Correlation functions of one-dimensional Bose-Fermi mixtures

Correlation functions of one-dimensional Bose-Fermi mixtures

Holger Frahm and Guillaume Palacios

Institut für Theoretische Physik, Universität Hannover, Appelstraße 2, 30167 Hannover, Germany 共Received 14 July 2005; published 16 December 2005兲

We calculate the asymptotic behavior of correlators as a function of the microscopic parameters for an integrable Bose-Fermi mixture with repulsive interaction in one dimension. For two cases, namely polarized and unpolarized fermions the singularities of the momentum distribution functions are characterized as a function of the coupling constant and the relative density of bosons.

DOI:10.1103/PhysRevA.72.061604 PACS number共s兲: 03.75.Mn, 05.70.Jk, 71.10.Pm In the past years, the advances in cooling and trapping of

atomic gases have opened the possibility to realize quasi- one-dimensional 共1D兲 systems with tunable strength of the interactions in optical lattices. This gives rise to opportuni- ties for the investigation of the striking phenomena appear- ing in correlated systems as a consequence of the enhanced quantum fluctuations in reduced spatial dimensions. The ob- servable signatures of these phenomena are encoded in the correlation functions of the system such as the momentum distribution function which can be measured directly in time- of-flight experiments or using Bragg spectroscopy 关1兴. A setup for the measurement of various density correlation functions has been proposed for the identification of domi- nant correlations in the atomic gas关2兴. Theoretical studies of correlation functions in cold atomic gases have been per- formed both using analytical methods, e.g., bosonization关3兴 combined with exact results from integrable models such as the Bose gas with repulsive ␦-function interaction 关4兴, and numerically. Additional correlation effects appear when the particles considered have internal degrees of freedom. In cold gases containing different constituent atoms Bose-Fermi mixtures can be realized 关5兴. Extensive theoretical results exist for 1D Fermi gases due to their equivalence with the Tomonaga-Luttinger共TL兲liquids realized by correlated elec- trons in 1D lattices关3,6兴. Only recently, theoretical investi- gations have been extended to Bose-Fermi mixtures: some correlation functions have been calculated numerically in the strong coupling limit关7,8兴where the problem simplifies due to the factorization of the many-particle wave function共see, e.g.,关9兴兲. For analytical results on these systems one has to go beyond mean-field approximations and use methods which can capture the strong quantum fluctuations in 1D systems. The phase diagram and certain correlation functions of atomic mixtures have been studied in the Luttinger liquid picture关10,11兴. Without further input, however, these results are limited to the weakly interacting regime since the TL parameters which determine the low-energy theory cannot easily be related to the microscopic parameters describing the underlying gas. Therefore, instabilities predicted within this approach may not appear in a specific realization 关7,12,13兴.

In this paper we establish the relation between the TL and the microscopic parameters for an integrable Bose-Fermi mixture关12兴. We employ methods from conformal quantum field theory 共CFT兲 to determine the asymptotic 共long- distance, low-energy兲behavior of correlation functions in the

model from a finite size scaling analysis of the exact spec- trum obtained by means of the Bethe ansatz. This approach gives the complete set of critical exponents of the model as a function of the parameters in the microscopic Hamiltonian 共see, e.g., 关6,14,15兴 for applications to 1D correlated elec- trons兲. As an application we compute the momentum distri- bution function of bosons and fermions in the atomic mixture as a function of their respective densities and the effective coupling constant. It should be emphasized that our results can be expected to describe the generic 共universal兲 low- energy behavior of atomic mixtures. Additional interactions—as long as they do not lead to a phase transition—will merely change the anomalous exponents but not the qualitative behavior of the correlation functions.

The 1D Bose-Fermi mixture ofN=Mf+Mbparticles with repulsive interaction共c⬎0兲 on a line of lengthL subject to periodic boundary conditions is described by the integrable Hamiltonian关12兴

H= −

兺

i=1 N ⳵²

⳵^xi2+ 2c

兺

⬍k

␦共xi−xk兲. 共1兲

HereMf=M_↑+M_↓of the particles are fermions carrying spin

␴⁼↑,↓ and Mb of them are bosons. Note that fixing the particle numbers breaks the apparent symmetry of the Hamiltonian arising from the equality of the particles’

masses and their mutual interactions. The many-particle eigenstates of 共1兲 are parametrized by the solutions of the Bethe ansatz equations共BAE兲 关12兴

exp共iq共j0兲L兲=

兿

k=1 M₁

ec共q共j0兲−q_k^共1兲兲,

兿

e_c共q_k^共1兲−q^共0兲_j 兲

兿

兿

k⬘⫽k M₁

e_2c共q_k^共1兲−q_k^共1兲_⬘兲,

兿

e_c共q_ᐉ^共2兲−q_k^共1兲兲= 1, 共2兲 where ea共x兲=共x+ia/ 2兲/共x−ia/ 2兲 and M₀=N, M₁=N−M_↑, M₂=Mb. The corresponding eigenvalue of 共1兲 is E

