Magnetism and quantum phase transitions in spin-1/2 attractive fermions with polarization

An extensive investigation is given for magnetic properties and phase transitions in one-dimensional Bethe ansatz integrable spin-1/2 attractive fermions with polarization by means of the dressed energy formalism. An iteration method is presented to derive higher order corrections for the ground state energy, critical fields and magnetic properties. Numerical solutions of the dressed energy equations confirm that the analytic expressions for these physical quantities and resulting phase diagrams are highly accurate in the weak and strong coupling regimes, capturing the precise nature of magnetic effects and quantum phase transitions in one-dimensional interacting fermions with population imbalance. Moreover, it is shown that the universality class of linear field-dependent behaviour of the magnetization holds throughout the whole attractive regime.


I. INTRODUCTION
Bosons and fermions reveal strikingly different quantum statistical effects at low temperatures. Bosons with integer spin undergo Bose-Einstein condensation (BEC), whereas fermions with half-odd-integer spin are not allowed to occupy a single quantum state due to the Pauli exclusion principle. However, fermions with opposite spin states can pair up to produce Bardeen-Cooper-Schrieffer (BCS) pairs to form a Fermi superfluid. Quantum degenerate gases of ultracold atoms open up exciting possibilities for the experimental study of such subtle quantum many-body physics in low dimensions [1,2,3,4]. In this platform, Feshbach resonance has given rise to a rich avenue for the experimental investigation of relevant problems, such as the crossover from BCS superfluidity to BEC [5], fermionic superfluidity and phase transitions, among others [6,7,8]. Particularly, pairing and superfluidity are attracting further attention from theory and experiment due to the close connection to high-T c superconductivity and nuclear physics. The study of pairing signature and fermionic superfluidity in interacting fermions has stimulated growing interest in Fermi gases with population imbalance [9,10,11,12], i.e., systems with different species of fermions [3] as well as multicomponent interacting fermions [13,14,15,16]. This gives rise to new perspectives to explore subtle quantum phases, such as a breached pairing phase [17] and a nonzero momentum pairing phase of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states [18,19] and colour superfluids [4].
One-dimensional (1D) atomic gases with internal degrees of freedom also provide tunable interacting many-body systems featuring novel magnetic properties and quantum phase transitions [20,21,22,23,24]. Although the FFLO state has not been fully confirmed experimentally, investigations of the elusive FFLO state in the 1D interacting Fermi gas with population imbalance are very promising [25,26,27,28,29,30,31,32,33,34,35,38,39,40,41,42]. Theoretical predictions for the existence of a FFLO state in the 1D interacting Fermi gas has emerged by a variety of methods including the Bethe ansatz (BA) solution [26], numerical methods [28,30,31,32,34,38,42] and field theory [35,43,44]. A powerful field theory approach [35,44] was used to describe a FFLO state in the 1D Fermi superliquid with population imbalance. Nevertheless, verification of the FFLO signature of polarized attractive fermions is still lacking via the Bethe ansatz solution. A recent thermodynamic Bethe ansatz (TBA) study of strongly attractive fermions [27] shows that paired and unpaired atoms form two Fermi liquids coupled to each other. The TBA equations indicate that spin wave fluctuations ferromagnetically couple to the unpaired Fermi sea. A full analysis of magnetic effects and low energy physics of spin-1/2 fermions with polarization in both the weak and strong coupling regimes, as well as a detailed discussion on the universality class of the magnetic behaviour in the whole attractive regime, are desirable in understanding such subtle paired states in 1D interacting fermions with polarization.
In this paper we provide an extended investigation of quantum phases and phase transitions for 1D interacting fermions with polarization in the presence of an external field. We analytically and numerically solve the dressed energy equations which describe the equilibrium state at zero temperature. We extend previous work on this model to derive higher order corrections (up to order 1/|γ| 3 ) for the ground state energy, magnetization, critical fields, chemical potentials and the external field-energy transfer relation. The phase diagrams in the weak and strong coupling regimes are obtained in terms of the external field, density and interaction strength. In the strong coupling regime, (i) the bound pairs in the homogeneous system form a singlet ground state when the external field is less than the lower critical value H c1 , (ii) a normal Fermi liquid phase without pairing occurs when the external field is greater than the upper critical value H c2 and (iii) for an intermediate range H c1 < H < H c2 , paired and unpaired atoms coexist. However, for weak coupling, a BCS-like pair scattering phase occurs only when the external field H = 0, while paired and unpaired fermions coexist when the field is less than a critical field. Significantly, we also show that the universality class of linear field-dependent behaviour of the magnetization remains throughout the whole attractive regime.
This paper is set out as follows. In section II, we present the Hamiltonian and discuss the pairing signature for the 1D fermions with population imbalance in the whole attractive regime. In section III, we present the dressed energy equations obtained from the thermodynamic Bethe ansatz equations (TBA) in the limit T → 0. In section IV, we present the magnetic properties for the model in the weak coupling regime. We solve the dressed energy equations in the strong coupling regime in section V. The explicit forms of the magnetic properties and the ground state energy are given in terms of the interaction strength, density and external field. In section VI, we present the full phase diagrams for the whole attractive regime. Section VII is devoted to concluding remarks and a brief discussion.

