We study dimensional crossover in Ising systems at complex temperatures by comparing three types of system: the infinite isotropic 2D Ising model; the infinite anisotropic 2D Ising model; and Ising ladders with a finite number of legs. In particular we present evidence, from both tensor-network calculations and numerical evaluations based on the exact solution of the model, that the infinite anisotropic 2D Ising model exhibits long-range spatially modulated magnetization in certain regions of the complex-temperature plane. We discuss the physics of the special unitary points that exists in the complex-temperature plane, and their connections to the theory of quantum information processing.
Non-local orders, entanglement entropy, and quantum fidelity are investigated in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya interaction by employing the infinite matrix product state representation with the infinite time evolving block decimation method. Directly computing two distinct types of finite string correlations for very large lattice distances, in contrast to an extrapolated extreme value for finite size chains, reveals two topologically ordered phases. As the bond alternation varies, a topological quantum phase transition with continuously variable …
Lattice-gas models for CO oxidation can exhibit a discontinuous nonequilibrium transition between reactive and inactive states, which disappears above a critical CO-desorption rate. Using finite-size-scaling analysis, we demonstrate a crossover from Ising to mean-field behavior at the critical point, with increasing surface mobility of adsorbed CO or with decreasing system size. This behavior is elucidated by analogy with that of equilibrium Ising-type systems with long-range interactions.
Hierarchical lattices that constitute spatially anisotropic systems are introduced. These lattices provide exact solutions for hierarchical models and, simultaneously, approximate solutions for uniaxially or fully anisotropic d=3 physical models. The global phase diagrams, with d=2 and d=1 to d=3 crossovers, are obtained for Ising, XY magnetic models and percolation systems, including crossovers from algebraic order to true long-range order.
"Swing" effects at the onset of crossover towards two dimensional behavior in thin Ising films are investigated close to Tc(D) by means of Monte Carlo calculations. We find that the effect is extremely large for the specific heat effective critical exponent, in comparison with the "swing" already noted by Capehart and Fisher for the susceptibility. These effects change considerably the system's evolution with thickness (D) from two-dimensional to three-dimensional behavior, forcing the effective exponents to pass near characteristic Tri Critical Point (TCP) values.
Recent numerical studies of the susceptibility of the three-dimensional Ising model with various interaction ranges have been analyzed with a crossover model based on renormalization-group matching theory. It is shown that the model yields an accurate description of the crossover function for the susceptibility.
We give simple expressions for the mean of the max and min bounds of the critical-to-classical crossover functions previously calculated [Bagnuls and Bervillier, Phys. Rev. E 65, 066132 (2002)] within the massive renormalization scheme of the F 4 d (n) model in three dimensions (d = 3) and scalar order parameter (n = 1) of the Ising-like universality class. Our main motivation is to get efficient theoretical expressions to coherently account for many measurements performed in systems where the approach to the critical point is limited but yield data which are still reproducible by the F 4 d (n…
We present an accurate numerical determination of the crossover from classical to Ising-like critical behavior upon approach of the critical point in three-dimensional systems. The possibility to vary the Ginzburg number in our simulations allows us to cover the entire crossover region. We employ these results to scrutinize several semi-phenomenological crossover scaling functions that are widely used for the analysis of experimental results. In addition we present strong evidence that the exponent relations do not hold between effective exponents.
The character of critical behavior in physical systems depends on the range of interactions. In the limit of infinite range of the interactions, systems will exhibit mean-field critical behavior, i.e., critical behavior not affected by fluctuations of the order parameter. If the interaction range is finite, the critical behavior asymptotically close to the critical point is determined by fluctuations and the actual critical behavior depends on the particular universality class. A variety of systems, including fluids and anisotropic ferromagnets, belongs to the three-dimensional Ising universal…
By means of extensive computer simulations we analyze in detail the two dimensional $\pm J$ Ising spin glass with ferromagnetic next-nearest-neighbor interactions. We found a crossover from ferromagnetic to ``spin glass'' like order both from numerical simulations and analytical arguments. We also present evidences of a second crossover from the ``spin glass'' behavior to a paramagnetic phase for the largest volume studied.
