Baryons in the Gross-Neveu model in 1+1 dimensions at finite number of flavors
Julian J. Lenz, Laurin Pannullo, Marc Wagner, Björn Wellegehausen, Andreas Wipf
BBaryons in the Gross-Neveu model in 1+1 dimensions at finite number of flavors
Julian J. Lenz, ∗ Bj¨orn H. Wellegehausen, † and Andreas Wipf ‡ Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, 07743 Jena, Germany
Laurin Pannullo § Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany
Marc Wagner ¶ Institut f¨ur Theoretische Physik, Goethe-Universit¨at Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, GermanyHelmholtz Research Academy Hesse for FAIR, Campus Riedberg, Max-von-Laue-Straße 12, 60438 Frankfurt am Main, Germany
In a recent work [1] we studied the phase structure of the Gross-Neveu (GN) model in 1 + 1dimensions at finite number of fermion flavors N f = 2 , ,
16, finite temperature and finite chemicalpotential using lattice field theory. Most importantly, we found an inhomogeneous phase at lowtemperature and large chemical potential, quite similar to the analytically solvable N f → ∞ limit.In the present work we continue our lattice field theory investigation of the finite- N f GN modelby studying the formation of baryons, their spatial distribution and their relation to the chiralcondensate. As a preparatory step we also discuss a linear coupling of lattice fermions to the chemicalpotential.
I. INTRODUCTION
In recent years experiments provided many interestinginsights concerning strongly interacting matter at highdensity (see e.g. Ref. [2] for a comprehensive review).On the theoretical side our present understanding of theQCD phase diagram at non-zero chemical potential µ isto a large extent based on conjectures relying on physicalintuition, on model calculations and on effective low en-ergy descriptions [3, 4], while reliable ab-initio results arestill missing, mostly due to the infamous sign problemin lattice-QCD. Even though there are a number of in-teresting approaches, which led to considerable progressto mitigate or solve the sign problem, such as using com-plex Langevin algorithms [5–8] or thimble methods [9–11],finding more suitable variables [12–14] or refining the den-sity of states approach [15–19], a better understandingof lattice-QCD at finite baryon density is certainly anurgent problem. Urgent, for example, since our colleaguesfrom gravitational wave astronomy and astrophysics arein need of more reliable equations of state of stronglyinteracting matter at baryon density n B up to severaltimes the nuclear density n ≈ .
17 fm − .It has been conjectured that in QCD at low tempera-ture and large baryon density there is an inhomogeneouscrystalline phase. This conjecture is based on mean fieldcalculations in various effective four Fermi theories in-dicating the existence of such an inhomogeneous phase[20–24]. The underlying mean-field (or Hartree-Fock like)approximation becomes exact in the limit of an infinite ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] number of fermion flavors N f , since in this limit quantumfluctuations are negligible.Mean field approximations are also common in con-densed matter physics. For example, for the GN modelconsidered in the present work, the mean field phase dia-gram with homogeneous and inhomogeneous phases hasbeen known in the condensed matter community [25, 26]long before it has been rediscovered in particle physics[20, 27].More recently, interesting models implementing thebreaking of translational invariance – for example bycharge density waves, dynamical defects or by magneticfields – have been proposed and studied within the holo-graphic framework [28, 29].At present it is largely unknown, whether crystallinephases exist in effective four Fermi theories at finite num-ber of fermion flavors, or whether quantum fluctuationslead to a qualitatively different phase structure. In a re-cent work [1] we performed lattice field theory simulationsof the GN model in 1 + 1 dimensions with N f = 2 , , N f → ∞ . In the present work we continue our investi-gation of the GN model in 1 + 1 dimensions at a finitenumber of fermion flavors and focus on baryonic excita-tions at low temperature and large chemical potential.We investigate their spatial distribution as well as theirrelation to the chiral condensate II. CHEMICAL POTENTIAL FOR LATTICEFERMIONS