II. THE MODEL
The Hamiltonian [47,48] we consider describes N δ-interacting spin-1 2 fermions of mass m constrained by periodic boundary conditions to a line of length L and subject to an external magnetic field H. In this formulation the field operators φ ↓ and φ ↑ describe fermionic atoms in the respective states | ↑ and | ↓ . The δ-type interaction between fermions with opposite hyperfine states preserves the spin states such that the Zeeman term in the Hamiltonian (1) is a conserved quantity. For convenience, we use units of = 2m = 1 and define c = mg 1D / 2 and a dimensionless interaction strength γ = c/n for the physical analysis, where n = N/L is the linear density. The inter-component interaction can be tuned from strongly attractive (g 1D → −∞) to strongly repulsive (g 1D → +∞) via Feshbach resonance and optical confinement. The interaction is attractive for g 1D < 0 and repulsive for g 1D > 0.
The solutions to the BA equations (2), as depicted in Figure 1, provide a clear pairing signature and the ground state properties of the model. The BA root distributions in the complex plane were studied recently [23,29]. The ground state for 1D interacting fermions with repulsive interaction has antiferromagnetic ordering [23,49,50]. Rather subtle magnetism for the model with repulsive interaction was recently studied [51,52]. For attractive interaction, fermions with different spin states can form BCS pairs with nonzero centre-of-mass momenta which might feature FFLO states [26,28,30,31,32,34,35].
In the weakly attractive regime, the weakly bound Cooper pairs are not stable due to thermal and spin wave fluctuations. The unpaired fermions sit on two outer wings in the quasimomentum space [23,27] due to the Fermi pressure (see Figure 1). The ground state can only have one pair of fermions with opposite spins having a particular quasimomentum k. The paired fermions occupy the central area in the quasimomentum k space. Indeed we find from the BA equations (2) that in the weak coupling limit, i.e. L|c| ≪ 1, the imaginary part of the quasimomenta for a BCS pair is proportional to |c|/L. However, for strongly attractive interaction, i.e. L|c| ≫ 1, the BCS pair has imaginary part ±i 1 2 |c|. In this regime, the lowest spin excitation has an energy gap which is proportional to c 2 . For the cross-over regime, i.e. L|c| ∼ 1, the imaginary part ±iy is asymptotically determined by the condition y tanh( 1 2 yL) ≈ 1 2 |c|. For this cross-over regime, the spin gap might be exponentially small. However, it is hard to analytically determine this small energy gap from the BA equations (2). Nevertheless, for the weak coupling regime L|c| ≪ 1, the bound state has a small binding energy ǫ b = 2 n|γ|/mL which has the same order of γ as the interacting energies of pair-pair and pair-unpaired fermions. In this limit, the real parts of the quasimomenta satisfy the Gaudin model-like BA equations [23,36] which describe BCS pair-pair and pairunpaired fermion scattering. They form a gapless superconducting phase. Using the above BA root configuration, the ground state energy per unit length is given by [23] in terms of the polarization P = (N − 2M)/N. The first term in Eq. (3) includes the collective interaction energy (pair-pair and pair-unpaired fermion scattering energy) and the binding energy (internal energy). We see clearly that for large polarization (P ≈ 1) the small portion of spin-down fermions are likely to experience a mean-field formed by the spin-up medium. This is consistent with the observation of Fermi polarons in an attractive Fermi liquid of ultracold atoms [37].
On the other hand, when the attractive interaction strength is increased, i.e. L|c| ≫ 1, the bound pairs gradually form hard-core bosons while the unpaired fermions can penetrate into the central region in the quasimomentum space (see Figure 1). The main reason for the unpaired fermions and BCS pairs having overlapping Fermi seas is that in 1D the paired and unpaired fermions have different fractional statistical signatures such that they are allowed to pass into each other in the quasimomentum space. In the thermodynamic limit, i.e., The dimensionless interaction strength γ = c/n is inversely proportional to the density n. This signature leads to different phase segments in 1D trapped fermions [25,26,27] than the phase separations in 3D trapped interacting fermions, where the Fermi gas has been separated into a uniformly paired inner core surrounded by a shell with the excess of unpaired atoms [6,7,8].
From the ground state energy for the model with strong attraction and arbitrary polarization [23], we find the finite-size corrections to the energy in the thermodynamic limit to be given by where the central charge C = 1 and the group velocities for bound pairs v b and unpaired Here the Fermi velocity is v F = πn/m. In the above equation E ∞ 0 is the ground state energy in the thermodynamic limit The For strongly attractive interaction, unpaired roots can penetrate into the central region, occupied by the bound pairs. However, for weakly repulsive interaction the roots with up-and down-spins separate gradually. In the strongly repulsive regime, the model forms an effective Heisenberg spin chain with the antiferromagnetic coupling constant J ≈ −4E F /γ [52], where E F = n 2 π 2 /3 is the Fermi energy.

