Berezinskii-Kosterlitz-Thouless Phase Research Articles

The interplay of topology and antiferromagnetic order in two-dimensional van der Waals crystals of (Ni x Fe1−x )2P2S6

The Mermin–Wagner theorem forbids spontaneous symmetry breaking of spins in one/two-dimensional (2D) systems at a finite temperature and rules out the stabilization of this ordered state. However, it does not apply to all types of phase transitions in low dimensions, such as the topologically ordered phase rigorously shown by Berezinskii–Kosterlitz–Thouless (BKT) and experimentally realized in very limited systems such as superfluids and superconducting thin films. Quasi-2D van der Waals magnets provide an ideal platform to investigate the fundamentals of low-dimensional magnetism. We explored the quasi-2D honeycomb antiferromagnetic single crystals of (Ni x Fe1−x )2P2S6 (x = 1, 0.7, 0.5, 0.3, and 0) using in-depth temperature-dependent Raman measurements supported by first-principles calculations of the phonon frequencies. Quite surprisingly, we observed renormalization of the phonon modes much below the long-range magnetic ordered temperature attributed to the topological ordered state, namely the BKT phase, which is also found to change as a function of doping. The extracted critical exponent of the order-parameter (spin–spin correlation length, ξ(T) ) evinces the signature of a topologically active state driven by vortex–antivortex excitations. As a function of doping, a tunable transition from paramagnetic to antiferromagnetic ordering is shown via phonons reflected in the strong renormalization of the self-energy parameters of the Raman active phonon modes. The extracted exchange parameter (J) is found to vary by ∼100% by increasing the value of doping, ranging from ∼6 meV (for x = 0.3) to 13 meV (for x = 1).

An exactly solvable model of randomly pinned charge density waves in two dimensions

The nature of the interplay between fluctuations and quenched random disorder is a long-standing open problem, particularly in systems with a continuous order parameter. This lack of a full theoretical treatment has been underscored by recent advances in experiments on charge density wave materials. To address this problem, we formulate an exactly solvable model of a two-dimensional randomly pinned incommensurate charge density wave, and use the large-N technique to map out the phase diagram and order parameter correlations. Our approach captures the physics of the Berezinskii–Kosterlitz–Thouless phase transition in the clean limit at large N. We pay particular attention to the roles of thermal fluctuations and quenched random field disorder in destroying long-range order, finding a novel crossover between weakly- and strongly-disordered regimes.

Berezinskii-Kosterlitz-Thouless transition from neural network flows.

We adopt the neural network (NN) flow method to study the Berezinskii-Kosterlitz-Thouless (BKT) phase transitions of the two-dimensional q-state clock model with q≥4. The NN flow consists of a sequence of the same units that proceed with the flow. This unit is a variational autoencoder trained by the data of Monte Carlo configurations in unsupervised learning. To gauge the difference among the ensembles of Monte Carlo configurations at different temperatures and the uniqueness of the ensemble of NN-flow states, we adopt the Jensen-Shannon divergence (JSD) as the information-distance measure "thermometer." This JSD thermometer compares the probability distribution functions of the mean spin value of two ensembles of states. Our results show that the NN flow will flow an arbitrary spin state to some state in a fixed-point ensemble of states. The corresponding JSD of the fixed-point ensemble takes a unique profile with peculiar features, which can help to identify the critical temperatures of BKT phase transitions of the underlying Monte Carlo configurations.

Flux-flow instability across Berezinskii Kosterlitz Thouless phase transition in KTaO3 (111) based superconductor

The nature of energy dissipation in 2D superconductors under perpendicular magnetic field at small current excitations has been extensively studied over the past two decades. However, dissipation mechanisms at high current drives remain largely unexplored. Here we report on the distinct behavior of energy dissipation in the AlOx/KTaO3 (111) system hosting 2D superconductivity in the intermediate disorder regime. The results show that below the Berezinskii Kosterlitz Thouless (BKT) phase transition temperature (TBKT), hot-spots and Larkin Ovchinnikov type flux-flow instability (FFI) are the major channels of dissipation, leading to pronounced voltage instability at large currents. Furthermore, such FFI leads to a rare observation of clockwise hysteresis in current-voltage characteristics within the temperature range TBKT < T < TC (TC is superconducting transition temperature). These findings deepen our understanding of how a BKT system ultimately transforms to a normal state under increasing current.