The continuum Lagrangian density of the (Euclidean)GN model with vanishing bare mass is given by L ψ = ¯ ψ i (cid:0) /∂ + µγ (cid:1) ψ + g N f ( ¯ ψψ ) , (1) a r X i v : . [ h e p - l a t ] J u l where µ denotes the chemical potential for the conservedbaryon number. To be able to perform the fermion inte-gration one follows Hubbard and Stratonovich by intro-ducing a fluctuating auxiliary scalar field σ to linearizethe operator ¯ ψψ in the interaction term L σ = ¯ ψ i Dψ + N f g σ , D = (cid:0) /∂ + σ + µγ (cid:1) . (2)The four Fermi term in Eq. (1) is recovered after elim-inating σ by its equation of motion or equivalently byintegrating over σ in the functional integral. Translationinvariance of d σ x in the (well-defined) functional integral (cid:81) d σ x for the lattice model implies the Ward identity N f g (cid:104) σ x (cid:105) = (cid:68) (cid:0) ¯ ψψ (cid:1) x (cid:69) . (3)Keeping as many global symmetries of the continuummodel as possible in a discretization can be crucial toobtain a lattice model with the correct continuum limit.Thus we shall discretize the operator D using the chiraland doubler-free SLAC derivative [30, 31]. While non-local SLAC fermions must not be used to discretize a fieldtheory with local gauge symmetries [32–34], they havebeen used successfully in various scalar-field theories andfermionic theories with global symmetries only [35–40].In addition to using SLAC fermions to simulate the GNmodel at finite N f , we have cross-checked our results witha discretization based on naive fermions. This fermionspecies is chiral as well but describes 2 d doublers in d dimensions. More details can be found in Ref. [1].Besides our preceding paper [1], we are not aware of anywork, in which SLAC fermions have been used to studyfermion systems at finite density. Thus, we begin withcomparing the thermodynamics of a gas of free massivefermions in a spatial box of size L in the continuumand on the lattice with different fermion discretiztations.A straightforward calculation yields the grand partitionfunction at inverse temperature β and chemical potential µ in the continuumln Z c = (cid:88) k (cid:16) βE k + ln (cid:16) e − β ( E k − µ ) (cid:17) + µ → − µ (cid:17) (4)with single-particle energies E k = k + m depending onthe spatial wave number k = 2 πn/L , n ∈ Z and mass m .The corresponding baryon density is n B,c = d ln Z c d µ = 1 L (cid:88) k (cid:18)
11 + e β ( E k − µ ) −
11 + e β ( E k + µ ) (cid:19) . (5)Note that the sum over all Matsubara frequencies hasalready been performed, i.e. the continuum limit in (imag-inary) time direction is already implied, while truncatingthe sum over k is conceptually similar to a finite latticespacing in spatial direction. Since the SLAC derivative discretizes the continuum dispersion relation up to themaximal momentum given by the inverse lattice spac-ing, the finite (truncated) sum is also the result for free,massive SLAC fermions discretized in spatial directiononly.This is not what is implemented in lattice Monte Carlosimulations where also the (imaginary) time is discretized.Asymptotically, the error in truncating the Matsubarasum after N t terms (with Matsubara frequencies symmet-rically about the origin) is ∼ β ( E k ± µ ) /π N t . Lettingthe lattice constant in time direction β/N t → fixed k ,we can neglect this error. However, E k will eventually be-come large in the sum over k and higher order correctionswill contribute, if the temporal “cutoff” N t /β is not sentto infinity before taking the limit L/N s →
0. In the partic-ular case of a uniform continuum limit N t = N s → ∞ , wepick up the following correction terms in 1 + 1 dimensions:lim N s = N t →∞ ln Z = ln Z c − µ π (6)lim N s = N t →∞ n B = n B,c − µ π . (7)As argued above, the expressions on the left correspondto the continuum limit of free massive SLAC fermions, for which the chemical potential enters the Lagrangianlinearly via i µ ¯ ψγ ψ as it does in the continuum, see e.g.Eq. (1). We conclude that introducing the chemical poten-tial linearly as in the continuum theory yields the correctpartition function up to a ( µ -dependent) constant and canthus be used in Monte Carlo simulations. In fact, sinceSLAC fermions couple fermion fields at well-separatedlattice sites it would be difficult to introduce an exponen-tially coupled µ for all hopping terms in the Lagrangian.In appendix A we show Eq. (7), i.e. that