Dimensional crossover in ultracold Fermi gases from Functional Renormalisation
Bruno M. Faigle-Cedzich, Jan M. Pawlowski, Christof Wetterich
DDimensional crossover in ultracold Fermi gases from Functional Renormalisation
Bruno M. Faigle-Cedzich, Jan M. Pawlowski,
1, 2 and Christof Wetterich Institute for Theoretical Physics, Heidelberg University, D-69120 Heidelberg, Germany ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung mbH, D-64291 Darmstadt, Germany (Dated: October 17, 2019)We investigate the dimensional crossover from three to two dimensions in an ultracold Fermigas across the whole BCS-BEC crossover. Of particular interest is the strongly interacting regimeas strong correlations are more pronounced in reduced dimensions. Our results are obtained fromfirst principles within the framework of the functional renormalisation group (FRG). Here, theconfinement of the transverse direction is imposed by means of periodic boundary conditions. Wecalculate the equation of state, the gap parameter at zero temperature and the superfluid transitiontemperature across a wide range of transversal confinement length scales. Particular emphasis isput on the determination of the finite temperature phase diagram for different confinement lengthscales. In the end, our results are compared with recent experimental observations and we discussthem in the context of other theoretical works.
I. INTRODUCTION
Lower-dimensional systems are of particular interestboth in condensed matter and statistical physics as theyfeature a pronounced influence of fluctuations. Further-more, they are of experimental and technological impor-tance with examples ranging from high temperature su-perconductors over layered semiconductors to graphene.To disentangle the effects of the dimensionality fromother many-body physics effects constitutes a key chal-lenge in the study of systems of reduced dimensionality.With the recent progress in trapping ultracold atomicgases in quasi-two-dimensional geometries [1, 2] both zero[3–5] as well as finite temperature effects [3, 6–11] havebeen measured. Hereby, strongly anisotropic trappingpotentials on the one hand and one-dimensional opticallattices one the other hand allow for the experimentalrealisation of quasi-two-dimensional quantum gases.For example, the algebraic correlations associated withthe BKT phase transition in (quasi-) two-dimensionalsystems have been observed in bosonic [9, 12–16], as wellas fermionic systems [6, 17]. In addition, (quasi-) two-dimensional systems exhibit the breaking of the scale in-variance in the strongly interacting regime of the BCS-BEC crossover. Here, extensive progress both in the-ory [18–25], as well as in experiment [26–28] has beenachieved in recent years.Due to an insufficient degree of anisotropy in the exper-imental setup one may not be restricted to a particulardimension, but find oneself in a dimensional crossoverwithout a well-defined dimensionality. Apart from be-ing an undesired effect for the investigation of pure two-dimensional systems, the crossover may also lead to newmaterials with physically interesting properties.We concentrate here on ultracold Fermi gases. A com-parable quasi-two-dimensional setup has been studied in[29, 30] in a mean-field approach, for a Fermi gas at uni-tarity and zero temperature in [31], using the Luttinger-Ward approach in two dimensions in [32] and using QMCcalculations in two dimensions in [33]. Furthermore, two-dimensional fermionic systems have been addressed in [34–36]. Fermi gases, typically a system of Li or K,constitute a rich physical system as their interatomic in-teractions may be altered via a Feshbach resonance. Fora large negative value of the three-dimensional scatteringlength a the fermions form large spatially overlappingCooper pairs below a critical temperature (BCS limit).On the other hand, for large positive scattering lengths,the fermions form tightly bound molecular dimers whichcondense into a Bose-Einstein condensate (BEC) at suf-ficiently low temperatures. The BCS-BEC crossover hasbeen studied extensively in three dimensions using func-tional renormalisation group (FRG) techniques [37–51].Finite size effects have been investigated in both coldatom systems [31, 52] and in QCD [53–55].In this work we study the dimensional crossover fromthree to two spatial dimensions for ultracold Fermi gasesby means of the functional renormalisation group, fora study of non-relativistic bosonic systems see [56]. Inparticular, we are interested in the critical temperaturefor the superfluid transition over the BCS-BEC-crossoverin dependence of the dimensionality.The dimensional crossover is achieved by compacti-fying the ‘transverse’ z -direction by a potential well oflength L . We discuss (anti-)periodic boundary condi-tions, as well as a confinement to a box with bound-aries fixed to zero. The compactification leads to a dis-crete momentum spectrum in z -direction. The choice ofthe boundary conditions is crucial for a well-defined two-dimensional limit. It also influences the mapping betweenthree- and two-dimensional parameters of the Fermi gas.Both aspects are discussed in detail in Section II C.This paper is organised as follows: In Section II we in-troduce the ultracold Fermi gas and the functional renor-malisation group (FRG) in the dimensional crossover. Inparticular, we discuss the aspect of boundary conditionsfor a dimensional reduction. The truncation used withinthe FRG, as well as the initial conditions are presented inSection III. In Section IV the results for the equation ofstate and the gap parameter in the dimensional crossoverat zero temperature are discussed. The finite tempera-ture phase diagrams with respect to the dimensionality a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t are addressed in Section V. We conclude in Section VI.Some technical details are deferred to Appendix A-C. II. MODEL AND FUNCTIONALRENORMALISATIONA. Model
Close to a broad Feshbach resonance, as found in quan-tum gases consisting of Li and K, details of the atomicinteractions in ultracold Fermi gases become irrelevantfor the description of the macrophysics. The system canthen be described by a universal action S [ ψ ] = (cid:90) X (cid:34) (cid:88) σ =1 , ψ ∗ σ ( ∂ τ − ∇ − µ ) ψ σ + λ ψ ψ ∗ ψ ∗ ψ ψ (cid:35) , (1)where ψ σ and ψ ∗ σ denote Grassmann fermions in the hy-perfine state σ = 1 ,
2. We introduce X = ( τ, (cid:126)x ) with τ being the Euclidean time and (cid:82) X = (cid:82) β dτ (cid:82) d d x with spa-tial dimension d . Moreover, the chemical potential µ andthe four-Fermi coupling λ ψ → λ ψ = 8 π a are related tothe physical chemical potential and the scattering lengththrough an appropriate vacuum renormalisation.We use (cid:126) = k B = 2 M = 1 with M being the mass ofthe fermionic atoms. For sufficiently low temperatures,the ultracold Fermi gas may develop many-body instabil-ities resulting in the formation of a macroscopic anoma-lous self-energy ∆ which is related to the non-vanishingexpectation value (cid:104) ψ ψ (cid:105) . This is signalled by a diver-gence of the frequency- and momentum-dependent four-Fermi vertex at lower momentum scales and causes thebreaking of the global U (1)-symmetry. In particular, inthe strongly coupled regime, i.e. for a diverging three-dimensional s-wave scattering length a , the quantita-tive determination of this phase transition is complicatedby the frequency and momentum dependence of the ver-tex. In oder to resolve this difficulty, a scale-dependenttreatment in the path integral formulation is appropriate.The starting point is the grand canonical partitionfunction of the system Z [ η, η ] = (cid:90) D ψ D ψ e − S [ ψ,ψ ] − η · ψ + ψ · η . (2)In order to exclude redundancies included in the grandcanonical partition function, the effective action may beintroduced as the Legendre transform of the Wigner func-tional W [ η, η ] = log Z [ η, η ]Γ[ ψ, ψ ] = (cid:90) X (cid:0) ψ X η X + η X ψ X (cid:1) − W [ η, η ] . (3) B. Functional renormalisation
For the scale-dependent treatment the integration isgrouped in frequency and momentum shells according to q + (cid:0) (cid:126)q − µ (cid:1) (cid:39) k (4)with external momentum scale k . The full grand canon-ical partition function is obtained by successively inte-grating over the corresponding frequency and momentumshells starting at k = ∞ and arriving in the end at k = 0.The microscopic action in (1) is related to an ultravi-olet (UV) momentum scale k = Λ at length scales muchsmaller than the van der Waals length (cid:96) vdW . However,the relevant physics takes place at scales (cid:28) Λ where thethermal and quantum fluctuations are included. To in-corporate these fluctuations and furthermore to obtainresults in the strongly coupled regime the above scale-dependent procedure is implemented via the functionalrenormalisation group (FRG), which includes these fluc-tuations successively at each momentum scale k . Intro-ducing the scale-dependent partition function Z k [ η, η ] = (cid:90) D ψ D ψ e − S [ ψ ] − ∆ S [ ψ ] − η · ψ + ψ · η (5)the suppression of the low momentum fluctuations ω, (cid:126)q (cid:28) k is incorporated via a mass-like infrared modifi-cation of the dispersion relation. In practice, a regulatoror cutoff term ∆ S k [ ψ ] is added to the microscopic action S [ ψ ] being quadratic in the fields∆ S [ ψ ] = (cid:90) Q (cid:88) σ =1 , ψ σ ( − Q ) R ψ ( Q ) ψ σ ( Q ) . (6)The regulator R k ( Q ) may be chosen freely with the re-quirementslim q /k → R k ( Q ) = k , lim q /k →∞ R k ( Q ) = 0 . (7)The scale-dependent effective action Γ k can be definedaccordingly. Starting at Γ Λ = S , the full effective actionis reached after the inclusion of all fluctuations where Γ k smoothly interpolates between the microscopic action Γ Λ and the full effective action Γ k =0 = Γ. Each infinitesimalchange of the average effective action is described by aflow equation ∂ k Γ k depending on the correlation functionof the theory and a way how to suppress infrared modeswith momenta smaller than k . In the end fluctuationswith large wavelengths are included. Since the functionalrenormalisation group includes the fluctuations stepwise,there are no infrared divergences when approaching theinclusion