=兺^N_j=1共q_j^共0兲兲². In the thermodynamic limit L→⬁ with Mi/L kept fixed, the root configurations 兵q^共i兲其 of Eqs.共2兲 can be PHYSICAL REVIEW A72, 061604共R兲 共2005兲

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described by distribution functions ␳i which, as a conse- quence of共2兲are solutions to关12兴

␳i共x兲=␳i共0兲+

兺

Here␳i

共0兲=共c/ 2␲兲␦i0andKˆ

ijare linear integral operators act- ing as共兰j⬅兰_−Q^Qj_jdy兲

共Kˆ_ij丢 f兲共x兲=

冕

kij共x−y兲f共y兲. 共4兲 The kernels of these integral operators are kij共x兲

=a₁共x兲␦_兩i−j兩,1−a₂共x兲␦i,1␦j,1 where 2␲^an共x兲= 4n/共4x²+n²兲.

The properties of the system are completely characterized by the densities m_i⬅M_i/L=兰i␳^共i兲共x兲 of the components of the mixture共these relations determine the boundariesQ_iof the above integral equations兲 and the dimensionless coupling strength ␥^=Lc/N. For later use we also introduce the frac- tion of bosons in the system␣^=Mb/N.

Generically, i.e., form_i⬎0, there are three modes of col- lective elementary excitations above the many-particle ground state of 共1兲. Their dispersion ⑀i共k兲 is linear at low energies with different sound velocitiesvi=⳵⑀i/⳵^k,i= 0 , 1 , 2 关13兴. These quantities determine the finite size scaling behav- ior of the ground-state energy

E₀−L⑀_⬁^{= −} ␲

6L

兺

冉

冊

Physical excitations of the system are combinations of the elementary ones. Due to the interacting nature of the system the different modes are coupled and excitations in one of the modes shift the energies in the other ones. In general, this effect can be described in terms of generalized susceptibili- ties which may be determined in an experiment or numeri- cally from studies of small systems关6兴. For the Bethe ansatz solvable models it is possible to describe the coupling of the modes in terms of the dressed charge matrix关16兴 which in this case reads

Z_ij=␰ij共Q_i兲. 共6兲 The functions ␰ij are given in terms of integral equations

␰ij共x兲=␦ij+兺k共Kˆ

ik丢␰kj兲共x兲.

Z determines the general form of the finite size correc- tions to the energies of low-lying excitations

⌬E共⌬M,D兲=2␲

L

冉

+

兺

冊

^冉

^冊

Here,V= diag共v0,v₁,v₂兲 is a 3⫻3 matrix of the sound ve- locities,N_k^±are nonnegative integers,⌬M is a vector of inte- gers denoting the change of M_i with respect to the ground state for charged excitations. TheD_iare integers or half-odd integers according to

D₀⬃ 共⌬M0+⌬M1兲/2 =⌬M_↑/2 mod 1,

D₁⬃ 共⌬M0+⌬M2兲/2 =⌬Mf/2 mod 1,

D₂⬃ 共⌬M₁+⌬M₂兲/2 =⌬M_↓/2 mod 1, 共8兲 and enumerate finite momentum transfer processes,

⌬P共⌬M,D兲=2␲

L

冉

^兺

冊

+ 2k_F,↓共D₀+D₁兲+ 2k_B

兺

Here k_F,␴=␲^M_␴^/L are the Fermi momenta of the fermion components, k_B=␲^Mb/L is the corresponding quantity for the interacting bosons.

