected norm of the order parameter to go to zero in the thermodynamic limit, provided its directional phase fluctuations are asymptotically smaller. We demonstrate this asymptotically slow directional mixing in the low-temperature BKT phase. This explicitly shows that the order parameter arbitrarily chooses some well-defined direction in the thermodynamic limit, predicting negligible phase fluctuations compared to the expected norm in arbitrarily large experimental BKT systems. Our results provide a model for directional mixing timescales across the diverse array of experimental BKT systems. We suggest various experiments. Published by the American Physical Society 2024

Quantum phase transition is one of the most interesting aspects in quantum many-body systems. Recently, geometric quantum discord has been introduced to signature the critical behavior of various quantum systems. However, it is well-known that topological quantum phase transition can not be described by the conventional Landau's symmetry breaking theory, and thus it is unknown that whether previous study can be applicable in this case. Here, we study the topological quantum phase transition in Kitaev's 1D p-wave spinless quantum wire model in terms of its ground state geometric quantum discord. The derivative of geometric quantum discord is nonanalytic at the critical point, in both zero temperature and finite temperature cases. The scaling behavior and the universality are verified numerically. Therefore, our results clearly show that all the key ingredients of the topological phase transition can be captured by the nearest neighbor and long-range geometric quantum discord.

1D topological systems for next-generation electronics

Order parameters make it possible to quantify the degree of structural ordering in a material and thus to apply as the reaction coordinates during the free-energy analysis of phase or structure transitions. Furthermore, order parameters are useful in determining the local structures of molecular groups during transition stages. However, identifying or developing local order parameters (LOPs) that are sensitive for specific materials and phases is a non-trivial task. In this study, the ability of LOPs to classify the solid and liquid structures of water at coexistence or triple points is investigated with the aid of supervised machine learning. The classification accuracy of a total of 179,738,433 combinations of 493 LOPs is automatically and systematically compared for water structures at the ice Ih-Ic-liquid coexistence point and the ice III-V-liquid and ice V-VI-liquid triple points. The optimal sets of two LOPs are found for each point, and sets of three LOPs are suggested for better accuracy.

We investigate Josephson currents in mesoscopic rings with a weak link which are in or near a topological superconducting phase. As a paradigmatic example, we consider the Kitaev model of a spinless p-wave superconductor in one dimension, emphasizing how this model emerges from more realistic settings based on semiconductor nanowires. We show that the flux periodicity of the Josephson current provides signatures of the topological phase transition and the emergence of Majorana fermions (MF) situated on both sides of the weak link even when fermion parity is not a good quantum number. In large rings, the MF hybridize only across the weak link. In this case, the Josephson current is h/e periodic in the flux threading the loop when fermion parity is a good quantum number but reverts to the more conventional h/2e periodicity in the presence of fermion-parity changing relaxation processes. In mesoscopic rings, the MF also hybridize through their overlap in the interior of the superconducting ring. We find that in the topological superconducting phase, this gives rise to an h/e-periodic contribution even when fermion parity is not conserved and that this contribution exhibits a peak near the topological phase transition. This signature of the topological phase transition is robust to the effects of disorder. As a byproduct, we find that close to the topological phase transition, disorder drives the system deeper into the topological phase. This is in stark contrast to the known behavior far from the phase transition, where disorder tends to suppress the topological phase.

We show that exotic spontaneous symmetry breaking appears in thermal topological phases by perturbing the exact solutions of quantum rotor models coupled to the three-dimensional toric code. The exotic Ising and XY transitions are shown to be in the same universality class, in drastic contrast to the conventional Wilson-Fisher classes without topological orders. Our results indicate that topological orders must be included to pin down the universality classes of thermal transitions in addition to order parameter symmetry and spatial dimension. We evaluate all the critical exponents and find that the exotic universality class is more stable under the couplings to acoustic phonons and disorder. Applying our results to experiments, we provide a plausible scenario in puzzlings of strongly correlated systems, including the absence of a specific heat anomaly in doped ${\mathrm{RbFe}}_{2}{\mathrm{As}}_{2}$.

