The microcanonical ensemble of the ideal relativistic quantum gas with angular momentum conservation

We derive the microcanonical partition function of the ideal relativistic quantum gas with fixed intrinsic angular momentum as an expansion over fixed multiplicities. We developed a group theoretical approach by generalizing known projection techniques to the Poincaré group. Our calculation is carried out in a quantum field framework and applies to particles with any spin. It extends known results in the literature in that it does not introduce any large volume approximation, and it takes particle spin fully into account. We provide expressions of the microcanonical partition function at fixed multiplicities in the limiting classical case of large volumes and large angular momenta and in the grand-canonical ensemble. We also derive the microcanonical partition function of the ideal relativistic quantum gas with fixed parity.     

On the Mean-Field Spherical Model

Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci/ of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N. The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor σ-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.     

Multiplicity distributions in canonical and microcanonical statistical ensembles

The aim of this paper is to introduce a new technique for the calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand canonical partition function. A Taylor expansion of the generating function is used to separate contributions to the partition function in their power in volume. We employ Laplace’s asymptotic expansion to show that any equilibrium distribution of multiplicity, charge, energy, etc. tends to a multivariate normal distribution in the thermodynamic limit. A Gram–Charlier expansion additionally allows for the calculation of finite volume corrections. Analytical formulas are presented for the inclusion of resonance decay and finite acceptance effects directly into the partition function of the system. This paper consolidates and extends previously published results of the current investigation into the properties of statistical ensembles.     

Percolation and fragmentation of excited atomic nuclei

We present an analysis which aims at explaining the similarities (and differences) which exist between a simple bond percolation process on a cubic lattice and the fragmentation of highly excited atomic nuclei. Emphasis is placed on discussing percolation in terms of concepts which are well known in nuclear physics such asQ-value and particle emission thresholds. Similarities and differences between the bond percolation process and nuclear fragmentation are discussed. An approximate expression for the microcanonical partition sum (number of microstates) corresponding to any given percolation partition is shown to provide a good starting point for predicting fragment size distributions.Communicated by: X. Campi     

Microcanonical ensemble study of a gas column under gravity

The study of a classical ideal gas column of finite height H in a uniform gravitational field g is made by the microcanonical ensemble at energy E. The primary functions of this ensemble, the phase volume and the density of states, are derived. Related statistical quantities, such as the entropy, the temperature and the heat capacity, are also reported. The equivalence in the thermodynamic limit between the calculated microcanonical expressions and those obtained from the canonical ensemble is shown numerically. The expression for the temperature is used to analyze the temperature change when the gas is permitted to expand into an evacuated region increasing the height of the column from H ₁ to H ₂. The microcanonical single-particle momentum and height distributions are also reported.     

Effect of rare fluctuations on the thermalization of isolated quantum systems

We consider the question of thermalization for isolated quantum systems after a sudden parameter change, a so-called quantum quench. In particular, we investigate the prerequisites for thermalization, focusing on the statistical properties of the time-averaged density matrix and of the expectation values of observables in the final eigenstates. We find that eigenstates, which are rare compared to the typical ones sampled by the microcanonical distribution, are responsible for the absence of thermalization of some infinite integrable models and play an important role for some nonintegrable systems of finite size, such as the Bose-Hubbard model. We stress the importance of finite size effects for the thermalization of isolated quantum systems and discuss two scenarios for thermalization.     

Partition function of a particle subject to Gaussian noise

We study the averaged partition function for a quantum particle subjected to Gaussian noise using the path integral representation. The noise is characterized by a covariance function with a strength and a range. It falls off rapidly with distance but the analytic form at short distances and the dimensionality are important. The remaining parameter is the thermal length of the particle. For a finite range we study the behavior of the partition function over the entire domain of strengths and thermal lengths. The techniques used are successively more accurate upper and lower bounds that include contributions from configurations involving traps. Particular attention is paid to a self-consistent field analysis lower bound and to a nonlocal quadratic action bound. We also study the white noise limit, i.e., vanishing range with finite values of the other parameters. In one dimension the white noise limit leads to convergent results. In three or higher dimensions the divergent terms can be isolated and computed. In two dimensions the degree of divergences changes at a finite value of the product of the strength and thermal length squared.     

Combinatorial and Topological Analysis of the Ising Chain in a Field

We present an alternative solution of the Ising chain in a field under free and periodic boundary conditions, in the microcanonical, canonical, and grand canonical ensembles, from a unified combinatorial and topological perspective. In particular, the computation of the per-site entropy as a function of the energy unveils a residual value for critical values of the magnetic field, a phenomenon for which we provide a topological interpretation and a connection with the Fibonacci sequence. We also show that, in the thermodynamic limit, the per-site microcanonical entropy is equal to the logarithm of the per-site Euler characteristic. The canonical and grand canonical partition functions are identified as combinatorial generating functions of the microcanonical problem, which allows us to evaluate them. A detailed analysis of the magnetic field-dependent thermodynamics, including positive and negative temperatures, reveals interesting features. Finally, we emphasize that our combinatorial approach to the canonical ensemble allows exact computation of the thermally averaged value <????> of the Euler characteristic associated with the spin configurations of the chain, which is discontinuous at the critical fields, and whose thermal behavior is expected to determine the phase transition of the model. Indeed, our results show that the conjecture <????>?(T _C)?=?0, where T _C is the critical temperature, is valid for the Ising chain.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Form factors in finite volume I: Form factor bootstrap and truncated conformal space

