Correlation Length Critical Exponent Research Articles

Influence of Anisotropy on the Study of the Critical Behavior of Spin Models by Machine Learning Methods

In this paper, we applied a deep neural network to study the issue of knowledge transferability between statistical mechanics models. The following computer experiment was conducted. A convolutional neural network was trained to solve the problem of binary classification of snapshots of the Ising model’s spin configuration on a two-dimensional lattice. During testing, snapshots of the Ising model spins on a lattice with diagonal ferromagnetic and antiferromagnetic connections were fed to the input of the neural network. Estimates of the probability of samples belonging to the paramagnetic phase were obtained from the outputs of the tested network. The analysis of these probabilities allowed us to estimate the critical temperature and the critical correlation length exponent. It turned out that at weak anisotropy the neural network satisfactorily predicts the transition point and the value of the correlation length exponent. Strong anisotropy leads to a noticeable deviation of the predicted values from the precisely known ones. Qualitatively, strong anisotropy is associated with the presence of oscillations of the correlation function above the Stefenson disorder temperature and further approach to the point of the fully frustrated case.

Random Quantum Ising Model with Three-Spin Couplings.

We apply a real-space block renormalization group approach to study the critical properties of the random transverse-field Ising spin chain with multispin interactions. First, we recover the known properties of the traditional model with two-spin interactions by applying the renormalization approach for the arbitrary size of the block. For the model with three-spin couplings, we calculate the critical point and demonstrate that the phase transition is controlled by an infinite disorder fixed point. We have determined the typical correlation-length critical exponent, which seems to be different from that of the random transverse Ising chain with nearest-neighbor couplings. Thus, this model represents a new infinite disorder universality class.

Quantum criticality of generalized Aubry-André models with exact mobility edges using fidelity susceptibility.

In this study, we explore the quantum critical phenomena in generalized Aubry-André models, with a particular focus on the scaling behavior at various filling states. Our approach involves using quantum fidelity susceptibility to precisely identify the mobility edges in these systems. Through a finite-size scaling analysis of the fidelity susceptibility, we are able to determine both the correlation-length critical exponent and the dynamical critical exponent at the critical point of the generalized Aubry-André model. Based on the Diophantine equationconjecture, we can determines the number of subsequences of the Fibonacci sequence and the corresponding scaling functions for a specific filling fraction, as well as the universality class. Our findings demonstrate the effectiveness of employing the generalized fidelity susceptibility for the analysis of unconventional quantum criticality and the associated universal information of quasiperiodic systems in cutting-edge quantum simulation experiments.

Dynamical scaling laws in the quantum q -state clock chain

We show that phase transitions in the quantum $q$-state clock model for $q\ensuremath{\le}4$ can be characterized by an enhanced decay behavior of the Loschmidt echo via a small quench. The quantum criticality of the quantum $q$-state clock model is numerically investigated by the finite-size scaling of the first minimum of the Loschmidt echo and the short-time average of the rate function. The equilibrium correlation-length critical exponents are obtained from the scaling laws which are consistent with previous results. Furthermore, we study dynamical quantum phase transitions by analyzing the Loschmidt echo and the order parameter for any $q$ upon a big quench. For $q\ensuremath{\le}4$, we show that dynamical quantum phase transitions can be described by the Loschmidt echo and the zeros of the order parameter. In particular, we find the rate function increases logarithmically with $q$ at the first critical time. However, for $q>4$, we find that the correspondence between the singularities of the Loschmidt echo and the zeros of the order parameter no longer exists. Instead, we find that the Loschmidt echo near its first minimum converges, while the order parameter at its first zero increases linearly with $q$.

Thermalization and prethermalization in the soft-wall AdS/QCD model

The real-time dynamics of chiral phase transition is investigated in a two-flavor ($N_f=2$) soft-wall AdS/QCD model. To understand the dynamics of thermalization, we quench the system from initial states deviating from the equilibrium states. Then, we solve the nonequilibrium evolution of the order parameter (chiral condensate $\langle \sigma\equiv\bar{q}q\rangle$). It is shown that the system undergoes an exponential relaxation at temperatures away from the critical temperature $T_c$. The relaxation time diverges at $T_c$, presenting a typical behavior of critical slowing down. Numerically, we extract the dynamic critical exponent $z$, and get $z\approx 2$ by fitting the scaling behavior $\sigma\propto t^{-\beta/(\nu z)}$, where the mean-field static critical exponents (order parameter critical exponent $\beta=1/2$, correlation length critical exponent $\nu=1/2$ ) have been applied. More interestingly, it is remarked that, for a large class of initial states, the system would linger over a quasi-steady state for a certain period of time before the thermalization. It is suggested that the interesting phenomenon, known as prethermalization, has been observed in the framework of holographic models. In such prethermal stage, we verify that the system is characterized by a universal dynamical scaling law and described by the initial-slip exponent $\theta=0$.

