Composite Fermions and their Pair States in a Strongly-Coupled Fermi Liquid

Our goal is to understand the phenomena arising in optical lattice fermions at low temperature in an external magnetic field. Varying the field, the attraction between any two fermions can be made arbitrarily strong, where composite bosons form via so-called Feshbach resonances. By setting up strong-coupling equations for fermions, we find that in spatial dimension $d>2$ they couple to bosons which dress up fermions and lead to new massive composite fermions. At low enough temperature, we obtain the critical temperature at which composite bosons undergo the Bose-Einstein condensate (BEC), leading to BEC-dressing massive fermions. These form tightly bound pair states which are new bosonic quasi-particles producing a BEC-type condensate. A quantum critical point is found and the formation of condensates of complex quasi-particles is speculated over.

scopic details. In this letter, as opposed to dilute Fermi gases, we study strongly interacting fermions in an optical lattice for ongoing experiments [2] and yet completely understood theoretical issues, such as quasi-particle spectra, phase structure and critical phenomena, as well as thermodynamical and transport properties, which can be very different from that of better-studied dilute Fermi gases. We use the approach of strong-coupling expansion to find the massive spectra of not only composite bosons but also composite fermions, and obtain the critical line and phase diagram in the strong-coupling region. Some preliminary discussions are presented on the relevance of our results to experiments.
Lattice fermions. We consider fermions in an underlying lattice with a spacing . In order to address strong-coupling fermions at finite temperature T , we incorporate the relevant s-wave scattering physics via a " 0 -range" contact potential in the Hamiltonian for spinor wave function ψ ↑,↓ (i), which represents a fermionic neutral atom of fermion number "e" that we call "charge", and ψ † ↑,↓ (i) represents its "hole" state "−e", β = 1/T , each fermion field ψ σ (i) of length dimension [ −d/2 ], mass m and chemical potential µ is defined at a lattice site "i". The index "i" runs over all lattice sites. The Laplace operator ∇ 2 is defined as (h = 1) whereˆ = 1, . . . , d indicate the orientations of lattice spacing to the nearest neighbors.
Tuned by optical lattice and magnetic field, the s-wave attraction between the up-and down-spins is characterized by a bare coupling constant and the range 0 < .
Inspired by strong-coupling quantum field theories [3,4], we calculate the two-point Green functions of composite boson and fermion fields to effectively diagonalize the Hamiltonian into the bilinear form of these composite fields, and find the composite-particle spectra in the strong-coupling phase .
Composite bosons. We first consider a composite bosonic field C(i) = ψ ↓ (i)ψ ↑ (i) and study its two-point function [4], We find (methods) that in the strong-coupling effective Hamiltonian, C = ψ ↓ ψ ↑ represents a massive composite boson with propagator with pole of mass M B and residue of form factor gR 2 B : From Eq. (4), the effective Hamiltonian of the composite boson field C can be written as, The chemical potential is µ B = −M B /2 and the wave function renremalization is Z B = gR 2 B /2M B . Provided Z B is finite, we renormalize the fermion field and the composite boson field as so that the composite boson C behaves like a quasi particle. This is a pair in a tightly bound Feshbach resonance, contrary to the loosely-bound Cooper pair in the weak-coupling region.
Analogously to C(i), we consider the composite field of fermion and hole, i.e., the plasmon field P(i) = ψ † ↓ (i)ψ ↑ (i). We perform a similar calculation to the two-point Green function G P (i) = P(0), P † (i) , and obtain the same result as (4) and (5), indicating a tightly bound state of plasmon field, whose Hamiltonian is (6) with C(i) → P(i). This is not surprised since the pair field C(i) with the plasmon field P(i) fields are symmetric in the strong-interacting Hamiltonian (1). However, the charged pair field C(i) and neutral plasmon P(i) field can be different up to a relative phase of field θ(i). We select the relative phase field as such that |P(i)| = |C(i)| . We also obtain the identically vanishing two-point Green function P(0), C † (i) , as C(i) is charged (2e) and P(i) is neutral (e − e = 0).
The bound states C are composed of two constituent fermions ψ ↓ (k 1 ) and ψ ↑ (k 2 ) around the Fermi surface, k 1 ≈ k 2 ≈ k F and k 2 − k 1 = q k F , see (q 1) in Eq. (4). The wavefunction renomalization Z B ∝ gT 2 (4) relates to the bound-state size ξ boson . As effective coupling gT 2 → 0, Z B decreases and the pair field C(i) describes loose Cooper pair. The vanishing form factor indicates that the bosonic bound state pole dissolves into two fermionic constituent cuts [5]. The transition to a normal Fermi liquid of unpaired fermions takes place at the dissociation scale of pseudogap temperature T * (g) that we shall study in future.
