Finite-temperature Critical Point Research Articles

Structure of quantum entanglement at a finite temperature critical point

We propose a scheme to characterize long-range quantum entanglement close to a finite temperature critical point using tripartite entanglement negativity. As an application, we study a model with mean-field Ising critical exponents and find that the tripartite negativity does not exhibit any singularity in the thermodynamic limit across the transition. This indicates that the long-distance critical fluctuations are completely classical, allowing one to define a `quantum correlation length' that remains finite at the transition despite a divergent physical correlation length. Motivated by our model, we also study mixed state entanglement in tight-binding models of bosons with $U(1)$ and time-reversal symmetry. By employing Glauber-Sudarshan `P-representation', we find a surprising result that such states have zero entanglement.

Entanglement and classical fluctuations at finite-temperature critical points

We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature T and of the quantum parameter g, which measures the strength of quantum fluctuations. While the von Neumann and the Rényi entropies exhibit a volume-law for any g and T, the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of g, T, the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any g, it exhibits the so-called sudden death, i.e. it is exactly zero for large enough T. The vanishing of the negativity defines a ‘death line’, which we characterise analytically at small g. Importantly, for any finite T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit . Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.

Thermal Critical Points and Quantum Critical End Point in the Frustrated Bilayer Heisenberg Antiferromagnet.

We consider the finite-temperature phase diagram of the S=1/2 frustrated Heisenberg bilayer. Although this two-dimensional system may show magnetic order only at zero temperature, we demonstrate the presence of a line of finite-temperature critical points related to the line of first-order transitions between the dimer-singlet and -triplet regimes. We show by high-precision quantum MonteCarlo simulations, which are sign-free in the fully frustrated limit, that this critical point is in the Ising universality class. At zero temperature, the continuous transition between the ordered bilayer and the dimer-singlet state terminates on the first-order line, giving a quantum critical end point, and we use tensor-network calculations to follow the first-order discontinuities in its vicinity.

The three-state Potts antiferromagnet on plane quadrangulations

We study the antiferromagnetic 3-state Potts model on general (periodic) plane quadrangulations Γ. Any quadrangulation can be built from a dual pair (G,G*). Based on the duality properties of G, we propose a new criterion to predict the phase diagram of this model. If Γ is of self-dual type (i.e. if G is isomorphic to its dual G*), the model has a zero-temperature critical point with central charge c = 1, and it is disordered at all positive temperatures. If Γ is of non-self-dual type (i.e. if G is not isomorphic to G*), three ordered phases coexist at low temperature, and the model is disordered at high temperature. In addition, there is a finite-temperature critical point (separating these two phases) which belongs to the universality class of the ferromagnetic 3-state Potts model with central charge c = 4/5. We have checked these conjectures by studying four (resp. seven) quadrangulations of self-dual (resp. non-self-dual) type, and using three complementary high-precision techniques: Monte-Carlo simulations, transfer matrices, and critical polynomials. In all cases, we find agreement with the conjecture. We have also found that the Wang–Swendsen–Kotecký Monte Carlo algorithm does not have (resp. does have) critical slowing down at the corresponding critical point on quadrangulations of self-dual (resp. non-self-dual) type.

Duality and the universality class of the three-state Potts antiferromagnet on plane quadrangulations.

We provide a criterion based on graph duality to predict whether the three-state Potts antiferromagnet on a plane quadrangulation has a zero- or finite-temperature critical point, and its universality class. The former case occurs for quadrangulations of self-dual type, and the zero-temperature critical point has central charge c=1. The latter case occurs for quadrangulations of non-self-dual type, and the critical point belongs to the universality class of the three-state Potts ferromagnet. We have tested this criterion against high-precision computations on four lattices of each type, with very good agreement. We have also found that the Wang-Swendsen-Kotecký algorithm has no critical slowing-down in the former case, and critical slowing-down in the latter.

