Effective field theory of one-dimensional polarized Fermi gas

Transcription

1 Journal of Low Temperature Physics manuscript No. (will be inserted by the editor) Effective field theory of one-dimensional polarized Fermi gas Erhai Zhao W. Vincent Liu Keywords Ultracold Fermi gas, spin imbalance, FFLO state, Luttinger liquid Abstract We derive an effective field theory describing the long wave length, low energy properties of one-dimensional (1D) attractive Femi gases with spin imbalance. Our theory is based on the exact solution (Bethe ansatz) of the microscopic Hamiltonian, the Gaudin-Yang model. We show that the 1D Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state is a novel quantum fluid, a two-component Luttinger liquid with spin-charge mixing. Applying the theory, we obtain the correlation functions and the universal low temperature thermodynamics. Our theory can also predict the phase diagram of weakly coupled 1D gases, a quasi-1d system recently realized in experiments at Rice university. PACS numbers: Fk,03.75.Ss, z 1 Introduction Rapid developments in trapping and cooling alkaline Fermi gases with spin imbalance, such as 6 Li atoms in hyperfine states F = 1/2,m f = ±1/2, have revived the interest in Fulde- Ferrel-Larkin-Ovchinnikov (FFLO) state, a superfluid with spatially modulated order parameter 7. Both theory 8 and experiments 3,4,5,6 indicate that the FFLO state only occurs in a tiny region in the phase diagram of 3D polarized Fermi gases, whereas phase separation is favored in energetics. Recently, it was argued by several authors 13,12,24 that a promising regime to search for FFLO is in lattice cold atom systems, especially in quasi-one dimension, i.e., a 2D lattice array of weakly coupled 1D gas tubes. It is thus highly desirable to have a theoretical phase diagram of such quasi-1d imbalanced Fermi gases. This turns out to be a hard problem, mainly because strong quantum fluctuations render mean field theories unreliable in the quasi-1d regime. To make progress, we first construct a field theory for the 1D Fermi gas, which enables us to compute the zero and finite temperature correlation functions of interest. Then, the instability of 1D FFLO can be analyzed by treating the interchain coupling as perturbation using standard renormalization group analysis. Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA

2 2 Fig. 1 (Color online) The zero temperature phase diagram of Gaudin-Yang model as function of magnetic field h for given chemical potential. The 1D FFLO phase can be viewed as a mixture of pairs and leftover fermions. 2 Model and Bethe ansatz phase diagram The Fermi gas confined in each tube is described by the Gaudin-Yang model 14,15, H = [ h dx ψ σ 2 (x) σ=, 2m 2 x 2 + µ σ ] ψ σ (x) c dxψ (x)ψ (x)ψ (x)ψ (x). (1) Here, two hyperfine species with equal mass m are labeled with spin up and down respectively. The contact attractive interaction strength is c. Experimentally, the chemical potentials for the spin up and spin down fermions can be controlled to be different, say µ > µ. The situation is referred to as spin imbalance. Following convention, we define the chemical potential µ = (µ + µ )/2, the effective magnetic field h = (µ µ )/2, the total density n = n + n, the magnetization M = n n, and the polarization p = M/n. Also, we introduce the dimensionless interaction strength γ = 2/(na 1 ), where a 1 = 2 h 2 /(mc) is the 1D scattering length. The Gaudin-Yang model can be solved exactly by Bethe ansatz. Recently, its zero temperature phase diagram has been worked out theoretically by several groups 9,10,11. There are three phases: the fully paired (BCS) phase which is a quasi-condensate with zero polarization (p = 0), the fully polarized (normal) phase with p = 1, and the partially polarized (1D FFLO) phase where 0 < p < 1. For given µ, the FFLO phase is separated from the BCS phase and the normal phase by two quantum critical points at magnetic field h = h c1 and h c2, respectively. Experimentally, the gas is dilute and strongly interacting (γ 1). In the limit of γ, the phase diagram significantly simplifies, as shown in Fig. 1. Although being exact and powerful, Bethe ansatz itself is unable to further clarify the nature of the 1D FFLO state, for example, its low energy collective excitations and correlation functions. This requires a field theory description of the 1D FFLO phase in the spirit of Haldane s effective field theory approach to Luttinger liquids 16. To guide the construction of the field theory, it is useful to develop physical intuitions in two limits, the weak coupling limit (γ 1) and strong coupling limit (γ 1). In the weakling coupling limit, we can apply the standard Abelian bosonization procedure. After linearizing the spectrum of each spin species around its Fermi points, the Gaudin-Yang model can be bosonized 24. We find that the charge and spin field, φ c/s = (φ ± φ )/ 2, are coupled together by density and current interactions terms, such as φ c φ s. These spincharge mixing terms exist for any 0 < p < 1. They are marginal operators (in the renormalization group sense) and cannot be simply dropped. This is in sharp contrast to the familiar case of Luther-Emery liquid of unpolarized attractive Fermi gases where the spin and charge