of long wavelength modes. Analogous to defin-ing the quantum theory by means of the classical actionin the path integral formulation, the initial effective ac-tion Γ Λ together with the flow equation (8) determinesthe full quantum theory.The infinitesimal change of the effective action Γ k withrespect to the momentum scale k is governed by the flowequation [44, 57–65] ∂ k Γ k = 12 STr (cid:104) (Γ (2) k + R k ) − ∂ k R k (cid:105) , (8)where Γ (2) k is the second functional derivative of of Γ k with respect to the fields. As the flow equation (8) is anintegro-differential equation, its full solution is in mostcases out of reach. One therefore relies on approximationschemes to the full effective action Γ k which should in-corporate the examined physics already at lower order ofthe approximation and reduce the number of flow equa-tions to a manageable set of couplings. Furthermore, itis convenient to rewrite the four-Fermi interaction λ ψ ata large cutoff Λ in terms of a bosonic degree of freedom φ via a Hubbard-Stratonovich transformation.In this work we choose a three-dimensional Litim-typeregulator [66–68] for the cutoff function R ( Q ) in threespatial dimensions. It is given for bosons and fermions,respectively, by R φ,k ( q ) = (cid:0) k − q / (cid:1) θ (cid:0) k − q / (cid:1) ,R ψ,k ( q ) = k (sgn ( z ) − z ) θ (1 − | z | ) , (9)where θ ( x ) represents the Heaviside-Theta function,sgn( x ) the sign function and we used z = ( q − µ ) /k .Note, that only spatial momenta q = | (cid:126)q | are regularisedfor this type of regulator. However, a particular neatproperty of (9) is that the finite temperature Matsubarasums can be performed analytically. C. Function space and boundary conditions
The choice of the boundary conditions plays a crucialrole in arriving at the correct two-dimensional physics.The dimensional crossover is implemented by compact-ifying the ‘transverse’ z -direction by a potential well oflength L , V box ( z ) = (cid:40) ≤ z ≤ L ∞ else . (10)One may choose (anti-)periodic boundary conditions ψ ( x, y, z = 0) = ± ψ ( x, y, z = L ) , (11)or restrict oneself to a box ψ ( x, y, z = 0) = ψ ( x, y, z = L ) = 0 . (12)The compactification leads to a discrete momentum spec-trum in z -direction. For periodic boundary conditionsthe respective energies, E z = (cid:126) q z M , are discrete with q z → k n k n = 2 π nL , n ∈ Z , (13) which includes a zero mode k = 0 with vanishing en-ergy E min = 0. In turn, for anti-periodic boundary con-ditions one finds k n = (2 n + 1) π/L with n ∈ Z andwith a lowest mode | k | = π/L with a finite energy E min = (cid:126) π / (2 M L ). Finally, confining the Fermi gasinside a box leads to k n = π n/L with a vanishing energy E min = 0.The non-vanishing zero point energy for anti-periodicboundary conditions results in a gap in the evaluation ofthe the (discrete) mode sum at zero temperature. Conse-quently, anti-periodic boundary conditions do not yieldthe two-dimensional limit for vanishing length L → L with theinverse temperature 1 /T in the evaluation of the discretemode sum at zero temperature. As a result, T = 0 and L = L gives the same result as T = 1 /L and L = 0,i.e. the zero length limit L → T = 0 corresponds to the limit of infinite temperature T → ∞ at zero length L = 0. For a non-relativistic sys-tem the situation is less simple, since the dispersion re-lation allows no clear mapping between the temperatureand the length of the system. Nevertheless, it is clearthat anti-periodic boundary conditions do not admit atwo-dimensional limit for L → L →
0. Since all modes with n (cid:54) = 0 have for L →
0a large gap they can be integrated out. In general, thethree-dimensional system with finite L can be viewed as atwo-dimensional system with infinitely many fermions as“modes”, one for each n . Integrating out the modes with n (cid:54) = 0 reduces the system to a single two-dimensionalfermion, the one for n = 0.The map from the three-dimensional system to thetwo-dimensional system proceeds by integrating out the n (cid:54) = 0 modes. This maps the parameters of the three-dimensional theory to the ones of an effective two-dimensional theory. For L → µ or the scattering length a . This canlead to shifts in fractions including ε F and T F , as wellas in the crossover parameter. For experiment, thethree-dimensional quantities are generally the ones avail-able, and we will typically use them for our discussion.When comparing to results obtained from computationsin two-dimensions, the matching between three- and two-dimensional parameters becomes important, however. Inthe present paper we do not deal with this issue, butthe reader should keep it in mind when comparing withtwo-dimensional results.Experimentally realistic confinement potentials, usedin most ultracold atom experiment, such as [6] and [10],are implemented by using harmonic trapping potentials.Here, the function space consists of Hermite polynomials.Heuristically, our choice is a limiting case. In particular,observables that do not show an impact of the differentboundary conditions studied here should be the same forthe harmonic trap. D. Dimensional reduction
In order to obtain a system within the dimensionalcrossover from three to two dimensions we initialise therenormalisation group (RG) flow at ultraviolet cutoffscale k = Λ where the effective action Γ Λ coincides withthe microscopic action of a three-dimensional ultracoldFermi gas. By delimiting the z -direction of the systemvia a potential well of length L we introduce an additionalscale to the three-dimensional system. By following theRG as a function of k for a given length scale L one ob-serves that the contribution of modes with k n (cid:29) k issuppressed by powers of k /k n . These modes decoupleand effective dimensional reduction is achieved automat-ically once k (cid:28) π/L . This is very similiar to the effec-tive dimensional reduction in finite temperature quantumfield theory realised by solutions of the flow equations[69]. Following the RG from k = Λ to k = 0 the flow al-ways makes a transition from a three-dimensional regimeto a two-dimensional one. For this purpose the UV scaleis always chosen such that Λ (cid:29) ( L − , µ / , T / ). Theflow equations become effectively two-dimensional for k (cid:28) π/L , while the physical system is effectively two-dimensional if L − is much larger than all other many-body scales [56].To incorporate the effects of the compactification intransversal z -direction given by the potential well in (10),the regulators in (9) are modified according to (cid:126)q = ˆ q + q z → ˆ q + k n , (14)where k n is chosen according to the boundary conditionsand ˆ q denotes the square of the x - and y -components ofthe momentum. III. RUNNING OF COUPLINGSA. Truncation
After a Hubbard-Stratonovich transformation the fullmicroscopic action is given by S = (cid:90) X (cid:34) (cid:88) σ =1 , ψ ∗ σ (cid:0) ∂ τ − ∇ − µ (cid:1) ψ σ + m φ φ ∗ φ − h ( φ ∗ ψ ψ − φ ψ ∗ ψ ∗ ) (cid:35) , (15)with λ ψ = − h /m φ , which can be seen via a Gaussian in-tegration over the bosonic field φ . The Feshbach coupling h accounts for the interconversion of two fermionic atoms ψ with different spin to a bosonic dimer φ . Connectingthe above action to the experimental setup we explicitly introduce the closed channel via the bosonic field φ . Thephysical detuning ν = ν ( B ), which depends on the exter-nal magnetic field of the trap in the experiment, denotesthe distance of the closed-channel bound state from thescattering threshold. In the kinetic term of the bosonicdimer φ the factor of ∇ / S [ ψ ∗ , ψ, φ ∗ , φ ] = (cid:90) X (cid:20) ψ ∗ (cid:0) ∂ τ − ∇ − µ (cid:1) ψ + φ ∗ (cid:18) ∂ τ − ∇ ν − µ (cid:19) φ − h ( φ ∗ ψ ψ − φ ψ ∗ ψ ∗ ) (cid:21) . (16)Our ansatz for the effective average action can be dividedinto a kinetic part and an interaction partΓ k = Γ kin + Γ int . (17)The kinetic part in terms of the renormalised fields ψ = A / ψ ψ and φ = A / φ φ describes the fermion and bo-son dynamics and is given byΓ kin [ ψ, φ ] = (cid:90) X (cid:88) σ = { , } ψ ∗ σ (cid:0) S ψ ∂ τ − ∇ − µ (cid:1) ψ σ + φ ∗ (cid:18) S φ ∂ τ − V φ ∂ τ − ∇ (cid:19) φ (cid:21) . (18)We normalised the coefficients of the gradient terms bymeans of the wave function renormalisations A ψ and A φ which enter the renormalisation group flow via theanomalous dimensions η ψ = − ∂ t log A ψ , η φ = − ∂ t log A φ . (19)Due to the renormalisation of the fields the expectationvalue ∆ = ( h ρ ) / can be non-zero, even in the two-dimensional limit [56], where the Mermin-Wagner the-orem [70, 71] forbids true long-range order. However,algebraically decaying correlation functions with a non-vanishing superfluid density can be found [72–75].The interactions can, after the Hubbard-Stratonovichtransformation, be written asΓ int [ ψ, φ ] = (cid:90) X (cid:20) U ( φ ∗ φ ) − h ( φ ∗ ψ ψ − φ ψ ∗ ψ ∗ ) (cid:21) . (20)The effective average potential U ( ρ ) depends only on the U (1)-invariant quantity ρ = φ ∗ φ and describes bosonicscattering processes. The U (1)-symmetry is sponta-neously broken for a non-zero minimum ρ of the effectiveaverage potential and thus describes superfluidity. In aTaylor-expansion we write U ( ρ ) = m φ ( ρ − ρ ) + λ φ ρ − ρ ) + N (cid:88) n =3 u n n ! ( ρ − ρ ) n . (21)where we need to include at least up to the second orderin ρ to reproduce the second order phase transition tosuperfluidity. In the symmetric regime we therefore have ρ = 0 and positive bosonic mass m φ >
0, whereas thesymmetry-broken regime is realised for ρ > m φ = 0. In the following we restrictthis work to order φ .The truncation can be classified by the diagrams in Fig.1 included on the right hand side of the flow equation(8). By including only fermionic diagrams (F) we arriveat the mean-field result. Bosonic fluctuations enter theflow equation by including diagrams with two internalbosonic lines (B). ∂ t ( ) − = F + B Figure 1: (F)- and (B)-truncation schemes of the flow equa-tions. The flow of the inverse boson propagator incorporatesboth fermionic and bosonic diagrams. Bosonic propagatorscorrespond to dashed and fermionic propagators to solid lines,while the distinct vertices are shown in different shapes. Theregulator insertion is denoted by a cross.