Critical scaling of the mutual information in two-dimensional disordered Ising models

Rényi mutual information, computed from second Rényi entropies, can identify classical phase transitions from their finite-size scaling at critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the ‘zero temperature only’ phase transitions are identified when there is no consistent finite-size scaling of the Rényi mutual information curves, while for finite temperature critical points, the curves can identify the critical temperature Tc by their crossings at Tc and .

Quantum critical local spin dynamics near the Mott metal-insulator transition in infinite dimensions

Finding microscopic models for metallic states that exhibit quantum critical properties such as $\omega/T$ scaling is a major theoretical challenge. We calculate the local dynamical spin susceptibility $\chi(T,\omega)$ for a Hubbard model at half filling using Dynamical Mean-Field Theory, which is exact in infinite dimensions. Qualitatively distinct behavior is found in the different regions of the phase diagram: Mott insulator, Fermi liquid metal, bad metal, and a quantum critical region above the finite temperature critical point. The signature of the latter is $\omega/T$ scaling where $T$ is the temperature. Our results are consistent with previous results showing scaling of the dc electrical conductivity and are relevant to experiments on organic charge transfer salts.

Emergence of a Fermionic Finite-Temperature Critical Point in a Kondo Lattice

The underlying Dirac point is central to the profound physics manifested in a wide class of materials. However, it is often difficult to drive a system with Dirac points across the massless fermionic critical point. Here by exploiting screening of local moments under spin-orbit interactions in a Kondo lattice, we show that below the Kondo temperature, the Kondo lattice undergoes a topological transition from a strong topological insulator to a weak topological insulator at a finite temperature T_{D}. At T_{D}, massless Dirac points emerge and the Kondo lattice becomes a Dirac semimetal. Our analysis indicates that the emergent relativistic symmetry dictates nontrivial thermal responses over large parameter and temperature regimes. In particular, it yields critical scaling behaviors both in magnetic and transport responses near T_{D}.

Detecting classical phase transitions with Renyi mutual information

By developing a method to represent the Renyi entropies via a replica trick on classical statistical mechanical systems, we introduce a procedure to calculate the Renyi mutual information (RMI) in any Monte Carlo simulation. Through simulations on several classical models, we demonstrate that the RMI can detect finite-temperature critical points, and even identify their universality class, without knowledge of an order parameter or other thermodynamic estimators. Remarkably, in addition to critical points mediated by symmetry breaking, the RMI is able to detect topological vortex-unbinding transitions, as we explicitly demonstrate on simulations of the $XY$ model.

Finite-Temperature Critical Behavior of Mutual Information

We study mutual information for Renyi entropy of arbitrary index n, in interacting quantum systems at finite-temperature critical points, using high-temperature expansion, quantum Monte Carlo simulations and scaling theory. We find that, for n>1, the critical behavior is manifest at two temperatures T(c) and nT(c). For the XXZ model with Ising anisotropy, the coefficient of the area law has a t lnt singularity, whereas the subleading correction from corners has a logarithmic divergence, with a coefficient related to the exact results of Cardy and Peschel. For T<nT(c) there is a constant term associated with broken symmetries that jumps at both T(c) and nT(c), which can be understood in terms of a scaling function analogous to the boundary entropy of Affleck and Ludwig.

Finite-size scaling critical behavior of randomly pinned spin-density waves

We have performed Monte Carlo studies of the three-dimensional $XY$ model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using $L$ values in the range 16--64, and with random anisotropy strengths of $D/2J=1$, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point, ${T}_{c}$, for each of these values of $D/2J$. We present results for the angle-averaged magnetic structure factor $S(k)$ at ${T}_{c}$ for $L=64$. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of $T$ is seen for $D/2J=6$ and $\ensuremath{\infty}$. For our finite values of $D/2J$ the scaled magnetization function increases with $L$ below ${T}_{c}$ and appears to approach an $L$-independent limit for large $L$. This suggests that the system is ferromagnetic below ${T}_{c}$.