Phase transition. On the other hand, the mass term M 2 B CC † changes its sign from M 2 B > 0 to M 2 B < 0 and the pole M B becomes imaginary, implying the second-order phase transition from the symmetric phase to the condensate phase [4]. M 2 B = 0 gives the critical line: . This result is qualitatively consistent with the strong-coupling behavior T c ∼ t 2 /U in the attractive Hubbard model [6].
The neighborhood of the critical line is the scaling domain, where the correlation length ξ is much larger than the lattice spacing , microscopic details of lattice are physically irrelevant. Therefore we adopt the continuum field theory describing Grand Canonical Ensembles at finite temperature [7,8] to approximately obtain the relation between the critical "bare" g c (Λ) and the s-wave scattering length a. As a result, the critical "bare" g c (Λ) is related to a "renormalized" coupling described by the s-wave scattering length a via the two-particle Here continuum spectrum k = |k| 2 /2m denotes the energy of the free fermions and ωn,|k|<Λ contains the phase-space integral and the sum over the Matsubara frequencies ω n = 2πT n, obtained in Eq. (3.273) of textbook [9]. This can be written in d dimensions as In d = 3 dimensions, we approximately adopt the half-filling fermion density n ≈ 1/ 3 ≈ . Moreover we introduce the dimensionless and optical lattice tunable parameter b = 2 −1 (3π 2 ) 1/3 0 / < 1, that measures the fraction of filled levels around F that contributes to the pairing, and find (2 ) and Eq. (9), we find for large 1/ak F or g c (Λ), where is the critical temperature at 1/ak F = 0. In the superfluid phase (T < T c ), the bosons fields C(i) develop a nonzero expectation value C(i) and undergo BEC. To illustrate the critical line (10) separating two phases, in Fig. 1, we plot numerical results of (10) in d = 3 for the parameters b = 0.02, 0.03 corresponding to the ratios 0 / = 0.013, 0.02. The "linear" critical line in Fig. 1 is due to the weak T c -dependence in the highly nonlinear relation (10).
Our result shows a decreasing critical temperature T c for large g c or 1/ak F , where the length a and size ξ boson can approach the lattice spacing and become even smaller. Taking the limit g c → ∞ at constant T c g c , we find T c → 0, implying a quantum critical point. We have to stress that our results are valid only in the very strong-coupling region (1/ak F > 0), since our approach of strong-coupling expansion to lattice fermions is particularly appropriate in this region. Nevertheless, we extrapolate the critical line in Fig. 1 to the critical temperature T u c / F ≈ 0.31, 0.2 at the unitarity limit 1/ak F = 0. The experimental value T u c / F ≈ 0.167 of dilute Fermi gas [10] can be achieved for b ≈ 0.037. However, such a simple extrapolation is not expected to be quantitatively correct, and high-order calculations are needed.
Composite Fermions. In this section, we show that the strong-coupling attraction between fermions forms not only composite bosons but also composite fermions. To exhibit the presence of composite fermions, using the Cooper field C(i) we calculate the two-point Green functions [4]: We find (methods) that in the strong-coupling effective Hamiltonian, the propagator represents a composite fermion that is the superposition of the fermion ψ ↑ and the three- where the three-fermion state C(i)ψ † ↓ (i) is made of a hole ψ † ↓ (i) "dressed" by a cloud of Cooper pairs. The associated two-point Green function reads Here µ F = −M F /2 is the chemical potential and Z F = g/M F the wave-function renormalization. Following the renormalization (7) of fermion fields, we renormalize composite fermion field Ψ ↑,↓ ⇒ (Z F ) −1/2 Ψ ↑,↓ , which behaves as a quasi-particle in Eq. (18), analogously to the composite boson (6). The negatively charged (e) three-fermion state is a negatively charged (2e) Cooper field C(i) = ψ ↓ (i)ψ ↑ (i) of two fermions combining with a hole ψ † ↓ (i). These negatively charged (e) composite fermions Ψ ↑↓ (i) are composed of three-fermion states Cψ † ↑ or Cψ † ↓ and a fermion ψ ↑ or ψ ↓ . Similarly, positively charged (−e) composite fermions Ψ † ↑ (i) or Ψ † ↓ (i) are composed by three-fermion states C † ψ ↑ or C † ψ ↓ combined with a hole ψ † ↑ or ψ † ↓ . Suppose that two constituent fermions ψ ↓ (k 1 ) and ψ ↑ (k 2 ) of the Cooper field, one constituent hole ψ † ↑ (k 3 ) are around the Fermi surface, k 1 ≈ k 2 ≈ k 3 ≈ k F , then the Cooper field for q = k 2 − k 1 k F and three-fermion state p = k 1 − k 2 + k 3 ≈ k 3 ≈ k F is around the Fermi surface. As a result, the composite fermions Ψ ↑↓ live around the Fermi surface as well.