We compute the crossover exponent $ϕ$ describing the crossover from the random-exchange to the random-field critical behavior in Ising systems. For this purpose, we consider the field-theoretical approach based on the replica method, and perform a six-loop calculation in the framework of a fixed-dimension expansion. The crossover from random-exchange to random-field critical behavior has been observed in dilute anisotropic antiferromagnets, such as Fe$_x$Zn$_{1-x}$F$_2$ and Mn$_x$Zn$_{1-x}$F$_2$, when applying an external magnetic field. Our result $ϕ=1.42(2)$ for the crossover exponent is in …
We present results of our Monte Carlo simulation of the Ising-O(3) model on the two-dimensional (2D) and quasi-2D lattices. This model is an effective classical model for the stacked square-lattice $J_1$-$J_2$ Heisenberg model where the nearest neighbor ($J_1$) and next-nearest neighbor ($J_2$) couplings are frustrated and we assume that $J_2$ is dominant. We find an Ising ordered phase where the O(3) spins remain disordered in a moderate quasi-2D region. There is a single first order transition for a sufficiently large 3D coupling. Our results can provide a qualitative explanation on the expe…
The 2-dimensional Ising model on a square lattice is investigated with a variational autoencoder in the non-vanishing field case for the purpose of extracting the crossover region between the ferromagnetic and paramagnetic phases. The encoded latent variable space is found to provide suitable metrics for tracking the order and disorder in the Ising configurations that extends to the extraction of a crossover region in a way that is consistent with expectations. The extracted results achieve an exceptional prediction for the critical point as well as agreement with previously published results …
We investigate by Monte Carlo simulations the critical properties of the three-dimensional bond-diluted Ising model. The phase diagram is determined by locating the maxima of the magnetic susceptibility and is compared to mean-field and effective-medium approximations. The calculation of the size-dependent effective critical exponents shows the competition between the different fixed points of the model as a function of the bond dilution.
We study the phase-ordering kinetics following a quench to a final temperature $T_f$ of the one-dimensional p-state clock model. We show the existence of a critical value $p_c=4$, where the properties of the dynamics change. At $T_f=0$, for $p\le p_c$ the dynamics is analogous to that of the kinetic Ising model, characterized by Brownian motion and annihilation of interfaces. Dynamical scaling is obeyed with the same dynamical exponents and scaling functions of the Ising model. For $p>p_c$, instead, the dynamics is dominated by a texture mechanism analogous to the one-dimensional XY model, and…
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both no…
Recently, a surprising low-temperature behavior has been revealed in a two-leg ladder Ising model with trimer rungs (Weiguo Yin, arXiv:2006.08921). Motivated by these findings, we study this model from another perspective and demonstrate that the reported observations are related to a critical phenomenon in the standard Ising chain. We also discuss a related curiosity, namely, the emergence of a power-law behavior characterized by quasicritical exponents.
We study through Monte Carlo simulations and finite-size scaling analysis the nonequilibrium phase transitions of the majority-vote model taking place on spatially embedded networks. These structures are built from an underlying regular lattice over which long-range connections are randomly added according to the probability, $P_{ij}\sim{r^{-α}}$, where $r_{ij}$ is the Manhattan distance between nodes $i$ and $j$, and the exponent $α$ is a controlling parameter [J. M. Kleinberg, Nature 406, 845 (2000)]. Our results show that the collective behavior of this system exhibits a continuous order-di…
Quantum many body systems with long range interactions are known to display many fascinating phenomena experimentally observable in trapped ions, Rydberg atoms and polar molecules. Among these are dynamical phase transitions which occur after an abrupt quench in spin chains with interactions decaying as $r^{-α}$ and whose critical dynamics depend crucially on the power $α$: for systems with $α<1$ the transition is sharp while for $α>1$ it fans out in a chaotic crossover region. In this paper we explore the fate of critical dynamics in Ising chains with long-range interactions when the transver…
Scaling nature of absorbing critical phenomena is well understood for the directed percolation (DP) and the directed Ising (DI) systems. However, a full analysis of the crossover behavior is still lacking, which is of our interest in this study. There are three different routes from the DI to the DP classes by introducing a symmetry breaking field (SB), breaking a modulo 2 conservation (CB), or making channels connecting two equivalent absorbing states (CC). Each route can be characterized by a crossover exponent, which is found numerically as $ϕ=2.1\pm 0.1$ (SB), $4.6\pm 0.2$ (CB), and $2.9\p…
Dynamical detection of quantum phases and phase transitions (QPT) in quenched systems with experimentally convenient initial states is a topic of interest from both theoretical and experimental perspectives. Quenched from polarized states, longitudinal magnetization decays exponentially to zero in time for the short-range transverse-field Ising model (TFIM) and hence, has a featureless steady state regime, which prevents it from exhibiting dynamical phase transitions of type-I. In this paper, we ask whether the transient regimes of such non-equilibrium processes probed by single-site observabl…
We consider an exactly solvable version of the quantum spin-1/2 orthogonal-dimer chain with the Heisenberg intra-dimer and Ising inter-dimer couplings. The investigated quantum spin system exhibits at zero temperature fractional plateaux at 1/4 and 1/2 of the saturation magnetization and it has a highly degenerate ground state at critical fields where the magnetization jumps. We study the field dependence of the specific heat at low temperature. The lattice-gas description is formulated in a vicinity of critical fields to explain the low-temperature behaviour of specific heat.
We performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional ST's applicability to dealing with phase transitions and to study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus, this simulation can be referred to as "Simulated Tempering and Magnetizing (STM)." We also performed the "Simulated Magnetizing" (SM) simulations, in which the external field was considered as a…
We study the crossover behaviors that can be observed in the high-temperature phase of three-dimensional dilute spin systems, using a field-theoretical approach. In particular, for randomly dilute Ising systems we consider the Gaussian-to-random and the pure-Ising-to-random crossover, determining the corresponding crossover functions for the magnetic susceptibility and the correlation length. Moreover, for the physically interesting cases of dilute Ising, XY, and Heisenberg systems, we estimate several universal ratios of scaling-correction amplitudes entering the high-temperature Wegner expan…
In this study, the internal energy and the specific heat of the one- dimensional Ising model obtained in the frame of the generalized statistical thermodynamics by Andrade [Physica A175 (1991) 285], have been extended somewhat and compared with the internal energy and the specific heat which have been calculated in the standard statistical thermodynamics.
Using the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since we can tune the critical point of each random sample automatically with the PCC algorithm, we succeed in studying the sample-dependent $T_c(L)$ and the sample average of physical quantities at each $T_c(L)$ systematically. Using the finite-size scaling (FSS) analysis for $T_c(L)$, we discuss the importance of corrections to FSS both in the strong-dilution and weak-dilution regions. The critical phenomena of the 2D site-diluted Ising model are s…
Driven quantum systems coupled to an environment typically exhibit effectively thermal behavior with relaxational dynamics near criticality. However, a different qualitative behavior might be expected in the weakly dissipative limit due to the competition between coherent dynamics and weak dissipation. In this work, we investigate a driven-dissipative infinite-range Ising model in the presence of individual atomic dissipation, a model that emerges from the paradigmatic open Dicke model in the large-detuning limit. We show that the system undergoes a dynamical crossover from relaxational dynami…
We investigate the relationship between a diffusive motion of an interface, heat conduction, and the roughening transition in the three-dimensional Ising model. We numerically compute the thermal conductivity and the diffusion constant and find that the diffusion constant shows a crossover in its temperature dependence. The crossover temperature is equal to the roughening transition temperature in equilibrium and deviates from it when heat flows in the system. From these results, we discuss the possibility that heat conduction causes a shift of the roughening transition temperature.
We investigate the surface critical behavior of two-dimensional multilayered aperiodic Ising models in the extreme anisotropic limit. The system under consideration is obtained by piling up two types of layers with respectively $p$ and $q$ spin rows coupled via nearest neighbor interactions $λr$ and $λ$, where the succession of layers follows an aperiodic sequence. Far away from the critical regime, the correlation length $ξ_\perp$ is smaller than the first layer width and the system exhibits the usual behavior of an ordinary surface transition. In the other limit, in the neighborhood of the c…
The idea that the dynamics of a spin is determined by the size of its neighbouring domains was recently introduced (S. Biswas and P. Sen, Phys. Rev. E {\bf 80}, 027101 (2009)) in a Ising spin model (henceforth, referred to as model I). A parameter $p$ is now defined to modify the dynamics such that a spin can sense domain sizes up to $R = pL/2$ in a one dimensional system of size $L$. For the cutoff factor $p$ \to 0$, the dynamics is Ising like and the domains grow with time $t$ diffusively as $ t^{1/z}$ with $z=2$, while for $p=1$, the original model I showed ballistic dynamics with $z \simeq…
The temperature dependence of the uniform susceptibility, chi, of diluted two-dimensional Ising antiferromagnets Rb2Co1-xMgxF4,0<=x<=0.4, is investigated in the limit of vanishing external field. Novel data for x = 0.15 are compared with those obtained for x = 0 and 0.4 by Breed et al. (1969) and Ikeda (1983), respectively. Whereas in the pure case, x = 0, Fishers (1962) "energetic contribution" dominates, Aharony et al.s (1979,1986) "random contribution" becomes increasingly important with increasing x. Taking into account both terms not only with respect to the global ordering temperature, T…
We use the mean crossover functions [Garrabos and Bervillier, Phys Rev. E 74, 021113 (2006)] estimated from the bounded results of the Massive Renormalization scheme applied to the $Φ_{d}^{4}(n)$ model in three dimensions ($d=3$) and scalar order parameter ($n=1$) [Bagnuls and Bervillier, Phys. Rev. E 65, 066132 (2002)], to represent the singular behavior of the isothermal compressibility of xenon along the critical isochore in the homogeneous preasymptotic domain. The validity range and the Ising nature of the crossover description are discussed in terms of a single scale factor whose value c…
A mixed Ising-Heisenberg spin system consisting of triangular XXZ-Heisenberg spin clusters assembled into a chain by alternating with Ising spins interacting to all three spins in the triangle is considered. The exact solution of the model is given in terms of the generalized decoration--iteration map and within the transfer-matrix technique. Exact expressions for thermodynamic functions are derived. Ground state phase diagrams, thermodynamic and magnetic properties of the system are examined.
