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arXiv:1504.06528 · Journal of Mathematical Physics

The total momentum of quantum fluids

András Sütő

Published Aug 1, 2015• 2 citations
arXivHTMLPDF

Abstract

The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids, this probability is 1, and in a crystal, it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase, so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid, we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and argue that Landau’s criterion cannot explain superfluidity and its breakdown at positive temperatures. We also comment on the relation between superfluidity and Bose-Einstein condensation.

DOI
10.1063/1.4927212

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The probability distribution of the total momentum P is studied in N-particle interacting homogeneous quantum systems at positive temperatures. Using Galilean invariance we prove that in one dimension, the asymptotic distribution of P/N is normal at all temperatures and densities, and in two dimensions, the tail distribution of P/N is normal. We introduce the notion of the density matrix reduced to the center of mass and show that its eigenvalues are N times the probabilities of the different eigenvalues of P. A series of results is presented for the limit of sequences of positive definite atomic probability measures, relevant for the probability distribution of both the single-particle and the total momentum. The P = 0 ensemble is shown to be equivalent to the canonical ensemble. Through some conjectures we associate the properties of the asymptotic distribution of the total momentum with the characteristics of fluid, solid, and superfluid phases. Our main suggestion is that in interacting quantum systems above one dimension, in infinite space, the total momentum is finite with a nonzero probability at all temperatures and densities. In solids, this probability is 1, and in a crystal, it is distributed on a lattice. Since it is less than 1 in two dimensions, we conclude that a 2D system is always in a fluid phase, so if the hexatic phase existed classically, it would be destroyed by quantum fluctuations. For a superfluid, we conjecture that the total momentum is zero with a nonzero probability and otherwise its distribution is continuous. We define a macroscopic wave function based on the density matrix reduced to the center of mass. We discuss how dissipation can give rise to a critical velocity, predict the temperature dependence of the latter, and argue that Landau’s criterion cannot explain superfluidity and its breakdown at positive temperatures. We also comment on the relation between superfluidity and Bose-Einstein condensation.
The total momentum of NNN interacting bosons or fermions in a cube equipped with periodic boundary conditions is a conserved quantity. Its eigenvalues follow a probability distribution, determined by the thermal equilibrium state. While in non-interacting systems the distribution is normal with variance ∼N\sim N∼N, interaction couples the single-particle momenta, so that the distribution of their sum is unpredictable, except for some implications of Galilean invariance. First, we present these implications which are strong in 1D, moderately strong in 2D, and weak in 3D. Then, we speculate about the possible form of the distribution in fluids, crystals, and superfluids. The existence of phonons suggests that the total momentum can remain finite when N→∞N\to\inftyN→∞. We argue that in fluids the finite momenta distribute continuously, but their integrated probability is smaller than 1, because the momentum can also tend to infinity with NNN. In the fluid-crystal transition we expect that the total momentum becomes finite with full probability and distributed over a lattice, and that in the fluid-superfluid transition a delta peak appears only at zero total momentum. Based on this picture, we discuss the superfluid flow in both the frictionless and the dissipative cases, and derive a temperature-dependent critical velocity. Finally, we show that Landau's criterion for excitations in moving superfluids is an in some cases correct result of an erroneous derivation.
Correlated Density Matrix (CDM) theory permits formal analyses of microscopic properties of strongly correlated quantum fluids and liquids at nonzero temperatures. Equilibrium properties, thermodynamic potentials, correlation and structure functions can be studied formally as well as numerically within the CDM algorithm. Here we provide the essential building blocks for studying the radial distribution function and the single-particle momentum distribution of the ingredients of the quantum systems. We focus on the statistical properties of correlated fluids and introduce the concept of renormalized bosons and fermions. These entities carry the main statistical features of the correlated systems such as liquid 4 He through their specific dependence on temperature, particle number density, and wavenumber encapsulated in their effective masses. The formalism is developed for systems of bosons and of fermions. Numerical calculations for fluid 4 He in the normal phase demonstrate the power of the renormalization concept. The formalism is further extended to analyze the Bose-Einstein condensed phases and gives a microscopic understanding of Tisza's two-fluid model for the normal and superfluid density components.
The hierarchy of equations for reduced density matrices that describes a thermodynamically equilibrium quantum system obtained earlier by the author is investigated in the momentum representation. In the paper it is shown that the use of the momentum representation opens up new opportunities in studies of macroscopic quantum systems both nonsuperfluid and superfluid. It is found that the distribution over momenta in a quantum fluid is not a Bose or Fermi distribution even in the limit of practically noninteracting particles. The distribution looks like a Maxwellian one although, strictly speaking, it is not Maxwellian. The momentum distribution in a quantum crystal depends upon the interaction potential and the crystalline structure. The momentum distribution in a superfluid contains a delta function. The momentum distribution for the condensate in a superfluid crystal consists of delta peaks that are arranged periodically in momentum space. The periodical structure remains if the condensate crystal is not superfluid.
The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum fluctuations, which favour disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of its "marginal" behaviour. In an infinite uniform 2d fluid thermal fluctuations at any non-zero temperature are strong enough to destroy the fully ordered state associated with Bose--Einstein condensation, but are not strong enough to suppress superfluidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual "quasi-long-range" order at low temperatures leads to an interesting interplay between superfluidity and condensation in all experimentally relevant finite-size systems. In these notes we give an introduction to the physics of 2d Bose fluids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultra-cold atomic gases confined to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other fields of physics.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
We discuss the superfluid properties of a uniform, weakly interacting Bose-Einstein condensed gas with spin-orbit coupling, realized recently in experiments. We find a finite normal fluid density ${\ensuremath{\rho}}_{n}$ at zero temperature which turns out to be a function of the Raman coupling. In particular, the entire fluid becomes normal at the transition point from the zero momentum to the plane wave phase, even though the condensate fraction remains finite. We emphasize the crucial role played by the breaking of Galilean invariance and by the gapped branch of the elementary excitations whose contribution to various sum rules is discussed explicitly. Our predictions for the superfluid density are successfully compared with the available experimental results based on the measurement of the sound velocities.