Zero-temperature equation of state of a two-dimensional bosonic quantum fluid with finite-range interaction

We derive the two-dimensional equation of state for a bosonic system of ultracold atoms interacting with a finite-range effective interaction. Within a functional integration approach, we employ an hydrodynamic parametrization of the bosonic field to calculate the superfluid equations of motion and the zero-temperature pressure. The ultraviolet divergences, naturally arising from the finite-range interaction, are regularized with an improved dimensional regularization technique.


Zero-temperature equation of state of a two-dimensional bosonic quantum fluid
2.1. Superfluid parametrization of the bosonic field. 30 We introduce the Euclidean Lagrangian L of a uniform quantum fluid of bosonic particles with mass m, described by the complex field ψ( r, t), namely [17] L =ψ( r, τ) h∂ τ −h 2 ∇ 2 2m − µ ψ( r, τ) whereh is Planck constant, µ is the chemical potential and we suppose that the particles interact with the isotropic two-body potential V(| r − r |). The imaginary time τ is introduced for uniformity with a functional integration approach, but real time t can be recovered in any moment performing the Wick rotation τ → it. According to the least action principle, the Euler-Lagrange equations of the system are obtained as the functional derivative of the action S[ψ, ψ], which reads where L D is the volume in D dimensions containing the particles and β = 1/(k B T), with T the absolute temperature and k B the Boltzmann constant. Until the end of this paper, for reasons connected to the dimensional regularization of the final results, we will not fix explicitly the spatial dimension to D = 2. The minimization of the action of Eq.
(2) gives the Gross-Pitaevski equation for the complex field ψ( r, τ), which constitutes the macroscopic wavefunction of the condensate [18]. In this work, however, we adopt a superfluid perspective through the following phase-amplitude parametrization of the bosonic field [19] ψ( r, τ) = ρ( r, τ) e iθ( r,τ) , where ρ( r, τ) = |ψ( r, τ)| 2 is a real field describing the system local density and θ( r, τ) is the phase field, which must be included due to the complex nature of the order parameter ψ( r, τ). This field transformation allows us to introduce the superfluid velocity v s , which is proportional to the gradient of the phase, namely v s =h m ∇θ.
We emphasize that the phase field θ( r, τ) is defined in the compact interval [0, 2π] and is therefore periodic of 2π: this fact constitutes the origin of many topological phenomena in condensed-matter physics. Indeed, here we focus on two-dimensional systems, where the singularities of the phase field -the vortices -are responsible for the Berezinski-Kosterlitz-Thouless (BKT) transition [20,21]. However, in the following we will study only the zero-temperature properties, for which the vortex-antivortex phenomenology does not play a fundamental role and can be neglected. We will then assume that the domain of definition of the phase field θ( r, τ) can be extended to R and that its spatial and time derivatives are well defined everywhere. In this case, the superfluid flow is irrotational, thus it has zero vorticity ∇ × v s = 0.
We now substitute the parametrization of Eq. (3) in the Lagrangian (1), obtaining where we omit for simplicity the dependence of the fields on their coordinates ( r, τ). The minimization of the action (2), which now becomes a functional of ρ and θ: S = S[ρ, θ], leads to the Euler-Lagrange equations for these fields. Recovering the real time t → −iτ, we get the hydrodynamic equations ∂ρ ∂t + ∇ · (ρ v s ) = 0 (7) and m ∂ v s ∂t   In the grand canonical ensemble we calculate the pressure of the bosonic fluid as P = −Ω/L D , where Ω is the grand potential and Z is the grand canonical partition function, which, within a functional integration perspective, can be calculated as To perform the explicit functional integration of the Lagrangian (6), we rewrite the local density field where ρ 0 is the condensate density of the system in the broken-symmetry phase and δρ( r, τ) is a real 41 field describing the density fluctuations. 42 We substitute the field transformation (11) in the Lagrangian of Eq. (6), obtaining where we keep only terms up to second order in the fluctuation fields δρ( r, τ) and θ( r, τ), thus making 43 a Gaussian (one-loop) approximation.
Considering the Lagrangian of Eq. (12) inside the action S, which now becomes the functional S = S[δρ, θ], it is particularly convenient to express it in terms of the Fourier series of the fluctuation fields, namely where ω n = 2πn/(βh) are the bosonic Matsubara frequencies. Notice that, since we are supposing that the phase field θ( r, τ) is defined on R, its Fourier components are non-numerable and can assume continuous values, thus they can be treated like ordinary functional integral variables. The action in the Fourier space is obtained by simply substituting these expressions in S and using the definition of the D + 1-dimensional delta function. Moreover, we also substitute the Fourier seriesṼ(k) of the real space interaction potential and we define with g 0 the zero-range interaction strength g 0 =Ṽ(k = 0). In this way, the action can be rewritten as the sum of two contributions The first is the action of the homogeneous system S 0 , namely which does not depend on the functional integration variables: using Eqs. (9) and (10) one can employ S 0 to calculate Ω 0 , the mean field contribution to the grand potential The second contribution to the action S is the Gaussian action S g , which is given by where, for simplicity of notation, we define δρ(±k) = δρ(± k, ±ω n ) and θ(±k) = θ(± k, ±ω n ). Since S g is quadratic in the fluctuation fields δρ(k) and θ(k), one can rewrite it in the following matricial form where M(k), the inverse of the propagator, is the 4 × 4 matrix The functional integral of the real fluctuation fields θ(k) and δρ(k) can be performed explicitly [23], obtaining the corresponding Gaussian grand canonical partition function Z g as which, considering the definition of the grand potential of Eq. (9), leads to the Gaussian contribution to the grand potential Here, we find the gapless excitation spectrum E k of the quantum fluid in the form where, within a perturbative approach, ρ 0 is determined by the saddle point condition ∂Ω 0 /∂ρ 0 = 0, which leads to and whose substitution in the excitation spectrum gives E B k , the renowned Bogoliubov spectrum [24] The sum over the Matsubara frequencies ω n in the Gaussian grand potential of Eq. (21) can be performed according to the prescriptions described in the Appendix, obtaining the grand potential as the sum of three contributions where Ω 0 = −L D µ 2 /(2g 0 ) due to Eqs. (16) and (23), and is the zero-temperature Gaussian grand potential encoding quantum fluctuations, while is the finite-temperature Gaussian grand potential, encoding thermal fluctuations. Finally, we explicitly write the zero-temperature equation of state, namely we calculate the pressure as the opposite of the grand potential of Eq. (25) at T = 0: In the thermodynamic limit of L → ∞, the sum over k can be rewritten as a D-dimensional integral in momentum space (2π/L) D ∑ k = d D k, and, substituting again the Bogoliubov spectrum (2.2), the equation of state becomes where the integral can be calculated after the explicit choice ofṼ(k). 45 2.3. Explicit implementation for finite-range interaction. 46 We now provide an explicit implementation of the zero-temperature equation of state (29) for a bosonic quantum fluid of particles interacting with the finite-range effective interactioñ where g 0 =Ṽ(k = 0) is the usual zero-range interaction coupling, and is the first nonzero correction in the gradient expansion of an isotropic interaction potential V(| r|). At zero temperature, we expect that the finite-range corrections to the equation of state are detectable, but small with respect to the zero-range result of Ref.
[25]. By using scattering theory in two spatial dimensions, these couplings can be linked with the s-wave scattering length a s and the characteristic range R of the real interatomic two-body interaction [12,26,27] where n is the number density of the system in D = 2.