3 3 degrees of freedom decouple. Previously, a field theory for 1D FFLO state was proposed by Yang to describe 1D superconductors in magnetic field 13. In Yang s model, spin and charge degrees of freedom are assumed decoupled throughout. Our analysis shows that his model does not apply to attractive Fermi gases with finite spin imbalance. The violation of spin-charge separation was also noticed in a recent DMRG study of 1D attractive Hubbard model 22. The strong coupling limit, on the other hand, is intuitively simpler. Due to strong attraction, any minority (spin ) fermion would like to pair up with a majority (spin ) fermion. Then, the FFLO phase can be viewed as a mixture of n tightly bound pairs and n n leftover fermions. From the expression of ground state energy as a function of density and polarization, zero temperature Bethe ansatz indeed shows that the system can be thought as a Tonks-Girardeau gas of pairs (hard-core bosons) and free Fermi gas of leftovers 9,10,11. This suggests the the 1D FFLO is equivalent to two coupled gases of spinless fermions. Yet, the correspondence remains to be firmly established. Thus our goal is to formulate a unified theory, applicable for arbitrary polarization and interaction strength, for the 1D FFLO based on the Bethe ansatz. 3 Nonperturbative bosonization To find the low energy, long wave length collective behavior of the Gaudin-Yang model, we take a detour and analyze its (regularized) lattice version, the 1D attractive Hubbard model H U. The basic idea is that in the continuum limit, the effective field theory of the two models are the same, whereas the parameters of these two models are related by m = h2 2ta 2, g = Ua, γ = U 2tn. (2) Here, t is the hopping, a is the lattice constant, and U is the on-site attraction. In this way, the theory has a well defined ultraviolet cutoff, and we can make use of several well established, nontrivial results (especially the dressed charge formalism) for the 1D Hubbard model. For technical convenience, we map the attractive Hubbard model H U for given (U/t,n, p) to a repulsive Hubbard model H U with onsite repulsion U, density n = 1 np, and imbalance p = (1 n)/(1 np) by the well-known staggered particle-hole transformation. Its coupled Bethe ansatz integral equations is solved for given (U/t,n, p ) to obtain the dressed energies, the dressed charge matrix elements, and the normal mode group velocities following the standard procedure 17,23. It is well known that H U can be mapped to a two-component Tomonaga-Luttinger Hamiltonian H T L, which yields the the same finite size spectrum and correlation functions at low energies 23. The various Luttinger parameters can be computed from the dressed energies and the dressed charge matrix. To obtain the effective Hamiltonian for H U, we particlehole transform H T L back to obtain a quadratic Hamiltonian with spin-charge mixing. We further diagonalize it by a canonical transformation using the dressed charge matrix 18,19. The end result is the effective Hamiltonian for the 1D FFLO phase, H FFLO shown below. The main steps outlined above is schematically shown in Fig. 2. Similar procedure has been employed for example to study the Mott transition of 1D repulsive Hubbard model in magnetic field 19.

