Correlation functions and momentum distribution of one-dimensional hard-core anyons in optical lattices
We address the problem of calculating the correlation functions in a system of one-dimensional hard-core anyons that can be experimentally realized in optical lattices. Using the summation of form factors we have obtained Fredholm determinant representations for the time-, space-, and temperature-dependent Green’s functions which are particularly suited to numerical investigations. In the static case we have also derived the large distance asymptotic behavior of the correlators and computed the momentum distribution function at zero and finite temperature. We present extensive numerical results highlighting the characteristic features of one-dimensional systems with fractional statistics.
pacs:05.30.Pr, 02.30.Ik, 71.10.Pm
One-dimensional (1D) systems possess certain characteristics which makes them extremely interesting to theoreticians and experimentalists alike. From the theoretical point of view they are special because their simplicity makes them amenable to exact solutions which, contrary to a naive expectation, are characterized by extremely rich physics. Furthermore, in recent years numerous experimental realizations of quasi-one-dimensional materials opened the way for the investigation of properties not present in the three-dimensional world. One such remarkable feature found only in low-dimensional systems is the presence of anyons LM (); GMS (); FW (), particles with statistics interpolating between fermions and bosons. One-dimensional anyons Kundu (); BGO (); BG (); SSC (); BGH (); CM (); PKA1 (); L2 (); L3 (); L4 (); SC (); HZC (); GHC (); SC2 (); C (); AN (); HZC1 (); HC (); WRDK (); AOE1 (); OAE1 (); IT (); LMP (); Gir (); BGK (); BCM (); BFGLZ (); Gr (); BB (); BS (); MSo (); YCG (); SPK (); Zin (); RFB () represent an active area of research with an increasing literature on the subject partly motivated by several proposals DDL (); ABVC (); JBGH (); KLMR (); MPMP () for experiments designed to find signatures of fractional statistics in ultracold gases.
The model studied in this paper AN (); HZC1 (); HC (); WRDK () can be viewed as the generalization to arbitrary statistics of hard-core bosons on the lattice. Unlike the continuum analog, which is represented by impenetrable Lieb-Liniger anyons Kundu (); BGO (); BG (); SSC (); BGH (); CM (); PKA1 (); L2 (); L3 (); L4 (); SC (); HZC (); GHC (); SC2 (); C () this model has not been subjected to a thorough analytical investigation of the correlation functions, especially at finite temperature. The need for such calculations has become even more pressing in the last decade as a result of the plethora of experimental techniques developed for the measurement of correlators. Here, we fill this gap in the literature, deriving Fredholm determinant representations for the time-, space-, and temperature-dependent Green’s functions. These representations not only represent the starting point for the rigorous investigation of the asymptotic behavior but also can be used to obtain efficiently extremely accurate numerical data. We should point out that the numerical implementation of the determinant formulas derived in this paper can be done in less than ten lines of code in Mathematica or Maple (see Bor ()) and can be used to obtain very precise results (errors smaller than ). This level of numerical efficiency (the running time on a personal computer is of the order of seconds) and precision cannot be achieved using other methods, such as the ones used in HZC1 (); HC (); WRDK (), which are also inapplicable at finite temperature. In the case of static correlators we also derive the large distance asymptotic behavior and compute the momentum distribution function at zero and finite temperature. Our results show that the Green’s function satisfies with for intermediate values of the statistics parameter which explains why the groundstate momentum distribution of anyons is asymmetric HZC (); SC (); HZC1 () with a peak whose location in momentum space depends on the statistics parameter. The location of this peak, which is present even at finite temperature, is extremely important from the experimental point of view due to the fact that the statistics of the particles can be inferred from its position.
The plan of the paper is as follows. In Sections II and III we present the anyonic model and the determinant representations which constitute one of the main results of our paper. Analytical and numerical data for the the asymptotic behaviour and momentum distribution can be found in Section IV. The derivation of the determinant representations is being presented in Section V. We end with some conclusions in Section VI. Some technical details of the calculations can be found in three Appendices.
