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arXiv:2106.15095

Statistic physics in bound state in 3D fermi gas system

Chen-Huan Wu

Published Jun 29, 2021• 2 citations
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Abstract

The bound state is treated as a long-lived quasiparticle with slow momenta and current relaxation in fermi liquid phase. In this paper, we discuss the realization of ensemble behavior in a 3D fermi gas system in non-fermi liquid phase. We reveal the relation between UV cutoff of relative momentum $Λ_{q}$ and its ensemble behavior. The behavior of a bound state system has rarely been investigated before, and the relation between the scattering momentum and the physics has not yet been explored before (to best of our knowledge). We found that the cutoff $Λ_{q}$ directly related to the distribution and statistical variance of coupling term, which becomes Gaussian variable (or Chi-square variable) in limit (i.e., with a large step number of a fractional distance $Λ_{q}^{-1}\rightarrow\infty$). Also, we show that the different $Λ_{q}^{-1}$ lending support to different phases, including non-fermi liquid phase and (disordered) fermi liquid phase, which correspond to ill-defined and well-defined bound states, respectively. The pair condensation induced by local coupling, which happen at critical temperature, would suppresses the non-fermi liquid for constant on-site coupling (it is not the case when the coupling be variant). % We deduce the free energy density and the divergent behaviors of anomalous propagator in this case. % Finally, we also present a discussion of ensembles of level statistic that depending on which symmetry class the system belongs to, and that can be controlled by the type of disorder. Further, when the range of condensation is larger, the homogeneous character will emerge and the thermalization can be observed where eigenstates with higher energy has lower overlap with the low-entangled states.

