Hard-core Gas Research Articles

Operator spreading in quantum hardcore gases

In this article we study a set of integrable quantum cellular automata, the quantum hardcore gases (QHCG), with an arbitrary local Hilbert space dimension, and discuss the matrix product ansatz based approach for solving the dynamics of local operators analytically. Subsequently, we focus on the dynamics of operator spreading, in particular on the out-of-time ordered correlation functions (OTOCs), operator weight spreading and operators space entanglement entropy (OSEE). All of the quantities were conjectured to provide signifying features of integrable systems and quantum chaos. We show that in QHCG OTOCs spread diffusively and that in the limit of the large local Hilbert space dimension they increase linearly with time, despite their integrability. On the other hand, it was recently conjectured that operator weight front, which is associated with the extent of operators, spreads diffusively in both, integrable and generic systems, but its decay seems to differ in these two cases (Lopez-Piqueres et al 2021 Phys. Rev. B 104 104307). We observe that the spreading of the operator weight front in QHCG is markedly different from chaotic, generic integrable and free systems, as the front freezes in the long time limit. Finally, we discuss the OSEE in QHCG and show that it grows at most logarithmically with time in accordance with the conjectured behaviour for interacting integrable systems (Alba et al 2019 Phys. Rev. Lett. 122 250603).

Fourier heat conduction as a strong kinetic effect in one-dimensional hard-core gases.

For a one-dimensional (1D) momentum conserving system, intensive studies have shown that generally its heat current autocorrelation function (HCAF) tends to decay in a power-law manner and results in the breakdown of the Fourier heat conduction law in the thermodynamic limit. This has been recognized to be a dominant hydrodynamic effect. Here we show that, instead, the kinetic effect can be dominant in some cases and leads to the Fourier law for finite-size systems. Usually the HCAF undergoes a fast decaying kinetic stage followed by a long slowly decaying hydrodynamic tail. In a finite range of the system size, we find that whether the system follows the Fourier law depends on whether the kinetic stage dominates. Our Rapid Communication is illustrated by the 1D hard-core gas models with which the HCAF is derived analytically and verified numerically by molecular dynamics simulations.

Efficiency at maximum power output for an engine with a passive piston

Efficiency at maximum power (MP) output for an engine with a passive piston without mechanical controls between two reservoirs is theoretically studied. We enclose a hard core gas partitioned by a massive piston in a temperature-controlled container and analyze the efficiency at MP under a heating and cooling protocol without controlling the pressure acting on the piston from outside. We find the following three results: (i) The efficiency at MP for a dilute gas is close to the Chambadal-Novikov-Curzon-Ahlborn (CNCA) efficiency if we can ignore the side wall friction and the loss of energy between a gas particle and the piston, while (ii) the efficiency for a moderately dense gas becomes smaller than the CNCA efficiency even when the temperature difference of reservoirs is small. (iii) Introducing the Onsager matrix for an engine with a passive piston, we verify that the tight coupling condition for the matrix of the dilute gas is satisfied, while that of the moderately dense gas is not satisfied because of the inevitable heat leak. We confirm the validity of these results using the molecular dynamics simulation and introducing an effective mean-field-like model which we call stochastic mean field model.

A solution to the combinatorial puzzle of Mayer’s virial expansion

Mayer’s second theorem in the context of a classical gasmodel allows us to write the coefficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with n vertices is –n(n–2)! . This paper addresses the question of achieving a purely combinatorial proof of this observation and extends the proof of Bernardi for the connected graph case.

Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

We point out a close connection between the Moser–Tardos algorithmic version of the Lovasz local lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard-core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard-core gas. Such an identification implies that the Moser–Tardos algorithm is successful in a polynomial time if the cluster expansion converges.

Finite-size effects on current correlation functions

We study why the calculation of current correlation functions (CCFs) still suffers from finite-size effects even when the periodic boundary condition is taken. Two important one-dimensional, momentum-conserving systems are investigated as examples. Intriguingly, it is found that the state of a system recurs in the sense of microcanonical ensemble average, and such recurrence may result in oscillations in CCFs. Meanwhile, we find that the sound mode collisions induce an extra time decay in a current so that its correlation function decays faster (slower) in a smaller (larger) system. Based on these two unveiled mechanisms, a procedure for correctly evaluating the decay rate of a CCF is proposed, with which our analysis suggests that the global energy CCF decays as ∼ t(-2/3) in the diatomic hard-core gas model and in a manner close to ∼ t(-1/2) in the Fermi-Pasta-Ulam-β model.