Topological quantum phase transition in a mixed-spin Heisenberg tetramer chain with alternating spin-1/2 and spin-5/2 dimers

We investigate magnetic quantum phases and the topological quantum phase transition between two distinct valence-bond solid (VBS) phases of a minimal interacting spin-chain model that captures a magnetic behavior of the novel low-dimensional quantum ferrimagnetic material Cu2Fe2Ge4O13. The studied mixed-spin Heisenberg tetramer chain consists of a 4-period array of spins involving two types of dimers sa−sa (sa=1/2) and Sb−Sb (Sb=5/2) under the assumption of unique intra- and inter-dimer coupling constants J1 and J2, respectively. Using the tensor-network formulation of the density-matrix renormalization group method, we have constructed the ground-state phase diagram in the parameter space magnetic field versus the interaction ratio J=J2/J1, whereby all ground states are in accordance with the Oshikawa–Yamanaka–Affleck theorem for a given period. In addition, we have implemented the recent tangential finite-size scaling method in order to characterize the critical behavior of the topological VBS transition through calculation of the critical point Jc and the critical exponent of the correlation length ν. Our results are consistent with the topological quantum phase transition from the SU(2) Wess–Zumino–Witten universality class with the critical exponent of correlation length ν=2/3 by assuming the proper logarithmic correction.

Entanglement transitions with free fermions

We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely, those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. We generalize this model using the language of fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators and numerically compute its phase diagram. We find that both the Goldstone and area-law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite nonzero measurement rate. This is in contrast with the CPLC, in which noncommuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry-based explanation for some differences in the phase transition line between the CPLC and generalized models.

Probing center vortices and deconfinement in SU(2) lattice gauge theory with persistent homology

We investigate the use of persistent homology, a tool from topological data analysis, as a means to detect and quantitatively describe center vortices in $\mathrm{SU}(2)$ lattice gauge theory in a gauge-invariant manner. We provide evidence for the sensitivity of our method to vortices by detecting a vortex explicitly inserted using twisted boundary conditions in the deconfined phase. This inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a simple $k$-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical $\beta$ and critical exponent of correlation length $\nu$ of the deconfinement phase transition.

Critical and noncritical non-Hermitian topological phase transitions in one-dimensional chains

In this work we investigate non-Hermitian topological phase transitions using real-space edge states as a paradigmatic tool. We focus on the simplest non-Hermitian variant of the Su-Schrieffer-Heeger model, including a parameter that denotes the degree of non-Hermiticity of the system. We study the behavior of the zero-energy edge states in the nontrivial topological phases with integer and semi-integer topological winding numbers, according to the distance to the critical point. We find that, depending on the parameters of the model, the edge states may penetrate into the bulk, as expected in Hermitian topological phase transitions. We also show that, using the topological characterization of the exceptional points, we can describe the intricate chiral behavior of the edge states across the whole phase diagram. Moreover, we characterize the criticality of the model by determining the correlation length critical exponent directly from numerical calculations of the penetration length of the zero-mode edge states.

Exploring unconventional quantum criticality in the p -wave-paired Aubry-André-Harper model

We have investigated scaling properties near the quantum critical point between the extended phase and the critical phase in the Aubry-Andr\'{e}-Harper model with p-wave pairing, which have rarely been exploited as most investigations focus on the localization transition from the critical phase to the localized phase. We find that the spectrum averaged entanglement entropy and the generalized fidelity susceptibility act as eminent universal order parameters of the corresponding critical point without gap closing. We introduce a Widom scaling ansatz for these criticality probes to develop a unified theory of critical exponents and scaling functions. We thus extract the correlation-length critical exponent $\nu$ and the dynamical exponent $z$ through the finite-size scaling given the system sizes increase in the Fibonacci sequence. The retrieved values of $\nu \simeq 1.000$ and $z \simeq 3.610$ indicate that the transition from the extended phase to the critical phase belongs to a different universality class from the localization transition. Our approach sets the stage for exploring the unconventional quantum criticality and the associated universal information of quasiperiodic systems in state-of-the-art quantum simulation experiments.