III. THERMODYNAMIC BETHE ANSATZ
The thermodynamic Bethe ansatz (TBA) provides a powerful and elegant way to study the thermal properties of 1D integrable systems. It also provides a convenient formalism to analyze quantum phase transitions and magnetic effects in the presence of external fields at zero temperature [45,53,54,55]. In the thermodynamic limit, the grand partition function which are the dressed energy equations [27,45,53] obtained from the TBA equations in the limit T → 0. The superscripts ± denote the positive and negative parts of the dressed energies, with the negative (positive) part corresponding to occupied (unoccupied) states.
The integration boundaries B and Q characterize the Fermi surfaces of the bound pairs and unpaired fermions, respectively.
The Gibbs free energy per unit length at zero temperature is given by The

IV. MAGNETIC PROPERTIES IN THE WEAK COUPLING REGIME
For weak coupling |c| → 0 caution needs to be taken in the thermodynamic limit. On solving the discrete BA equations (2) in the regime L|c| ≪ 1 the imaginary part of the BCS-like pairs tends to |c|/L [23]. However, the TBA equations [27,53] usually follow from the root patterns in the thermodynamic limit, i.e. L, N → ∞ with N/L is fixed. Under this limit, we naturally have the BA root patterns k j = Λ j ± i 1 2 |c| with j = 1, . . . , M for the charge degree and the string patterns with equally spaced imaginary distribution for spin rapidity Λ n α,j = Λ (n) α + i 1 2 (n + 1 − 2j)c, with j = 1, . . . , n. Here the number of strings α = 1, . . . , N n . Λ n α is the position of the centre for the length-n string on the real axis in Λspace. Therefore, in the weak coupling limit, i.e., |c| → 0, the integral BA equations and the TBA equations do not properly described the true solutions to the discrete BA equations (2) unless under the thermodynamic limit conditions. Nevertheless, the discrepancy is minimal, i.e., it is O(γ 2 ).
The BA equations (2) in principle give complete states of the model. However, at finite temperatures, the true physical states become degenerate. The dressed energies in the TBA equations (8) characterize excitation energies above the Fermi surfaces of the bound pairs and unpaired fermions. All physical quantities, for example, free energy, pressure and magnetic properties can be obtained from the TBA equations withought deriving the spectral properties of low-lying excitations. In the weak coupling limit, the interaction energy is proportional to |c| which is much less than the kinetic energy. Therefore, in this regime, the exact ground state energy with leading term of order |c| is precise enough to capture the nature of phase transitions and magnetic ordering.
where m z = M z /n and the magnetization is defined by M z = nP /2. A linear field-dependent behaviour of the magnetization is observed. Figure 2 shows the magnetization vs the field H for different interaction values |γ|. We observe that the analytic results plotted from (11) are in excellent agreement with the numerical curves evaluated directly from the dressed energy equations (8). We also find that a fully paired ground state only occurs in the absence of the external field. However, for H ≥ H c where the fully polarized phase occurs. Paired and unpaired fermions coexist in the intermediate range 0 < H < H c . The phase diagram for weak coupling is illustrated in Figure 3.

