{'fraction_non_alphanumeric': 0.04571679418035871, 'fraction_numerical': 0.0221372130941929, 'mean_word_length': 3.988113856740694, 'pattern_counts': {'":': 1, '<': 7, '<?xml version=': 0, '>': 13, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Supercurrent Flow in NJL_{2+1} at High Baryon Density

We present results of numerical simulations of the 2+1d Nambu -- Jona-Lasinio model with non-zero baryon chemical potential mu and spatially-varying complex diquark source strength j. By choosing arg(j) to vary smoothly through 2 pi across the spatial extent of the lattice, a baryon number current is induced which in the high density phase remains non-vanishing as |j|->0; we are hence able to extract a quantity characteristic of a superfluid known as the helicity modulus. We also study supercurrent flow at non-zero temperature and estimate the critical temperature at which the normal phase is restored, which is consistent with the conventional picture for thin-film superfluids in which the transition is viewed in terms of vortex -- anti-vortex unbinding.


Introduction
There are unfortunately rather few quantum field theories amenable to study using lattice Monte Carlo techniques in the presence of a non-zero chemical potential µ, or more specifically with µ/T ≫ 1. Many important theories, including QCD, cannot be studied because their path integral measure with µ = 0 is not real on analytic continuation to Euclidean metric, making Monte Carlo importance sampling inoperative. Of those with positive definite measure, the Nambu Jona-Lasinio (NJL) model with N f = 2 quark flavors [1] is one of the most interesting. At µ = 0 the theory exhibits dynamical chiral symmetry breaking, with the generation of a constituent quark mass scale Σ much larger than the bare mass m. For µ > µ c ≈ Σ, chiral symmetry is restored, and the ground state is a degenerate fermi system with µ = E F ≃ k F [2]. In d + 1 dimensions the baryon density in this case is . The precise nature of the ground state at high density depends on d. For the realistic case d = 3, lattice simulations suggest that condensation of diquark pairs at the Fermi surface takes place leading to spontaneous breakdown of the U(1) B baryon number symmetry [3]. An energy gap ∆ > 0 to excite fermionic quasiparticles develops; for phenomenologically-motivated lattice parameters the simulations predict ∆/Σ ≃ 0.15, in good agreement with self-consistent model calculations of the gap in superconducting quark matter [4]. In this case the NJL model appears to behave as an orthodox BCS superfluid; there is long-range ordering of the ground state signalled by the non-vanishing condensate qq = 0, and a dynamically-generated mass scale ∆. Since a U(1) B symmetry has been spontaneously broken, there is a massless scalar qq bound state in the spectrum, which is associated with long-range interactions between vortex excitations in the superfluid, and with a collective propagating mode for T > 0 known as second sound.
However, both for obvious numerical convenience, and for a more formal reason, namely the existence of an interacting continuum limit, lattice simulations were first applied to the NJL model with µ = 0 in 2+1 dimensions [5,6,7]. In this case the physics appears radically different. Whilst there is evidence for long-range coherence of diquark correlation functions [5], there is no long-range order, and apparently no gap. Rather, the condensate vanishes non-analytically with the diquark source strength, qq ∝ j α , with 0 < α(µ) < 1 [6]. The results were interpreted in terms of a critical phase for all µ > µ c , in which the diquark correlator decays algebraically, qq(0)qq( r) ∝ r −η [7]. Since all simulations are performed on finite systems, and therefore necessarily at T > 0, the absence of long-range order is consistent with the Coleman-Mermin-Wagner theorem for 2d systems [8]. The situation is analogous to the low-T phase of the 2d X-Y model, one of whose physical applications is the description of superfluidity in thin films [9].
The defining property of a superfluid is that the flux density of conserved charge, or supercurrent J , is related to the spatial variation of the phase angle θ of the U(1)-valued order parameter field (in this case qq ) via The constant of proportionality Υ is known as the helicity modulus. For a textbook non-relativistic superfluid such as 4 He it is given by where M is the mass of the helium atom and n s is a parameter called the superfluid density, which need not coincide with the charge density of the condensate. For a relativistic system Υ is best thought of as a phenomenological parameter in its own right, rather like f 2 π in (d + 1)-dimensional chiral model [10]. One way of understanding superfluidity in the absence of long-range order in a 2d system is to observe that the only way to change the quantised circulation J.d l around one direction of a finite torus is to excite a vortex -anti-vortex pair, and transport one member of the pair around the other direction of the torus before re-annihiliation. The energy required to do this scales as ln L where L is the size of the system [9]: hence in the thermodynamic limit circulation patterns are topologically stable.
This Letter will present further support for this scenario in NJL 2+1 by extracting Υ via a calculation of the induced baryon number current J = ψ γψ in response to a spatially varying diquark source. As well as providing direct verification of superfluid behaviour in a fermionic model, we will also study the behaviour of J as temperature T is increased, and find the transition to "normal" behaviour at a critical T c consistent with analytic expectations.