Quantum criticality driven by the cavity coupling in the Rabi-dimer model

The superradiant phase transition (SPT) controlled by the interacting strength between the two-level atom and the photons has been a hot topic in the Rabi model and the Rabi-dimer model. The latter describes two Rabi cavities coupled with an inter-cavity hopping parameter. Moreover, the SPT in the Rabi-dimer model is found to be the same universal class that in the Rabi model by investigating the correlation-length critical exponent. In this paper, we are concerned about whether the inter-cavity hopping parameter between two Rabi cavities (i.e., the Rabi-dimer model) will induce the SPT and to which the universal class of the phase transition belongs. We analytically derive the phase boundary of the SPT and investigate the ground-state properties of the system. We uncover that the inter-cavity induced SPT can be apparently understood from the ground-state energy and the ground-state photon population, as well as the ground-state expectation value of the squared anti-symmetric mode. From the scaling analysis of the fidelity susceptibility, we numerically verify that the SPT driven by the cavity coupling belongs to the same universal class as the one driven by the atom–cavity interaction. Our work enriches the studies on the SPT and its critical behaviors in the Rabi-dimer model.

Characterizing quantum criticality and steered coherence in the XY -Gamma chain

In this paper, we show that an effective spin Hamiltonian with various types of couplings can be engineered using quantum simulators in atomic-molecular-optical laboratories, dubbed the $XY$-Gamma model. We analytically solve the one-dimensional short-range interacting case with the Jordan-Wigner transformation and establish the phase diagram. In the gapless phase, an incommensurate spiral order is manifested by the vector-chiral correlations. Between distinct gapped phases, a logarithmic scaling behavior of local measures, including spin correlations and the steered quantum coherence, is identified for the quantum critical points, yielding a compelling value of the correlation-length critical exponent. We derive explicit scaling forms of the excitation gap near the quantum critical points. The extracted critical exponents reveal the quantum phase transition on the boundary of Tomonaga-Luttinger liquid belongs to Lifshitz universality class. Our results may provide useful insights into the underlying mechanism in quantum criticality for state-of-the-art experiments of quantum simulation.

Dynamical scaling of Loschmidt echo in non-Hermitian systems

We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanced decay behavior of Loschmidt echo. Our studies show how to detect equilibrium many-body phase transitions with biorthogonal Loschmidt echo that can be observed in future experiments via quantum dynamics after a quench.

Quantum Criticality Driven by the Cavity Coupling in the Rabi-Dimer Model

In the previous researches, the superradiant phase transition (SPT) controlled by the interacting strength between the two-level atom and the photons has been a hot topic in the Rabi model and the Rabi-dimer model which describes two Rabi cavities coupled by an inter-cavity hopping parameter. Moreover, the SPT in the Rabi-dimer model is found to be the same universal class as that in the Rabi model by investigating the correlation-length critical exponent. In our research, we concern about that whether the inner-cavity hopping parameter between two Rabi cavities (i.e., the Rabi-dimer model) will induce the SPT and which universal class of the phase transition belongs to. By analyzing the ground-state energy and the ground-state photon population, we verify that the SPT can be induced by the inter-cavity hopping parameter. From the scaling analysis of the fidelity susceptibility, we verify that the SPT driven by the cavity coupling belongs to the same universal class as the one driven by the atom-cavity interaction. Our work enriches the studies on the SPT and its critical behaviors in the Rabi-dimer model.

Anisotropic scaling for 3D topological models

A proposal to study topological models beyond the standard topological classification and that exhibit breakdown of Lorentz invariance is presented. The focus of the investigation relies on their anisotropic quantum critical behavior. We study anisotropic effects on three-dimensional (3D) topological models, computing their anisotropic correlation length critical exponent nu obtained from numerical calculations of the penetration length of the zero-energy surface states as a function of the distance to the topological quantum critical point. A generalized Weyl semimetal model with broken time-reversal symmetry is introduced and studied using a modified Dirac equation. An approach to characterize topological surface states in topological insulators when applied to Fermi arcs allows to capture the anisotropic critical exponent theta =nu _{x}/nu _{z}. We also consider the Hopf insulator model, for which the study of the topological surface states yields unusual values for nu and for the dynamic critical exponent z. From an analysis of the energy dispersions, we propose a scaling relation nu _{bar{alpha }}z_{bar{alpha }}=2q and theta =nu _{x}/nu _{z}=z_{z}/z_{x} for nu and z that only depends on the Hopf insulator Hamiltonian parameters p and q and the axis direction bar{alpha }. An anisotropic quantum hyperscaling relation is also obtained.

