Excitation spectra of one-dimensional spin-1/2 Fermi gas with an attraction

Jia-Feng Pan ,Jia-Jia Luo and Xi-Wen Guan,4,?

1 State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics,Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences,Wuhan 430071,China

2 Department of Fundamental and Theoretical Physics,Research School of Physics,Australian National University,Canberra ACT 0200,Australia

3 University of Chinese Academy of Sciences,Beijing 100049,China

4 NSFC-SPTP Peng Huanwu Center for Fundamental Theory,Xi'an 710127,China

Abstract Using an exact Bethe ansatz solution,we rigorously study excitation spectra of the spin-1/2 Fermi gas(called Yang–Gaudin model)with an attractive interaction.Elementary excitations of this model involve particle-hole excitation,hole excitation and adding particles in the Fermi seas of pairs and unpaired fermions.The gapped magnon excitations in the spin sector show a ferromagnetic coupling to the Fermi sea of the single fermions.By numerically and analytically solving the Bethe ansatz equations and the thermodynamic Bethe ansatz equations of this model,we obtain excitation energies for various polarizations in the phase of the Fulde–Ferrell–Larkin–Ovchinnikov-like state.For a small momentum (long-wavelength limit) and in the strong interaction regime,we analytically obtained their linear dispersions with curvature corrections,effective masses as well as velocities in particle-hole excitations of pairs and unpaired fermions.Such a type of particle-hole excitations display a novel separation of collective motions of bosonic modes within paired and unpaired fermions.Finally,we also discuss magnon excitations in the spin sector and the application of Bragg spectroscopy for testing such separated charge excitation modes of pairs and single fermions.

Keywords: Yang–Gaudin model,quantum integrability,the Bethe ansatz,excitation spectrum

1.Introduction

The quantum many-body systems manifest abundant physical phenomena,such as Bose–Einstein condensation (BEC),superfluidity,superconductivity,and quantum phase transition,etc,which are regarded as emergent phenomena in modern physics.However,the complexity of many-body systems,involving huge internal degrees of freedom,quantum statistics and interaction,always brings us a formidable task to access their physics of interest.Among the known many-body theories,there are two universal low energy effective theories,i.e.Fermi liquid and Tomonaga–Luttinger liquid (TLL) that capture significant different features of many-body correlations in one dimension (1D) and higher dimensions.Landau Fermi liquid theory [1–3] remarkably describes the metallic behaviour of interacting fermions in 2D and 3D.In this theory the concept of quasiparticles reveals the essence of individual particle excitations close to the Fermi surface in the interacting fermions.In contrast,when the degrees of freedom are reduced to 1D,the elementary excitations in 1D systems strikingly form collective motions of bosons,which are named the TLL in the long-wavelength limit [4,5].In the TLL theory,the low energy behaviour of 1D systems can be universally described by the bosonic fields of quantized sound waves or phonons.The early theoretical framework of the TLL for various 1D problems was developed by Lieb and Mattis[6],Haldane[7]and others,see reviews [8,9].The TLL has become the main theme in the study of the critical behaviour of 1D many-body systems.

On the other hand,integrable models provide significant insight into the emergent phenomena driven by interactions and quantum statistics in low and higher dimensions [9–11].The quantum integrability should trace back to H Bethe’s work in 1931 when he solved the eigenvalue problem of 1D spin-1/2 Heisenberg chain[12]by writing the wave function of the model as the superposition of all possible plane waves.It was over 30 years after Bethe’s work that this method was coined as the BA(Bethe ansatz)by C N Yang and C P Yang in their series of publications in the mid-60s [13–15].Using Bethe’s method,Lieb and Liniger [16] solved the 1D Bose gas with a δ-function interaction,which is now called the Lieb–Liniger model.In 1964 Mcguire [17] studied the 1D Fermi gas with the δ-function interaction by considering the exact solution of one spin-up fermion interacting withN-1 spin-down fermions.The exact solution of the 1D Fermi gas with arbitrary numbers of spin-up and spin-down particles was solved by Yang [18,19],while at that time Gaudin [20]obtained the BA solution for the model with a spin balance.This model is now named the Yang–Gaudin model.In C N Yang’s seminal work [18],a key discovery of the necessary condition for the BA solvability was surprisingly found.In 1972,Baxter [21] independently showed that such a factorization relation also occurred as the conditions for commuting transfer matrices in 2D vertex models in statistical mechanics.It is now known as the Yang–Baxter equation,i.e.the factorization condition.The Yang–Baxter equation has laid out a profound legacy in a variety of fields in mathematics and physics.