The same results (34)- (18) are obtained for the plasmon field P(i) = ψ † ↓ (i)ψ ↑ (i) combined with another fermion or hole, and the associated composite fermion whose two-point Green function, The same is for Ψ P P(i)ψ ↑ (i) the spin-down field. They can be represented in the effective Hamiltonian (18) with Ψ σ (i) → Ψ P σ (i), following the renormalization (7) of fermion fields, and renormalization Ψ P ↑,↓ ⇒ (Z F ) −1/2 Ψ P ↑,↓ . The charged three-fermion states Pψ ↑↓ or P † ψ † ↑↓ are composed of a fermion or a hole combined with a neutral plasmon field P(i) = ψ † ↓ (i)ψ ↑ (i) or P † (i) = ψ † ↑ (i)ψ ↓ (i) of a fermion and a hole. The composite fermions Ψ P ↑,↓ (i) are composed of a three-fermion state Pψ ↑,↓ in combination with a fermion ψ ↑ or ψ ↓ . The same thing is true for its charge-conjugate state. Suppose that constituent fermion ψ ↓ (k 1 ) and hole ψ † ↓ (k 2 ), another constituent fermion ψ ↑ (k 3 ) are all around the Fermi surface, k 1 ≈ k 2 ≈ k 3 ≈ k F , and the plasmon field q = k 2 − k 1 k F and composite fermion p = k 1 − k 2 + k 3 ≈ k 3 ≈ k F is around the Fermi surface as well. The three-fermion states in Eqs. (16) and (19) are related, C(i)ψ † ↓ (i) = −P(i)ψ ↓ (i). This implies that the three-fermion states C(i)ψ † ↓ (i) and P(i)ψ ↓ (i) are the same up to a definite phase factor e iπ . Thus the composite fermions Ψ σ (i) (16) and Ψ P σ (i) (19) are indistinguishable up to a definite phase factor.
Conclusion and Remarks. We present some discussions of the critical temperature (10), effective Hamiltonians (6) which undergo a BEC condensate Φ B = 0 to minimize the ground-state energy. This may be the origin of high-T c superconductivity and a similar form of composite superfluidity. If there is a smooth crossover transition from BSC to BEC, these features, although discussed for 1/ak F > 0, are expected to be also true in 1/ak F 0 with much smaller scale M F,B (T ). Due to the presence of composite fermions in addition to composite bosons, we expect a further suppression of the low-energy spectral weight for single-particle excitations and the material follows harder equation of state. Its observable consequences include a further T -dependent suppression of heat capacity and gap-like dispersion in the density-of-states and spin susceptibility. The measurements of entropy per site [2], vortexnumber [11], and shift in dipole oscillation frequency [12] can be probes into composite-boson and -fermion spectra, and pairing mechanism.
It is known that the limit 1/ak F 0 produces an IR-stable fixed point, and its scaling domain is described by an effective Hamiltonian of BCS physics with the gap scale ∆ 0 = ∆(T c ) in T ∼ T c < ∼ T * . This is analogous to the IR-stable fixed point and scaling domain of an effective Lagrangian of Standard Model (SM) with the electroweak scale in particle physics [13,14].
The unitarity limit 1/ak F → 0 ± representing a scale invariant point [15] was formulated in a renormalization group framework [16], implying a UV-stable fixed point of large coupling, where the running coupling g approaches g UV , as the energy scale becomes large. In the scaling domain of this UV-fixed point 1/ak F → 0 ± and T → T u c , an effective Hamiltonian of composite bosons and fermions is realized with characteristic scale with a critical exponent ν = 1 [17]. This shows the behavior of the correlation length ξ ∝ M −1 B,F , and characterizes the size of composite particles via the wave function renormalization factor Z B,F ∝ M −1 B,F . This domain should be better explored experimentally. The analogy was discussed in particle physics with anticipations of the UV scaling domain at TeV scales and effective Lagrangian of composite particles made by SM elementary fermions [18].
In the limit h → 0 for g → ∞ and finite T , the kinetic terms (22) are neglected, and the partition function (24) has a nonzero strong-coupling limit where N is the total number of lattice sites, i↑ ≡ [dψ † ↑ (i)dψ ↑ (i)] and i↓ ≡ [dψ † ↓ (i)dψ ↓ (i)]. The strong-coupling expansion can now be performed in powers of the hopping parameter h.
Going to momentum space we approximately obtain G(q) = 1 βg + 2 Here we have resumed the original lattice spacing by setting back β → β 3 and 2m → 2m 2 .