Unitary fermions and Luscher's formula on a crystal
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Unitary fermions and L¨uscher’s formula on a crystal
Manuel Valiente
SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, EdinburghEH14 4AS, United Kingdom
Nikolaj T. Zinner
Department of Physics & Astronomy, Aarhus University, 8000 Aarhus C, Denmark
Abstract
We consider the low-energy particle-particle scattering properties in a pe-riodic simple cubic crystal. In particular, we investigate the relation be-tween the two-body scattering length and the energy shift experienced bythe lowest-lying unbound state when this is placed in a periodic finite box.We introduce a continuum model for s-wave contact interactions that respectsthe symmetry of the Brillouin zone in its regularisation and renormalisationprocedures, and corresponds to the na¨ıve continuum limit of the Hubbardmodel. The energy shifts are found to be identical to those obtained in theusual spherically symmetric renormalisation scheme upon resolving an im-portant subtlety regarding the cutoff procedure. We then particularize to theHubbard model, and find that for large finite lattices the results are identicalto those obtained in the continuum limit. The results reported here are validin the weak, intermediate and unitary limits, and can be used for the ex-traction of scattering information ,via exact diagonalisation or Monte Carlomethods, of two-body systems in realistic periodic lattices.
Keywords:
Scattering theory, Effective field theory, Lattice fermions,Finite-size effects
1. Introduction
The notion of low-energy universality in particle-particle collisions is apowerful concept that has been around, in one form or another, for a very longtime. The general idea consists of replacing realistic, complicated interactionswith much simpler ones and renormalising their bare coupling constants in
Preprint submitted to XX May 25, 2016 avour of exact (either experimental or theoretical) scattering properties. Themodel interaction, irrelevant at the two-body level, can then be used toinvestigate the effect of interactions with higher particle numbers withoutthe extra complications of the realistic interactions. For example, in 1957,Huang and Yang introduced their s-wave regularised pseudopotential [1],which aimed at reproducing the exact scattering length of a realistic two-bodyprocess by means of a very simple model interaction. The idea, however, wasput forward much earlier by Fermi in the 1930s [2], who studied neutron-nucleon scattering, and was able to fit the scattering length in the first Bornapproximation.The calculation of two-body scattering properties with rather realistictwo-body interactions, if these are spin-independent, spherically symmet-ric, and single-channel, is not too complicated, especially with the comput-ers we have access to nowadays. It is genuine multichannel collisions thatcan become quite challenging to approach directly from the correspondingLippmann-Schwinger equation. In nuclear physics, there is a fundamental in-terest in obtaining nucleon-nucleon scattering properties from first principlesusing lattice QCD (see [3] and references therein). In condensed matter andcold atomic physics, particle-particle scattering theory in a crystal, where theincident waves are Bloch waves, and with a realistic (e.g. screened Coulomb)interaction, constitutes a formidable problem: all bands of the periodic po-tential are coupled by the interaction, Galilean relativity does not hold sothat the collisional properties depend on the total center of mass momentum,and spherical symmetry is not present and cannot be exploited.In situations like those described above, it is of great appeal to be ableto extract scattering properties without having to directly use scatteringtheory. In massive quantum field theories, L¨uscher showed how to extractpartial-wave scattering amplitudes, i.e. phase shifts and scattering lengths,from the energy shifts due to the interactions when the system is placed ina finite volume [4], thereby generalising an older result from non-relativisticquantum mechanics [1]. However, neither L¨uscher’s nor Huang-Yang’s resultsfor the scattering lengths hold when these are unnaturally large, i.e. nearthe unitary limit. This problem was elegantly solved using pionless effectivefield theory [5] by Beane et al. in ref. [6]. In this way, an analysis oflow-energy nucleon-nucleon collisions, directly from QCD, was possible soonafter [7]. In the case of particle-particle scattering in a crystal, finite-sizeeffects in the forms given in refs. [4] and [6], however, do not appear to beknown. In this article we derive the relevant formulas from effective field2heory on a tight-binding lattice, that is, the Hubbard model. We will focuson the three-dimensional (space) case, for the one-dimensional case is wellunderstood [8] and L¨uscher’s formula in one dimension [4] holds. We derivethe pertinent expressions using Bethe-Goldstone equations, and we beginby proving these via the introduction of a more general class of methodsconsisting of kernel subtractions. We then introduce the Hubbard modeland analyse, in first instance, its na¨ıve continuum limit. We then return tothe Hubbard model itself and show that, at low energy and large volume,the energy shifts are completely analogous to those obtained from the na¨ıvecontinuum limit, which shows once more how low-energy physics, at least inthe two-particle case, is universal.