In 1969,Yang and Yang [22] obtained the thermodynamics of the Lieb–Liniger Bose gas at finite temperatures.They found that the thermodynamics of the model can be determined by the minimisation conditions of the Gibbs free energy in terms of the microscopic states determined by the BA equations.Takahashi generalized Yang and Yang’s method to deal with the thermodynamics of the 1D Heisenberg spin chain and 1D Hubbard model though introducing string hypotheses [23–27].He coined the method as the Yang–Yang thermodynamic Bethe ansatz (TBA),see a feature review [28].Further developments of the TBA approach have been made in the study of universal thermodynamics,Luttinger liquid,spin-charge separation,transport properties and critical phenomena for a wide range of low-dimensional quantum many-body systems,see reviews [11,29].

In the attractive regime,the Yang–Gaudin model exhibits novel Fulde–Ferrell–Larkin–Ovchinnikov (FFLO)pairing correlation [30,31],i.e.coexistence of paired and unpaired fermions [32].Understanding the FFLO pairing behavior in 1D and higher dimensions is still an open challenge in condensed matter physics.The phase diagram of the attractive Fermi gas,consisting of a fully-paired state for an external field is less than the lower critical fieldHc1,a fully-polarized state for the magnetic field is greater than an upper critical fieldHc2and the FFLO-like state lies in between the two critical fields,was predicted in[33–35]and experimentally confirmed by R.Hulet’s group in [36].This novel phase diagram reveals striking features of thermodynamics,for instance,universal behaviour of the specific heat [37],the dimensionless ratios,such as the Grüneisen parameter[38]and Wilson ratios[39,40].The dark-solitonlike excitations in the Yang–Gaudin gas of attractively interacting fermions and the Lieb–Liniger gas[41,42]shed light on the nonlinear effects of many-body correlation.Nevertheless,one expects that the elementary excitations in this attractive Yang–Gaudin model would provide significant collective nature of multi-component TLLs with pairing and depairing in thermodynamics and dynamic response functions.This is the major research of the following study in this paper.

In section 2,we will introduce the BA equations and TBA equations,which will be used to accomplish our study of the excitation spectra of the Yang–Gaudin model with an attraction.In section 3,we will present the particle-hole excitation spectra in paired and unpaired fermi seas.In section 4,we will analytically derive dispersion relations with band curvature corrections,effective mass and sound velocities of pairs and unpaired fermions for the Yang–Gaudin model with polarizations in a strong coupling regime.In section 5,we will discuss multiple particle-hole excitations and the magnon excitations in the FFLO-like phase.The last section remains for our conclusion and discussion.

2.Yang–Gaudin model with an attractive interaction

2.1.Bethe ansatz equations and string hypotheses

The 1D two-component Fermi gas with a delta-function interaction is called the Yang–Gaudin model [18–20].Its Hamiltonian is given by

whereNis the number of fermions,N↓is the number of down-spin fermions,His the external magnetic field,c<0 for attractive interaction,and herewith we set ?2=2m=1.We always choose an upward magnetic field so that spindown fermions are less than spin-up ones.As a result,each spin-down fermion can be paired with a spin-up fermion and form a bounded state.While the remaining spin-up fermions are unpaired and in a polarized state.Denote the number of paired fermions asM,then that of unpaired fermions isN-2M.

The quasimomenta{kj}of the fermions and the rapidities{Λα} of the spin-down fermions are given by the BA equations

forj=1,2,…,Nand α=1,2,…,N↓,wheree(x)=exp (iπ-2i arctanx).

Takahashi [23–27] introduced the following spin string hypotheses for the root patterns of the BA equations (2):

Supposing that there areMcharge bound pairs and(N-2M)single fermions with the quasi-momenta{kj} withj=1,2,…,N-2M.Then the BA equations (2) can be rewritten in the following form [26]

whereθ(x)=2 arctanx,-π<θ<π,and

In the thermodynamic limit,i.e.L,N→∞,these quantum numbers could be treated as functions of continuous variablesks,which satisfy

where [n] is an operator defined by

2.2.Thermodynamical Bethe ansatz equations and thermodynamic quantities

Building on microscopic state energies,we may further define dressed energies of pairsε=Tlnη,single fermionsκ=Tlnζ,and length-nstringsεn=Tlnζn,respectively.Here we denoted the quantities ζ=ρh/ρ,η=σh/σ,andMinimizing thermodynamic potential Ω ≡ETS-AN-HSZwith respect to the densities through equation(9),i.e.δΩ=0,we may obtain the following TBA equations [26]

wheren=1,…,∞.In the above equations,T,AandHstand for temperature,chemical potential and magnetic field,respectively.Accordingly,we can give the equation of state,namely the pressurepis given by

The pressure of this system is simply the sum of two terms,the term regarding the paired Fermi sea and another regarding the unpaired sea,respectively.Other thermodynamic quantities can be calculated through the usual thermodynamics relations:

where the last three quantities are the compressibility,magnetic susceptibility,and specific heat.