We describe the volume dependence of matrix elements of local fields to all orders in inverse powers of the volume (i.e., only neglecting contributions that decay exponentially with volume). Using the scaling Lee–Yang model and the Ising model in a magnetic field as testing ground, we compare them to matrix elements extracted in finite volume using truncated conformal space approach to exact form factors obtained using the bootstrap method. We obtain solid confirmation for the form factor bootstrap, which is different from all previously available tests in that it is a non-perturbative and direct comparison of exact form factors to multi-particle matrix elements of local operators, computed from the Hamiltonian formulation of the quantum field theory. We also demonstrate that combining form factor bootstrap and truncated conformal space is an effective method for evaluating finite volume form factors in integrable field theories over the whole range in volume.     

Topological charge from thermodynamics in two dimensions

It is shown that in the weak coupling limit the partition function of massless (QED)₂ in a finite volume is identical to that of a bose field if the boson topological charge is identified with the fermion number. Generalizations to theories with U(N) gauge symmetry, fractional fermion number, and the Schwinger model for any coupling are also discussed.     

One-way quantum deficit and quantum coherence in the anisotropic <Emphasis Type="Italic">XY</Emphasis> chain

In this study, we investigate pairwise non-classical correlations measured using a one-way quantum deficit as well as quantum coherence in the XY spin-1/2 chain in a transverse magnetic field for both zero and finite temperatures. The analytical and numerical results of our investigations are presented. In the case when the temperature is zero, it is shown that the one-way quantum deficit can characterize quantum phase transitions as well as quantum coherence. We find that these measures have a clear critical point at λ = 1. When λ ≤ 1, the one-way quantum deficit has an analytical expression that coincides with the relative entropy of coherence. We also study an XX model and an Ising chain at the finite temperatures.     

Nonthermal statistics in isolated quantum spin clusters after a series of perturbations

We show numerically that a finite isolated cluster of interacting spins 1/2 exhibits a surprising nonthermal statistics when subjected to a series of small nonadiabatic perturbations by an external magnetic field. The resulting occupations of energy eigenstates are significantly higher than the thermal ones on both the low and the high ends of the energy spectra. This behavior semiquantitatively agrees with the statistics predicted for the so-called "quantum microcanonical" ensemble, which includes all possible quantum superpositions with a given energy expectation value. Our findings also indicate that the eigenstates of the perturbation operators are generically localized in the energy basis of the unperturbed Hamiltonian. This kind of localization possibly protects the thermal behavior in the macroscopic limit.     

Volume dependence of Schwinger functions in the Yukawa2 quantum field theory

We prove upper bounds on the partition function and Schwinger functions for the Euclidean Yukawa₂ quantum field theory which depend on the interaction volume Λ only through a term of the form (const)^|Λ|. We also prove a lower bound of the form (const)^|λ| for the partition function. We work throughout in the Matthews-Salam representation with the fermions integrated out.     

Spontaneous supersymmetry breaking in matrix models from the viewpoints of localization and Nicolai mapping

In the previous work, it was shown that, in supersymmetric (matrix) discretized quantum mechanics, inclusion of an external field twisting the boundary condition of fermions enables us to discuss spontaneous breaking of supersymmetry (SUSY) in the path-integral formalism in a well-defined way. In the present work, we continue investigating the same systems from the points of view of localization and Nicolai mapping. The localization is studied by changing of integration variables in the path integral, which is applicable whether or not SUSY is explicitly broken. We examine in detail how the integrand of the partition function with respect to the integral over the auxiliary field behaves as the auxiliary field vanishes, which clarifies a mechanism of the localization. In SUSY matrix models, we obtain a matrix-model generalization of the localization formula. In terms of eigenvalues of matrix variables, we observe that eigenvalues' dynamics is governed by balance of attractive force from the localization and repulsive force from the Vandermonde determinant. The approach of the Nicolai mapping works even in the presence of the external field. It enables us to compute the partition function of SUSY matrix models for finite N (N is the rank of matrices) with arbitrary superpotential at least in the leading nontrivial order of an expansion with respect to the small external field. We confirm the restoration of SUSY in the large-N limit of a SUSY matrix model with a double-well scalar potential observed in the previous work.     

Quantum spin glass and the dipolar interaction

Anisotropic dipolar systems are considered. Such systems in an external magnetic field are expected to be a good experimental realization of the transverse field Ising model. With random interactions, this model yields a spin glass to paramagnet phase transition as a function of the transverse field. We show that the off-diagonal dipolar interaction, although effectively reduced, induces a finite correlation length and thus destroys the spin-glass order at any finite transverse field. We thus explain the behavior of the nonlinear susceptibility in the experiments on LiHo(x)Y(1-x)F(4), and argue that a crossover to the paramagnetic phase, and not quantum criticality, is observed.