2. Kernel subtractions and Bethe-Goldstone theory
For general interactions, one way of establishing the relation between thescattering states of a system (i.e. the positive energy states in the infinitesize limit) and the energy shifts in a finite box is to use Bethe-Goldstone(BG) theory [9] in vacuum, that is, without the static Fermi sea background.In fact, even if it looks very different from it, L¨uscher rederived, and used theexact same old Bethe-Goldstone’s equation in his seminal paper [4]. Here wewill relate BG theory to a much more general method – kernel subtractions –for solving homogeneous integral equations , and show that it corresponds toa particular choice of kernel subtraction. The method is easy to implement.Let | ψ i be the solution to the integral equation | ψ i = ˆ K ( E ) | ψ i , (1)where ˆ K ( E ) is the kernel (operator), with E some parameter, which in therelevant case here will correspond to an energy eigenvalue. The above equa-tion (1) is homogeneous and therefore only has a solution for particular valuesof E . Consider now a vector | ¯ ψ i that satisfies the following integral equation | ¯ ψ i = | F i + h ˆ K ( E ) − | F ih Γ | i | ¯ ψ i , (2)where | F i and h Γ | are arbitrary vectors. Then, for the values of the energyfor which h Γ | ¯ ψ i = 1, it holds that | ¯ ψ i = | ψ i . Here, integral equation is taken in the most general sense, and applies equally todiscrete kernels.
3e now particularize to the physically relevant case, and derive the BGequation. Let | ψ i satisfy the stationary Schr¨odinger equation H | ψ i = E | ψ i . (3)We assume the Hamiltonian is of the usual form H = H + V . Then, if E is not in the spectrum of H , the wave function satisfies the homogeneousLippmann-Schwinger equation | ψ i = G ( E ) V | ψ i , (4)where G ( E ) = ( E − H ) − . The kernel, Eq. (1), is therefore given byˆ K ( E ) = G ( E ) V . We choose the following vectors | F i and | Γ i , | F i = | k i , (5) h Γ | = h k | G ( E ) V, (6)where | k i is an eigenstate of H , i.e. H | k i = ǫ ( k ) | k i . Then, | ψ i satisfies | ψ i = | k i + [ G ( E ) V − | k ih k | G ( E ) V ] | ψ i , (7) E = ǫ ( k ) + h k | V | ψ i . (8)Eq. (7) together with the auxiliary condition (8) correspond exactly to BGtheory in operator form. In order to relate it to the finite-size reaction matrix,we introduce the identity 1 = P k | k ih k | and define V | ψ i ≡ ˆ r | k i . Afterstraightforward algebra, we find h k ′ | ˆ r | k i = h k ′ | V | k i + X q = k h k ′ | V | q ih q | ˆ r | k i E − ǫ ( q ) , (9) E = ǫ ( k ) + h k | ˆ r | k i . (10)
3. Hubbard model
We shall focus on one specific model, namely the Hubbard model on asimple cubic lattice. This is the minimal model for interacting electrons (orany other kind of spin-1 / H = − J X h i,j i ,σ = ↑ , ↓ c † iσ c jσ + U X i n i ↑ n i ↓ + 6 J N. (11)4bove,
J > h i, j i denotes nearest neighbours, with i, j ∈ Z , c iσ ( c † iσ ) are fermionic creation and annihilation operators, U is theon-site interaction strength, n iσ = c † iσ c iσ is the local number operator and N = P iσ n iσ is the total number operator (we have set the lattice spacing d ≡ ǫ ( k ) of the Hubbard model (11) isgiven by ǫ ( k ) = − J X α = x,y,z [cos( k α ) − , (12)where the quasi-momenta k α ∈ [ − π, π ), and the Brillouin zone is given byBZ ≡ [ − π, π ) . If the system is placed on a finite cube of side length L = L s (recall the lattice spacing d ≡
1) with periodic boundary conditions, thequasi-momenta are restricted to take values k α = 2 πn α /L (mod 2 π ), with n α ∈ { , , . . . , L − } .