At the zero temperatureT→0,due to the ferromagnetic ordering,we observe that ηn→∞and σn=0 forn=1,2,…and the TBA equations (12) reduce to

where the superscript‘-’means the corresponding quantities are the negative parts.Since→0 forn≥1 in the limitT→0,these equations are referred to the zero temperature dressed energy equations.In this case,the Fermi pointsB,Qare determined byε(B)=0 andκ(Q)=0.It turns out thatε(k)andκ(k)are monotonically increasing functions ofk2.Therefore,BandQare Fermi surfaces referring to the continuous quasimomentum of paired and unpaired fermions.They can also be determined by the relations

Thus the distribution functions are given by

where σn=0 for the ground state and spin sector is completely gapped.

Moreover,at zero temperature limitT→0,there exists the phase transition among fully paired phase,fully-polarized phase,partially-polarized phase,and vacuum phase,which will be further studied in section 4.1.The fully paired phase can be regarded as the Bardeen–Cooper–Schrieffer (BCS) like phase,which is expected to manifest the first-type superconductivity.The phase diagram are given in [34,35,37,43].Fulde and Ferrell [30] and Larkin and Ovchinnikov [31] predicted the exotic superfluid phase,which is now called the FFLO phase.The partially polarized phase in the attractive Yang–Gaudin model is composed of BCS-like pairs and unpaired fermions,presenting a 1D-analogy of the FFLO phase.The FFLO-like phase diagram of the attractive Fermi gas was experimentally confirmed by Hulet’s group [44].

3.Spectra of one-particle-hole excitations

3.1.Ground state

Consider the partially polarized phase,where paired and unpaired fermions coexist.The total momentum of the system is given by

Due to the ferromagnetic ordering in the spin sector,the distribution functions σn=0 withn=1,2,…at the ground state.In other words,there is no particle for the lengthnstrings,i.e.Mn=0.Therefore,the spin sector of unpaired fermions does not contribute to the total momentum,i.e.Ks,u=0 at the ground state.

For the ground state,both paired and unpaired fermions should occupy those lowest-energy states,i.e.states with the lowest absolute quantum numbers (quasi-momenta).Consequently,most contributions of particles to total momentum are neutralized in the sum of positive and negative quantum numbers.The total momentum is not zero only if the distribution of quasi-momenta is not symmetric around the zero quasi-momentum.Figure 1 presents an example of non-zero total momentum,which essentially shows a two-fold degeneration.

Figure 1.Schematic illustration of the two-fold degeneration.IF,a,-and IF,a,+,(orIF ,b,′- andIF ,b,′+)give respectively the Fermi points of the unpaired (paired) sector.In this case,the particle number is odd while the corresponding quantum number Ijs take half-odd integers.It appears that these quantum numbers cannot distribute symmetrically.Therefore,the total momentum is non-zero and the ground state is twofold degenerate.

Figure 2.(a)Density distribution functions and(b)dressed energies at zero temperature.(a)For k →±∞,the density distributions of paired fermions σ0 and unpaired fermions ρ0 tend to 1/π or 1/(2π),respectively.(b) The dressed energies of paired and unpaired fermions are monotonically increasing with |k|.Therefore,the Fermi points referring to the continuous quasi-momenta can be well defined by the conditions ε0(±B0)=0 and κ0(±Q0)=0,respectively.

Figure 3.Schematic illustration of the one-particle-hole excitation:moving one particle within the two Fermi points to outside the Fermi sea.

Figure 4.One-particle-hole excitation spectra of paired (blue) and unpaired(dark brown)fermions.kF,b(vb)and kF,u(vu)are the Fermi momenta (velocities) of paired and unpaired fermions,respectively.For small ΔK,the paired and unpaired excitation spectra tend to the linear dispersions with different velocities vb and vu,which are indicated by the white and green dashed lines,respectively.The curvature effects of these dispersions can be conceived from the effective masses which were further studied in section 4.In this figure,we set c=-1.00,A=-0.9996,μ0H=1.0020 for our numerical calculation with the dispersions represented by equations (24)–(26).

Figure 5.One-particle-hole excitation spectra of paired(blue)and unpaired(dark brown)fermions with kF,b=kF,u(a)and vu=2vb(b).In(a),we set c=-1.00,A=-0.9999,μ0H=1.0003 for our numerical calculation with the equations (24)–(26);in (b),we set c=-1.00,A=-0.9999,μ0H=1.0003.These parameters are artificially selected,which we will explain in section 4.

Figure 6.Phase diagrams represented by dimensionless particle densities 1/γ,1/γb and 1/γu.Here dimensionless quantities A/(2c2) and μ0H/(2c2)are used as variables,where 2c2 is the binding energy.The black lines indicate the phase boundaries,which are obtained from the two diagrams in (a).For the partially polarized phase,the strong coupling γ ?1 near the quartet point.Clearly,according to (b),the boundaries of each phase(black lines)in(a)are given at the boundaries of zero-or nonzero-particle densities of paired and unpaired sections.The phase boundaries in (a) can also be determined by equation (15).

Figure 7.Phase diagram is represented by polarization.In (a),the white lines give the phase boundaries.In (b),the polarization varies drastically in the vicinity of the quartet point.