Non-existence of phase transitions in continuous hard core gases with nonnegativepotentials

It is shown that for all dimensional classical continuous hard core gases with nonnegativeregular potentials, the pressure is an analytic, increasing function of the activity. Thus insuch models phase transitions do not occur.

Quantum Monte Carlo calculation of the zero-temperature phase diagram of the two-component fermionic hard-core gas in two dimensions

Motivated by potential realizations in cold-atom or cold-molecule systems, we have performed quantum Monte Carlo simulations of two-component gases of fermions in two dimensions with hard-core interactions. We have determined the gross features of the zero-temperature phase diagram by investigating the relative stabilities of paramagnetic and ferromagnetic fluids and crystals. We have also examined the effect of including a pairwise, long-range ${r}^{\ensuremath{-}3}$ potential between the particles. Our most important conclusion is that there is no region of stability for a ferromagnetic fluid phase, even if the long-range interaction is present. We also present results for the pair-correlation function, static structure factor, and momentum density of two-dimensional hard-core fluids.

Dynamics of localization phenomena for hard-core bosons in optical lattices

We investigate the behavior of ultracold bosons in optical lattices with a disorder potential generated via a secondary species frozen in random configurations. The statistics of disorder is associated with the physical state in which the secondary species is prepared. The resulting random potential, albeit displaying algebraic correlations, is found to lead to localization of all single-particle states. We then investigate the real-time dynamics of localization for a hardcore gas of mobile bosons which are brought into sudden interaction with the random potential. Regardless of their initial state and for any disorder strength, the mobile particles are found to reach a steady state characterized by exponentially decaying off-diagonal correlations and by the absence of quasicondensation; when the mobile particles are initially confined in a tight trap and then released in the disorder potential, their expansion is stopped and the steady state is exponentially localized in real space, clearly revealing Anderson localization.

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup Pt. A fundamental and still largely open problem is the understanding of the long time behavior of δηPt when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree Open image in new window, we study the above problem for the Ising and hard core gas (independent sets) models on Open image in new window. If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δηPt to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.

Long-Range Frustration in a Spin-Glass Model of the Vertex-Cover Problem

In a spin-glass system on a random graph, some vertices have their spins changing among different configurations of a ground-state domain. Long-range frustrations may exist among these unfrozen vertices in the sense that certain combinations of spin values for these vertices may never appear in any configuration of this domain. We present a mean field theory to tackle such long-range frustrations and apply it to the NP-hard minimum vertex-cover (hard-core gas condensation) problem. Our analytical results on the ground-state energy density and on the fraction of frozen vertices are in good agreement with known numerical and mathematical results.

Perfect sampling using bounding chains

Bounding chains are a technique that offers three benefits to Markov chain practitioners: a theoretical bound on the mixing time of the chain under restricted conditions, experimental bounds on the mixing time of the chain that are provably accurate and construction of perfect sampling algorithms when used in conjunction with protocols such as coupling from the past. Perfect sampling algorithms generate variates exactly from the target distribution without the need to know the mixing time of a Markov chain at all. We present here the basic theory and use of bounding chains for several chains from the literature, analyzing the running time when possible. We present bounding chains for the transposition chain on permutations, the hard core gas model, proper colorings of a graph, the antiferromagnetic Potts model and sink free orientations of a graph.

Entropy puzzle in small exploding systems

We use a simple hard-core gas model to study the dynamics of small exploding systems. The system is initially prepared in a thermalized state in a spherical container and then allowed to expand freely into the vacuum. We follow the expansion dynamics by recording the coordinates and velocities of all particles until their last collision points (freeze-out). We have found that the entropy per particle calculated for the ensemble of freeze-out points is very close to the initial value. This is in apparent contradiction with the Joule experiment in which the entropy grows when the gas expands irreversibly into a larger volume.