Anisotropic deformation of the 6-state clock model: Tricritical-point classification

The two-dimensional q-state clock models exhibit the Berezinskii–Kosterlitz–Thouless (BKT) transition for q≥5 since they are a subset of the isotropic XY model. We examine the 6-state clock model with an anisotropic deformation. Selecting the 6-state Potts model as a source of the deformation, the model naturally violates the discrete rotational symmetry of the clock model. We introduce the anisotropic deformation parameter α in the clock model interpolating the clock (α=1) and the Potts (α=0) models. We employ the corner transfer matrix renormalization group method to analyze the phase transitions on the square lattice in the thermodynamic limit. Three different phases and phase transitions are identified. The phase diagram is constructed, and we determine a tricritical point at αc=0.21405(4) and Tc=0.834017(5). Analyzing the latent heat and the entanglement entropy in the vicinity of the Tc(αc), we observe a single discontinuous phase transition and two BKT phase transitions meeting in the tricritical point. The tricritical point exhibits a phase transition of the second order with the critical exponents β≈1/10 and δ≈14. We conjecture that an infinitesimal surrounding of the tricritical point consists of the three fundamental phase transitions, in which the first and the BKT orders gradually weaken into the second-order tricritical point.

Emergent Halperin–Saslow mode, gauge glass and quenched disorders in quantum Ising magnet TmMgGaO4

Quenched disorders could bring novel quantum excitations and models to certain quantum magnets. Motivated by recent experiments on the quantum Ising magnet TmMgGaO4, we explore the effects of the quenched disorder and the interlayer coupling in this triangular lattice Ising antiferromagnet. It is pointed out that the weak quenched (nonmagnetic) disorder would convert the emergent 2D Berezinskii–Kosterlitz–Thouless (BKT) phase and the critical region into a U(1) gauge glass. There will be an emergent Halperin–Saslow mode associated with this gauge glass. Using the Imry-Ma’s renormalization group result, we explain the fate of the finite-field [Formula: see text] symmetry breaking transition at the low temperatures. The ferromagnetic interlayer coupling would suppress the BKT phase and generate a tiny ferromagnetism. With quenched disorders, this interlayer coupling changes the 2D gauge glass into a 3D gauge glass, and the Halperin–Saslow mode persists. This work merely focuses on addressing a phase regime in terms of emergent U(1) gauge glass behaviors and hope to inspire future works and thoughts in weakly disordered frustrated magnets in general.

Two-dimensional superconducting nature of Bi2Sr2CaCu2O8+δ thin films revealed by BKT transition.

High-quality Bi2Sr2CaCu2O8+δ superconducting thin films are successfully grown on a SrTiO3 substrate by the Pulsed Laser Deposition technique. Superconducting critical transition temperatures Tc,zero have reached up to 85 K by using optimized growth parameters. In addition, we demonstrated the two-dimensional nature of the superconductivity of thin films by virtue of exhibiting Berezinskii-Kosterlitz-Thouless (BKT) physics and anisotropic magnetic response. Furthermore, three distinct regimes are identified based on the analysis of direct current resistance. The non-Fermi liquid phase and BKT phase fluctuation zone almost perfectly merge together, which implies that the system undergoes a unique topological state that is determined by the BKT phase fluctuation preceding the onset of the superconducting state. The emergence of such a topological state radically differentiates from the three-dimensional superconducting transition, which spontaneously breaks the gauge symmetry. The current studies on the Bi2Sr2CaCu2O8+δ superconducting thin films provide some new insights for understanding the rich quantum states of matter that emerge in the vicinity of the superconducting phase transition and highlight the significant role of BKT fluctuation on two-dimensional superconducting transition.