Figure 8.Variation of sound velocities,vb and vu,and effective masses,versus the dimensionless interaction strengths γ.The polarization is fixed to be P=1/5 with γ=10.The analytical (blue)equation(49)are agreeable with the numerical(red)results which are plotted with respect to equations (33) and (36).

Figure 9.Two-particle-hole excitations spectra.The numbers of excited particle holes in paired and unpaired sectors.Figures (a)–(c) show particle-hole excitations for different Nb and Nu.Here we denote [αb+βu]≡:2(αkF,b+βkF ,u).In this figure,we set c=-1.00,A=-0.9960,μ0H=0.9976 for our numerical calculation with the dispersions represented by equation (52).

Figure 10.Schematic illustration of forming or breaking a pair.

There are totallyMpaired fermions,whereasJαs take integers (half-odd integers) forN-Modd (even).Therefore,Jαs are symmetrically distributed if and only ifNis even.Similarly,there areN-2Munpaired fermions,whereasIjs take integers (half-odd integers) forM+ΣMn=Mare even(odd).Therefore,Ijis symmetrically distributed if and only ifN+Mis odd,see equation (5).Consequently

wherenb=M/Landnu=(N-2M)/Lare particle densities of paired and unpaired sectors,respectively.

Introduce the superscript ‘0’ to denote the ground state.Accordingly,the dressed-energies equation (15) and the density distribution functions and equation (17) are respectively written as

both of which are plotted in figure 2 with certain values of the interaction strength,chemical potential and magnetic field,i.e.{c,A,μ0H}.

In terms of the additivity of the total momentum in equation(18),we can naturally define the Fermi points,kF,b,±andkF,u,±,referring to the discrete quasimomenta of both paired and unpaired sectors,by maximal and minimal quantum numbers within occupied{Jα}and{Ij},respectively.ForJαsymmetrically distributed,∣kF,b,±∣=2π∣JF,±∣L=π(M-1)L≈πnbfor a thermodynamical limit,i.e.N,L→∞;whereas for the asymmetrical case,we still have ∣kF,b,+∣=π(M-1 ±1)L≈πnband∣kF,b,-∣=π(M-1 ?1)L≈πnb,where the additional terms±π/Lemerging due to the two-fold degeneration are negligible for the thermodynamical limit.With similar regard,we havekF,u,±=2πIF,±≈πnufor the Fermi points referring to the unpaired fermions.The Fermi points are referred to the conditionsε0(Q0)=0andσ0(B0)=0,whereB0andQ0are different from the quasimomentakF,b,±andkF,u,±of pairs and unpaired fermions.

3.2.Sound velocity and effective mass for one-particle-hole excitation

Consider one particle-hole excitation of unpaired fermions,where one quasiparticle with a quasimomentumkhinside the Fermi points is excited outside the Fermi points with a quasimomentumkp(see figure 3),i.e.

whereB±andQ±are the fermi surfaces of the excited system.In the TBA approach,the equilibrium state of the system is determined by minimizing free energy Ω=E-μN-TS,so,the excited free energy is

For the Yang–Gaudin model with repulsive interactions,it is proved [45] that the excited energy can be expressed in the dressed energies.Using a similar method and after a lengthy calculation,we may obtain excitation spectra in the attractive Fermi gas (see appendix A.1),namely

Since the particle numbersN,M,andMnare conserved,the configurations of quasimomenta remain unchanged.In other words,if the positions of quasi-momenta are originally symmetrically(asymmetrically)distributed with respect to the zero-point,they are still symmetrically (asymmetrically) distributed after an excitation.Therefore,the excited momentum of the systems is equal to the quasi-momentum of the excited particle,namely,

We have similar results for the one-particle-hole excitation of paired fermions,namely,

By giving all possiblekpandkh(λpand λh),one can obtain one-particle-hole excitation of unpaired (paired) fermions.Accordingly,the spectrum can be plotted for certain choices of parameters {c,A,μ0H},see figure 4.

Apparently,there exists charge–charge separation at least for small ΔK.The particle-hole continuum in the longwavelength limit (ΔK→0) manifests the free fermion-like dispersion.Two thresholds of the excitation spectrum (black solid lines in figure 4)reveal a curvature effect.The first-order corrections to the linear dispersions of paired and unpaired fermions can define the effective massesmb*andmu*as well as the sound velocities,vbandvu,see cyan and white dashed lines in figure 4,respectively.We will analytically calculate them in section 4.This is meaningful because it is much easier to measure specific signals referring to specific quantities,such as the sound velocities,which characterize a spectrum than to measure a whole spectrum in the experiment.By tuning the parameters {c,A,μ0H},the ratio of sound velocities and Fermi surface of the two sectors are controllable,as shown in figure 5.