In the framework of CFT关17兴the finite size spectrum共5兲 and共7兲can be understood as that of a critical theory based on the product of three Virasoro algebras each having central chargeC= 1关6,14兴. Correlation functions of a general opera- tor in the theory—characterized by the quantum numbers

⌬MiandDi—will contain contributions from these three sec- tors. The simplest ones, analogues of primary fields in the CFT, have correlation functions共in Euclidean time␶兲

具␾_⌬共x,␶兲␾_⌬共0,0兲典

=

exp关2iD0k_F,↑x+ 2i共D0+D₁兲k_F,↓x+ 2i共

兺

兿

共10兲 The operators␾_⌬ are characterized by their scaling dimen- sions⌬k

±in the chiral共left and right moving兲components of all three constituent theories. The latter are uniquely deter- mined from the finite size energies共7兲and momenta共9兲and form towers starting at

2⌬k

±=

冉 ^兺

^兺

冊

The asymptotic exponential decay of correlation functions in a large but finite system or at finite temperature T can be obtained from共10兲by conformal invariance. For example, at T⬎0 the denominators in 共10兲 have to be replaced by 共v_k␶^±ix兲^−2⌬k±→关␲^T/vksin␲^T共␶^±ix/vk兲兴^2⌬k±.

With 共11兲 the critical exponents which determine the long-distance asymptotics of any correlation function are known as soon as we have computed the dressed charge matrix共6兲. To calculate the correlation functions of a given local operatorOin the microscopic theory共1兲one needs to know its expansion in terms of the fields ␾_⌬ of the CFT.

Usually, this expansion is not known butO and␾_⌬have to generate the same set of selection rules in calculating the correlation function. This drastically reduces the number of possible terms in the expansion: As an example consider the bosonic Green’s function Gb共x,␶兲=具⌿b共x,␶兲⌿b

†共0 , 0兲典:

clearly ⌿b

† generates a state with ⌬Mb= 1 which implies

⌬Mj⬅1 in共11兲. By共8兲the quantum numbersDjare further restricted to integers: the uniform part of the Gb共x兲

⬃兩x兩^−1/2Kb is described by the operator with D_j⬅0 which allows to identify the TL parameterK_b关18兴 from共11兲. The

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interactions lead to additional contributions toGboscillating with wave numbersk₀= 2k_F␴, 2k_B, . . ..

For a comparison with experimental data one is often in- terested in Fourier transforms of the two-point correlation functions given above. The large distance behavior of 共10兲 determines the singularities of spectral functions near ␻

⬇±v_k共k−k₀兲 共see, e.g.,关6兴兲. Quantities accessible in experi- ments with cold gases 关1兴 are the momentum distribution functions of the constituent particles. For the bosons this is the Fourier transform of the equal time Green’s function Gb共x兲. From 共10兲 its singularities at wave numbers k₀ are then nb共k兲⬃兩k−k₀兩^␯b near k⬇k₀. The exponent ␯b is the minimal value of 2兺k共⌬k

++⌬k

−兲− 1 compatible with the quan- tum numbers⌬M and the selection rules for the D for the givenk₀, e.g., 1 / 2Kb=␯b+ 1 = 1 / 4兺k关兺j共Z−1jk兲兴² fork₀= 0.

Using the same procedure for the fermionic Green’s func- tion G_␴共x,␶兲=具⌿_␴共x,␶兲⌿_␴^†共0 , 0兲典 we find that their asymptotic behavior is determined by the conformal fields with⌬M0= 1,⌬M1=⌬M2= 0, half-odd integersD₀,D₁, and integerD₂for G_↑ and⌬M0= 1 =⌬M1,⌬M2= 0, half-odd in- tegersD₁,D₂, and integerD₀forG_↓. Again, the singularities of the fermions’ distribution functions n_␴共k兲 follow from 共10兲. Neark₀−k_F␴= 0 , ± 2k_B, . . . they are given by

n_␴共k兲 ⬃sgn共k−k₀兲兩k−k₀兩^␯f fork⬇k₀. 共12兲

␯fis related to the dimensions共11兲for the quantum numbers

⌬M andD as␯b above. The Fermi distribution of noninter- acting particles corresponds to␯f共kF␴兲= 0.

In the following we consider two cases of particular rel- evance关11–13兴, namely 共i兲 the unpolarized case where M_↑

=M_↓=Mf/ 2 and the ground state of the system is invariant under rotations in the spin index of the fermions and共ii兲the fully polarized case where there is only one spin component of the fermions.

The unpolarized gas. For Q₁=⬁ one obtains M_↑=M_↓ from共3兲, i.e., with vanishing net magnetization. In this case the dressed charge matrix共6兲takes the form

Z=1

2

冢

^冑

冣

Here the Wiener-Hopf method has been used to determine Z₁₁= 1 /

冑

␨ij共x兲=␦ij+

冕

R共x−y兲␨0j共y兲+

冕

R共x−y兲␨1j共y兲 共14兲 with R共x兲=共1 /␲兲兰0⬁d␻^{e−兩␻兩/2cos}共␻^x兲/ cosh共␻^{/ 2}兲. Using Eq.