4 4 Fig. 2 (Color online) The main steps to derive the effective Hamiltonian H FFLO for the Gaudin-Yang model. 4 Main results The final result of the above analysis can be summarized as follows. The 1D FFLO state is a two component Tomonaga-Luttinger liquid described by the effective Hamiltonian H FFLO = i=1,2 dx 2π u [ i ( ϑi ) 2 + ( ϕ i ) 2]. (3) Here, the two normal modes of the system (i = 1,2) have different group velocities, u 1 and u 2. Each mode, described by the boson field ϕ i and its dual ϑ i, is a superposition of the spin up and down fields φ σ and θ σ, ( ϕ1 ϕ 2 ) ( ) = ( Z T ) 1 φ, φ ( ϑ1 ϑ 2 ) = Z ( θ θ ). (4) The superscript T means matrix transpose, and the dressed charge matrix is given by [ ] Zcc Z Z = sc Z sc. (5) Z ss Z cs Z ss This clearly shows that in general each normal mode ϕ i is a mixture of the spin field φ s and the charge field φ c. The degree of spin-charge mixing is determined by the dressed charge matrix Z 18,17,19,20. All the parameters in the effective bosonized Hamiltonian H FFLO, such as { Z i j,u i }, can be computed from the parameters of the microscopic Hamiltonian H GY exactly. Pair correlation functions. The bosonized effective Hamiltonian Eq. (3) makes it straightforward to compute the correlation functions, for example the pair susceptibility, χ ( x,τ) = T τ (x,τ) (0,0).

5 5 Here, (x) = ψ R, (x)ψ L, (x), and L,R are the chiral indices for slow fermion fields. For x at zero temperature, we find χ(x) eiq x. (6) x2δ The FFLO wave vector q = k f, k f,, with k f,σ = n σ π. Here δ is the scaling dimension of the pair operator. It is directly related to the dress charge matrix elements that describe the effect of spin-charge mixing. Its value can be computed from the Bethe ansatz following the procedure above. Eq. (6) is the hallmark of 1D FFLO phase: the s-wave pair correlation function oscillates in space at the FFLO wave vector q on top of the power law decay. The oscillation shows the tendency of the system to develop crystalline order which is prevented by strong quantum fluctuation in 1D. Therefore, this quasi-condensate which has only algebraic order can be viewed as a quantum melted FFLO state. Our results can also shed more light on the nature of the magnetic field driven BCS-FFLO transition (see Fig. 1). Under a particle-hole transformation, it is equivalent to the Mott transition of repulsive Fermi gas in magnetic field (driven by varying chemical potential). As shown recently by Frahm and Vekua 19, due to the intrinsic spin-charge coupling, it is not in the same universality class as the single mode commensurate-incommensuarate transition. Low temperature thermodynamics. The two gapless excitations in the 1D FFLO phase, described by H FFLO, have relativistic dispersion (in the long wavelength limit) and govern the thermodynamics at low temperatures. Using the effective field theory and the standard conformal mapping 26,25, we find the low temperature thermodynamic potential (per unit length), g(t ) = g 0 π 6 h ( 1 u u 2 )T (7) From g, it is straightforward to find the density n = g/ µ, the magnetization M = g/ h, the entropy s = g/ T, and the compressibility κ = n/ µ. These results can be used along with local density approximation to make predictions about trapped 1D Fermi gas and directly compare to the experimental column density data. Note that at higher temperatures, the dispersion crosses over from relativistic (linear in momentum) to nonrelativistic. Then, Eq. (7) becomes inadequate. In particular, the crossover happens at vanishingly low temperatures in the vicinity of quantum critical points h = h c1 and h c2. The two normal modes reduce to two coupled gases of spinless fermions in the limit of strong coupling. The first Fermi gas describes the pair degree of freedom. (The transmutation in statistics from bosonic pairs to spinless fermions is analogous to the well known case of a Tonks-Girardeau gas where hard-core bosons are mapped onto spinless fermions.) The other Fermi gas describes the leftover fermions. The two gases are coupled by residue scattering, so the effective chemical potential of each gas depends on the Fermi pressure of the other gas. In the strong coupling limit, the two gases become decoupled asymptotically. Thus, at zero temperature, the 1D FFLO phase is characterized by two Fermi surfaces. The normal modes correspond to density fluctuations (phonons) of each gas, and their group velocities are simply the respective Fermi velocities. 5 Conclusions We developed an effective field theory for the 1D FFLO state based on the Bethe ansatz exact solution. The 1D FFLO phase is shown to be a novel type of two-component Tomonaga-

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