4.Analytical results

4.1.Strong coupling limit in partially polarized phase

In general,analytical expressions of physical quantities are hardly obtained from the BA equations unless in certain extreme limits.Usually,one considers the weak and strong coupling limits,∣c∣?nand∣c∣?nin 1D ultracold atomic systems.The coupling strengthcan be experimentally tuned from weak to strong by controlling the 3D scattering length near the Feshbach resonance,i.e.

whereg1Dis the effective inter-component interaction,a3Dis the 3D scattering length,a⊥is the transverse oscillator length.For our convenience,we denote the dimensionless coupling strengthsWhat below are the derivations of analytical results of physical quantities for the strong coupling regime,γ ?1.

In the grand canonical ensemble,quantum phase transitions take place at zero-temperature,which have already been studied in both the canonical ensemble and the grand canonical ensemble [34,35,37,43].Here we discuss analytical results in the grand canonical ensemble,where the parameters{c,A,μ0H} are our driving parameters.For example,the phase diagram and interaction strengths can be given in μ–Hplane,see figures 6 and 7.

From figure 6,we observe that the strong coupling γ ?1(regions plotted in dark blue)can be reached in the vicinity of the edge of the vacuum phase.In the partially polarized(FFLO-like) phase,the strong coupling regime locates near the quartet pointA/c2=-1 and μ0H/c2=1.If we define the effective chemical potentials of both sectorAb=A+c2andAu=A+μ0H,the strong coupling regime meets the conditionAb?c2andAu?c2.

In figure 7(a),we plot the polarization in the μ–Hplane for a fixed value of interaction strength.From figure 7(b),we observe the polarizationP=nu/(2nb+nu) varies drastically in the vicinity of the quartet point.Therefore,it might come into the expression of the band curvature corrections of sound velocities and effective masses for the strong coupling.

4.2.Sound velocity and effective mass in one-particle-hole excitation spectrum

In this section,we will derive the exact dispersions of particle-hole excitations in both paired and unpaired fermions.We first derive the dispersion of bound pairs in the long-wavelength limit,i.e.we derive the linear dispersion with a curvature correction.Consequently,we can obtain the sound velocities and effective masses in terms of dressed energies and the density distribution functions.For a one-particle-hole excitation near the Fermi pointBof the bound pairs,the excited energy and momentum are given by

In the long-wavelength limit,the excitation is near the Fermi point,i.e.Δk?1.Therefore,the excited energy and momentum can be expanded atk=B.Namely,

wheref′(x) andf″(x) denote the first-and second-order derivatives off(x) with respect tox,respectively.From the relation

we directly obtain Δk,

Subsequently,we have the following dispersion relation for the bound pairs

expressed up to the second order of ΔKb.From this relation,we read off the sound velocity and effective mass of the oneparticle-hole excitation in the paired Fermi sea as

which holds true for arbitrary coupling strength and thus will be used in our numerical calculations.

Similarly,a one-particle-hole excitation for unpaired fermions near the Fermi point can be calculated via the excited energy and momentum,

The excited energy can be expressed by the excited momentum as well as equation (32),

Therefore,by comparison,the sound velocity and effective mass of the one-particle-hole excitation in the unpaired Fermi sea are given by

The obtained relations for velocities and effective masses(33)and (36) are very convenient to carry out numerical calculations,see figure 4.

For a strong coupling,i.e.|c|?1,we may directly use the dressed energies equation (15) and the distribution functions equation (17) to determine the dispersion relations characterized by the sound velocities and effective masses.To this end,we first calculate Fermi momentaBandQin terms of particle densitiesnu,nbwith our chosen parameters {c,A,μ0H}(i.e.[Ab,Au]).For the strong coupling,we may expand the kernelsa1(k) anda2(k) up to the first order of 1/c,the distribution functions equation (17) thus become

The particle densitiesnbandnuare related to Fermi momentaBandQvia

which suggest the following relations between particle densities (nuandnb) and Fermi momenta (BandQ)

For a further derivation of the velocities,we need the effective chemical pressures of pairs and unpaired fermions

From the dressed energy equation (15),we may obtain the following relations between effective chemical potentials and Fermi momenta [33]

Consequently,according to equations (39) and (41),we built up the relations between the particle densities and our chosen parameters {c,A,μ0H} through the Fermi pointsBandQ.

Now we can rewrite the expressions of the sound velocities and effective masses following the expressions equations (33) and (36) for the strong coupling regime.To determinevbandgiven by equation(33),we first calculateε′ (B)and σ(B) and their derivatives.Equation (15),ε′ (B)is given by

Furthermore,owing to equations (39) and (41),we replace γband γuin equations (45) and (47) with γ and the polarizationPor the ratio between two effective chemical potentialsAbandAu.Since the polarization is defined byP=nu/n,we have γ/γb=nb/n=(1-P)/2 and γ/γu=nu/n=P,which allows us to express the sound velocities and effective masses as a function ofP.Alternatively,we would like to express the sound velocities and effective masses in terms ofAbandAu.Since they are directly related to our controllable parameters {c,A,μ0H}.According to equations (39) and (41),we have