Artificial spin ices are arrays of nano-scale magnetic islands correlated by the interactions of their associated macrospins. They have proven an excellent playground in which to study phase transitions and non-equilibrium phenomena. Originally envisaged as a two-dimensional analogue to the frustrated rare-earth pyrochlores, they are now seen in their own right as promising candidates for a wide range of applications, including nanomagnetic computation and magnonics. At the same time, the capability of tuning their behaviour---whether by means of the constituent material, the fabrication pattern, or the application of external stimuli---enables the realisation of unusual aspects of statistical physics. This thesis comprises a combined numerical and experimental study of artificial spin ice. The aims are twofold. First, it seeks to address how magnetic order and defect textures are influenced by the choice of lattice geometry. Second, it considers one route towards making artificial spin ice configurable via a coupling to a site-specific exchange bias. In the initial segment of this thesis, the recently studied pinwheel form of artificial spin ice is described. This is created by rotating each island in the square lattice about 45 degrees through its centre. The rotation angle of the islands acts as a proxy for a mechanism to vary the interactions between spins. A transition between antiferromagnetism in the square lattice and ferromagnetism in the pinwheel lattice is predicted. The phase diagram and critical exponents of the transition are obtained numerically. The nature of this transition is confirmed experimentally using in-situ Lorentz transmission electron microscopy on thermally annealed cobalt arrays. Varying degrees of thermalisation are observed across the samples, as well as an apparent change in the nature of the defects: from one-dimensional strings in square ice to two-dimensional vortex-like structures for geometries similar to pinwheel. The numerical scaling of these quantities is consistent with that predicted by the Kibble-Zurek mechanism. Finally, a two-dimensional hybrid artificial spin ice is outlined. In this, exchange bias is inserted at specific sites to constrain the magnetisation dynamics of individual islands. By examining correlations, a model for the influence of this pinning is constructed. As the density of constrained spins is varied, different magnetic textures are observed following a simulated field demagnetisation. These simulations show good agreement with results obtained experimentally. In this manner, local control over individual islands provides a route to tuning the global response of the array, thus making the system configurable. This is an essential step towards device-based applications. Taken together, these results illustrate the interplay between topology and magnetic order in artificial spin structures, and enable the exploration of critical phenomena in frozen and glassy systems. The findings presented here demonstrate conclusively that artificial spin ice is an excellent test bed with which to probe out-of-equilibrium dynamics. They will also underpin its potential use in fields which are reliant on adressing specific microstates, such as neuromorphic computing. .

Topological quantum phases cannot be characterized by local order parameters in the bulk. In this work however, we show that signatures of a topological quantum critical point do remain in local observables in the bulk, and manifest themselves as non-analyticities in their expectation values taken over a family of non-equilibrium states generated using a quantum quench protocol. The signature can be used for precisely locating the critical points in parameter space. A large class of initial states can be chosen for the quench (including finite temperature states), the sufficient condition being existence of a finite occupation-gradient with respect to energy for the single-particle critical mode. We demonstrate these results in tractable models of non-interacting fermions exhibiting topological phase transitions in one and two spatial dimensions. We also show that the non-analyticities can be absent if the gap-closing is non-topological, i.e., when it corresponds to no phase transition.

An extended Bose-Hubbard model (EBHM) with three- and four-body constraints can be feasible in cold atoms in an optical lattice. A rich phase structure including various symmetry-protected topological (SPT) phases is obtained numerically with suitable parameter settings and particle filling. The SPT phase is characterized by the Berry phase as a local topological order parameter and the structure of the entanglement spectrum (ES). Based on the presence of various topological phases, separated by gapless phase boundaries, the EBHM exhibits various bosonic topological pumps, which are constructed by connecting the different SPT phases without gap closing. The bulk topological pumps exhibit the plateau transitions characterized by many-body Chern numbers. For the system with boundary, the center of mass (CoM) under grand canonical ensemble elucidates the contributions of multiple edge states and reveals the topology of the system. We demonstrate that the interacting bosonic pumps obey the bulk-edge-correspondence.

Recent experiments [Science Advances 4 eaao4513 (2018)] have revealed the evidence of nodal-line superconductivity in half-Heusler superconductors, e.g., YPtBi. Theories have suggested the topological nature of such nodal-line supercon-ductivity and proposed the existence of surface Majorana flat bands on the (111) surface of half-Heusler superconductors. Due to the divergent density of states of the surface Majorana flat bands, the surface order parameter and the surface impurity play essential roles in determining the surface properties. We study the effect of the surface order parameter and the surface impurity on the surface Majorana flat bands of half-Heusler superconductors based on the Luttinger model. To be specific, we consider the topological nodal-line superconducting phase induced by the singlet-quintet pairing mixing, classify all the possible translationally invariant order parameters for the surface states according to irreducible representations of C3v point group, and demonstrate that any energetically favorable order parameter needs to break the time-reversal symmetry. We further discuss the energy splitting in the energy spectrum of surface Majorana flat bands induced by different order parameters and non-magnetic or magnetic impurities. We propose that the splitting in the energy spectrum can serve as the fingerprint of the pairing symmetry and mean-field order parameters. Our theoretical prediction can be examined in the future scanning tunneling microscopy experiments.

The topological one-way waveguide mode of the acoustic wave has recently been demonstrated in various meta-structure systems. Here, we show that in a topological phononic crystal with a symmetry-broken acoustic unit cell, the topological state possesses not just a “spin-like” pseudospin mode but also real spin angular momentum. By rotating the double-square units, the band of the phononic crystal will become inverse and induce both topological phase transition and spin angular momentum reversal. As such, by putting two topologically different systems together, the spin angular momentum dependent one-way interface modes can be selectively excited by acoustic spin sources, exhibiting robust transport protected by tight spin-momentum locking. The spin angular momentum density distribution in the unit cell and edge states shows that in addition to the pseudospin, there is a strong correlation between the real spin angular momentum and topological properties in this acoustic system, producing the topologically selective acoustic torque. Revealing the real spin angular momentum and associated acoustic spin torque properties of these topological phononic and acoustic systems will give people a more general understanding about symmetric breaking wave systems and help people to explore more potential applications of acoustic spins in various topological systems.