Furthermore, the flow of the density of the Fermi gasis calculated via a derivative of the effective action withrespect to the chemical potential ∂ k n k = − ∂ k ∂ U ( ρ ) ∂ µ . (22)In practice, we approximate the dependence of the ef-fective average action on the chemical potential by anexpansion in ρ and µ [45] U ( ρ ) = (cid:88) n =1 u n n ! ( ρ − ρ ) n − n k δµ + α k ( ρ − ρ ) δµ . (23)Here the chemical potential is split into a reference part µ and an offset δµ , such that µ = µ + δµ . B. Initial conditions and universality
In three dimensions the running couplings approachfixed points in the renormalisation group flow of theFermi gas. As a result, the macrophysics (on the lengthscales of the inter-particle spacing) becomes independent of the microphysics (on the molecular scales) to a largeextent, cf. e.g. [45, 47].When reaching the fixed points the system loses itsmemory of the microphysics with its initial conditions.Consequently, the initial conditions of the running cou-plings are irrelevant and we may essentially start at thefixed point values in the ultraviolet. Even if we had notdone so, they would be immediately generated.An exception constitutes the bosonic mass term m φ whose fixed point is unstable towards the infrared.Hence, for the effective potential we set as initial con-dition in the ultraviolet U Λ ( ρ ) = ( ν Λ − µ ) ρ . (24)Herein the chemical potential µ can be artificially splitinto a vacuum component µ v and a many-body contri-bution µ mb such that the vacuum part µ v equals half thebinding energy of a bosonic dimer ε B / ν Λ is related to the physicaldetuning via an appropriate vacuum renormalisation [45].Since the RG flow for a system in reduced dimensionis initialised at an UV scale where the Fermi gas is de-scribed by the three-dimensional classical action, theseconsiderations can be applied to the study of systems in-side the dimensional crossover. We therefore choose thefixed point values of the three-dimensional Fermi gas asour initial conditions. IV. DIMENSIONAL CROSSOVER AT ZEROTEMPERATURE
The flow equations underlying the results at zero andat finite temperature shown below are obtained analyti-cally with periodic boundary conditions for both bosonicand fermionic fields inside the potential well. They aregiven in Appendix A. Imposing anti-periodic boundaryconditions for fermionic fields ψ ( x ) we find, as expectedin Section II C, that for small confinement length scales L √ µ mb ∼ (cid:0) ρ h (cid:1) / /ε F on the final scale of the RG-flow t f ina system with reduced dimensionality. This dependenceon t f starts at a confinement length of L √ µ mb (cid:39)
10 andfollows through until we arrive at the two-dimensionallimit.In order to display the the confinement in transver-sal direction we introduce the dimensionless lengthparamenter L √ µ mb of the potential well, where µ mb = µ − ε B / ε B / - - Figure 2: Comparison of the equation of state at zero tem-perature for different confinement length scales and the threedimensional case with respect to the 3D crossover parame-ter 1 / ( k F a ). From top to bottom: 3D limit (solid-red), L √ µ mb = 1000 (dashed-black), L √ µ mb = 2 (dashed-blue), L √ µ mb = 1 (dashed-red), L √ µ mb = 0 . L √ µ mb . At zero temperature a reduction of the dimensionlessconfinement length parameter L √ µ mb leads to an in-creased density and thereby to an increased Fermi en-ergy ε F = k F . As a consequence the equation of state( µ − ε B / /ε F in Figs. 2 and 3 is lowered for more con-fined systems.Here the Fermi momentum is calculated using thethree-dimensional definition k F = (3 π n ) / as the ini-tial condition for the flow of the density is explicitelygiven for a three-dimensional system. This means thatthe Fermi momentum k F of the (quasi-) two-dimensionalsystem has to be calculated by using the functional formgiven in the ultraviolet, where the reduced dimension en-ters via the flow of the density.In Fig. 2 the equation of state is shown as afunction of the three-dimensional crossover parameter c − = ( k F a ) − , which can be interpreted as the in-verse concentration of the Fermi gas. For large confine-ment length scales L √ µ mb the three-dimensional resultis recovered, while the equation of state in dependence ofthe transversal extension starts to saturate only at theorder of L √ µ mb = 10 − for a two-dimensional limit.For better comparison to experiment the equation ofstate is also displayed in Fig. 3 with respect to the two-dimensional crossover parameter ln ( k F a ). Here the(quasi-) two-dimensional scattering length a is calcu-lated by [56] a (pbc)2D = L exp (cid:26) − La (cid:27) (25)for our setup with periodic boundary conditions. Com-paring the result in 2 with the experimental data found in[3] for a (quasi-) two-dimensional setup we find a qual-itatively good agreement. Especially on the BEC-side, - - Figure 3: Comparison of the equation of state for differentconfinement length scales to the experimental data from [3]with respect to the 2D crossover parameter ln( k F a ). Herebe show: L √ µ mb = 9 (blue), L √ µ mb = 6 (red), L √ µ mb =2 . T /T F ≈ .