We denote the ratio of effective chemical potential asAu/Ab=α2for our convenience in analysis.Using the polarizationPand the ratio α,we can rewrite equations (45)and (47) as

which are confirmed by numerical results in figure 8,where we set the polarizationP=1/5 (i.e.α=1).According to equation(48)and the last equation(49),the ratio between two sound velocities is simply given byvu/vb=2nu/nb=α up to the order of 1/c2,which provides physical insight into the sound velocities in figure 5,where our chosen parameters{c,A,μ0H} indicate a strong coupling with α=2 and α=1,respectively.Furthermore,sincekF,b≈πnbandkF,u≈πnu(illustrated at the end of section 3.1),we havekF,u/kF,b=α/2,which gives a special ratio between two Fermi points in figure 5.

So far,we derived the sound velocities and effective masses of paired and unpaired fermions starting from the zero-temperature TBA equations.For the repulsive case,it is proved [45] that these quantities can be derived from the BA equations;by a similar method,one can prove that this works for the attractive case as well (see appendix A.2 for detailed procedure).This is reasonable since the TBA equations are derived from the BA equations.

The sound velocities and effective masses can effectively characterize the spectra of one-particle-hole excitation.Therefore,these quantities can be regarded as evidence and provide convenience to verify the separation of collective motions within paired and unpaired charges at zero temperature.This kind of charge–charge separation can even be characterized at non-zero but low temperature.According to the relations given by equations (33) and (36),the zerotemperature subtraction for low-temperature free energy density can be expressed by (see appendix A.3 for proof)

which is simply the addition of contribution of paired and unpaired sections.This additivity manifests the independent character and separation of paired and unpaired bosonic modes.Moreover,theT2dependence given by the low-temperature pressure correction shows a typical linear dispersions feature.The specific heat can also be obtained by its definition,i.e.

which preserves the additivity and describes the novel charge–charge separation as well at low temperature.

5.Excitations other than single particle-hole

In this section,we will consider excitations other than the one particle-hole excitation,such as excitations of multiple particle-holes,breaking and forming pairs,as well as length-nstring excitations.

5.1.Multiple particle-hole excitations

The calculation is essentially similar to that for one-particle-hole dispersion.Here we do not wish to present the calculation in detail.Figure 9 shows three examples for various choices of the numbersNbandNu.

Figure 11.Excited spectrum of breaking or forming one pair.The excitation spectra show two-fold degeneracy in breaking a pair and forming a pair.Here we denote[αb+βu]≡:2(αkF,b+βkF ,u).We set c=-1.00,A=-0.9000,μ0H=1.0100 for our numerical calculation with the dispersions equations (54) and (55).

Figure 12.(a)Dressed energiesand distribution functionsfor n=1,2,3.The dressed energiestend to 2nμ0H for k →±∞.For the ground state,gives the distribution of holes because of Mn=0.(b) Excitation spectra of the spinon-bound-states.For a single length-n string excitation,double degeneracy spectrum is observed (left).The right figure shows the continuum spectra of two length-1 spinons.

5.2.Excitations by pairing and depairing without exciting any n-string

For the case of breakingNubpairs,we createNubholes in the paired Fermi sea while adding 2Nubfermions in the unpaired sea,see figure 10.For the case of formingNbupairs,we create 2Nbuholes in the unpaired Fermi sea and addNbupairs in the paired Fermi.In comparison with the ground state,for breakingNubpairs,while formingNbupairs,the particle numbers are given by

We introduce δ-functions for representing the excited pairs and unpaired particles and holes as

respectively,wherekp1,i,kp2,i,kh1,i,andkh2,istand for the excited quasimomenta of unpaired particles and holes.As we did in equation (22),substitute the δ-functions (53) into the distribution functions equation (17).By taking a similar calculation to that of the single particle-hole excitation,the excited energy of multiple pair-forming and breaking is given by

The total momentum for this excitation depends on not only the quasimomenta of exited holes and particles but also the numbers of breaking pairs and forming pairsNubandNbu.The parity of quasimomenta of pairs in such a type of excitations does not change because it only depends onN=NG(see equation (19)).Meanwhile the parity of quasimomenta of unpaired fermions is changed since it depends onN+M=NG+MG-Nub+Nbu.Therefore,an additional phase shift is caused due to the change of parity,namely,

We see that odd{Nub-Nbu} gives a two-fold-degenerate excited state.In figure 11 we present excitation spectra of breaking or forming one pair.Obviously,the spectra are more complicated due to the two-fold degeneracy.

5.3.n-strings excitations

Now we consider excitations by flipping spins in the unpaired fermions without forming pairs.According to equation (12),the dressed energy and distribution function in the spin sector at the ground state are given by

For the ground state,all spin strings are unoccupied at zero temperature.The dressed energies and distribution functions forn-strings withn=1,2,3 are plotted in figure 12(a).