A thorough numerical investigation of the slow dynamics in the d=1 random field Ising model in the limit of an infinite ferromagnetic coupling is presented. Crossovers from the preasymptotic pure regime to the asymptotic Sinai regime are investigated for the average domain size, the autocorrelation function and staggered magnetization. By switching on an additional small random field at the time tw the linear off equilibrium response function is obtained, which displays as well the crossover from the nontrivial behavior of the d=1 pure Ising model to the asymptotic behavior where it vanishes i…
To understand the collinear-magnetism-driven ferroelectricity in multiferroic Ca3CoMnO6 compound, we have established an elastic diatomic Ising spin-chain model with axial-next-nearest-neighbor interaction to describe its magnetoelectric properties. By employing magneto-phonon decoupling and transfer-matrix method, the possible ground-state configurations and thermodynamic behaviors of the system have been exactly determined. From the perspective of the ground-state configuration, we analyze the computational results and make a detail comparison with experimental data. The parameter relation f…
We calculate the fractal dimension $d_{\rm f}$ of critical curves in the $O(n)$ symmetric $(\vec ϕ^2)^2$-theory in $d=4-\varepsilon$ dimensions at 6-loop order. This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and XY lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\rm f}^{\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\rm f}^{\rm tot} = 1/ν$. The combination $ϕ_{\rm c}= d_{\rm f}/d_{\rm f}^{\rm tot} = νd_{\rm f}$ is the crossover e…
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, generally we expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has non-zero long-range interactions, it shows criticality in the mean-field univers…
In this work we formulate the nonequilibrium dynamical renormalization group (ndRG). The ndRG represents a general renormalization-group scheme for the analytical description of the real-time dynamics of complex quantum many-body systems. In particular, the ndRG incorporates time as an additional scale which turns out to be important for the description of the long-time dynamics. It can be applied to both translational invariant and disordered systems. As a concrete application we study the real-time dynamics after a quench between two quantum critical points of different universality classes.…
The two-dimensional Ising model on a distorted Kagomé lattice is studied by means of exact solutions and the tensor renormalisation group (TRG) method. The zero-field phase diagrams are obtained, where three phases such as ferromagnetic, ferrimagnetic and paramagnetic phases, along with the second-order phase transitions, have been identified. The TRG results are quite accurate and reliable in comparison to the exact solutions. In a magnetic field, the magnetization ($m$), susceptibility and specific heat are studied by the TRG algorithm, where the $m=1/3$ plateaux are observed in the magnetiz…
We investigate the global persistence properties of critical systems relaxing from an initial state with non-vanishing value of the order parameter (e.g., the magnetization in the Ising model). The persistence probability of the global order parameter displays two consecutive regimes in which it decays algebraically in time with two distinct universal exponents. The associated crossover is controlled by the initial value m_0 of the order parameter and the typical time at which it occurs diverges as m_0 vanishes. Monte-Carlo simulations of the two-dimensional Ising model with Glauber dynamics d…
In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width $m$ connected by sequences of vertical strings of length $n$ mutually separated by distance $N$, with $N=1,2$ and $3$. We find that the critical temperature $T_c(N,m,n)$, arising from the collective effects, decreases as $n$ and $N$ increase, and increases as $m$ increases, as it should be. The amplitude $A(N,m,n)$ of the logarit…
Calculation of thermodynamic functions of the three-dimensional Ising ferromagnet above and below critical temperature is performed in the approximation of sixfold basis distribution ($ρ^6$ model). Comparison with the results for the $ρ^4$ model indicates that dependence of the thermodynamic functions on the renormalization group parameter $s$ becomes weaker. The optimal interval of the renormalization group parameter values is determined.
We study phase transitions and thermodynamic properties in the two-dimensional antiferromagnetic Ising model with next-nearest-neighbor interaction on a Kagome lattice by Monte Carlo simulations. A histogram data analysis shows that a second order transition occurs in the model. From the analysis of obtained data, we can assume that next-nearest-neighbor ferromagnetic interactions in two-dimensional antiferromagnetic Ising model on a Kagome lattice excite the occurrence of a second order transition and unusual behavior of thermodynamic properties on the temperature dependence.