Method
The lattice NJL model studied is identical to that of [5,7]: where χ,χ are isodoublet staggered lattice fermion fields, Φ = σ1 1 + i π. τ is an auxiliary bosonic field defined on the dual lattice sitesx, and the matrix M is (4) Here <x, x > denotes the set of 8 dual sitesx surrounding x, η µx = (−1) x 0 +···+x µ−1 is the Kawamoto-Smit phase required for a Lorentz covariant continuum limit, and ε x = (−1) x 0 +x 1 +x 2 . A full description of the symmetries of (3) and the numerical simulation method is given in [7]. The only novelty in the current study is that the diquark source strengths j, are now specified to be spatially varying, or "twisted": with j 0 a real constant. To ensure homogeneity and single-valuedness on an L 2 s × L t lattice we demand A constant supercurrent of the form (1) is therefore specified by a pair of integers (n 1 , n 2 ). It remains to define the conserved baryon number current J ν : The timelike component of (7) is none other than the baryon charge density n B reported in [5,7]. Here we shall use the same stochastic technique to estimate the quantum expectation value of the spacelike components J(j, µ) = (L 2 to demonstrate behaviour of the form (1). The strategy will be to compute J for fixed (n 1 , n 2 ) for a range of j 0 and extrapolate j 0 → 0. Behaviour consistent with (1) in this limit is deemed to be superfluid.
We used the same simulation parameters as [5,7], namely g 2 = 2.0a, ma = 0.01, which at µ = 0 yields a dynamically-generated constituent mass, which in effect sets the scale, of Σa = 0.71. As µ is raised, there is a sharp transition from a chirally broken vacuum with χχ ≃ 2 g 2 Σ, n B ≃ 0 to a chirally restored phase with n B > 0 at µ c a ≃ 0.65. Studies of the fermion dispersion relation in the phase µ > µ c are consistent with a sharp Fermi surface with k F < ∼ µ and vanishing gap ∆ ≃ 0 [6, 7].

Results
µ=0.2 µ=0.8 In Fig. 1 we plot J 2 (strictly its imaginary part) as a function of j 0 for lattices of various temporal extent L t at two representative values of µ: µa = 0.2 lies in the chirallybroken low density phase, and µa = 0.8 in the high density phase, where n B a 2 ≃ 0.25 [7]. In all the plots shown here we have chosen (n 1 , n 2 ) = (0, 1) to minimise lattice artifacts, and from now on we set the lattice spacing a = 1.
The contrast between the two phases is quite dramatic. For µ = 0.2 J 2 appears to vary approximately quadratically with j 0 , and extrapolate to zero as j 0 → 0. There is no significant effect as L t → ∞, or alternatively as T → 0. At µ = 0.8 the small-j 0 behaviour depends very sensitively on L t ; as T → 0 the data accumulate on a straight line which clearly extrapolates to a non-zero value as j 0 → 0.
This behaviour is readily explained by writing the order parameter (diquark) field as φ = φ 0 e iθ , with φ 0 approximately constant. A natural effective Hamiltonian for long wavelength order parameter fluctuations at low temperature is then The corresponding Noether current is it is natural to postulate φ proportional to j, leading to J 2 (j 0 ) ∝ j 2 0 . For µ > µ c , if we assume that lim j 0 →0 φ 0 = 0 we recover (1) with Υ = φ 2 0 .  With confidence that L t = 64 suffices to determine the phase, in Fig. 2 we plot J 2 (j 0 ) for various µ. There is a sharp change between values of µ ≤ 0.65, which display the low density quasi-quadratic behaviour and smoothly extrapolate to zero as j 0 → 0, and µ ≥ 0.68 which show a negative curvature characteristic of the high density phase. We thus determine the critical chemical potential for the onset of superfluidity 0.65 < µ c < 0.68, in good agreement with the critical value for chiral symmetry restoration. Since as yet we have no systematic method of extrapolating to j 0 → 0 for µ > ∼ µ c to obtain an estimate for J 2 (µ) as an "order parameter", we can make no decisive statement about the nature of the transition, but note that the behaviour of J 2 (j 0 ) varies much more sharply across the transition than the diquark condensate qq + (j) , either in this model (Cf. Figure 2 of [7]), or even in NJL 3+1 (Cf. Figure 4 of [3]). This matches the sharp drop in the chiral order parameter χχ and corresponding rise in n B at µ = µ c [5,7], and is consistent with the analytic prediction of a strong first order transition in the large-N f limit [11].
From now on we work exclusively at µ = 0.8 in an attempt to understand the superfluid phase further. In Fig. 3 we plot Υ = J 2 L s /2π versus j 0 on L 2 s ×64 lattices. Just as for the qq + (j) data [7], it turns out that the data are well-fitted by Υ(j 0 ) = A+B/L s , resulting in the L s → ∞ extrapolation shown in the plot. Recall that as well as genuine finite-size effects in this case there may also be some discretisation artifacts, since as L s increases the gradient operator in (6) becomes better-approximated by the finite difference. Finally the Υ(j 0 ) data in the thermodynamic limit are extrapolated to j 0 → 0 with a remarkably simple linear fit, resulting in Υ = 0.1413(14). We thus quote a result for the helicity modulus of Υ/Σ = 0.200(2) at µa = 0.8. It is interesting to pause and ask what value might be expected for Υ in a conventional symmetry-breaking scenario. Let us define diquark operators qq ± = 1 2 (χ tr τ 2 χ ±χτ 2χ tr ) and source strengths j ± = j ±, so that the diquark terms in the action (3) read j + qq + + j − qq − . In the limit j − = 0 the equation of motion for the current is then In the same limit the U(1) B -equivalent of the axial Ward identity reads where the second equality assumes that the correlation function is dominated by a pseudo-Goldstone pole of the form (k 2 + M 2 − ) −1 , and |− denotes a one-Goldstone state. We now introduce the U(1) B -equivalent form of the PCAC hypothesis: where we have used the relation Υ = f 2 π derived in [10], and the second equality follows from (9). Combining (10) and (11) leads to the equivalent of the "Gell-Mann-Oakes-Renner" relation: This can be compared with numerical data for qq + (j) and M − (j) in [7]. At j = 0.3, M − = 0.95, qq + = 0.72 yielding Υ GM OR ≈ 1.9; at j = 0.1, M − = 0.4, qq + = 0.52 yielding Υ GM OR ≈ 2.6. We conclude Υ ≪ Υ GM OR , consistent with our hypothesis that no symmetry breaking occurs, but that the dynamics are dominated by long-range phase fluctuations of the order parameter field, described by a strongly-interacting scalar diquark field rather than a weakly-interacting Goldstone mode.