For simplicity,consider the case of exciting one length-lstring by flipping unpaired fermions,i.e.Ml=Ml,G+1=1 andMn=0 forn≠l.The quantum numbers of the spin setorJαnare integer(half-odd integer)forN-Mnodd(even),and satisfy equation (7).Therefore,for the ground state,Jαnare confined by the condition

which indicatesN-2Mholes and no occupation of allnstrings.For the excited state withM?=1,Jαnare confined by the condition

that indicatesN-2M-2nholes forn≤?,N-2M-2?holes forn>? and one occupation of length-?string forn=?.By introducingδ(k-λ?)Lrepresenting the excited ?-string into the distribution functions,we have

where the last term equals zero since all strings are unoccupied at the ground state.By the calculations similar to that of the excited energy of one particle-hole excitation (see appendix A.1),the excited energy of the length-?string excitation is given by

In the case of exciting one length-?string,the parity of quasimomentumkjfor the unpaired charge sector is changed due to its dependence on the quantum numbersTherefore,apart from the contribution of the excited string,there is an additional term in the total excited momentum attributed to the momentum change of the unpaired sector,namely,

From equation(18),we see that the momentum is confined in an interval given by taking λ?→±∞,i.e.

The excitation spectra of one and twon-strings withn=1,2,3 are given in figure 12(b),respectively.These spectra are gapped with a magnitude of 2nμ0Hover the ground state.Since a certain number of unpaired fermions are flipped,these excitations should be regarded as magnon excitations that show a ferromagnetic coupling in the unpaired sector.The string structure of these kinds has been confirmed in the experiment [46].

6.Conclusion and discussion

We have rigorously studied the excitation spectra of the Yang–Gaudin model with an attractive interaction.For the one particlehole excitations in paired and unpaired fermions,the spectra apparently manifest a novel separation of collective motions of bosonic modes,i.e.the pairs-unpaired-fermions separation,which can be regarded as evidence for the existence of the FFLO-state.We further analytically characterized this separation by calculating the curvature corrections to the linear dispersions of both paired and unpaired sections for a small momentum and a strong coupling regime.We have determined the sound velocities and effective masses in these dispersions,and compared them with numerical results.We have shown that the free energy and the specific heat of the system can be simply expressed in a sum of inverse sound velocities of paired and unpaired Fermi seas at low temperatures,reflecting the universal thermodynamics of TLLs.We have also studied other types of excitations,such as breaking or forming a pair and length-nstring excitations in the spin sector,in order to understand subtle novel pairing and depairing states.Here the gapped length-nstring excitations manifest a ferromagnetic coupling in the unpaired Fermi sea,i.e.magnon excitations.

Building on our results obtained,we expect to further study dynamical correlation functions for the charge–charge separation theory of FFLO states.According to the linear response theory JFP thanks the XWG?s team members for helpful discussion and suggestion in completing his Masters project and thank the Innovation Academy for Precision Measurement Science and Technology,Chinese Academy of Sciences for the kind hospitality.

Appendix

A.1.Express excited energies in dressed energies

First,introduce two useful formulas.

Formula 1:Any two groups of the analytical functions of the form[47],one can measure the spectra of pairing and depairing in the system by imposing a perturbation onto the system and observing their linear responses.Bragg spectroscopy[48]is one of the common experimental tools used to measure the many-body correlation.Using Bragg spectroscopy,the recent experiment[49] conducted by Hulet’s group at Rice University has confirmed the spin-charge separation theory of TLLs in the 1D repulsive Fermi gas.In Bragg spectroscopy,6Li atoms are trapped in 1D-tubes and two beams of different orientations are imposed onto the tubes.Due to the disturbance of the two beams,initially symmetrically distributed particles distribute asymmetrically in the tube.The dynamic structure factor(DSF)S(q,ω),usually defined as the density-density correlation,is related to the change of the total momentum as ΔP∝[S(q,ω)-S(-q,-ω)].Therefore,the DSF is a measurable quantity in the experiment with both repulsive and attractive Fermi gases.Moreover,at low-energy excitations,the maximum(peak)of the DSF is related to the sound velocity,namely,the corresponding peak frequency is given by ωp=vcq,and it is independent of the effective mass.Accordingly,different sound velocities of the paired and unpaired fermions can be observed by the Bragg spectroscopy,showing the subtle nature of the FFLO states in 1D.

Acknowledgments

JJL and XWG are supported by the NSFC key grant No.12134015,the NSFC grant No.11874393 and No.12121004.satisfy

One can prove it by multipyη(k) andf(k) .