Ii The model
where is the hopping parameter, is the number of lattice sites which we will consider to be even and is the chemical potential. The operators satisfy anyonic commutation relations
and the statistics parameter. For we have (hard-core condition) and . Varying the statistics parameter the commutation relations (2) interpolate between the ones for spinless fermions () and hard-core bosons (). Some of the ground state properties and relaxation dynamics of the model were studied in AN (); HZC1 (); HC (); WRDK (). We should point out that the Hamiltonian (1) is a particular case () of the XXZ spin chain with fractional statistics first considered by Amico, Osterloh and Eckern in AOE1 ().
Defining the Fock vacuum in the usual fashion the eigenstates of the Hamiltonian (1) with particles are
with the -body wavefunction and the momenta of the particles. Similar to the case of Lieb-Liniger anyons PKA1 () the order in which the creation operators appear in (4) is important as we will see in the subsequent calculation of form factors. A direct consequence of this ordering and commutation relations (2) is that the exchange symmetry of the wavefunction is given by
The solutions of the Schrödinger equation with the appropriate symmetry (5) are
with the group of permutations of elements and we denoted by the signature of the permutation . The factor is added for convenience so that in the bosonic limit () the wavefunctions reduce to those used in CIKT () for hard-core bosons (XX0 spin chain). The eigenspectrum of the system is
Due to the exchange symmetry (5), imposing periodic boundary conditions (PBC) in an anyonic system has nontrivial consequences as it was first noticed by Averin and Nesteroff AN () (for a detailed discussion see Appendix A of PKA1 ()). Let us consider the simple case of two particles. If we impose PBC on the first particle using (5) we obtain . This shows that if the wavefunction is periodic in the first variable then, due to the anyonic symmetry, the wavefunction will have twisted boundary conditions in the second variable. The generalization in the case of particles is PKA1 ()
Imposing PBC on (6) we obtain the Bethe Ansatz Equations (BAEs) satisfied by the momenta
The BAEs (8) reproduce the well known equations for hard-core bosons (XX0 spin chain) and spinless fermions for and , respectively. Using the commutation relations (2) and the BAEs it is easy to show that the eigenstates (4) satisfy the orthogonality condition if and if . The normalization is 111 More precisely if where we have
Introducing the notation the values of the momenta in the groundstate with particles, odd, are
In the thermodynamic limit the momenta fill densely the Fermi zone with the Fermi momentum. We remind the reader that and the system is gapless. For the structure of the ground-state is different and will not be considered in this paper. For the thermodynamic behavior of hard-core anyons and spinless fermions is the same.
Iii Determinant representation for the correlation functions
We are interested in calculating the thermodynamic limit of the time-, space-, and temperature- dependent Green’s functions
where , and . Using the summation of form-factors we were able to express these correlators as Fredholm determinants which, as we will show in Section IV, can be used to obtain extremely precise numerical data. We would like to stress the fact that our result is exact and does not employ any approximations. The derivation of the determinant representation is quite involved, therefore, in this section, we are going to present only the final results. The interested reader can find the full derivation in Section V.
The correlation function has the following representation in terms of Fredholm determinants (for the definition of a Fredholm determinant see Section V.4):
with and integral operators acting on the interval
The functions appearing in (14) are defined as
with P.V. denoting the principal value of the integral and is the Fermi function
A similar representation is obtained for the correlation function
In (III), is the same integral operator which appears in the determinant representation of . is an integral operators which acts on
The main difference in this case is that the function which enters the definition of the kernel (14a) is now defined as (note the sign change)
In the limit (III) and (III) reproduce the well-known results for hard-core bosons CIKT () (note that in our notation, and correspond to and of CIKT ()). At we obtain the results for spinless free fermions on the lattice.
iii.1 Static limit
Certain simplifications occur in the static limit . Due to the relation
it will be sufficient to consider only . In this limit we have and
The static limit of (III) for is
with and integral operators acting on like in Eq. (13) and kernels
In the case of the representation (24) remains valid but now the kernels of the integral operators are
This shows that for intermediate statistics and
with the bar denoting complex conjugation. As we will see in Section IV.3 the direct consequence of this relation is that the momentum distribution of anyons is no longer symmetric around the origin.
iii.2 Zero temperature limit
The zero temperature limit results can be derived easily noticing that at the Fermi function (19) becomes the characteristic function of the interval with . Therefore, the groundstate Green’s functions have the same determinant representations (III),(III), (24) with integral operators acting on
and similar expressions for and .