An Elementary Proof of Phase Transition in the Planar XY Model

Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer (Commun Math Phys 81(4):527–602, 1981) and establishes a Berezinskii–Kosterlitz–Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers (Probab Relat Fields, 2021) on delocalisation of general integer-valued height functions, and classical correlation inequalities.

Berezinskii-Kosterlitz-Thouless phases in ultrathin PbTiO3/SrTiO3 superlattices

We study the emergence of Berezinskii-Kosterlitz-Thouless (BKT) phases in ${({\mathrm{PbTiO}}_{3})}_{3}/{({\mathrm{SrTiO}}_{3})}_{3}$ superlattices by means of second-principles simulations. Between a tensile epitaxial strain of $\ensuremath{\epsilon}=0.25--1\phantom{\rule{0.16em}{0ex}}%$, the local dipole moments within the superlattices are confined to the film-plane, and thus the polarization can be effectively considered as two-dimensional. The analysis of the decay of the dipole-dipole correlation with the distance, together with the study of the density of defects and its distribution as a function of temperature, supports the existence of a BKT phase in a range of temperature mediating the ordered ferroelectric (stable at low $T$), and the disordered paraelectric phase that appears beyond a critical temperature ${T}_{\mathrm{BKT}}$. This BKT phase is characterized by quasi-long-range order (whose signature is a power-law decay of the correlations with the distance), and the emergence of tightly bound vortex-antivortex pairs whose density is determined by a thermal activation process. The proposed ${\mathrm{PbTiO}}_{3}/{\mathrm{SrTiO}}_{3}$ superlattice model and the imposed mechanical boundary conditions are both experimentally feasible, making it susceptible for an experimental observation of these new topological phases in ferroelectric materials.

The Double-Peak Specific Heat in the Two Dimensional J1–J2 3-State Clock Model

By the extensive tensor network algorithms, the J1–J2 3-state clock model is investigated. We focus on the case of J1 > 0, J2 < 0, in which the double-peak structure appears in the curve of the specific heat Cv versus the temperature T. The four parameters \(J_{2} = - 0.2, - 0.8, - 1.4, - 2\) are chosen for the detailed numerical simulation in unit of J1 = 1. The mismatch of peak position between entanglement entropy (EE) and Cv suggests the existence of two Berezinskii–Kosterlitz–Thouless (BKT) phase transitions in case of \(J_{2} = - 0.2, - 0.8\), where the peaks of EE lie inbetween the double peaks of Cv. In the case of J2 = −1.4, the first-peak temperature of Cv and of EE are very close. With further increasing of |J2|, the sequence of the first peak swaps, i.e., the first peak of EE arises at a lower temperature than of Cv. We believe that there is a critical |J2|, above which the first peak of Cv does not correspond to the BKT phase transition. The location offset of the second peak between EE and Cv becomes smaller with |J2| increasing. In addition, the double-peak structure of the specific heat still holds when |J2| is large enough.

Berezinskii–Kosterlitz–Thouless transition – A universal neural network study with benchmarks

Using a supervised neural network (NN) trained once on a one-dimensional lattice of 200 sites, we calculate the Berezinskii–Kosterlitz–Thouless phase transitions of the two-dimensional (2D) classical XY and the 2D generalized classical XY models. In particular, both the bulk quantities Binder ratios and the spin states of the studied systems are employed to construct the needed configurations for the NN prediction. By applying semiempirical finite-size scaling to the relevant data, the critical points obtained by the NN approach agree well with the known results established in the literature. This implies that for each of the considered models, the determination of its various phases requires only a little information. The outcomes presented here demonstrate convincingly that the employed universal NN is not only valid for the symmetry breaking related phase transitions, but also works for calculating the critical points of the phase transitions associated with topology. The efficiency of the used NN in the computation is examined by carrying out several detailed benchmark calculations.