The phase transition in the polarization switching (PS) of vertical-cavity surface-emitting lasers (VCSELs) was recently reported to be a second-order phase transition (SOPT). However, some features of this phase transition indicate that the VCSEL’s PS (VPS) is different from the traditional SOPTs. Most of the phase transition investigations of the laser employ the laser’s intensity as the order parameter. In Landau’s paradigm, that parameter evolutes from zero to non-zero values, or vice versa, during SOPTs, corresponding to a transition between a disordered phase and an ordered phase. Nevertheless, in the VPS, the laser’s intensity remains constant before and after the PS, revealing an order-to-order transition. Furthermore, the laser’s transverse modes cannot transfer to each other through continuous deformations in geometry. That feature attributes a topological characteristic to the laser’s transverse modes. The spatial coherence of the laser also implements a globally geometric characteristic to the laser’s output. Accordingly, there are two similarities between the VPS and quantum phase transitions (QPTs) with topological order. First, both of them belong to the orderto- order phase transitions. Second, in both transitions, two ground states are orthogonal, and are degenerate at the critical point. This paper investigated the analogy between the QPT with topological order and the VPS, exploring that the VPS has a potential to simulate the QPTs of other physical systems.

Topological states of matter are characterized by global topological invariants which change their value across a topological quantum phase transition. It is commonly assumed that the transition between topologically distinct noninteracting gapped phases of fermions is necessarily accompanied by the closing of the band gap as long as the symmetries of the system are maintained. We show that such a quantum phase transition is possible without closing the gap in the case of a three-dimensional topological band insulator. We demonstrate this by calculating the free energy of the minimal model for a topological insulator, the Bernevig-Hughes-Zhang model, and show that as the band curvature continuously varies, a jump between the band gap minima corresponding to the topologically trivial and nontrivial insulators occurs. Therefore, this first order phase transition is a generic feature of three-dimensional topological band insulators. For a certain parameter range we predict a re-entrant topological phase transition. We discuss our findings in connection with the recent experimental observation of a discontinuous topological phase transition in a family of topological crystalline insulators.

We investigated quantum critical behaviours in the non-equilibrium steady state of a $XXZ$ spin chain with boundary Markovian noise using the Fisher information. The latter represents the distance between two infinitesimally close states, and its superextensive size scaling witnesses a critical behaviour due to a phase transition, since all the interaction terms are extensive. Perturbatively in the noise strength, we found superextensive Fisher information at anisotropy $|\Delta|\leqslant1$ and irrational $\frac{\arccos\Delta}{\pi}$ irrespective of the order of two non-commuting limits, i.e. the thermodynamic limit and the limit of sending $\frac{\arccos\Delta}{\pi}$ to an irrational number via a sequence of rational approximants. From this result we argue the existence of a non-equilibrium quantum phase transition with a critical phase $|\Delta|\leqslant1$. From the non-superextensivity of the Fisher information of reduced states, we infer that this non-equilibrium quantum phase transition does not have local order parameters but has non-local ones, at least at $|\Delta|=1$. In the non-perturbative regime for the noise strength, we numerically computed the reduced Fisher information which lower bounds the full state Fisher information, and is superextensive only at $|\Delta|=1$. Form the latter result, we derived local order parameters at $|\Delta|=1$ in the non-perturbative case. The existence of critical behaviour witnessed by the Fisher information in the phase $|\Delta|<1$ is still an open problem. The Fisher information also represents the best sensitivity for any estimation of the control parameter, in our case the anisotropy $\Delta$, and its superextensivity implies enhanced estimation precision which is also highly robust in the presence of a critical phase.

The topological photonics plays an important role in the fields of fundamental physics and photonic devices. The traditional method of designing topological system is based on the momentum space, which is not a direct and convenient way to grasp the topological properties, especially for the perturbative structures or coupled systems. Here, we propose an interdisciplinary approach to study the topological systems in real space through combining the information entropy and topological photonics. As a proof of concept, the Kagome model has been analyzed with information entropy. We reveal that the bandgap closing does not correspond to the topological edge state disappearing. This method can be used to identify the topological phase conveniently and directly, even the systems with perturbations or couplings. As a promotional validation, Su–Schrieffer–Heeger model and the valley-Hall photonic crystal have also been studied based on the information entropy method. This work provides a method to study topological photonic phase based on information theory, and brings inspiration to analyze the physical properties by taking advantage of interdisciplinarity.Graphical