47
The equation of state (29) becomes, with the finite-range interaction of Eq. (30) where we define the zero temperature Gaussian pressure P (0) g as with Since the integrand function depends only on the modulus of the momentum | k|, we rewrite the integral in P (0) g using D-dimensional spherical coordinates, namely where S D = 2π D/2 /Γ[D/2] is the solid angle in D-dimensions and Γ[D/2] is the Euler Gamma function. In order to integrate this equation, we introduce the adimensional variable t = h 2 k 2 λ/(4mµ), obtaining As a consequence of the substitution of the real interatomic potential with an effective interaction, the zero-temperature Gaussian pressure P (0) g is ultraviolet divergent. In our framework, an efficient way to regularize P (0) g is constituted by the technique of dimensional regularization [28]. The basic idea of this approach is to rewrite a diverging integral in terms of the Euler beta and gamma functions, whose integral representation for x, y, z > 0 is given by Thank to the properties B(x, y) = Γ(x)Γ(y)/Γ(x + y) and Γ[z + 1] = z Γ[z], one can extend the domain of definition of the gamma and beta functions by analytic continuation of their arguments x, y, z also to negative values, which usually appear in many physical problems. However, despite this dimensional regularization procedure can be successfully used to regularize many ultraviolet diverging integrals, in our peculiar two-dimensional case the procedure described above would lead to a result containing the gamma function evaluated for negative integer values, which is again a diverging quantity. To avoid this residual divergence, we extend the dimension of the system to the complex value D = D − ε, and we formally perform the integration of Eq. (37). We obtain in which the wavevector κ is introduced for dimensional reasons. Notice how in D = 2 and for ε = 0 the Gaussian pressure is still divergent. To regularize it, we rely on the following small-ε expansion of the gamma function [29] where Ψ(n + 1) is the digamma function and Ψ (n + 1) is its derivative. Moreover, we express the exponentiation of a generic coefficient x ε for ε → 0 as With this recipe, the Gaussian pressure P (0) g in D = 2 gives 2π 3/2h 2 λ 3/2 π 1/2 2 where γ ≈ 0.55722 is the Euler-Mascheroni constant. Finally, we delete the o(ε −1 ) divergence in the square bracket [30] and we rewrite the zero-temperature equation of state P(µ, T = 0) of Eq. (29) as where we define the energy cutoff 0 as

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We illustrate here the procedure to calculate the summation over the bosonic Matsubara frequencies ω n , which are defined as where n ∈ Z are integer numbers. The most common sum that one has to perform is in the form Using the properties of the logarithm and considering that the summation involves all n ∈ Z integers, both positive and negative, I[ξ k ] can also be rewritten in the useful form Taking the derivative of I[ξ k ] with respect to ξ k in the Eq. (47) we get In the zero temperature limit, the difference becomes infinitesimal and we can substitute the sum over n with an integral over ω, obtaining which is the zero-temperature contribution to I[ξ k ]. If the temperature is relatively low, but non-zero, we cannot substitute the sum in Eq. (49) with an integral, but we can rewrite it as we obtain We integrate this equation on ξ k and, setting the arbitrary constant resulting from the indefinite integral to zero (it is not dependent on physical parameters), we finally obtain the result of the summation over the Matsubara frequencies which is used in this article to obtain Eq. (21).