Formula 2: For any even analytical functionf(k) integrated between[k-,k+],wherek+=k0+ΔQ+,k-=-k0+ΔQ-(for ΔQ±small enough),it can be expanded with ΔQ±,i.e.

so that

Ifk+=-k-=k0+ΔQ,namely,ΔQ+=-ΔQ-=ΔQ,the expansion can be expressed as

and

Introducing-δ(k-kh)/Land δ(k-kp)/Linto the distribution functions of unpaired fermions for one-particle hole excitations,we obtain

According to equation (23),the excited energy density is given by

Substitute the last equation into the excited energy density,and we obtain

According to formula 2,the change of distribution functions is given by

While according to formula 1,we obtain the following relation,

Therefore,the last three terms of R.H.S.of equation(A8)can be substituted by the above relation (A9),namely,

Substitute the last equation into the excited energy,it follows that

Together with equation (25),the dispersion of one-particlehole excitation of the unpaired fermions is obtained.

The dispersion of paired fermions can be acquired in the same way.In brief,we again introduce two δ-functionsAccordingly,the distribution functions are

According to equation (23),the excited energy density is given by

Using the same technique as that for unpaired fermions,one can essentially acquire the excited exergy given by equation (26).

In general,we can have excitations of 2Nbparticle-holes in the paired sector and 2Nuin the unpaired sector simultaneously.One can easily prove that the multiple particle-hole excitations are simply a summation of single-particle-hole excitations with 2Nu+2Nbδ-functions introduced into the distribution functions,i.e.

For solely breaking and forming pairs,since there is no excitation of length-nstrings,the dispersions are similar to the above ones.The difference is likely that the excited state might be degenerate in this case due to the change of particle numbers of each sector,as described by equation (55).

For excitations of one length-lstring,the distribution functions are given by equation (59).Since there is exactly one occupied string,Then the excited free energy density equation (60) can be rewritten as

Since the particle number is fixed,i.e.N-NG=0,we directly obtain

According to formula 2,the changes in distribution functions are

According to formula 1 and the expression of depressed energies,we have

Substituting the last equation into equation (A15) we obtain the excited energy equation (61) expressed in the length-?-string dressed energy.

A.2.Sound velocity and effective mass from Bethe ansatz

In section 3.2,we derived sound velocities and effective masses of paired and unpaired fermions from the zerotemperature TBA equations.Here we apply the method of[45] to derive the results from the BA equations in detail.

SinceMn=0 at the ground state,the logarithm of the BA equations of the two charge sectors are

For the ground state,at the thermodynamic limit,we can expand both equations with coupling strength up to orderthe first of which gives

Ifkjand Λβare both symmetrically distributed to the zeropoint,the arctan-terms drop out in the sum with respect tojand β.Namely,we have

ifkjor Λβis asymmetrically distributed to the zero-point,there is an additional but negligible term

respectively.According to equations (A20) and (A21),the approximation of Λαis independent of the parity ofkj.By substituting equation (39) into (A20),we have

where the first term of RHS of the last equation πJα/Lequals the quasimomentum of free paired fermions,while the remaining terms emerge from the interactions among fermions.

The total energy and momentum of the paired fermions are given by

where β is chosen to be much smaller thanMso that the excitation is taken near the Fermi surface.Therefore,the excited momentum of free paired fermions is

whereas the excited energy of attractively paired fermions is

We would like to express this dispersion in free-fermion approximation,with the mass of free-fermionm(set to be 1/2 previously) replaced by the effective massm?due to the interactions.Namely,

By comparison of equations (A26) and (A27),the sound velocities and effective masses are given by

which is the same as the sound velocity and effective mass of paired fermions in equation (45).

Similarly,we can calculate these quantities of unpaired sectors.By expanding the second equation of equation(A19),we have

The dispersion of unpaired fermions can be expressed as

Therefore,the sound velocity and effective mass of unpaired fermions are

which is the same as the sound velocity and effective mass of unpaired fermions in equations (47).

A.3.Free energy at low temperature

To distinguish quantities at zero-temperature and low-temperature,we introduce the subscript ‘0’ indicating zero-temperature.Accordingly,dressed energies at zero-temperature ε0and κ0given by equation(15);while dressed energies at lowtemperature can be expanded with respect to zero-temperature as ε=ε0+β and κ=κ0+γ.Namely,we have

where we omitted the integral intervals (-∞,∞) for convenience of notation.Since for low-temperature,the integrals on the R.H.S.of the last two equations mostly attribute to the values of integrands near the Fermi momentaBandQ,we replace ε(k) and κ(k) with their linear expansions (k-B)uband(k-Q)uu,respectively,whereubanduudenoteε′ (B) andκ′ (Q),respectively.Namely,we have

According to equation (13),the free energy density (i.e.pressure) at zero-temperature is given by

while the free energy density at low temperature can be expanded with respect to zero-temperature by

Below we will prove that the last four terms on the R.H.S can be expressed in sound velocitiesvbandvu.

where the last term on the L.H.S should cancel with the fourth term on the R.H.S,and the third and last terms on the R.H.S should cancel with each other.By rearranging the last equation and substituting β0and γ0into it,we have

where we used the relation between sound velocities and distribution functions and dressed energies given by equations(33)and(36),which is the same as equation(50).