Dimensional Reduction Formulas for Branched Polymer Correlation Functions

In [BI01] we have proven that the generating function for self-avoiding branched polymers in D+2 continuum dimensions is proportional to the pressure of the hard-core continuum gas at negative activity in D dimensions. This result explains why the critical behavior of branched polymers should be the same as that of the iϕ3 (or Yang–Lee edge) field theory in two fewer dimensions (as proposed by Parisi and Sourlas in 1981). In this article we review and generalize the results of [BI01]. We show that the generating functions for several branched polymers are proportional to correlation functions of the hard-core gas. We derive Ward identities for certain branched polymer correlations. We give reduction formulae for multi-species branched polymers and the corresponding repulsive gases. Finally, we derive the massive scaling limit for the 2-point function of the one-dimensional hard-core gas, and thereby obtain the scaling form of the 2-point function for branched polymers in three dimensions.

Low-temperature smoothness of the pressure for antiferromagnets and other models

Under certain conditions on a grandcanonical Hamiltonian, it will be proven that at low temperature the pressure is infinitely differentiable with respect to the inverse temperature and other parameters in the Hamiltonian, when the parameters are chosen so that the number of extremal Gibbs states is at least equal to the number of ground states. Applications are made to antiferromagnets and hard-core gases.

NMR relaxation times for a dilute polarized Fermi gas

We calculate the longitudinal relaxation time T1 for a polarized Boltzmann spin−12 Fermi gas. We show that T1 is independent of polarization of the gas. At high T, where the thermal wavelength a is small compared to the scattering length a, T1 is proportional T12, while at low T, such that λ is greater than a, T1 is proportional to T−12. T1 thus has a minimum at some intermediate temperature confirming the numerical results of Shizgal. The existence of the minimum does not depend on the presence of an attractive part of the potential. As an example of the expected temperature dependence we calculate T1 numerically for a hard core gas.

NMR relaxation times for a dilute polarized fermi gas

Abstract We calculate the longitudinal relaxation time T1 for a polarized Boltzmann spin − 1 2 Fermi gas. We show that T1 is independent of polarization of the gas. At high T, where the thermal wavelength a is small compared to the scattering length a, T1 is proportional T 1 2 , while at low T, such that λ is greater than a, T1 is proportional to T − 1 2 . T1 thus has a minimum at some intermediate temperature confirming the numerical results of Shizgal. The existence of the minimum does not depend on the presence of an attractive part of the potential. As an example of the expected temperature dependence we calculate T1 numerically for a hard core gas.

An approximate adsorption isotherm for physical adsorption. I. Lattice gas

The matrix method of statistical mechanics has been used to obtain an approximate adsorption isotherm for a lattice gas in the presence of a wall which interacts with adjoining sites. The excess number adsorbed per surface site has the form [ρII(z′)−ρIII(z)](1 +∂ ln α/∂ ln z), where the densities ρII and ρIII are two- and three-dimensional (bulk) functions of activity z; α is defined by ρII(α2z)=ρIII(z); and z′ =z⋅α⋅exp(−βεB). Here, εB is the interaction energy of a particle with the wall. Surface viral coefficients were calculated as a test of the theory. For nearest neighbor lattice gases, the first and second coefficients are exact. The third coefficient is exact in the special case of a hard wall, and it is also exact for a hard core gas with coordination number 12 in three dimensions and 6 on the surface layer (corresponding to the geometry of the closest packed spheres). The isotherm was also applied to the continuum model of hard spheres against a hard wall. Virial coefficients indicate results slightly inferior to scaled particle theory.

Spin and isospin stability of dense nuclear matter

The possibility of spin instability of dense neutron matter, and of spin and isospin instability of dense nuclear matter is investigated using a simplified model of a pure hard core gas, with hard core radius c. Expansion in powers of x = k F c and the variational approach are applied. The model predicts spin stability of neutron matter, and for χ ≳ 1 it suggests a possible spin, isospin, and spin-isospin instability of nuclear matter.

Ferromagnetism of dense neutron matter

The possibility of a ferromagnetic transition in dense neutron matter is investigated using a simplified model of a pure hard core gas, with core radius c. Previous estimates of the density at which the ferromagnetic transition would occur are shown to be incorrect. The inclusion of terms cubic in k/sub F/c, in particular the P state interaction, leads to the conclusion that neutron matter in the liquid phase should not exhibit ferromagnetism.