T > 0
In this section for the first time we explore the superfluid phase at non-zero temperature.
We expect a restoration to the normal phase at some critical T c . In a comparable numerical study of NJL 3+1 which exhibits superfluidity via orthodox symmetry breaking [3], the value of T c could be estimated from the zero temperature gap ∆ via the BCS prediction ∆/T c ≃ 1.76. Since this implied that L t had to exceed 35a for the system to be superfluid, an unambiguous extrapolation j → 0 to permit a systematic study of T > 0 was not possible. In the current case we shall see that although the j → 0 extrapolation still remains a problem, attaining T < T c is well within reach of the simulation. First let us review a heuristic argument for the expected value of T c , starting from the Hamiltonian H ef f (8) with φ 2 0 = Υ [9]. The phase field θ( x) may be disrupted by topologically non-trivial vortex excitations of the form θ = qψ, | ∇θ| = q/r, where q is integer and x is written (r, ψ). The energy of a single vortex is thus Since a vortex can be located on any of L 2 s lattice sites, the entropy The free energy F = E−T S thus changes sign for q = 1 vortices at a critical temperature The interpretation is that a phase transition separates a low-T superfluid phase in which vortices are confined to bound dipole pairs, and a high-T normal phase in which the vortex anti-vortex plasma screens the long-range interactions responsible for the divergent energy in (13). The relation (15) remains valid in a more sophisticated renormalisation group treatment, except that Υ is now T -dependent and should be replaced by its value Υ(T c ) exactly at the transition [12]. Combining our result for Υ with (15) yields an estimate L t ≈ 4.5 for the temporal lattice extent where a transition to the normal phase might be expected at µ = 0.8. Fig. 4 shows J 2 (j 0 ) on 32 2 × L t lattices with L t ranging from 64 all the way down to 2. At the extremes L t ≥ 32, L t ≤ 4 the data are reminiscent of those characterising respectively the  persists is unclear.
In Fig. 5 we try a different tactic, plotting J 2 for every even L t ∈ [2, 64] for fixed j 0 = 0.025. A linear fit J 0 = a 0 L t +a 1 through data with L t ≤ 42 seems quite reasonable, yielding a 0 = 0.00212(25), a 1 = −0.01537 (9). If we identify the intercept on the L taxis with the transition, we deduce L tc = 7.25(95) and hence Υ/T c = 1.02(13), to be compared with the theoretical value 0.637 from (15).

Summary
In this short study of the response of the system to a twisted diquark source forcing a baryon number current, we have provided direct evidence for the superfluid nature of the ground state of NJL 2+1 at high baryon density, and quantified it at one representative value of µ via the helicity modulus Υ. It should be stressed that the "physical" value Υ/Σ ≃ 0.2 quoted is still to be extrapolated to the continuum limit. It is probably more important to note that the numerical value of Υ is an order of magnitude smaller than might be expected in an orthodox symmetry breaking scenario, and is consistent with the non-Goldstone, strongly self-interacting nature of the scalar diquark excitations above the ground state.
We also studied the response of the system to non-zero temperature. Whilst we were unable to extrapolate to the zero source limit in a controlled way, by studying fixed j 0 we were able to estimate a critical temperature T c for breakdown of superfluidity of the same order of magnitude as, and only slightly smaller than, the Kosterlitz-Thouless prediction for a 2d system with U(1) global symmetry, which follows from characterising the superfluid/normal transition as arising from vortex pair unbinding. More refined simulations would be required to determine whether T c actually has the KT value, or whether NJL 2+1 , which in addition to the scalar diquark excitations contains massless fermion degrees of freedom, actually lies in a different universality class, as suggested by estimates of the critical exponent δ [7].