05 on the BEC-side and
T /T F ≈ . where the measurements were obtained in the superfluidphase, the equation of state for lower values of the con-finement length L √ µ mb our result yields the correct be-haviour. However, on the BCS-side the equation of statefor confinements L √ µ mb (cid:46) h ρ ) / with re-spect to the Fermi energy ε F in Fig. 4 for different con-finement length scales one finds a flattening of the curvefor lower dimensionality, while the three-dimensional caseis recovered for large length scales L √ µ mb . Interestingly,the gap saturates much faster for small length scales, al-ready around L √ µ mb (cid:39) . a , regions of an increased gap ∆ /ε F canbe found at intermediate length scales within the dimen-sional crossover. This dip-like structure is a characteris-tic of the modes given by the boundary conditions chosenand is also found at finite temperature. - - Figure 4: Comparison of the gap parameter for different con-finement length scales and the three dimensional case withrespect to the 3D crossover parameter 1 / ( k F a ). From topto bottom: 3D limit (solid-red), L √ µ mb = 1000 (dashed-black), L √ µ mb = 2 (dashed-blue), L √ µ mb = 1 (dashed-red), L √ µ mb = 0 . L √ µ mb . V. SUPERFLUID TRANSITIONA. Dimensional crossover of the criticaltemperature
At finite temperature we study the behaviour of thecritical temperature T c /T F with respect to the spa-tial extension in transversal z -direction L √ µ mb . TheFermi temperature T F = k F is, as in the zero tem-perature case, calculated using the three-dimensional re-lation between the Fermi momentum and the density k F = (3 π n ) / . The order parameter for the superfluidtransition is the (finite-temperature) gap ∆ = ( h ρ ) / .As shown exemplary for a − = 0 in Fig. 5 one canidentify a dimensional crossover from three to two dimen-sions for all values of the three-dimensional scatteringlength. The limiting case of three dimensions is reachedfor large confinement scales L √ µ mb . Moreover, a dis-tinct two-dimensional limit is obtained where the criticaltemperature in units of the Fermi temperature saturatesand is significantly reduced with respect to the three-dimensional case.Furthermore, one can clearly discern dips in the di-mensional crossover of the critical temperature where wefind an increased T c /T F at intermediate stages betweenthe two- and three-dimensional limit. Interestingly, theirappearance and amplitude seem to be related to the scat-tering length a chosen in the ultraviolet. Moreover, wefind a larger amplitude for more confined systems. Thisbehaviour is caused by the mode structure of a confinedsystem specified by the chosen boundary conditions. Asa consequence, the density of states for a confined systemhas a step-like structure and the dips can be found at thepostions of the discontinuities. The dip structure for the Figure 5: Critical temperature T c /T F as a function of theconfinement length scale L √ µ mb at an exemplary three di-mensional fermion scattering length of a − = 0. Similar plotscan be found for different scattering lengths with the differ-ence being the amplitude and the position of the dips. Theseresult from the mode structure caused by the chosen bound-ary conditions and are related to the step-like structure of thedensity of states for a confined system. Similiar dips were alsofound in a mean-field analysis with a harmonic confinement[29]. critical temperature T c /T F emerge at the same confine-ment length scales L √ µ mb as for the zero temperaturegap parameter ∆. In a mean-field analysis with a con-finement in transversal z -direction induced by a harmonicpotential on the weakly-interacting BCS-side of the BCS-BEC crossover a similiar dip-like structure of the criticaltemperature was found [29]. B. Finite temperature phase diagram
In Figs. 6 and 7 the critical temperature T c /T F asa function of the three dimensional inverse concentra-tion c − = ( k F a ) − and the two-dimensional crossoverparameter ln( k F a ) is shown for different confine-ment length scales over the whole BCS-BEC crossover.The phase diagram in Fig. 6 approaches the three-dimensional limit for large confinement length scales,while the critical temperature is reduced for lower di-mensionality over the BCS-BEC crossover. On the otherhand, we find an increased critical temperature on theBCS-side of the crossover around L √ µ mb = (0 . . . . T c /T F continues to be reduced for moreconfined systems.In Figs. 7 and 8 we find the expected exponen-tial decrease on the BCS-side of the crossover, whereln( k F a ) (cid:29)
1, for small confinement scales in a quasi-two-dimensional geometry. Here it was found [76] that T c T F = 2 e γ π k F a (26) - - Figure 6: Phase diagram in terms of T c /T F for different con-finement length scales and the three dimensional case withrespect to the 3D crossover parameter 1 / ( k F a ). From topto bottom: 3D limit (solid-red), L √ µ mb = 1000 (dashed-black), L √ µ mb = 10 (dashed-blue), L √ µ mb = 5 (dashed-red), L √ µ mb = 2 (dashed-green) and L √ µ mb = 1 (dashed-orange). with the Euler number γ (cid:39) . e when including theGorkov-Melik contribution [77].Furthermore, the BKT-transition temperature on theBEC-side, where ln( k F a ) (cid:28)
1, is approximatelyreached for these length scales. However, for smaller L √ µ mb , we obtain a smaller value than the predictedBKT transition temperature [77, 78] T c T F = 12 (cid:20) log (cid:18) B π log (cid:18) πk F a (cid:19)(cid:19)(cid:21) − , (27)with B (cid:39) z -direction may lead to a shiftin the parameters of the Fermi gas. This shift can alsobe differently pronounced depending on the scatteringlength. The observation that T c /T F decreases towardszero on the BEC-side for L → L -dependence in the map from three-dimensional to two-dimensional parameters in this regionof the phase diagram and range of L .In the region of strong correlations, whereln( k F a ) (cid:39)
1, we find a substantial increase inthe critical temperature T c /T F which cannot be foundin a mean-field analysis by extrapolation of the knownBCS- and BEC-limits.Comparing our results for L √ µ mb = 2 . L √ µ mb is ap-proximately of the order 0 . . . .
5, we find a qualitativelysimilar phase diagram. Here the increased critical tem-perature in the strong coupling regime can also be found, - - Figure 7: Phase diagram in terms of T c /T F for different con-finement length scales with respect to the 2D crossover pa-rameter ln( k F a ). From top to bottom L √ µ mb = 10 (blue), L √ µ mb = 5 (red), L √ µ mb = 2 (green) and L √ µ mb = 1 (or-ange). The low critical temperature on the BEC-side is causedby our choice of boundary conditions, see Section II C. yet slightly less pronounced.In Fig. 9 we show our result for a confinement lengthof L √ µ mb = 2 . - - Figure 8: Phase diagram in terms of T c /T F for a confinementlength of L √ µ mb = 2 . k F a ). Here weshow the experimental data from [6] with the correspondingstatistical errors in orange, as well as both the perturbativeBKT- and BCS-transition temperature as dashed red linesin the appropriate regimes, i.e. ln( k F a ) (cid:28) − k F a ) (cid:29) - - Figure 9: Phase diagram in terms of T c /T F for a confinementlength of L √ µ mb = 2 . k F a ).Here we show the experimental data from [6]. The exper-imental critical temperature T c /T F with the correspondingstatistical errors is depicted in white, while the colour scaledenotes the non-thermal fraction which signals the onset of apresuperfluid phase. VI. CONCLUSIONS AND OUTLOOK
In this paper we have studied the dimensional crossoverin an ultracold Fermi gas from three to two dimensions,thus extending the work on non-relativistic bosons car-ried out in [56], as well as the mean-field analysis in [29]for fermions. Particular emphasis was put on the su-perfluid phase transition calculated over the whole BCS-BEC crossover in dependence on different confinementlength scales. A comparision to recent experiments in[3] and [6] found a qualitative good agreement. More-over, we find a non-trivial behaviour of the finite tem-perature phase diagram when confining the Fermi gasin reduced dimensionality. Here for small confinementlength scales a substantial reduction of the critical tem-perature T c /T F on the one hand is found on the BEC-side of the crossover, while on the other hand the criticaltemperature on the BCS-side is moderately increased.Notably, in the strongly-coupled regime a substantiallyhigher critical temperature is found which is on par withrecent measurements [6].Within the dimensional crossover from three to twodimensions a dip-like structure with regions of increasedand reduced critical temperature T c /T F were found. Thisdip-like structure is more or less pronounced dependingon the scattering length chosen in the ultraviolet a andis an artefact of the boundary conditions chosen for theconfinement. For a harmonic confinement similiar dipswere seen in [29] for a mean-field study of the criticaltemperature on the BCS-side for quasi-two dimensionalFermi gases.These results suggest that a geometry lying betweenthree and two dimensions might be beneficial in findingsystems with increased critical temperature and thus inadvancing in the quest for high- T c superconductors.The above procedure of confinement from three to two dimensions can in general be extended to confinementsfrom three to one and from two to one dimensions. More-over, for a more realistic confinement scenario a har-monic trapping potential V ( z ) = m ω z z , as it is ap-proximately realised in most ultracold atom experiments,should be implemented instead of the periodic conditionsused in this work in order to account for the correct trap-ping geometry. However, already the periodic bound-ary conditions yield qualitatively similiar features in the L -dependence of the critical temperature as a harmonictrap.A further quantitative improvement, within the dimen-sional crossover as well as in three dimensions, concernsthe calculation of the density by which every quantity isnormalised, by means of the Fermi momentum k F . Asdetailed in Appendix C, the initial conditions for observ-ables g i with scaling dimension d g i ≥ µ . As a consequence, the flow ofthe density, calculated by an µ -derivative of the effectivepotential, is not UV-finite. In Appendix C we outline aniterative safe way of calculating the density whose resultswill be presented in future work. In addition, the trunca-tion may be extended to include also the renormalisationof the fermion propagator, as well as higher orders in thederivative expansion.Another interesting aspect would be the study of spin-and mass-imbalanced Fermi gases within the dimensionalcrossover, since here the influence of mismatching Fermisurfaces and stronger fluctuations in lower dimensionsmight result in competing effects concerning pairing [79–83]. This may shed further physical insight, for examplein the search for high temperature superconductors.Already at the present stage our beyond-mean-fieldanalysis is an advancement in the study of the inter-play between many-body physics and dimensionality ofultracold Fermi gases. It reveals that the dependence offluctuation effects on the effective dimensionality leadsto new characteristic features that can be exploited inexperiment and serve as a test for theoretical methods. VII. ACKNOWLEDGEMENTS
We thank Igor Boettcher, S¨oren Lammers, StefanFl¨orchinger, Selim Jochim, Luca Bayha, Marvin Holtenand Ralf Klemt for discussions. We furthermore thankthe group of Selim Jochim for providing their experimen-tal data to us.The work is supported by EMMI and by the DFG Col-laborative Research Centre SFB 1225 (ISOQUANT) aswell as by the DFG under Germany’s Excellence Strat-egy EXC - 2181/1 - 390900948 (the Heidelberg ExcellenceCluster STRUCTURES).0
Appendix A: Flow equations
In this appendix we derive the flow equations for an ul-tracold Fermi gas in the dimensional crossover. By defin-ing a Master equation all flow equations of the individualcouplings can be obtained by suitable projection descrip-tions. Furthermore, we consider only the isotropic casewhere the flow of the couplings in transeversal directionequal the ones in the plane g i = g i,z , since this distinctionis negligible [56]. Our procedure is based on [47].The ansatz for the effective average action can be di-vided in an kinetic part which consists of the fermion andboson dynamics and and interaction partΓ k = Γ kin + Γ int . (A1)The kinetic part in terms of the renormalised fields ψ = A / ψ ψ and φ = A / φ φ is given byΓ kin [ ψ, φ ] = (cid:90) X (cid:88) σ = { , } ψ ∗ σ (cid:0) S ψ ∂ τ − ∇ + m ψ (cid:1) ψ σ + φ ∗ (cid:0) S φ ∂ τ − V φ ∂ τ − ∇ / (cid:1) φ . (A2)We normalised the the coefficients of the gradient termsby means of the wave function renormalisations A ψ and A φ which enter the renormalisation group flow via theanomalous dimensions η ψ = − ∂ t log A ψ , η φ = − ∂ t log A φ . (A3)Unrenormalised quantities are in the following denotedwith an overbar, while renormalised ones are overbar-less. The interactions can, after a Hubbard-Stratonovichtransformation, be written asΓ int [ ψ, φ ] = (cid:90) X ( U ( φ ∗ φ ) − h ( φ ∗ ψ ψ − φ ψ ∗ ψ ∗ ))(A4)neglecting the RG-flow of the four-fermion vertex λ ψ,k .The effective average potential depends only on the U (1)-invariant quantity ρ = φ ∗ φ and describes bosonicscattering processes. The U (1)-symmetry is sponta-neously broken for a non-zero minimum ρ of the effectiveaverage potential and thus describes superfluidity.In a Taylor-expansion we write U ( ρ ) = m φ ( ρ − ρ ) + λ φ ρ − ρ ) − n k δµ + α k ( ρ − ρ ) δµ , (A5)where we need to include at least up to the second or-der in ρ to reproduce the second order phase transitionto superfluidity. In the symmetric regime we thereforehave ρ = 0 and m φ >
0, whereas the symmetry-brokenregime is realised for ρ > m φ = 0.
