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arXiv:1603.07207 · Physical Review A

Universal properties of Fermi gases in one dimension

Wenbin He, Yang-Yang Chen, Shizhong Zhang, Xi‐Wen Guan

Published Sep 21, 2016• 12 citations
arXivHTMLPDF

Abstract

In this Rapid Communication, we investigate the universal properties of a spin-polarized two-component Fermi gas in one dimension (1D) using the Bethe ansatz. We discuss the quantum phases and phase transitions by obtaining exact results for the equation of state, the contact, the magnetic susceptibility, and the contact susceptibility, giving a precise understanding of the 1D analog of the Bose-Einstein condensation and Bardeen-Cooper-Schrieffer crossover in three dimensions (3D) and the associated universal magnetic properties. In particular, we obtain the exact form of the magnetic susceptibility $\ensuremath{\chi}\ensuremath{\sim}1/\sqrt{T}exp(\ensuremath{-}\mathrm{\ensuremath{\Delta}}/T)$ at low temperatures, where $\mathrm{\ensuremath{\Delta}}$ is the energy gap and $T$ is the temperature. Moreover, we establish exact upper and lower bounds for the relation between polarization $P$ and the contact $C$ for both repulsive and attractive Fermi gases. Our findings emphasize the role of pair fluctuations in strongly interacting 1D fermion systems that can shed light on higher dimensions.