Scaling of current-voltage characteristics without and with finite superfluid phase stiffness below [formula omitted

Superfluid phase stiffness (SPS) of NCBCO superconductor as a function of T has been extracted by using the Ambegaokar–Halperin–Nelson–Siggia (AHNS) theory at the zero magnetic field. The Berezinskii–Kosterlitz–Thouless (BKT) phase transition temperature, TBKT has been extracted to be 54.0 K. The idea of the vortex glass (VG) transition has been applied to the resistive states of the BKT phase transition observed at 54.0 K. We have extracted the static exponent required in the Fisher, Fisher and Huse (FFH) scaling equation.The scaling of the current-voltage (IV) curves around 54.0 K with and without finite SPS is possible within the framework of the FFH theory.

Short-Range Berezinskii-Kosterlitz-Thouless Phase Characterization for the q-State Clock Model.

Beyond the usual ferromagnetic and paramagnetic phases present in spin systems, the usual q-state clock model presents an intermediate vortex state when the number of possible orientations q for the system is greater than or equal to 5. Such vortex states give rise to the Berezinskii-Kosterlitz-Thouless (BKT) phase present up to the XY model in the limit . Based on information theory, we present here an analysis of the classical order parameters plus new short-range parameters defined here. Thus, we show that even using the first nearest neighbors spin-spin correlations only, it is possible to distinguish the two transitions presented by this system for q greater than or equal to 5. Moreover, the appearance at relatively low temperature and disappearance of the BKT phase at a rather fix higher temperature is univocally determined by the short-range interactions recognized by the information content of classical and new parameters.

Percolation of the two-dimensional XY model in the flow representation.

We simulate the two-dimensional XY model in the flow representation by a worm-type algorithm, up to linear system size L=4096, and study the geometric properties of the flow configurations. As the coupling strength K increases, we observe that the system undergoes a percolation transition K_{perc} from a disordered phase consisting of small clusters into an ordered phase containing a giant percolating cluster. Namely, in the low-temperature phase, there exhibits a long-ranged order regarding the flow connectivity, in contrast to the quasi-long-range order associated with spin properties. Near K_{perc}, the scaling behavior of geometric observables is well described by the standard finite-size scaling ansatz for a second-order phase transition. The estimated percolation threshold K_{perc}=1.1053(4) is close to but obviously smaller than the Berezinskii-Kosterlitz-Thouless (BKT) transition point K_{BKT}=1.1193(10), which is determined from the magnetic susceptibility and the superfluid density. Various interesting questions arise from these unconventional observations, and their solutions would shed light on a variety of classical and quantum systems of BKT phase transitions.

Evidence of the Berezinskii-Kosterlitz-Thouless phase in a frustrated magnet

The Berezinskii-Kosterlitz-Thouless (BKT) mechanism, building upon proliferation of topological defects in 2D systems, is the first example of phase transition beyond the Landau-Ginzburg paradigm of symmetry breaking. Such a topological phase transition has long been sought yet undiscovered directly in magnetic materials. Here, we pin down two transitions that bound a BKT phase in an ideal 2D frustrated magnet TmMgGaO4, via nuclear magnetic resonance under in-plane magnetic fields, which do not disturb the low-energy electronic states and allow BKT fluctuations to be detected sensitively. Moreover, by applying out-of-plane fields, we find a critical scaling behavior of the magnetic susceptibility expected for the BKT transition. The experimental findings can be explained by quantum Monte Carlo simulations applied on an accurate triangular-lattice Ising model of the compound which hosts a BKT phase. These results provide a concrete example for the BKT phase and offer an ideal platform for future investigations on the BKT physics in magnetic materials.