4. Na¨ıve continuum limit
Before studying the Hubbard model in detail, it is very instructive toconsider its na¨ıve continuum limit. The bare continuum Hamiltonian is givenby H = X σ = ↑ , ↓ ~ m Z r ∈ [0 ,L ) d r ∇ ψ † σ ( r ) · ∇ ψ σ ( r ) + g Z r ∈ [0 ,L ) d r ψ †↑ ( r ) ψ †↓ ( r ) ψ ↓ ( r ) ψ ↑ ( r ) . (13)Above, m is the effective mass, g is the bare interaction strength, and ψ σ ( ψ † σ ) is the fermionic annihilation (creation) operator in the continuum.The above Hamiltonian, Eq. (13), is obtained as follows. Firstly, werestore the lattice spacing d (which was set to d = 1 in the previous section),in such a way that the action of the non-interacting Hamiltonian on a single-particle wave function, in the first quantisation, is given by( H ψ )( r ) = − J X α = x,y,z [ ψ ( r + ˆ e α d ) + ψ ( r − ˆ e α d ) − ψ ( r )] , (14)where r = ( x, y, z ) = ( n x , n y , n z ) d , with n α ∈ Z , and where we have definedthe unit vectors ˆ e α such that ˆ e α · r = α ( α = x, y, z ). The continuum limitof Eq. (14) is attained by setting J = ~ / md . The continuum limit of thebare interaction strength is attained by identifying U = g/d as d → et al. in ref.[6], and we will show that L¨uscher’s formula does change significantly whenthe lattice symmetry is respected in the regularisation process, unless specialcare is taken in the renormalisation process. In this subsection, we shall obtain the reaction matrix ( R -matrix), byregularising the contact interaction in such a way that the symmetry of theBrillouin zone is manifestly preserved at all steps. The regularised contact in-teraction V ( k , k ′ ) in the momentum representation for a spin-singlet fermionpair with ultraviolet (UV) cutoff Λ is therefore given by h k ′ | V | k i ≡ V ( k , k ′ ) = gθ (Λ , k ′ ) θ (Λ , k ) , (15)where we have defined θ (Λ , k ) = Y α = x,y,z θ (Λ − k α ) . (16)Above, θ ( q ) is the Heaviside step function. The above interaction, Eq. (15),has to be contrasted with the widely used spherically symmetric regularisa-tion [5, 6, 11], which reads gθ (Λ − k ).After separation of center of mass and relative coordinates, and at van-ishing center of mass momentum, the bare R -matrix ˆ R ∗ with the interactionin Eq. (15) is straightforward to calculate, and reads h k ′ | ˆ R ∗ ( E ) | k i = θ (Λ , k ′ ) θ (Λ , k )1 /g + I ( E ) / (2 π ) , (17)6here I ( E ) = P Z q ∈ [ − Λ , Λ) d q E − ~ q /m , (18)with P denoting Cauchy’s principal value. The UV structure of I ( E ), Eq.(17), can be extracted from its value at zero energy. Numerically, we haveobtained I (0) = − . . . . Λ + O (1 / Λ). The zero-energy on-shell R -matrix can be renormalised in favour of the scattering length a bymeans of minimal subtraction, i.e. by setting1 g = 1 g R − I (0)(2 π ) , (19)in which case the renormalised coupling constant takes the value g R =4 π ~ a/m . The R -matrix at arbitrary energy is renormalisable if I ( E ) − I (0)is finite for all energies E , that is, if the linearly divergent part of the integrals I ( E ) is energy-independent. To show that this is indeed the case, write theintegral I ( E ) as I ( E ) = P Z q< Λ d q E − ~ q /m + Z q ∈ [ − Λ , Λ) d q θ ( q − Λ ) E − ~ q /m . (20)The second integral on the right hand side of Eq. (20) can be calculated asa series expansion that is convergent for any finite energy. We expand theintegrand as θ ( q − Λ ) E − ~ q /m = − θ ( q − Λ ) ~ q /m ∞ X n =0 (cid:18) mE ~ q (cid:19) n . (21)The resulting integral in Eq. (20) is therefore given by I ( E ) = I (0) − m ~ ∞ X n =1 Z q ∈ [ − Λ , Λ) d q θ ( q − Λ ) k n q n +1) , (22)where k = mE/ ~ . The integral of the n -th term in the series is boundedas 0 ≤ Z q ∈ [ − Λ , Λ) d q θ ( q − Λ ) 1 q n +1) ≤ π (2 n − n − . (23)Therefore, the following bound holds0 ≤ Z q ∈ [ − Λ , Λ) d q θ ( q − Λ ) E − ~ q /m ≤ π Λ ∞ X n =1 (cid:18) k Λ (cid:19) n = 4 π Λ ( k/ Λ) − ( k/ Λ) . (24)7n the limit Λ → ∞ , the integral is bounded by zero from above and below,and it therefore converges to zero. Consequently, I ( E ) = I (0) and therenormalised R -matrix in this regularisation scheme, for finite energies, isconstant and identical to that obtained by means of spherically symmetricregularisation.There is a subtlety that needs to be considered in order to obtain mean-ingful results. The integer cutoff in lattice sums with L s lattice sites (whichwe consider odd for convenience) is λ = ( L s − /