DOI
10.48550/arxiv.2106.15095

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While bound states are typically treated as long-lived quasiparticles with slow momentum and current relaxation in a Fermi liquid (FL) phase, this paper investigates their properties within a non-Fermi liquid (NFL) phase exhibiting Sachdev–Ye–Kitaev (SYK)-like ensemble behavior. We establish a fundamental relation between the UV cutoff of the relative scattering momentum, [Formula: see text], and the emergence of this SYK-like physics, a connection which has not been previously explored. We find that [Formula: see text] directly controls the statistical properties of the coupling term. In the limit [Formula: see text], the large number of momentum steps causes the coupling to behave as a Gaussian or Chi-square random variable, fulfilling the conditions for SYK-like chaos. This reveals [Formula: see text] as a tuning parameter for a phase transition: the [Formula: see text] limit corresponds to the NFL phase with ill-defined bound states, while a finite [Formula: see text] corresponds to a (disordered) FL phase with well-defined bound state quasiparticles. Furthermore, we show that pair condensation, which emerges at a critical temperature, competes with and suppresses the NFL phase. Finally, we note that in the case of large-range condensation, a homogeneous character emerges and thermalization is observed, where high-energy eigenstates show low overlap with low-entanglement states.
The behavior of Fermi systems which approach the fermion condensation quantum phase transition (FCQPT) from the disordered phase is considered. We show that the quasiparticle effective mass M ∗ diverges as M ∗ ∝ 1/|x − xFC | where x is the system density and xFC is the critical point at which FCQPT occurs. Such a behavior is of general form and takes place in both three dimensional (3D) systems and two dimensional (2D) ones. Since the effective mass M ∗ is finite, the system exhibits the Landau Fermi liquid behavior. At |x −xFC|/xFC ≪ 1, the behavior can be viewed as a highly correlated one, because the effective mass is large and strongly depends on the density. In case of electronic systems the Wiedemann-Franz law is held and Kadowaki-Woods ratio is preserved. Beyond the region |x−xFC|/xFC ≪ 1, the effective mass is approximately constant and the system becomes conventional Landau Fermi liquid.
The appearance of the fermion condensation, which can be compared to the Bose-Einstein condensation, in different Fermi liquids is considered; its properties are discussed; and a large amount of experimental evidence in favor of the existence of the fermion condensate (FC) is presented. We show that the appearance of FC is a signature of the fermion condensation quantum phase transition (FCQPT), which separates the regions of normal and strongly correlated liquids. Beyond the FCQPT point, the quasiparticle system is divided into two subsystems, one containing normal quasiparticles and the other, FC, localized at the Fermi level. In the superconducting state, the quasiparticle dispersion in systems with FC can be represented by two straight lines, characterized by effective masses M FC * and M L * and intersecting near the binding energy E 0, which is of the order of the superconducting gap. The same quasiparticle picture and the energy scale E 0 persist in the normal state. We demonstrate that fermion systems with FC have features of a “quantum protectorate” and show that strongly correlated systems with FC, which exhibit large deviations from the Landau Fermi liquid behavior, can be driven into the Landau Fermi liquid by applying a small magnetic field B at low temperatures. Thus, the essence of strongly correlated electron liquids can be controlled by weak magnetic fields. A reentrance into the strongly correlated regime is observed if the magnetic field B decreases to zero, while the effective mass M* diverges as $$M^ * \propto {1 \mathord{\left/ {\vphantom {1 {\sqrt B }}} \right. \kern-\nulldelimiterspace} {\sqrt B }}$$ . The regime is restored at some temperature T∗∝BT^ * \propto \sqrt B T∗∝B​ . The behavior of Fermi systems that approach FCQPT from the disordered phase is considered. This behavior can be viewed as a highly correlated one, because the effective mass is large and strongly depends on the density. We expect that FCQPT takes place in trapped Fermi gases and in low-density neutron matter, leading to stabilization of the matter by lowering its ground-state energy. When the system recedes from FCQPT, the effective mass becomes density independent and the system is suited perfectly to be conventional Landau Fermi liquid.
The behavior of Fermi systems that approach the fermion condensation quantum phase transition (FCQPT) from the disordered phase is considered. We show that the quasiparticle effective mass M * diverges as M * ∝ 1/¦ x − x FC ¦, where x is the system density and x FC is the critical point at which FCQPT occurs. Such behavior is of general form and takes place in both three-dimensional (3D) and two-dimensional (2D) systems. Since the effective mass M * is finite, the system exhibits the Landau Fermi liquid behavior. At ¦ x − x FC ¦/ x FC ≪1, the behavior can be viewed as highly correlated, because the effective mass is large and strongly depends on the density. In the case of electronic systems, the Wiedemann-Franz law is valid and the Kadowaki-Woods ratio is preserved. Beyond the region ¦ x – x FC ¦/ x FC ≪1, the effective mass is approximately constant and the system becomes a conventional Landau Fermi liquid.
law is enhanced by a logarithmic factor. In Ref. [1], we use multi-dimensional bosonization to provide a simple derivation of this result, and show that the logarithimic factor has a 1D origin. More importantly the bosonization technique allows us to take into account the Fermi liquid interactions, and obtain the leading scaling behavior of the entanglement entropy of Fermi liquids. The central result of our work is that Fermi liquid interactions do not alter the leading scaling behavior of the entanglement entropy, and the logarithmic enhancement of area law is a robust property of the Fermi liquid phase. In sharp contrast to the fermioic systems with Fermi surfaces, quantum critical (or gapless) bosonic systems do not violate the area law above 1D (except for the case discussed below). The fundamental difference lies in the fact that gapless excitations live near a single point (usually origin of momentum space) in such bosonic systems, while they live around an (extended) Fermi surface in Fermi liquids. In Ref. [2], we studied entanglement properties of some specific examples of the so called Bose metal states, in which bosons neither condense (and become a superfluid) nor localize (and insulate) at T=0. The system supports gapless excitations around “Bose surfaces,” instead of isolated points in momentum space. We showed that similar to free Fermi gas and Fermi liquids, these states violate the entanglement area law in a logarithmic fashion. Compared to ground states, much less is known concretely about entanglement in (highly) excited states. Going back to free fermion systems, in Ref. [3] we show that there exists a duality relation between ground and excited states, and the area law obeyed by ground state turns into a volume law for excited states, something widely expected but hard to prove. Most importantly, we find in appropriate limits the reduced density matrix of a subsystem takes the form of thermal density matrix, providing an explicit example of the eigenstate thermalization hypothesis. Our work [3] explicitly demonstrates how statistical physics emerges from entanglement in a single eigenstate.
We discuss in this paper the low energy properties of a liquid of fermions coupling to a U(1) gauge field at wavevectors q<Λ≪k F at dimensions larger than one, where Λ≪k F is a high momentum cutoff and k F is the Fermi wave vector. In particular, we shall consider the e2→∞ limit where charge and current fluctuations at wave vectors q<Λ are forbidden, and the problem reduces to the problem of imposing constraint that no charge and current fluctuations are allowed in the liquid of fermions. Within a bosonization approximation, we show that the low energy properties of the system can be described as a Fermi liquid of chargeless quasiparticles which has vanishing wavefunction overlap with the bare fermion's in the system. The case of a two component system (t–J model) will also be discussed.