Influence of Topological Phase Transition on Entanglement in the Spin-1 Antiferromagnetic XX Model in Two Dimensions

In this paper, we study the von Neumann entanglement entropy as a measure of the quantum entanglement in the spin-1 two-dimensional XX model with single-ion anisotropy. We use the bond operator formalism and consider the range of large anisotropy D and in the neighborhood of the critical point $$D_\mathrm{c}$$ . One discusses the influence of the Berezinskii–Kosterlitz–Thouless phase transition (BKT) that occurs at critical anisotropy point $$D_\mathrm{c}$$ , or the transition of the topological order of vortices and disordered phase on quantum entanglement.

Machine-learning study using improved correlation configuration and application to quantum Monte Carlo simulation.

We use the Fortuin-Kasteleyn representation-based improved estimator of the correlation configuration as an alternative to the ordinary correlation configuration in the machine-learning study of the phase classification of spin models. The phases of classical spin models are classified using the improved estimators, and the method is also applied to the quantum Monte Carlo simulation using the loop algorithm. We analyze the Berezinskii-Kosterlitz-Thouless (BKT) transition of the spin-1/2 quantum XY model on the square lattice. We classify the BKT phase and the paramagnetic phase of the quantum XY model using the machine-learning approach. We show that the classification of the quantum XY model can be performed by using the training data of the classical XY model.

Phases of translation-invariant systems out of equilibrium: iterative Green’s function techniques and renormalization group approaches

We introduce a method to evaluate the steady-state non-equilibrium Keldysh–Schwinger Green’s functions for infinite systems subject to both an electric field and a coupling to reservoirs. The method we present exploits a physical quasi-translation invariance, where a shift by one unit cell leaves the physics invariant if all electronic energies are simultaneously shifted by the magnitude of the electric field. Our framework is straightaway applicable to diagrammatic many-body methods. We discuss two flagship applications, mean-field theories as well as a sophisticated second-order functional renormalization group approach. The latter allows us to push the renormalization-group characterization of phase transitions for lattice fermions into the out-of-equilibrium realm. We exemplify this by studying a model of spinless fermions, which in equilibrium exhibits a Berezinskii–Kosterlitz–Thouless phase transition.

BKT transition of the XXZ ferromagnetic model on the spherical surface

The Berezinskii-Kosterlitz-Thouless (BKT) transition is a well-known phenomenon on planar surfaces; however, for curved spaces, the situation is not clear. In this work, we applied the Self-consistent Harmonic Approximation (SCHA) to study the XXZ ferromagnetic model on the spherical surface. The Hamiltonian is written in terms of the canonically conjugate fields (operators in the quantum formalism) Sz and Φ that provide a renormalized spin stiffness depending on temperature. We consider both semiclassical and quantum approaches to the problem. The BKT transition is then determined by the temperature that provides abrupt vanishing of the spin stiffness. Our results show TBKT values close to that obtained for the planar model, and they are in agreement with the general aspects of the BKT phase transition theory.

Dynamics of the Berezinskii–Kosterlitz–Thouless transition in a photon fluid

In addition to enhancing confinement, restricting optical systems to two dimensions gives rise to new photonic states, modified transport and distinct nonlinear effects. Here we explore these properties in combination and experimentally demonstrate a Berezinskii–Kosterlitz–Thouless phase transition in a nonlinear photonic lattice. In this topological transition, vortices are created in pairs and then unbind, changing the dynamics from that of a photonic fluid to that of a plasma-like gas of free (topological) charges. We explicitly measure the number and correlation properties of free vortices, for both repulsive and attractive interactions (the photonic equivalent of ferromagnetic and antiferromagnetic conditions), and confirm the traditional thermodynamics of the Berezinskii–Kosterlitz–Thouless transition. We also suggest a purely fluid interpretation, in which vortices are nucleated by inhomogeneous flow and driven by seeded instability. The results are fundamental to optical hydrodynamics and can impact two-dimensional photonic devices if temperature and interactions are not controlled properly. A topological transition in a nonlinear photonic lattice results in new vortex dynamics and a change from photonic fluid behaviour to that of a plasma-like gas.