Correlation length critical exponent as a function of the percolation radius for one-dimensional chains in bond problems

A one-dimensional lattice percolation model is constructed for the problem of constraints flowing along non-nearest neighbors. In this work, we calculated the critical exponent of the correlation length in the one-dimensional bond problem for a percolation radius of up to 6. In the calculations, we used a method without constructing a covering lattice or an adjacency matrix (to find the percolation threshold). The values of the critical exponent of the correlation length were obtained in the one-dimensional bond problem depending on the size of the system and at different percolation radii. Based on original algorithms that operate on a computer faster than standard ones (associated with the construction of a covering lattice), these results are obtained with corresponding errors.

Tangential finite-size scaling at the Gaussian topological transition in the quantum spin-1 anisotropic chain

Scaling aspects of Gaussian topological phase-transitions in quantum spin chains are investigated using the prototypical one-dimensional spin-1 XXZ Heisenberg model with uniaxial single-ion anisotropy $D$. This model presents a critical line separating the gaped Haldane and large-$D$ phases, with the relevant energy gap closing at the transition point. We show that a proper tangential finite-size scaling analysis is able to accurately locate the Gaussian critical line and to probe the continuously varying set of correlation length critical exponents. The specific features of the tangential scaling are highlighted in contrast with the standard scaling holding in the Ising-like transition between the gapless AF-N\'eel and gaped Haldane phases. Our results are compared with field-theoretic predictions and available high-accuracy data for specific points along the Gaussian line.

Local integrals of motion and the quasiperiodic many-body localization transition

We study the many-body localization (MBL) transition for interacting fermions subject to quasiperiodic potentials by constructing the local integrals of motion (LIOMs) in the MBL phase as time-averaged local operators. We study numerically how these time-averaged operators evolve across the MBL transition. We find that the norm of such time-averaged operators drops discontinuously to zero across the transition; as we discuss, this implies that LIOMs abruptly become unstable at some critical localization length of order unity. We analyze the LIOMs using hydrodynamic projections and isolating the part of the operator that is associated with interactions. Equipped with these data we perform a finite-size scaling analysis of the quasiperiodic MBL transition. Our results suggest that the quasiperiodic MBL transition occurs at considerably stronger quasiperiodic modulations and has a larger correlation-length critical exponent than previous studies had found.

Quantum Phase Transition of a Quantum Mixed Spin Chain by Employing Density Matrix Renormalization Group Method

AbstractThe fidelity susceptibility and trace distance in the valence‐bond‐solid (VBS) transition of a quantum mixed spin chain is investigated, which consists of unit cells arrayed as 1/2‐1/2‐1‐1 with alternating Heisenberg antiferromagnetic exchange couplings, by using the density matrix renormalization group (DMRG) method in the matrix product state (MPS) form. It is observed that the fidelity susceptibility and the first derivative of the trace distance display explicit divergence near the quantum critical point. The information about the quantum criticality, such as the quantum critical point and correlation length critical exponent, is extracted from the finite size scaling behaviors. The obtained quantum critical point is more accurate than previous work, and the existence of the scaling function of the trace distance near the critical point is observed numerically for the first time. In addition, it is also found that the trace distance between two sites can supplement the VBS picture for describing the change of the ground state as the control parameter is varied through the quantum critical point.

Critical point determination from probability distribution functions in the three dimensional Ising model

In this work we propose a new numerical method to evaluate the critical point, the susceptibility critical exponent and the correlation length critical exponent of the three dimensional Ising model without external field using an algorithm that evaluates directly the derivative of the logarithm of the probability distribution function with respect to the magnetisation. Using standard finite-size scaling theory we found that correction-to-scaling effects are not present within this approach. Our results are in good agreement with previous reported values for the three dimensional Ising model.