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arXiv:1112.4098 · Physical Review C

Higher order corrections to density and temperature of fermions from quantum fluctuations

Hua Zheng, Aldo Bonasera A

Published Aug 31, 2012• 24 citations
arXivHTMLPDF

Abstract

A novel method to determine the density and temperature of a system based on quantum Fermionic fluctuations is generalized to the limit where the reached temperature $T$ is large compared to the Fermi energy ${\ensuremath{\varepsilon}}_{f}$. Quadrupole and particle multiplicity fluctuation relations are derived in terms of $\frac{T}{{\ensuremath{\varepsilon}}_{f}}$. The relevant Fermi integrals are numerically solved for any values of $\frac{T}{{\ensuremath{\varepsilon}}_{f}}$ and compared to the analytical approximations. The classical limit is obtained, as expected, for large temperatures and low densities. The entropy can also be easily derived from quantum fluctuations and gives important insight into the behavior of the system near a phase transition.

DOI
10.1103/physrevc.86.027602

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A novel method to determine the density and temperature of a system based on quantum Fermionic fluctuations is generalized to the limit where the reached temperature TTT is large compared to the Fermi energy \ensuremathεf{\ensuremath{\varepsilon}}_{f}\ensuremathεf​. Quadrupole and particle multiplicity fluctuation relations are derived in terms of T\ensuremathεf\frac{T}{{\ensuremath{\varepsilon}}_{f}}\ensuremathεf​T​. The relevant Fermi integrals are numerically solved for any values of T\ensuremathεf\frac{T}{{\ensuremath{\varepsilon}}_{f}}\ensuremathεf​T​ and compared to the analytical approximations. The classical limit is obtained, as expected, for large temperatures and low densities. The entropy can also be easily derived from quantum fluctuations and gives important insight into the behavior of the system near a phase transition.
We suggest a new method of calculation of the equilibrium correlation functions of an arbitrary order for the interacting Fermi-gas model in the framework of the static fluctuation approximation method. This method based only on a single and controllable approximation allows obtaining the so-called far-distance equations. These equations connecting the quantum states of a Fermi particle with variables of the local field operator contain all necessary information related to the calculation of the desired correlation functions and basic thermodynamic parameters of the many-body system. The basic expressions for the mean energy and heat capacity for the electron gas at low temperatures in the high-density limit were obtained. All expressions are given in the units of r s , where r s determines the ratio of a mean distance between electrons to the Bohr radius a 0. In these expressions, we calculate terms of the respective order r s and r 2 . It is also shown that the static fluctuation approximation allows finding the terms related to higher orders of the decomposition with respect to the parameter r s .
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
A transition in the hole probability at finite temperature for free fermions in ddd dimensions
In a free Fermi gas at temperature TTT much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability P(R,T)P(R,T)P(R,T), i.e. the probability that a region of size RRR is devoid of particles, in dimension ddd, and on the case of a spherical region of large radius RRR. We show that at low temperature it takes the scaling form P(R,T)∼exp⁡[−(kFR)d+1Φd(u=2R T/kF)],P(R,T)\sim \exp\big[-(k_F R)^{d+1}Φ_d(u=2R\,T/k_F)\big],P(R,T)∼exp[−(kF​R)d+1Φd​(u=2RT/kF​)], where kFk_FkF​ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function Φd(u)Φ_d(u)Φd​(u) in any dimension. Remarkably, it exhibits a transition of order 32(d+1)\tfrac{3}{2}(d+1)23​(d+1) at the universal critical value uc=2/πu_c=2/πuc​=2/π, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.
Abstract In this paper, we study the chaotic behavior of the unitary Fermi gas in both high- and low-temperature limits by calculating the quantum Lyapunov exponent defined in terms of the out-of-time-order correlator. We take the method of generalized Boltzmann equations, derived from the augmented Keldysh approach (Aleiner et al 2016 Ann. Phys. 375 , 378). At high temperatures, the system is described by weakly interacting fermions with two spin components and the Lyapunov exponent is found to be λ L = 21 n T 1 / 2 . Here n is the density of fermions for a single-spin component. In the low-temperature limit, the system is a superfluid and can be described by phonon modes. Using the effective action derived in (Son and Wingate 2006 Ann. Phys. 321 , 197), we find λ L = 9 × 10 3 T T F 4 T where T F is the Fermi energy. By comparing these to existing results of heat conductivity, we find that D E ≪ v 2 / λ L where D E is the energy diffusion constant and v is some typical velocity. We argue that this is related to the conservation law for such systems with quasi-particles.