Iv Asymptotic behavior of static correlators
Similar Fredholm determinant representations for the correlation functions of other integrable systems were previously obtained in the case of impenetrable bosons Lenard (); IIK1 (), XX0 spin chain CIKT (), two-component bosons and fermions IP () and Lieb-Liniger anyons L2 (). In all these cases, including ours, the relevant integral operators belong to the so-called “integrable” class of integral operators IIKS2 (); KBI (); HI () and as a result the correlation functions satisfy a completely integrable classical system of differential equations. The determinant representation and the associated differential equations represent the basis for the rigorous investigation of the large time and distance asymptotic behavior of the correlators via the solution of a certain matrix Riemann-Hilbert problem. This rather involved program was implemented for the impenetrable Bose gas in IIK1 (); IIKV (); KBI (), for the XX0 spin chain in IIKS (), for the two-component fermions in BL (); GIKP (); CZ () and for Lieb-Linger anyons in L3 (); L4 ().
It would be naturally to expect a comparable number of papers in the literature devoted to the numerical exploration of the determinant representations which would allow for a better understanding of the intermediate distance regime which is inaccessible by analytical methods. This is unfortunately not true (two notable exceptions being CSZ (); Z ()). While this might be attributed to the possible uncontrollable errors in the evaluation of an infinite determinant, recently Bornemann Bor () provided a simple and easily implementable method which allows for very precise numerical evaluations of such representations. This method, which is based on the Nyström solution of the Fredholm integral equations of the second kind with the Gauss-Legendre as the quadrature rule, will be used to calculate the short and intermediate distance static correlation functions and also to check the validity of the large distance asymptotics which we will derive below.
iv.1 Asymptotic behavior at zero temperature
At zero temperature we expect the system to be critical and the large distance asymptotic behavior of the correlation functions can be derived using Conformal Field Theory ideas. More precisely we are going to use Cardy’s result Cardy () relating the conformal dimensions of the conformal fields present in the theory from the finite size corrections of the low-lying excitations of the system (a detailed presentation of the method can be found in Chap. XVIII of KBI ()).
For our system we have three type of low-lying excitations: addition of one or more particles into the system with quantum number , backscattering of particles over the Fermi ”sea” with quantum number and ”particle-hole” excitations close to the left or right Fermi surface with quantum numbers . The asymptotic behavior of the correlation functions is then
where , are amplitudes which cannot be obtained using this method, is the macroscopic part of the momentum gap and the conformal dimensions can be obtained from the finite size corrections of the energy and momentum using ( is the Fermi velocity)
The derivation of the finite size corrections in our model is very similar with the one performed for the XX0 spin chain 222Ref. KBI () considers the more general case of the XXZ spin chain. The results relevant for us are obtained considering and the dressed charge . (Chap. II of KBI ()) and can be found in Appendix A. The central charge of the model is equal to one and
Neglecting the contributions coming from the terms we obtain the following asymptotic expansion for the Green’s function
where is the number of backscattered particles. For this expansion reduces to the Green’s function asymptotic expansion of hard-core bosons with leading term At the free-fermionic point, , neglecting all the terms except we get which is in fact the exact result modulo the amplitude. For intermediate values of the statistics parameter the leading term of the expansion, given by , is oscillating with a wavevector proportional to . This is a general characteristic of 1D anyonic systems first observed by Calabrese and Mintchev CM () and it can be seen in Fig. 1. The asymptotic behavior plotted is
with real parameters obtained using a fitting procedure and the relative errors defined in the usual fashion and a similar expression for the imaginary part. The correlation function was computed from the zero temperature limit of Eq. (24). Using the method presented in Bor () for the numerical implementation of Fredholm determinants we were able to obtain extremely accurate values (absolute errors smaller than ).