1. Truncation
By including only the fermionic diagrams (F) of Fig.1 we arrive at the mean-field result and the bosonic fluc-tuations are taken care of by the diagrams including twobosonic lines (B).The inverse propagators G − φ ( Q ) and G − ψ ( Q ) are cal-culated byΓ (2) φ i ,φ j ( X, Y, ρ ) = δ Γ δφ i ( X ) δφ j ( Y ) [ φ ]Γ (2) ψ ( ∗ ) α ,ψ ( ∗ ) β ( X, Y, ρ ) = −→ δδψ ( ∗ ) α ( X ) Γ ←− δδψ ( ∗ ) β ( Y ) [ φ ] (A6)where the boson background field φ is assumed to bereal valued and the direction of the arrow for the inversefermion propagator denotes derivatives acting from leftand right on the effective potential. In momentum spacewe arrive atΓ (2) BB ( Q, Q (cid:48) ) = δ ( Q + Q (cid:48) ) G − φ ( Q ) , Γ (2) F F ( Q, Q (cid:48) ) = δ ( Q + Q (cid:48) ) G − ψ ( Q ) (A7)After performing the functional derivatives we obtain inthe { φ , φ } -basis for a constant bosonic background field φ = √ ρG − φ ( Q ) = A φ (cid:32) P S,Qφ + U (cid:48) + 2 ρ U (cid:48)(cid:48) i P A,Qφ − i P A,Qφ P S,Qφ + U (cid:48) (cid:33) G − ψ ( Q ) = A ψ (cid:32) − h √ ρ ε − P − Qψ P Qψ h √ ρ ε (cid:33) (A8)with = G − = A φ G − ( Q ), being the 2-dimensionalunity matrix, ε = ((0 , , ( − , ρ . The regulators in the { φ , φ } -basis are given by R Qφ = A φ R Qφ = A φ (cid:32) R Sφ ( Q ) i R Aφ ( Q ) − i R Aφ ( Q ) R Sφ ( Q ) (cid:33) R Qψ = A ψ R Qψ = A ψ (cid:32) − R − Qψ R Qψ (cid:33) (A9)Moreover we defined the symmetrised and anti-symmetrised components of the propagators and regu-lator functions as f S,A ( Q ) = f ( Q ) ± f ( − Q )2 . (A10)By introducing short-hand notations for the sum of prop-1agator and regulator, as well as the determinants L Qψ = P Qψ + R Qψ det QF = L Qψ L − Qψ + h ρL Qφ = P Qφ + R Qφ + U (cid:48) + ρ U (cid:48)(cid:48) ˜ L Qφ = P Qφ + R Qφ det QB = L Qφ L − Qφ − ( ρU (cid:48)(cid:48) ) . (A11)we may write the regularised propagators as G Qφ = A φ G Qφ = 1det Q B (cid:32) ˜ L S,Qφ + U (cid:48) − i ˜ L A,Qφ i ˜ L A,Qφ ˜ L S,Qφ + U (cid:48) + 2 ρ U (cid:48)(cid:48) (cid:33) G Qψ = A ψ G Qψ = 1det Q F (cid:32) ( h ρ ) / ε L − Qψ − L Qψ − ( h ρ ) / ε (cid:33) (A12)We can also represent the boson propagator in the con-jugate field basis { φ, φ ∗ } where the corresponding matrixwill be labeled by a hat.For φ = ( φ + i φ ) / √ (cid:32) φφ ∗ (cid:33) = 1 √ (cid:32) − i (cid:33) (cid:18) φ φ (cid:19) (A13)and thus arrive at (cid:98) G − φ = U G − φ U t (A14)with the definitions U = 1 √ (cid:32) − i1 i (cid:33) , U t = 1 √ (cid:32) − i i (cid:33) . (A15)Thus we obtain for the inverse boson propagator in the { φ, φ ∗ } -basis (cid:98) G − φ = (cid:32) ρ U (cid:48)(cid:48) L − Qφ L Qφ ρ U (cid:48)(cid:48) (cid:33) , (cid:98) R φ ( Q ) = (cid:32) R − Qφ R Qφ (cid:33) (A16)and (cid:98) G Qφ = 1det Q B (cid:32) − ρ U (cid:48)(cid:48) L − Qφ L Qφ − ρ U (cid:48)(cid:48) (cid:33) (A17) To generate higher n-point functions further functionalderivatives have to be applied, once again paying atten-tion to the correct ordering for fermionic derivatives.Since we assume momentum and frequency independentvertices to close our set of equation, the complexity ofthe system of differential equations is drastically reducedΓ ( n> k ( Q , . . . , Q n ) = γ ( n ) k δ ( Q , . . . , Q n ) . (A18)
2. Master equations
In order to solve the Wetterich equation in practice weneed to convert it into a set of coupled differential equa-tions of the correlation functions. We therefore start froma few Master equations, namely for the inverse fermionand boson propagators, the effective average potentialand the Feshbach coupling.In the next step these equation are projected appropri-ately to arrive at flow equations for the running couplings { g k } .For general regulators the flow equation of the effectiveaverage potential is then given by˙ U k ( ρ ) = 12 Tr (cid:90) Q G Qφ ˙ R Qφ −
12 Tr (cid:90) Q G Qψ ˙ R Qψ = 12 (cid:90) Q A φ L Qφ ˙ R − Qφ + L − Qφ ˙ R Qφ det QB − (cid:90) Q A ψ L Qψ ˙ R − Qψ + L − Qψ ˙ R Qψ det QF (A19)Our flow equations can be divided into a bosonic anda fermionic contribution resulting from bosonic (B) andfermionic (F) diagrams, respectively.˙ U ( ρ ) = ˙ U ( B ) + ˙ U ( F ) (A20)Including the additional term of the anomalous dimen-sion we find the flow for the renormalised quantities, e.g.˙ U ( ρ ) = ˙ U ( B ) + ˙ U ( F ) + η φ ρ U (cid:48) ( ρ ) . (A21)2For the flow of the inverse boson propagator in the { φ , φ } -basis we find˙ G − φ i φ j ( P ) = 12 Tr (cid:90) Q G φ ( Q ) γ (3) φ i BB G φ ( Q + P ) γ (3) φ j BB G φ ( Q ) ˙ R φ ( Q )+ 12 Tr (cid:90) Q G φ ( Q ) γ (3) φ j BB G φ ( Q − P ) γ (3) φ i BB G φ ( Q ) ˙ R φ ( Q ) −
12 Tr (cid:90) Q G φ ( Q ) γ (4) φ i φ j BB G φ ( Q ) −
12 Tr (cid:90) Q G ψ ( Q ) γ (3) φ i F | F G ψ ( Q + P ) γ (3) φ j F | F G ψ ( Q ) ˙ R ψ ( Q ) −
12 Tr (cid:90) Q G ψ ( Q ) γ (3) φ j F | F G ψ ( Q − P ) γ (3) φ i F | F G ψ ( Q ) ˙ R ψ ( Q ) . (A22)Likewise the flow of the inverse fermion propagator is obtained, taking the Grassmannian nature of fermions inaccount, ˙ G − ψ α ψ β ( P ) = 12 Tr (cid:90) Q G φ ( Q ) γ (3) ψ α B | F G ψ ( Q + P ) γ (3) F | Bψ β G φ ( Q ) ˙ R φ ( Q ) −
12 Tr (cid:90) Q G φ ( Q ) γ (3) BF | ψ β G ψ ( Q − P ) γ (3) ψ α | F B G φ ( Q ) ˙ R φ ( Q ) −
12 Tr (cid:90) Q G ψ ( Q ) γ (3) ψ α | F B G φ ( Q + P ) γ (3) BF | ψ φ G ψ ( Q ) ˙ R ψ ( Q )+ 12 Tr (cid:90) Q G ψ ( Q ) γ (3) F | Bψ β G φ ( Q − P ) γ (3) ψ α B | F G ψ ( Q ) ˙ R ψ ( Q ) . (A23)
3. Projection description for the running couplings