共13兲the scaling dimensions ⌬₁^± in Eq.共11兲 are independent on the remaining system parameters, i.e., the effective cou- pling␥and the bosonic fraction␣. This a consequence of the SU共2兲invariance of the system in this case. The mode⑀1共k兲 is the spinon mode of the unpolarized system, the CFT de- scribing its low-energy properties is an 关SU共2兲兴1 Wess- Zumino-Witten model.

Additional simplifications arise in the strong coupling limit ␥→⬁ 共i.e., Q₀→0兲 where ␨00= 1, ␨10= 0, ␨01=␨11共0兲

− 1 =␣ ^and ␨11共x兲 is given by a scalar integral equation re- sulting from共14兲. In Fig. 1 we present results obtained from the numerical solution of these integral equations for the exponents which determine the singularities of the momen- tum distribution functions for bosons atk= 0 and fermions at k=kF as a function of the bosonic fraction ␣ for various values of␥. The exponents␯b,f at the other wave numbers are always larger than 1. Note that the system is in a different universality class for ␣= 0 or 1. At ␣= 0, all particles are fermionic and the critical exponents are those of the 1D Fermi gas关6兴. Here the exponent␯f for the singularity at the Fermi point varies between 0 and 1 / 8 as a function of␥^{. On} the other hand, the limit of ␯b as ␣→1 gives exactly the exponent of the 1D Bose gas with␦ interaction关18兴.

The spin-polarized gas. SettingQ₂=⬁ in共3兲corresponds toM_↓= 0. This case has been discussed recently in Ref.关7兴 where some correlation functions have been computed nu- merically in the strong coupling limit. In this case, the finite size spectrum and the scaling dimensions are determined by two gapless modes. Again, the equations simplify in the strong coupling limit where all exponents can be given as a function of ␣ directly, e.g., ␯b共0兲=␣^{2/ 2 −}␣ ^and ␯f共kF兲=␣²

−␣+ 1 / 2 for the dominant singularities of the bosonic and fermionic momentum distribution functions, respectively 共see Fig. 2 for the␥ dependence兲. While the dependence of

␯bon␣ is similar to the one found in the unpolarized case, the strong coupling behavior of␯fat a small bosonic fraction is seen to be very different. Note that the singularity at k_F + 2k_B becomes very pronounced for sufficiently small␣ 共␯f

FIG. 1. Exponents characterizing the singularities in the bosonic 共upper panel兲 and fermionic共lower panel兲momentum distribution function for the unpolarized gas atk= 0 andk_F, respectively, as a function of the bosonic fraction ␣ in the mixture for ␥

= 0.2, 1.0, 5.0, 25.0,⬁共bottom to top兲.

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=␣²⁺␣+ 1 / 2 at strong coupling兲. This feature of the fermi- onic distribution function is a direct signature of the interac- tion and should be observable in experiments.

In summary we have used predictions from CFT on

the finite size scaling of the low-energy spectrum to study the critical properties of a 1D Bose-Fermi mixture.

Generically—not limited to the integrable model共1兲consid- ered in this Rapid Communication—the low-energy effective theory of such systems and therefore the asymptotic behavior of its correlation functions is determined by three linearly dispersing modes. Combined with the exact solution of共1兲 this approach allows to relate the critical exponents directly to the parameters in the microscopic Hamiltonian, i.e., the coupling strength, the fraction of bosons and polarization of the fermions. For two special cases we have studied this relation for the momentum distribution functions and ob- tained simplified expressions valid in the strong coupling limit. These quantitative predictions indicate how to tune the parameters of a given system for enhanced signatures of in- teraction which can be detected in experiments such as关1兴.

Our approach opens new possibilities to investigate the phase diagram of the 1D mixture by identifying the order parameter with the slowest long-distance decay of its corre- lation functions共smallest exponent兲 关10,11兴. Within the inte- grable model 共1兲 there is no instability leading to a phase transition关12,13兴. We emphasize, however, that our expres- sion共11兲for the exponents is valid in more general systems 共with different coupling constants or longer-ranged interac- tion兲and therefore allows for the study of the phase diagram based, e.g., on numerical data on the spectrum of finite systems.

This work has been supported by the Deutsche Forschungsgemeinschaft.

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Cazalilla, J. Phys. B 37, S1共2004兲. FIG. 2. Same as Fig. 1 for a mixture with polarized fermions.

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