Quantum Mott Transition and Superconductivity

The gas-liquid transition is a first-order transition terminating at a finite-temperature critical point with diverging density fluctuations. Mott transition, a metal-insulator transition driven by Coulomb repulsion between electrons, has been identified with this textbook transition. However, the critical temperature of the Mott transition can be suppressed, resulting in unusual quantum criticality. It accounts for non-Fermi-liquid-like properties, and strongly momentum dependent quasiparticles as in many materials near the Mott insulator. Among all, the mode-coupling theory of the density fluctuations supports d-wave superconductivity at the order of a hundred Kelvin for relevant parameters of the copper oxide superconductors.

Pairing and superconductivity from weak to strong coupling in the attractive Hubbard model

The finite-temperature phase diagram of the attractive Hubbard model is studied by means of the dynamical mean-field theory. We first consider the normal phase of the model by explicitly frustrating the superconducting ordering. In this case, we obtain a first-order pairing transition between a metallic phase and a paired phase formed by strongly coupled incoherent pairs. The transition line ends in a finite temperature critical point, but a crossover between two qualitatively different solutions still occurs at higher temperature. Comparing the superconducting- and the normal-phase solutions, we find that the superconducting instability always occurs before the pairing transition in the normal phase, i.e. Tc > Tpairing. Nevertheless, the high-temperature phase diagram at T > Tc is still characterized by a crossover from a metallic phase to a preformed pair phase. We characterize this crossover by computing different observables that can be used to identify the pseudogap region, like the spin susceptibility, the specific heat and the single-particle spectral function.

Investigation into the Itinerant Metamagnetism of Sr3Ru2O7 for the Field Parallel to the Ruthenium Oxygen Planes

We report a detailed investigation into the metamagnetism of Sr3Ru2O7 at low temperatures for the magnetic field parallel to the ruthenium oxygen planes. The metamagnetism is studied as a function of temperature, magnetic field and sample quality using magnetisation, magnetotransport and specific heat as probes. From hysteretic behaviour in the magnetisation, we confirm earlier work and observe a finite temperature critical point at (5 T, >0.25 K). In our highest quality samples two-step metamagnetic transitions are additionally observed at 5.8 T and at 6.3 T, which coincide with a range of broad maximum in the magnetoresistance. At low temperatures, these two metamagnetic features each further split in two. Such behaviour of the multiple transitions are qualitatively different from the first order transition at 5.1 T.

On the finite-temperature generalization of theC-theorem and the interplay between classical and quantum fluctuations

The behaviour of the finite-temperature C-function, defined by Neto and Fradkin (1993 Nucl. Phys. B 400 525), is analysed within a d -dimensional exactly solvable lattice model, recently considered by Vojta (1996 Phys. Rev. B 53 710), which is of the same universality class as the quantum nonlinear O(n) sigma model in the limit n. The scaling functions of C for the cases d = 1 (absence of long-range order), d = 2 (existence of a quantum critical point), d = 4 (existence of a line of finite-temperature critical points that ends up with a quantum critical point) are derived and analysed. The locations of regions where C is monotonically increasing (which depend significantly on d) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d = 4.

Multifractal critical phenomena in traffic and economic processes

Multifractal critical phenomena with infinite-temperature critical point and with complex coexistence of the infinite and finite temperature critical points are considered and it is shown that strange attractors generated by cascades of period-doubling bifurcations (Feigenbaum scenario) as well as fields of velocity differences in fluid turbulence belong to the former subclass of the multifractal critical phenomena, while the real traffic processes and real currency exchange processes belong to the last (complex) subclass of the multifractal critical phenomena. Data obtained by different authors are used for this purpose.

Quantum-classical crossover in critical dynamics

The stochastic quantization of Nelson is extended to the finite-temperature equilibrium states and is applied to the study of the quantum effects in the critical dynamics of the time-dependent Ginzburg-Landau model in the large-$n$ limit. A temperature-dependent (non-Markovian) stochastic process for the order parameter is derived which contains the crossover from the ground-state dynamics dominated by the quantum fluctuations, to classical critical dynamics in the neighborhood of the finite-temperature critical points. The crossover is exhibited by means of the effective dynamical critical exponent.