In this section we derive suitable projection descrip-tions for the flow equations of the running couplings { g k } and expansion coefficients of the effective average poten-tial U ( ρ ). We use a derivative expansion of the inversefermion and boson propagators P ψσ ( Q ) = Z ψσ i q + A ψσ q − µ = A ψσ (cid:0) S ψσ i q + q − µ (cid:1) P φ ( Q ) = Z φ i q + A φ q / A φ (cid:0) S φ i q + q / (cid:1) . (A24)Expanding the effective potential in a Taylor series wecan easily project the flow equation (A19) onto the coef-ficients U k ( ρ ) = m φ ( ρ − ρ ) + λ φ ρ − ρ ) + N (cid:88) n> u n n ! ( ρ − ρ ) n . (A25)There are several candidates for projection descriptionsfor the running couplings which may at a first glanceseem equal. However, as the Wetterich equation is an exact equation incorporating all orders of the effectiveaverage action, every projection neglects certain higherorder couplings and thus resultes in different flows. Weexpect though that our truncation includes the most im-portant effects and a precise projection would only yieldnegligible modifications. The distinction between differ-ent projection descriptions may also be used for an errorestimate.In the symmetric regime of the flow we have ˙ m φ =˙ U (cid:48) ( ρ = 0) which makes place for the flow of ˙ ρ = − ˙ U (cid:48) ( ρ ) /λ φ in the symmetry broken regime. For theflow of higher expansion coefficients one finds˙ u n = ∂ t (cid:16) U ( n ) ( ρ ) (cid:17) = ˙ U ( n ) ( ρ ) + u n +1 ˙ ρ . (A26)We obtain the flow of the renormalised couplings m φ = m φ A φ , ρ = A φ ρ , u n = u n A nφ (A27)3by ˙ m φ = η φ m φ + ˙ m φ A φ , ˙ ρ = − η φ ρ + A φ ˙ ρ , ˙ u n = η φ u n n + ˙ u n A nφ . (A28)Since we restrict ourselves to purely fermionic andbosonic diagrams, we have no running of the couplingsentering the fermionic propagator.For the couplings associated with the boson propagatorwe obtain ˙ S φ = − ∂ p ˙ G − φ φ ( P, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) P =0 ,ρ , ˙ A φ = 2 ∂ p ˙ G − φ φ ( P, ρ ) (cid:12)(cid:12)(cid:12)(cid:12) P =0 ,ρ , (A29)and for the renormalised quantities with the anomalousdimension η φ = − ˙ A φ /A φ ˙ S φ = η φ S φ + ˙ S φ A φ . (A30)In the flow equations for the running couplings we ne-glected a term proportional to ˙ ρ which would be gener-ated if one took the RG-time derivative after performingthe projections.
4. Flow equations using the optimised regulator
In this section we use the optimised regulator (9) forderiving the flow equations of the running couplings.These equations will be our main starting point in study-ing the BCS-BEC crossover in dimensions 2 ≤ d ≤ q = | (cid:126)q | .The procedure may, however, further be simplified by in-terchanging the order of the derivative projection and theMatsubara summation. We therefore start again fromthe general form of the flow of the inverse propagatorswith the trace not being evaluated so that we can ex-pand the inverse propagators G ( Q ± P ) in powers of p and p and perform the projections afterwards.For the fermionic contributions we arive at the generalflow equations with the loop integration still unevaluated˙ S ( F ) φ = − h S ψ (cid:90) Q ˙ R ψ ( q ) A ψ (cid:18) − h ρ det (cid:19) ,η ( F ) φ = 8 h d (cid:90) Q ˙ R ψ ( q ) A ψ q R (2) ψ det . (A31)The expansion for the bosonic contributions results inthe flow equations˙ S ( B ) φ = − S φ ρ U (cid:48)(cid:48) (cid:90) Q ˙ R φ ( q ) A φ U (cid:48)(cid:48) + ρ U (3) det ( Q ) + 2 ρ U (cid:48)(cid:48) (cid:104) ρ U (cid:48)(cid:48) (cid:0) U (cid:48)(cid:48) + ρ U (3) (cid:1) − (cid:0) U (cid:48)(cid:48) + ρ U (3) (cid:1) L Sφ ( Q ) (cid:105) det ( Q ) ,η ( B ) φ = 4 ρ ( U (cid:48)(cid:48) ) (cid:90) Q ˙ R φ ( q ) A φ R (1) φ + 4 q x R (2) φ det ( Q ) − q x (cid:16) R (1) φ (cid:17) L Sφ ( Q )det ( Q ) . (A32)After performing the Matsubara sums and the momen-tum integrations the overall flow equations in our trun-cation can be cast into the form˙ U ( F ) ( ρ ) = − v d d k d +2 (cid:96) (1 , F ˙ U ( B ) ( ρ ) = 8 v d d/ d k d +2 (cid:96) (1 , B . (A33)The fermionic contributions to the boson propagator are found to be˙ S ( F ) φ = − h v d d k d − (cid:16) (cid:96) (0 , F − w (cid:96) (0 , F (cid:17) ,η ( F ) φ = 16 h v d d k d − (cid:96) (0 , F, , (A34)4while the bosonic contributions are given by˙ S ( B ) φ = − S φ d ρ U (cid:48)(cid:48) v d d/ k d − (cid:20) (cid:16) U (cid:48)(cid:48) + ρ U (3) (cid:17) (cid:96) (0 , B + 2 ( ρ U (cid:48)(cid:48) ) (cid:16) U (cid:48)(cid:48) + ρ U (3) (cid:17) k − (cid:96) (0 , B − ρ U (cid:48)(cid:48) (cid:16) U (cid:48)(cid:48) + ρ U (3) (cid:17) k − (cid:96) (1 , B (cid:21) ,η ( B ) φ =8 ρ ( U (cid:48)(cid:48) ) v d d/ d k d − (cid:96) (0 , B, . (A35)Here we used the definitions for fermionic contributions (cid:96) ( n,m ) F (cid:16) ˜ µ, ˜ T , w (cid:17) = (cid:40) (cid:96) (˜ µ ) F m (cid:0) √ w (cid:1) n even (cid:96) (˜ µ ) F m (cid:0) √ w (cid:1) n odd(A36) and (cid:96) ( n,m ) F, (cid:16) ˜ µ, ˜ T , w (cid:17) = (cid:40) (cid:96) (˜ µ ) F m (cid:0) √ w (cid:1) n even (cid:96) (˜ µ ) F m (cid:0) √ w (cid:1) n odd , (A37)where we made use of w = h ρ/k , as well as w = U (cid:48) /k and w = ρ U (cid:48)(cid:48) /k . For bosonic diagrams we de-fined (cid:96) ( n,m ) B (cid:16) ˜ T , w , w (cid:17) = 1 S mφ (cid:18) − η φ d + 2 (cid:19) (1 + w + w ) n B m (cid:16)(cid:112) (1 + w )(1 + w + 2 w ) /S φ (cid:17) (A38)and (cid:96) (0 ,m ) B, (cid:16) ˜ T , w , w (cid:17) = 1 S mφ B m (cid:16)(cid:112) (1 + w )(1 + w + 2 w ) /S φ (cid:17) = (cid:96) (0 ,m ) B (cid:12)(cid:12)(cid:12) η φ =0 . (A39) F m ( z ) and B m ( z ) label the fermionic and bosonic Mat-subara sums of order m , respectively. The functions (cid:96) i are defined as (cid:96) ( x ) = θ ( x + 1) ( x + 1) d/ − θ ( x −
1) ( x − d/ ,(cid:96) ( x ) = θ ( x + 1) ( x + 1) d/ + θ ( x −
1) ( x − d/ ,(cid:96) ( x ) = (cid:96) ( x ) − θ ( x ) x d/ (A40)and the d-dimensional volume integral is given by v − d =2 d +1 π d/ Γ( d/ n = n k → we may split the chemical potential into a reference part µ and an offset δµ such that µ = µ + δµ . We thenexpand our effective potential (21) with respect to theoffset chemical potential δµ according to U k ( ρ ) = (cid:88) n =1 u n n ! ( ρ − ρ ) n − n k δµ + α k ( ρ − ρ ) δµ (A41)The differentiation with respect to µ acts rather on δµ as the reference chemical potential is fixed˙ n k = − ∂ ˙ U∂ δµ . (A42)According to our master equation for the effective aver-age potential (A19) we now expand L S,Qψ and det Q F interms of δµ while the fermionic cutoff still regularisesaround the Fermi surface, i.e. the reference chemical po-tential µ .