Another interesting feature specific to correlation functions of 1D anyonic systems (see CM ()) is the presence of fermionic beats for values of the statistics parameter close to . In this region we can see from the r.h.s. of (31) that we have two oscillations with almost equal amplitudes and wavevectors producing beating effects. This phenomenon can be seen in Fig. 2 for .
iv.2 Asymptotic behavior at finite temperature
At low-temperatures the asymptotic behavior of can be derived from the zero temperature result (30) by replacing with with the Fermi velocity. However, at higher temperatures the CFT description is no longer valid and we need to use a different method. Here, we present an heuristic derivation of the temperature dependent asymptotic expansion of the Green’s function which is based on the similar results obtained rigorously for impenetrable Lieb-Liniger anyons L3 (). Even though the considerations below are nothing more than an educated guess, the numerical data presented in Fig. 3 show that our result, Eq. (35), is nevertheless correct. In the case of Lieb-Liniger impenetrable anyons, the main term responsible for the exponential decay of the correlators, denoted by in Eq. 3 of L3 (), contained the logarithm of the ratio between the Fermi distribution and an “anyonic” distribution function interpolating between the Fermi and Bose distributions. We will assume that for our model the lattice equivalent of this anyonic distribution function is given by
in terms of which we can define the lattice analog of
An important role is played by the zeroes of closest to the real axis and situated in the upper half plane denoted by
with real parameters. For the second term in the r.h.s. of Eq. (35) is much smaller than the first term and we obtain the known result for hard-core bosons IIKS (). In the fermionic limit and we have with . This result reproduces the first two terms of the asymptotic expansion of free fermions which can also be derived starting from and moving the integration contour in the upper half-plane for . Plots of the correlation function and the asymptotic expansion Eq. (35) for are shown in the top panels of Fig. 3. Additional numerical checks of our asymptotic expansion for intermediate values of the statistics parameter are presented in the lower panels of Fig. 3. For values of in the interval it is sufficient to fit the data using only the first term in the r.h.s. of (35) obtaining extremely accurate results (relative errors of for and ). As approaches the fermionic point, we need to use both terms in order to obtain accurate results (relative errors of for and ). The oscillatory behavior of the correlation functions is still present but at finite temperature the damping is exponential compared to the algebraic decay at zero temperature. The rate of exponential decay is a decreasing function of being maximum for spinless fermions.
iv.3 Momentum distribution at zero and finite temperature
The accurate knowledge of the large distance asymptotic behavior allows us to compute the momentum distribution function defined as
A direct consequence of Eq. (27) is that and which means that
For hard-core bosons and spinless fermions and, therefore, the momentum distribution is symmetric with respect to the axis. However, in the case of anyonic systems and the momentum distribution is asymmetric (see also SC (); HZC (); GHC ()). Also, from the asymptotic behavior (31) we expect that at zero temperature the momentum distribution will have a singularity at and a weaker singularity at . The numerical results presented in Fig. 4 confirm these theoretical predictions. At zero temperature we can see clearly how the peak present at for (hard-core bosons) decreases and moves to the right as and becomes the discontinuity at of the momentum distributions for spinless fermions. The weak singularity at for manifests itself in the derivative of which becomes sharper with decreasing and becomes the discontinuity of the momentum distribution at (see also SC ()). At finite temperature the momentum distribution gets smoother and wider. Even so, the peaks at still remain visible and can be experimentally detected.
V Derivation of the determinant representation for the correlation functions
The determinant representations presented in Section III were derived using the method known as the summation of form-factors IIK1 (); IP (); L2 (); KBI () which is briefly sketched below (an alternative method based on Wick’s theorem can be found in ZCG ()). We start from the finite lattice expression of the Green’s function (we focus on , the case can be treated along similar lines)
where we have used the short notation Inserting a resolution of identity we obtain
where the sum is over all sets of allowed values for the momenta with , and we have introduced the form factors (note that )
Even though the summation appearing in Eq. (V) seems daunting, as we will show in the next sections, it can be done exactly in the form of a finite size determinant. Using this representation in Eq. (38) the thermodynamic limit can be performed explicitly obtaining the representations given in Section III. Summarizing, the derivation of the Fredholm determinant representation involves three steps: a) calculation of the form factors on the finite lattice, b) derivation of a determinant formula for the normalized mean value of bilocal operators via summation of the form factors, and c) taking the thermodynamic limit.
v.1 Form factors on the finite lattice
The implementation of the program sketched above starts with the calculation of the form factors. Using we have
where we have introduced the static form-factor which in terms of the wavefunctions can be written as (see Appendix B)