2, which implies that thelattice site number-dependent real cutoff Λ( L s ) is given byΛ( L s ) = πd (cid:18) − L s (cid:19) . (25)The interaction must be renormalised by using the L → ∞ limit of the cutoff,Λ( ∞ ), since the limit L → ∞ is taken at the start of the calculation whendoing scattering theory, that is, the renormalisation prescription reads1 g = 1 g R − π ) πbd m ~ = 1 g R − m/ ~ (2 π ) Λ( ∞ ) , (26)where b = − . . . . . Obviously, in the limit L → ∞ , we recoverthe result from minimal subtraction. We place the system in a cubic box of side length L with periodic bound-ary conditions, at vanishing pair momentum K = 0. We will consider herethe lowest unbound state. From BG theory, Eq. (9) and Eq. (10), andsetting E = ǫ (0) + ∆ E = ∆ E , we obtain∆ E = gL + gL ∆ E X q =0 E − ǫ ( q ) , (27)where the sum is assumed to be regularised and is renormalised to any givenorder by using the renormalisation prescription for the bare coupling constant g , Eq. (26). To next-to-leading order in g/L , the energy shift reads∆ E ≈ gL − (cid:16) gL (cid:17) S ( L ) , (28)where we have defined S n ( L ) = X q =0 ǫ ( q )] n . (29)8n the above sums, S is regularised with a cutoff Λ( L s ) (given by Eq. (25),and will be seen to be renormalised by subtracting I (0)), calculated usingthe cutoff Λ( ∞ ), while for n > S n are regular and convergent. Ifwe write S ( L ) L = m (2 π ) ~ (cid:18)Z [ − Λ( L ) , Λ( L )] d q q + δ ( L ) L (cid:19) , (30)we obtain, to lowest order S ( L ) L ≈ m ~ (2 π ) (cid:16) b Λ( L ) + σL (cid:17) , (31)where σ ≈ − . E ≈ π ~ m aL h . aL i . (32)The above Equation (32) is in perfect agreement with L¨uscher’s formula [4],and emphasizes the importance of the subtlety regarding the renormalisationof the effective interaction in the naive continuum limit, as discussed in theprevious section.There are several advantages of working with Eq. (32) with respect to theway one obtains L¨uscher’s formula in the continuum limit of a cubic lattice [4]. Firstly, there is no need to project the interaction and the R -matrix ontotheir s -wave components, and since Eq. (32) only depends on the scatteringlength, this can be extracted with ease. Secondly, ”brute force” summations(i.e. direct computation of the sums, equivalent to exact diagonalisation)provide smooth convergence for all the sums S n ( L ) in Eq. (29), and equallyfor the energy-dependent sums in Eq.(27). This is to be contrasted withthe direct computation of the sum analogous to S ( L ) in the s -wave channel[4, 6]. In that case, the sums are restricted to q < Λ, while in the presentcase these are restricted to | q α | < Λ ( α = x, y, z ). In Fig. 1 we plot the valueof the sums s ( λ ) = X n =0 θ ( λ, n ) | n | − bλ, (33) s Luescher1 ( λ ) = X n =0 θ ( λ − | n | ) | n | , (34)as functions of the cutoff λ , where we have subtracted λ = λ ( L ) = λ ( ∞ ) − /
2, while the second sum is renormalised with the spherically-symmetric9
00 200 300 400 500-9.1-9.05-9-8.95-8.9