5. Flow equations for finite volume
When confining our system by means of a compactifa-tion of one spatial dimension in a dimensional crossoverfrom 3d to 2d with a confinement length scale L . Byadopting periodic boundary conditions we restrict oursystem to a torus in one spatial direction ψ ( L ) = ψ (0) (A43)such that we obtain a ‘spatial Matsubara sum’ over dis-crete momenta k n = 2 πn/L with n ∈ Z . Accompany-ing this quantisation of energy levels the bosonic andfermionic regulators defined are modified accordingly.5For the optimised regulator they become R φ,k ( q ) = (cid:18) k − q + k n (cid:19) θ (cid:18) k − q + k n (cid:19) ,R ψ,k ( q ) = k (cid:104) sgn (cid:16) z + ˜ k n (cid:17) − (cid:16) z + ˜ k n (cid:17)(cid:105) × θ (cid:16) − | z + ˜ k n | (cid:17) , (A44)where we again used z = ( q − µ ) /k and ˜ k n = k n /k .Hence the d -dimensional spatial integration splits up intoa sum over the discrete momenta k n and a momentumintegral in d − (cid:90) d d q (2 π ) d = 1 L (cid:88) k n (cid:90) d d − q (2 π ) d − . (A45)Due to the inclusion of the discrete momenta in the reg-ulator the evaluation of the spatial boils down to countingthe modes within the potential well. For periodic bound-ary conditions we hereby encounter the following type ofsums N (cid:88) n = − N α = α (1 + 2 N ) ( α ∈ R ) , N (cid:88) n =1 n = 16 N (1 + N ) (1 + 2 N ) , (A46) and N (cid:88) n =1 n = 130 N (1 + N ) (1 + 2 N ) ( − N + 3 N ) . (A47)As a result of the periodic boundary conditions the reg-ulator function restricts the Matsubara-type summationin the transversal direction to | k n | = | πn/L | < √ k orequivalently | n | < ˜ L/ √ π .For bosonic contributions we define N ( B ) = (cid:36) ˜ L √ π (cid:37) (A48)with (cid:98) x (cid:99) being the largest integer smaller than x . Inthree dimensions we find C L = 1 L (cid:88) k n (cid:18) − k n k (cid:19) d/ (cid:18) − η φ d + 2 (cid:18) − k n k (cid:19)(cid:19) θ (cid:18) k − k n (cid:19) = k ˜ L (cid:16) N ( B ) (cid:17) (cid:34) − η φ − (cid:18) − ηφ (cid:19) (cid:18) π ˜ L (cid:19) N ( B ) (cid:16) N ( B ) (cid:17) − η φ (cid:18) π ˜ L (cid:19) N ( B ) (cid:16) N ( B ) (cid:17) (cid:18) − N ( B ) + 3 (cid:16) N ( B ) (cid:17) (cid:19)(cid:35) . (A49)Thus all bosonic flow equations still hold with the re-placements (cid:18) − η φ d + 2 (cid:19) → C L , d → d − . (A50) The fermionic momentum integrals can be generalisedby the transformation z → (cid:98) z = ( q + k n − µ ) /k . Allresults can then be transferred by the transformation µ → (cid:98) µ = ˜ µ − ˜ k n . For periodic boundary conditions itcan be easily shown in d = 3 dimensions61 L (cid:88) k n θ (ˆ µ + 1) (ˆ µ + 1) ( d − / = 1 L (cid:34) (˜ µ + 1) (cid:16) N ( F )1 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )1 (cid:16) N ( F )1 (cid:17) (cid:16) N ( F )1 (cid:17)(cid:35) θ (˜ µ + 1) , L (cid:88) k n θ (ˆ µ −
1) (ˆ µ − ( d − / = 1 L (cid:34) (˜ µ − (cid:16) N ( F )2 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )2 (cid:16) N ( F )2 (cid:17) (cid:16) N ( F )2 (cid:17)(cid:35) θ (˜ µ − , L (cid:88) k n θ (ˆ µ ) (ˆ µ ) ( d − / = 1 L (cid:34) ˜ µ (cid:16) N ( F )3 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )3 (cid:16) N ( F )3 (cid:17) (cid:16) N ( F )3 (cid:17)(cid:35) θ (˜ µ ) . (A51)Here we defined N ( F )1 = (cid:36) ˜ L (˜ µ + 1) / π (cid:37) ,N ( F )2 = (cid:36) ˜ L (˜ µ − / π (cid:37) ,N ( F )3 = (cid:36) ˜ L ˜ µ / π (cid:37) . (A52) Hence for the spatial threshold function with explicitMatsubara summation we obtain for periodic boundaryconditions in d = 31 L (cid:88) k n (cid:96) a (ˆ µ ) = k ˜ L (cid:40)(cid:34) (˜ µ + 1) (cid:16) N ( F )1 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )1 (cid:16) N ( F )1 (cid:17) (cid:16) N ( F )1 (cid:17)(cid:35) ( − a (cid:20) (˜ µ + 1) → (˜ µ −
1) & (cid:16) N ( F )1 → N ( F )2 (cid:17) (cid:21) − (1 + ( − a ) (cid:34) ˜ µ (cid:16) N ( F )3 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )3 (cid:16) N ( F )3 (cid:17) (cid:16) N ( F )3 (cid:17)(cid:35) (cid:41) (A53)for a = 1 , L (cid:88) k n (cid:96) (ˆ µ ) = k ˜ L (cid:40)(cid:34) (˜ µ + 1) (cid:16) N ( F )1 (cid:17) − (cid:18) π ˜ L (cid:19) N ( F )1 (cid:16) N ( F )1 (cid:17) (cid:16) N ( F )1 (cid:17)(cid:35) + (cid:20) (˜ µ + 1) → (˜ µ −
1) & (cid:16) N ( F )1 → N ( F )2 (cid:17) (cid:21)(cid:27) (A54)Thus all fermionic flow equations can be transferred tothe case of finite volume with periodic boundary condi-tions with the replacement (cid:96) i → (cid:96) i,L = k ˜ L (cid:88) k n (cid:96) i , d → d − . (A55) Appendix B: Numerical procedure
The set of coupled differential equations for the pro-jected flow equations from A are numerically evaluatedfor both zero and finite temperature. However, it is a use-ful feature of the functional renormalisation group thatfor larges scales k (cid:29) T the finite temperature flow can7be approximated by the zero temperature system [47].For a practical computation we choose k switch ,T = 10 π T ,i.e. we follow the zero temperature flow until k switch ,T where the temperature starts to become an importantscale and we switch to the finite temperature flow equa-tions.Likewise, the Fermi gas confined to a trap can be re-garded as an unconfined system for large scales k (cid:29) L − .Here we choose k switch ,L = 100 /L , which significantly de-creases the runtime of the computation. The agreementof the results with and without splitting the flow in zeroand finite temperature, as well as unconfined and con-fined flow equations was checked numerically. Appendix C: µ -dependence In this appendix we discuss the potential µ -dependences of initial conditions as well as an iterativesafe way of how to extract related observables such asthe density and higher µ -derivatives of the free energy.A similar procedure can be found in [84].It is well-known that thermal fluctuations decay expo-nentially with the infrared cut-off scale, f ( k/T, R ) e − c ( R ) k/T , (C1)where f ( k/T, R ) rises not more than polynomially oreven decays, depending on the (canonical) dimension ofthe observable under consideration, see [85]. The formof the prefactor as well as the coefficient c ( R ) depend onthe shape of the regulator. In particular, for non-analyticcut-offs (in frequency) such as the sharp cut-off and theoptimal cut-off we have c ( R ) = 0 and the thermal be-haviour at large cut-off scales relates to the dimensionof the observable. Note that (C1) can be shown to holdto any order of a given approximation scheme and henceis a formal, exact property of thermal fluctuations. It isintimitely linked to the fact that thermal sums can berepresented as contour integrals and the infrared cut-offscale serves as a mass parameter which shifts poles tomomenta p ∝ ik . This also hints at the fact that it isnot present for non-analytic regulators, where the Mat-subara sum cannot be represented as a contour integral,and a naive dimensional analysis prevails.In contradisctinction, the chemical potential µ as wellas other external tuning parameters only lead to a poly-nomial decay or rise in the dimensionless ratioˆ k = kµ , ˆ k = k √ µ , (C2)for the relativistic case and non-relativistic case respec-tively. In most cases this behaviour is related to the(canonical) dimension of the observable at hand. Forexample, the free energy or effective action has a vanish-ing canonical dimension. However, it relates to (nega-tive) pressure times space-time volume V and hence hasa scaling dimension d p = d with the cut-off scales in the relativistic case and scaling dimension d p = d + 1 in thenon-relativistic case.The above arguments entail that the flow of the ther-mal pressure, ∂ t p ( T, µ ) := − (cid:18) ∂ t Γ k [ φ EoS ,k ; T, µ ] V T − ∂ t Γ k [ φ EoS ,k ; 0 , µ ] V (cid:19) , (C3)in general decays exponentially for large cut-off scales, ∂ t p ( T, µ ) = ∝ e − c ( R ) k/T , (C4)while the free energy density, f , normalised in the vac-uum, ∂ t f ( T, µ ) := (cid:18) ∂ t Γ k [ φ EoS ,k ; T, µ ] V T − ∂ t Γ k [ φ EoS ,k ; 0 , V (cid:19) , (C5)has polynomial growth with k , ∂ t f ( T, µ ) → c d f − k d f ˆ k − + c d f − k d f ˆ k − + k d f O (ˆ k − ) , (C6)Here, vanishing exponents (in the relativistic case) in-clude logarithms.