V. SOLUTIONS TO THE DRESSED ENERGY EQUATIONS
In this section we solve the dressed energy equations (8)  the interaction strength γ. We present a systematic way to obtain these physical properties up to order 1 |γ| 3 which gives a very precise phase diagram for finite strong interaction. Here we note that Iida and Wadati [29] have presented a different method to solve the dressed energy equations. We have solved the dressed energy equations (8) numerically in the whole attractive regime to compare with the analytic results. Excellent agreement between numerical and analytical results is found.
First consider the ground state P = 0. Following the method developed in [27] where the dressed energy equations (8) are asymptotically expanded in terms of 1/|c|, the ground state dressed energy equation for P = 0 is given by withμ = µ + c 2 4 . For convenience, we introduce the notation for the pressure of bound pairs and un-paired fermions. Since the Fermi point B is finite, we can take an expansion with respect to Λ ′ in the integral in Eq. (13). By a straightforward calculation, the pressure p b is found to be We obtained this equation by iteration in terms of p b andμ. In such a way, the accuracy of physical quantities can be controlled to order 1/|c|. This provides a systematic way to obtain accurate results from dressed energy equations. It is free from restriction on the integration boundaries B and Q. Furthermore, from Eq. (16) and the condition ǫ b (±B) = 0 we find Finally, using the above equations and the relation ∂p b /∂µ = n, the pressure per unit length follows as ) (18) and the energy per unit length is The dressed energy equations (8) can also be solved analytically for 0 < P < 1. Following [27], we defineμ = µ + H/2. We notice that the Fermi points Q and B are still finite in the presence of an external field H. Similar to the case P = 0, using the conditions ǫ b (±B) = 0 and ǫ u (±Q) = 0, we obtain the relations and After eliminating B and Q, we have Obviously, the pressures p b and p u are functions ofμ,μ and the interaction strength c, i.e., p b = p b (μ,μ, |c|) and p u = p u (μ,μ, |c|). Furthermore, taking into account the relations ∂p b ∂H + ∂p u ∂H = P/2 and ∂p b ∂µ + ∂p u ∂µ = n, after a tedious calculation we find the effective chemical potentials for the pairs µ b = µ + ǫ b /2 and for the unpaired fermions µ u =μ = µ + H/2.

Explicitly,
This result provides higher order corrections in terms of the interaction strength 1 |γ| compared to previous studies [27,29].

VI. QUANTUM PHASE TRANSITIONS
In section IV we examined magnetic effects and phase transitions for spin-1/2 weakly attractive fermions with polarization. As the attractive interaction strength |γ| increases, the In the vicinity of the critical field H c1 , the system exhibits a linear field-dependent magne- with a finite susceptibility This universality class of linear field-dependent magnetization behaviour is also found for the multicomponent Fermi gases with attractive interaction [57]. However, it differs subtly from the case of a Fermi-Bose mixture due to the different statistical signature of a boson and a bound pair of fermions with opposite spin states [52]. For the model under consideration here the magnetic properties in this gapless phase can be exactly described by the external field-magnetization relation where µ u and µ b are given by (25) and (26)  superconducting to normal states [58]. Figure 5 shows the magnetization vs external field for different values of the interaction strength γ. Numerical solution of the dressed energy equations (8) shows that the analytic results are highly accurate in the strong and finitely strong coupling regimes.
A similar configuration occurs for the external field exceeding the upper critical field H c2 , given by where a phase transition from the mixed phase into the normal Fermi liquid phase occurs. Figure 6 shows this configuration in the dressed energy language. From the relation (31), we obtain the linear field-dependent magnetization as with a finite susceptibility A typical phase diagram in the n − H plane for finite strong interaction is shown in Figure 7. Smooth magnetization curves at the critical fields H c1 and H c2 indicate second order phase transitions. Very good agreement is observed between the curves obtained from the numerical solution of the dressed energy equations and the analytical predictions (28) and (32) for the critical fields.  (28) and (32).

VII. CONCLUSION
In summary, we have studied magnetic properties and quantum phase transitions for the 1D Bethe ansatz integrable model of spin-1/2 attractive fermions. Previous work on this model has been extended to derive higher order corrections for the ground state energy, pressure, chemical potentials, magnetization, susceptibility and critical fields in terms of the external magnetic field, density and interaction strength. The range and applicability of the analytic results have been compared favourably with numerical solutions of the dressed energy equations. The universality class of linear field-dependent behaviour of the phase transitions in the vicinity of the critical field values has been predicted for the whole attractive regime. This universal behaviour is consistent with the prediction for the 1D Hubbard model [59]. However, it appears not to support the argument [45,46] for a van Hove-type singularity of quantum phase transition for 1D attractive fermions. Finite temperature properties of 1D interacting fermions will be considered elsewhere.
We further confirm that 1D strongly attractive fermions with population imbalance exhibit three quantum phases, subject to the value of the external field H  Figure 3. We have shown that the mixed phase in 1D interacting fermions with polarization can be effectively described by two coupled Fermi liquids. Our exact phase diagrams for the weak and strong coupling regimes also provide a space segment signature for an harmonically trapped Fermi gas in 1D geometry. These quantum phases and magnetic properties may also possibly be observed in experiments with ultracold fermionic atoms [60,61].