DOI
10.1103/physreva.94.031604

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We investigate magnetic properties and statistical effects in one-dimensional (1D) strongly repulsive two-component fermions and in a 1D mixture of strongly repulsive polarized fermions and bosons. Universality in the characteristics of phase transitions, magnetization, and susceptibility in the presence of an external magnetic field HHH are analyzed from the exact thermodynamic Bethe ansatz solution. We show explicitly that polarized fermions with a repulsive interaction have antiferromagnetic behavior at zero temperature. A universality class of linear field-dependent magnetization persists for weak and finite strong interaction. The system is fully polarized when the external field exceeds the critical value ${H}_{c}^{F}\ensuremath{\approx}\frac{8}{\ensuremath{\gamma}}{E}_{F}$, where EF{E}_{F}EF​ is the Fermi energy and $\ensuremath{\gamma}$ is the dimensionless interaction strength. In contrast, the mixture of polarized fermions and bosons in an external field exhibits square-root field-dependent magnetization in the vicinities of H=0H=0H=0 and the critical value $H={H}_{c}^{M}\ensuremath{\approx}\frac{16}{\ensuremath{\gamma}}{E}_{F}$. We find that a pure boson phase occurs in the absence of the external field, fully polarized fermions and bosons coexist for 0<H<HcM0<H<{H}_{c}^{M}0<H<HcM​, and a fully polarized fermion phase occurs for $H\ensuremath{\geqslant}{H}_{c}^{M}$. This phase diagram for the Bose-Fermi mixture is reminiscent of weakly attractive fermions with population imbalance, where the interacting fermions with opposite spins form singlet pairs.
This article briefly reviews recent theoretical developments in quantum critical phenomena in one-dimensional (1D) integrable quantum gases of cold atoms. We present a discussion on quantum phase transitions, universal thermodynamics, scaling functions and correlations for a few prototypical exactly solved models, such as the Lieb–Liniger Bose gas, the spin-1 Bose gas with antiferromagnetic spin-spin interaction, the two-component interacting Fermi gas as well as spin-3/2 Fermi gases. We demonstrate that their corresponding Bethe ansatz solutions provide a precise way to understand quantum many-body physics, such as quantum criticality, Luttinger liquids (LLs), the Wilson ratio, Tan's Contact, etc. These theoretical developments give rise to a physical perspective using integrability for uncovering experimentally testable phenomena in systems of interacting bosonic and fermonic ultracold atoms confined to 1D.
Abstract It has been shown (Yang and You 2011 Chin. Phys. Lett. 28 020503 ) that at zero temperature the ground state of the one-dimensional (1D) w -component Fermi gas coincides with that of the spinless Bose gas in the limit . This behavior was experimentally evidenced through quasi-1D tightly trapping ultracold 173 Yb atoms in a recent paper (Pagano et al 2014 Nat. Phys. 10 198 ). However, understanding of low-temperature behavior of Fermi gases with a repulsive interaction requires spin-charge separated conformal field theories of an effective Tomonaga–Luttinger liquid and an antiferromagnetic SU ( w ) Heisenberg spin chain. Here we analytically derive universal thermodynamics of 1D strongly repulsive fermionic gases with SU ( w ) symmetry via the Yang–Yang thermodynamic Bethe ansatz method. The analytical free energy and magnetic properties of the systems at low temperature in a weak magnetic field are obtained through the Wiener–Hopf method. In particular, the free energy essentially manifests the spin-charge separated conformal field theories for high-symmetry systems with arbitrary repulsive interaction strength. We also find that the sound velocity of the Fermi gases in the large w limit coincides with that for the spinless Bose gas, whereas the spin velocity vanishes quickly as w becomes large. This indicates strong suppression of the Fermi exclusion statistics by the commutativity feature among the w -component fermions with different spin states in the Tomonaga–Luttinger liquid phase. Moreover, the equations of state and critical behavior of physical quantities at finite temperature are analytically derived in terms of the polylogarithm functions in the quantum critical region.
This work studies zero-temperature Fermi gases with short-ranged and attractive interactions with focus on the dimensionality aspects by confining the system from three to two dimensions. Interacting Fermi gases are superfluids characterized as being strongly correlated if the scattering length is much larger than the other scales present. In the many-body context, this category of systems requires a careful treatment since mean-field approaches are not expected to hold under the conditions which define the strong pairing. For this reason, we employ quantum Monte Carlo methods to study the dimensionality effects in two-component Fermi gases in the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover. To be able to investigate such aspects, we consider a three-dimensional gas with periodic boundary conditions in the xy plane and a confining potential in the z direction capable of reducing the dimensionality of the system, which defines a dimensional BCS-BEC crossover. We calculate the equation of state as the energy per particle as a function of the dimensionality and as a function of the interaction parameter, which are characterized by the oscillator length in the z direction and the scattering length respectively. The greater confinement in the system lowers the interparticle distances, which increases the effects of the attractive interactions between antiparallel-spin fermions.We also calculate the density profile in the z direction over the crossover for different confinements and different interaction parameters to quantify their influence on the spatial distribution of the particles. This study connects the perspectives between theoretical and experimental works in low-dimensional ultracold Fermi gases by utilizing an approach that is similar to what it is realized in modern experiments.
Fermions in one dimension are vastly different from their higher-dimension counterparts. At the same time, many theoretical models in one dimension have the luxury of being integrable and exactly solvable, which makes them excellent test beds for quantum simulation. In using these test beds, we can understand and improve on the techniques of quantum simulation so that we move towards solving difficult problems that cannot be solved using other methods due to the sheer complexity of these systems. In this work, we prepare one-dimensional fermions using cold atom techniques and probe low-energy excitations in such systems. We do so using the technique of Bragg spectroscopy, which enables us to probe the dynamical structure factor of the system directly, which in turn tells us about the density-density and spin-density correlations in the system. Utilizing the extreme tunability of cold atom experiments, we can study different regimes of these 1D Fermi gases. In particular, moving on from our previous investigations in the repulsive interactions, we investigate the attractive interaction side, where the attraction of the fermions causes bound states to form. The binding energy of these bound states causes a gap in the spin excitation, a hallmark of a new class of model: Luther-Emery liquid. We investigate weakly attractive Fermi gases close to the Luther-Emery limit and observe an inversion of the spin-charge separation hierarchy compared to the repulsive side, confirming expectations from exact Bethe ansatz solutions for the homogeneous gas at zero temperature, in which weakly bound fermion pairs are predicted. The result presented here provides insight into physics in one dimension, an important realm that is well known to host many exciting physics and exotic phases.
In this thesis, we investigate thermodynamic properties, magnetic phase transitions and correlation functions of integrable quantum many-body systems in the field of ultracold atomic gases. The scope of our investigation encompasses three models in one-dimension that are integrable by means of the Bethe ansatz method. They are the multi-component 5-interacting Fermi gas, the spin-l Bose gas with antiferromagnetic spin interaction and the SU(2) spinor Bose gas with finite-range Gaussian interaction. We derive the thermodynamic Bethe ansatz equations for the general SU(k) Fermi gas. We illustrate the root patterns and give a quantitative description of the ground state properties for the strongly and weakly interacting regimes. The thermodynamics and magnetic phase transitions are described quantitatively by means of the thermodynamic Bethe ansatz equations. We find that all phase transitions in the attractive regime are of second order with a linear field dependent magnetization in the vicinities of critical points. The SU(2) Fermi gas is studied in more detail. The low temperature thermodynamics for the repulsive case is derived by means of the Wiener-Hopf technique. Finite-size corrections and correlation functions for both the repulsive and attractive cases are derived from the nested Bethe ansatz equations and conformal field theory. In the attractive case, we show that spatial oscillations of the leading terms in the pair correlation function and spin correlation function solely depend on the mismatch between Fermi surfaces of spin-up and spin-down fermions. Such spatial modulations are characteristic of a Fulde- Ferrell-Larkin-Ovchinnikov (FFLO) state. For the spin-l Bose gas with antiferromagnetic spin interaction, we find that at zero temperature, the system exhibits three quantum phases: a phase with singlet pairs of bosons, a fully-polarized Tonks-Girardeau gas phase of unpaired ms = 1 bosons, and a mixed phase of singlet pairs and unpaired ms = 1 bosons. Phase transitions in the vicinities of the critical fields are also of second order with a linear field dependent magnetization. Finally, we use the asymptotic Bethe ansatz method to derive the quasimomenta distribution and ground state energy for the SU(2) spinor Bose gas with finite-range Gaussian interaction. We show that finite range potentials are more likely to lead to quasi Bose-Einstein condensation than zero range potentials by studying the spatial density profiles as the interaction width increases.