1. Initial conditions
Evidently, the initial conditions for observables or cou-plings λ i with scaling dimension d λ i ≥ µ -dependent.In turn, for sufficiently large cut-off scales ˆ k (cid:29) d λ i < ∂ t Γ k [ φ ] = 12 Tr G k,ϕ ∂ t R k,ϕ − Tr G k,ψ ∂ t R k,ψ , (C7)where the field ϕ stands for bosonic fields while ψ standsfor fermionic ones. Every observable and coupling canbe derived directly from (C7) and its solution. Indeed,if different definitions of observables such as the densityexist, the one directly using the flow (C7) has the smallestsystematic uncertainty.For our investigation we write the effective action asΓ k = Γ k [ φ ; (cid:126)g ] , (cid:126)g = ( m ψ , m ϕ , Z ψ , Z ϕ , h, λ ϕ , λ ψ , ... ) , (C8)where (cid:126)g encodes all couplings (expansion coefficients) ofthe effective action, ordered in decaying mass dimension.We conclude that in d = 4 dimensions the only couplingsthat potentially require µ -dependent initial conditions8are the mass parameters (including µ itself). However,the flow of the dimer mass reads asymptotically ∂ t m ϕ ∝ k h k (1 + µ/k ) / (C9)and hence its µ -derivative tends towards zero, and theonly coupling to be taken care of is the fermionic mass(and chemical potential).
2. Density
As already mentioned above, the equation for the den-sity with the smallest systematic error is its flow. For thenon-relativistic case it reads ∂ t n = 1Vol d∂ t Γ k dµ → c n, k + c n, µ k + O (ˆ k − ) , (C10)and a similar equation holds for the relativistic case. Theflow of the susceptibility reads ∂ t ∂ µ n = 1Vol d ∂ t Γ k dµ → c n, k + O (ˆ k − ) , (C11)while the flow of the second µ -derivative of the densitytends towards zero for large cut-off scales, ∂ t ∂ µ n = ∂ µ ∂ t Γ k Vol → O (ˆ k − ) , (C12)We conclude that we can represent the density, and thesusceptibility at vanishing cut-off, k = 0, by n ( µ ) = (cid:90) µ dµ (cid:48) ∂ µ (cid:48) n ( µ (cid:48) ) , with n (0) = 0 , (C13)and ∂ µ n ( µ ) = (cid:90) µ dµ (cid:48) ∂ µ (cid:48) n ( µ (cid:48) ) , with ∂ µ n (0) = 0 , (C14)It is left to determine ∂ µ n k ( µ ). To that end we rewritethe flow of the density as ∂ t n k = d∂ t Γ k dµ = ∂ µ | (cid:126)g ∂ t Γ k + dg i dµ ∂ g i ∂ t Γ k . (C15)Both terms follows analytically from the master equa-tion, (C7), and each partial µ -derivatives and dg i /dµ ∂ g i -derivative lowers the effective k -dimension by two. Thecoefficients g (1) i = dg i /dµ with g ( n ) i = d n g i dµ n (C16)follow from their flow ∂ t g (1) i = ddµ ∂ t g i = ∂ µ ∂ t g i + g (1) j ∂ g j ∂ t g i . (C17) Eq. (C17) is a coupled differential equation for (cid:126)g (1) , ∂ t (cid:126)g (1) = (cid:126)A + B · (cid:126)g (1) (C18)with coefficients A ,i = ∂ µ ∂ t g i , B ,ij = ∂ g j ∂ t g i . (C19)The coefficients A ,i and B ,ij can be read-off from theflow (C7), and hence (C18) is a so-called derived flow: itdoes not feed back into the flow of the effective action.Naturally, this can be iteratively extended to the higherderivatives w.r.t. µ . For g (2) i it reads ∂ t g (2) i = ddµ (cid:16) A ,i + B ,ij g (1) j (cid:17) = ∂ µ A ,i + g (1) j ∂ g j A ,i + g (1) j (cid:16) ∂ µ + g (1) m ∂ g m (cid:17) B ,ij + B ,ij g (2) j . (C20)Again this can be conveniently rewritten in terms of asystem of linear differential equations ∂ t (cid:126)g (2) = (cid:126)A + B · (cid:126)g (2) , (C21)with A ,i = (cid:16) ∂ µ + g (1) m ∂ g m (cid:17) A ,i + g (1) j (cid:16) ∂ µ + g (1) m ∂ g m (cid:17) B ,ij ,B ,ij = B ,ij . (C22)More explicitly we have A ,i = ∂ µ ∂ t g i + 2 g (1) j ∂ g j ∂ µ ∂ t g i + g (1) j g (1) m ∂ g m ∂ g j ∂ t g i ,B ,ij = ∂ g j ∂ t g i . (C23)This already allows us to put down the general structure.At a given order g ( n ) i the matrix B n is simply B . Thevector A n depends on (cid:126)g, (cid:126)g (1) , ..., (cid:126)g ( n − . Hence it can bedetermined iteratively with A n,i = (cid:32) ∂ µ + n − (cid:88) m =1 g ( m ) j ∂ g ( m − j (cid:33) A n − ,i + g ( n − j (cid:16) ∂ µ + g (1) m ∂ g m (cid:17) B ij (C24)with g (0) i = g i and (cid:16) ∂ µ + g (1) m ∂ g m (cid:17) B ij = ∂ µ ∂ g j ∂ t g i + g (1) m ∂ g m ∂ g j ∂ t g i . (C25)For n = 3 this explicitly yields A ,i = (cid:104) ∂ µ + 3 g (1) j ∂ g j ∂ µ + 3 g (1) j g (1) m ∂ g m ∂ g j ∂ µ + g (1) k g (1) j g (1) m ∂ g m ∂ g j ∂ g k + 3 g (2) m ∂ g m ∂ µ + 3 g (2) j g (1) m ∂ g m ∂ g j (cid:105) ∂ t g i . (C26)9Note that there are various forms for the coefficients A n and B n . The above forms have the advantage that allderivatives w.r.t. µ and g ( n ) i can be performed ana-lytically. Finally we write down the flow for higher µ -derivatives of Γ k ∂ ( n − µ ˙ n ( µ ) = d n ∂ t Γ dµ n = (cid:32) ∂ µ + n (cid:88) m =1 g ( m ) j ∂ g ( m − j (cid:33) C n − , (C27)with C = ∂ t Γ k . (C28)For n = 2 this explicitly yields ∂ µ ∂ t n k = d ∂ t Γ k dµ = (cid:104) ∂ µ (cid:12)(cid:12) (cid:126)g + 2 g (1) i ∂ g i ∂ µ (C29)+ g (1) j g (1) i ∂ g i ∂ g j + g (2) i ∂ g i (cid:105) ∂ t Γ k , while the second µ -derivative of the flow for the densityis found to be ∂ µ ∂ t n k = d ∂ t Γ k dµ = (cid:104) ∂ µ | (cid:126)g + 3 g (1) i ∂ g i ∂ µ + 3 g (1) j g (1) i ∂ g i ∂ g j ∂ µ + g (1) m g (1) j g (1) i ∂ g i ∂ g j ∂ g m + 3 g (2) i ∂ g i ∂ µ + 3 g (2) i g (1) j ∂ g j ∂ g i + g (3) i ∂ g i (cid:105) ∂ t Γ k . Hence, overall the density at vanishing cutoff k = 0 isobtained by integrating twice over the chemical potential n ( µ ) = (cid:90) µ dµ (cid:48) (cid:34)(cid:90) µ (cid:48) dµ (cid:48)(cid:48) ∂ µ (cid:48)(cid:48) n ( µ (cid:48)(cid:48) ) + ∂ µ (cid:48) n (0) (cid:35) + n (0) , (C30)where n (0) and ∂ µ n (0) are vanishing.Moreover, we have ∂ µ n k =0 ( µ ) = (cid:90) dkk ∂ µ ˙ n k ( µ ) (C31)for a UV vanishing flow ∂ µ ˙ n k →∞ → [1] I. Bloch, J. Dalibard, and W. 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