100 200 300 400 500-1.238-1.236-1.234-1.232100 200 300 400 500 s L u e s c h e r λ λ s Figure 1: Left: Blue dots are the numerically computed sums s Luescher1 as a function of theinteger cutoff λ (see Eq. (34)), and the dashed red line is its asymptotic value ( λ → ∞ )quoted in refs. [4, 6]. Right: Blue dots are the numerically computed sums s as a functionof λ (see Eq. (33)), and the dashed red line is the extrapolated value for λ → ∞ . prescription of ref. [6]. Notice that the standard result for L¨uscher’s sumis obtained here as − . − b/ − . . . . . As is clearly observedin Fig. 1, direct computation of the sums using projections of the interac-tions onto the s -wave channel gives rise to potentially high numerical errors.These would be inevitable in any computation relying on exact diagonalisa-tion techniques, and would result in poor estimates of the scattering length. In the unitary limit, i.e. for a → ±∞ , Eq. (32) is obviously incorrect.In the spherically symmetric case, Beane et al. [6] used pionless effectivefield theory to derive the finite-size correction to the ground state energywhen the scattering length is unnaturally large. This corresponds to thelimit a/L → ±∞ . Here, we derive the corresponding expressions for thecontinuum limit of the lattice model in two different cases: (i) Beane et al. ’scase ( a/L → ±∞ ) and (ii) the more natural case of a = O ( L ) in the limit L → ∞ .For unnaturally large scattering lengths, Eq. (27) gives∆ E = − "X q =0 E − ǫ ( q ) − . (35)Clearly, we have ∆ E = O (1 /L ) in this case, and therefore ∆ E is of thesame order as the dispersion ǫ ( q ) at low momenta. The energy shift in this10ase is given by ∆ E ≈ − π ~ mL [0 . . . . + O (1 /aL )] . (36)Again, the leading order term is identical to the one obtained in the spheri-cally symmetric case [6].In the case of large scattering lengths that scale as the length of thesystem, a = αL , with α = O (1), Eq. (27) gives∆ E = 4 π ~ αmL − π ~ αmL P q =0 [∆ E − ǫ ( q )] − . (37)Once more, ∆ E = O (1 /L ), and by setting ∆ E ≡ (2 π ) β/L , Eq. (37)becomes β = απ − ( α/π ) P n =0 ( β − n ) − . (38)The weak-coupling expansion in this case has the form β ≈ απ [1 − . α ] . (39)Comparing Eqs. (36) and (39), it is easy to observe that it is not alwayspossible to distinguish the cases (i) and (ii) referred to above, correspondingto unnaturally large and ”naturally” large scattering lengths, respectively,based on only one box size in the lowest energy shift. It can be tested, how-ever, by choosing two different lattice sizes without changing the interactionparameters.
5. Lattice case
In the previous section we introduced the analog of L¨uscher’s formula inthe weak-coupling [4] and unitary [6] limits for the na¨ıve continuum limitof the Hubbard model. In this section we study these expressions in theHubbard model itself.
The Hubbard model, Eq. (11), is a one-parameter theory, since thephysics of the model only depends on the ratio
U/J . At zero energy, its R -matrix gives the scattering length. It does, however, have non-zero shape11arameters and, of course, higher (even) partial waves, too. On the otherhand, since it only depends on U/J all shape parameters depend exclusivelyon the scattering length , and so do the energies. In this way, scatteringlengths can be extracted from energy shifts in finite lattices.The on-shell zero-energy R -matrix is trivially shown to take the value R = 11 /U − W (0) , (40)where W ( E ) = 1(2 π ) P Z BZ d q E − ǫ ( q ) . (41)Above, the energy dispersion becomes that of the Hubbard model, Eq. (12),for the relative motion of two particles at zero total momentum, i.e. ǫ ( q ) = − J X α = x,y,z [cos( q α ) − . (42)The zero-energy integral is known exactly and was computed by Watson [12].It has the numerical value J W (0) = − . . . . ≡ ω . Therefore, thescattering length is given by m ~ a = 4 π (cid:20) U − W (0) (cid:21) , (43)where m is the effective mass of the particles, related to the tunnelling rateby J = ~ / m ( d = 1). The energy shift of the lowest unbound state is readily obtained from BGtheory, Eq. (9) and Eq. (10), and reads in this case∆ E = U/L − UL P q =0 1∆ E − ǫ ( q ) . (44)In terms of the scattering length, Eq. (43), Eq. (44) becomes∆ E = 4 π ~ amL " πa (2 ω ) − π ~ amL X q =0 E − ǫ ( q ) − . (45) This is analogous to what happens with a hard-sphere interaction [1].