We numerically compute the temperature dependence of spin structure factor and thermodynamic quantities in the antiferromagnetic quantum Ising model in the pyrochlore lattice. This model exhibits spin disorder ground state with exponentially-decayed spin correlation. We reproduce the temperature dependence of the pinch point structure in the neutron scattering experiment and correct entropy obtained from the measurement of the specific heat.
While the Ising model on the Cayley tree has no spontaneous magnetization at nonzero temperatures in the thermodynamic limit, we show that finite systems of astronomical sizes remain magnetically ordered in a wide temperature range, if the symmetry is broken by fixing an arbitrary single (bulk or surface) spin. We compare the behavior of the finite size magnetization of this model with that of the Ising model on both the Sierpinski Gasket, and the one-dimensional linear chain. This comparison reveals the analogy of the behavior of the present model with the Sierpinski Gasket case.
Noisy mean field annealing (NMFA) is an algorithm that mimics a coherent Ising machine (CIM), which is an optical system for solving Ising problems. The NMFA has reproduced the solver performance of the CIM for systems of limited size even though it simplifies the interaction between spins with a mean-field approximation. However, recent experiments observing various thermodynamic quantities have revealed that the CIM can capture the phase transitions of the two-dimensional Ising models that the mean field cannot capture. This situation leads to a fundamental question as to how well the NMFA c…
We study the two-dimensional fully frustrated XY (FFXY) model and two related models, a discretization of the Landau-Ginzburg-Wilson Hamiltonian for the critical modes of the FFXY model and a coupled Ising-XY model, by means of Monte Carlo simulations on square lattices L x L, L=O(10^3). We show that their phase diagram is characterized by two very close chiral and spin transitions, at T_ch > T_sp respectively, of the Ising and Kosterlitz-Thouless type. At T_ch the Ising regime sets in only after a preasymptotic regime, which appears universal to some extent. The approach is nonmonotonic for m…
Random bond Ising systems on a general hierarchical lattice are considered. The inequality between the specific heat exponent of the pure system, $α_p$, and the crossover exponent $ϕ$, $α_p<=ϕ$, gives rise to a possibility of a negative $α_p$ along with a positive $ϕ$, leading to random criticality in disagreement with the Harris criterion. An explicit example where this really happens for an Ising system is presented and discussed. In addition to that, it is shown that in presence of full long-range correlations, the crossover exponent is larger than in the uncorrelated case.
We present a self-consistent theory for the thermodynamics of the BCS-BEC crossover in the normal and superfluid phase which is both conserving and gapless. It is based on the variational many-body formalism developed by Luttinger and Ward and by DeDominicis and Martin. Truncating the exact functional for the entropy to that obtained within a ladder approximation, the resulting self-consistent integral equations for the normal and anomalous Green functions are solved numerically for arbitrary coupling. The critical temperature, the equation of state and the entropy are determined as a function…
We show that a nominal temperature can be consistently and uniquely defined everywhere in the phase diagram of large classes of nonequilibrium kinetic Ising spin models. In addition, we confirm the recent proposal that, at critical points, the large-time ``fluctuation-dissipation ratio'' $X_\infty$ is a universal amplitude ratio and find in particular $X_\infty \approx 0.33(2)$ and $X_\infty = 1/2$ for the magnetization in, respectively, the two-dimensional Ising and voter universality classes.
We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a c…
We study thermodynamic properties of the one-dimensional Heisenberg ferrimagnet with antiferromagnetically exchange-coupled two kinds of spins 1 and 1/2. The specific heat and the magnetic susceptibility are calculated employing a modified spin-wave theory as well as a quantum Monte Carlo method. The specific heat is in proportion to $T^{1/2}$ at low enough temperatures but shows a Schottky-like peak at mid temperatures. The susceptibility diverges as $T^{-2}$. We reveal that at low temperatures the model is regarded as a ferromagnet, while at mid temperatures it behaves like a gapped antiferr…
Three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorate the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat arbitrary number of…
A hallmark of a thermodynamic phase transition is the qualitative change of system thermodynamic properties such as energy and heat capacity. On the other hand, no phase transition is thought to operate in the supercritical state of matter and, for this reason, it was believed that supercritical thermodynamic properties vary smoothly and without any qualitative changes. Here, we perform extensive molecular dynamics simulations in a wide temperature range and find that a deeply supercritical state is thermodynamically heterogeneous, as witnessed by different temperature dependence of energy, he…