12n the weak-coupling limit, we therefore obtain∆ E ≈ π ~ amL " − πa (2 ω ) + 4 π ~ amL X q =0 E − ǫ ( q ) . (46)The sum on the right hand side of Eq. (46) can be estimated as follows.Consider a cubic region Ω = [ − η, η ] − { (0 , , } , with η > η ≪ π ). The sum can be split intothe two regions BZ − Ω and Ω. The sum over BZ − Ω can be approximatedby an integral, and therefore1 L X q =0 ǫ ( q ) ≈ π ) Z BZ − Ω d q ǫ ( q ) + 1 L X q ∈ Ω ǫ ( q ) . (47)In the region Ω, the full lattice dispersion can be approximated by ǫ ( q ) ≈ J q = (cid:18) πL (cid:19) J | n | . (48)setting η = (2 π ¯ N /L ), we obtain in the limit ¯ N → ∞ with ¯ N /L → X q ∈ Ω ǫ ( q ) ≈ (cid:18) L π (cid:19) J X n =0 θ ( ¯ N , n ) | n | = (cid:18) L π (cid:19) J (cid:2) b ¯ N + s (cid:3) , (49)where b = 15 . . . . and s = − . . . . , as calculated in theprevious section. The integral in the region BZ − Ω is easily calculated as Z BZ − Ω d q ǫ ( q ) ≈ Z BZ d q ǫ ( q ) − J Z Ω d q q = Z BZ ǫ ( q ) − (cid:18) L π (cid:19) b ( ¯ N + 1 / J . (50)Using the above results in Eq. (46), the energy shift becomes∆ E ≈ π ~ m aL h . aL i . (51)The above expression coincides with the corresponding formula in the na¨ıvecontinuum limit, Eq. (32), as expected. However, Eq. (51) has been obtainedwithout assuming the continuum limit at all energies, and it proves that thena¨ıve continuum limit – with its associated subtleties – provides an adequatedescription of low-energy scattering in the weak-coupling limit.The corresponding expressions in the unitary limits (both the natural andunnatural cases) are obtained in a fashion completely analogous to the oneleading to Eq. (51) and are identical to the expressions obtained using thena¨ıve continuum limit in the previous section.13 . Conclusions and outlook Besides purely theoretical implications, the results reported in this workhave interesting applications. For instance, two-particle scattering in a real-istic three-dimensional lattice is of great relevance for condensed matter andcold atomic systems in optical lattices, as it is necessary in order to renor-malise the coupling constant (
U/J ) in the Hubbard model. Unfortunately,the calculation of low-energy scattering properties of this system via directuse of collision theory is generally very involved, the reason being the genuinemulti-channel structure of these systems. One way to estimate the scatter-ing properties is to solve for the ground state energy of the realistic modelconsisting of only a few lattice wells (the number of wells being L s ). Withthe formulas derived in this article, one can fit the scattering length, andtherefore the interaction coupling constant, to match the energy shift in therealistic calculation in a small box. Generalisations of the current work tostudy, e.g. effective interparticle interactions in graphene [13] and graphene-like optical lattices [14], are possible by using the methods presented here. Inintrinsic graphene, as opposed to the cold atomic case and extrinsic graphene[15], however, the inclusion of the Coulomb interaction is necessary and theproblem is more complicated, but can be done by extending the methodsfor continuous theories [16]. In particular, the rigorous construction of thecontinuum quasi-relativistic (it is not Lorentz invariant) effective field theoryaround the Dirac points is still an open problem .It will be interesting to study the so-called Bertsch parameter [18], definedas the ratio between the ground-state energy of a resonantly interacting three-dimensional Fermi gas in the s-wave channel to the non-interacting groundstate energy, but for the Hubbard model itself. This can be done usingquantum Monte Carlo simulations [19, 20]. Given that the energy shift,considered here, for two fermions at the lattice resonance is the same as thatobtained from the pure s-wave results, it is to be expected that the latticeversion of Bertsch parameter will be identical, or similar, with its continuum,pure s-wave counterpart. Its calculation would actually be much less involvedsince the elimination of higher-partial waves and effective range, cleverly donein ref. [19], is not necessary at all. The two-body problem in the na¨ıve continuum limit has been studied in, e.g. ref. [17].There, however, the interaction is purely phenomenological. cknowledgements We thank L. G. Phillips for illuminating discussions. The authors ac-knowledges support from EPSRC grant No. EP/J001392/1 and from theDanish Council for Independent Research under the Sapere Aude program.
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