the baryon den-sity computed from a lattice action with linearly coupled µ has to be corrected by the constant + µ/ π . This isactually not a defect of SLAC fermions but just expressesthe fact that conventionally one first performs the contin-uum limit in time direction and afterwards in the spatialdirections to arrive at the well-known expression for thethermodynamic potentials at finite temperature and den-sity. A comparison of free massive baryon densities forvarious commonly used lattice discretizations is depictedin Fig. 1. The (properly corrected) SLAC result is almostindistinguishable from the continuum result. In passing,we note the following: • A similar analysis for naive fermions reveals thatintroducing a chemical potential as additive lin-ear term requires the same correction as for SLACfermions (when β/N t and L/N s approach zero si-multaneously) and thus could also be used. When letting N t = N s → ∞ at fixed box size. A detailedcalculation of the correction term at finite lattice spacing, possi-bly different in temporal and spatial direction, can be found inappendix A. m n B / m continuumSLAC (lin)Wilson (exp)staggered / naive (exp)naive (lin) Figure 1: Baryon density n B as a function of µ for freemassive fermions discretized on an N t × N s = 64 × Lm = 10, βm = 64. For SLAC fermionsthe additive linear coupling according to Eq. (7) hasbeen used, for Wilson fermions and for staggeredfermions only exponential coupling and for naivefermions we distinguish between conventionalexponential coupling (exp) and additive linear coupling(lin). Note that staggered fermions and naive fermionswith exponential coupling lead to identical results. • For d > c via0 = d ( n B − cµ )d µ (cid:12)(cid:12)(cid:12)(cid:12) µ =0 (8)( c is finite at finite lattice spacing a ) and subtracting cµ from the partition function. It has been demon-strated in Ref. [42] that this method of divergenceremoval works in (quenched) QCD. This is differentfrom earlier attempts to eliminate the divergenciesby suitably modifying the lattice action [43–45]. • The standard exponential coupling of the chemicalpotential for naive fermions has significantly largerdiscretization errors than the linear coupling (ifcorrected properly). In particular, a linearly cou-pled chemical potential yields quite accurately theposition of the first step, i.e. the fermion mass. • In appendix A we explicitly calculate the correctionterm for N t ≈ N s (cid:29) µ -dependent correctionterms are not lattice artifacts – they may also appear incontinuum theories, depending on how divergent integralsare treated (cf. the detailed discussion below Eq. (7)).We have crosschecked results obtained within this projectusing naive fermions with a conventional exponentiallycoupled chemical potential (see Ref. [1]) as this is theestablished method of introducing a chemical potentialin lattice field theory. Since the exponentially coupled µ couples to the exactly conserved charge on the lattice, nocorrection terms are needed [43]. III. BARYONIC MATTER
In a recent paper [1] we studied the phase structureof the GN model in 1 + 1 dimensions for N f = 8 usinglattice field theory and SLAC fermions with exact chiralsymmetry. The resulting phase diagram is shown in Fig. 2in units of σ = lim L →∞ (cid:104)| σ |(cid:105) (cid:12)(cid:12)(cid:12) µ =0 ,T =0 (9)with σ = 1 N t N s (cid:88) t,x σ ( t, x ) . (10)As expected, we identified a homogeneously broken phasewith non-zero constant chiral condensate at small chemicalpotential and low temperature and a symmetric phase athigh temperature. Most interestingly, however, we alsofound a phase, where the spatial correlator C ( x ) = (cid:10) c ( x ) (cid:11) , c ( x ) = 1 N t N s (cid:88) t,y σ ( t, y + x ) σ ( t, y )(11)is an oscillating function (see e.g. Fig. 4b, bottom). As asimple observable to distinguish the three phases we used C min = min x C ( x ) , (12)where C min (cid:29) ≈ < Note that our numerical results did not allow to decide whetherthese regions in the µ - T plane are phases in a strict thermody-namical sense or rather regimes, which strongly resemble phases.In any case, throughout this paper we denote these regions as“phases”. T / N f phase boundaries 0.20.00.20.40.60.81.0 C m i n / Figure 2: Phase diagram of the 1 + 1-dimensional GNmodel for N f = 8 (SLAC fermions, a ≈ . /σ , N s = 63; figure taken from Ref. [1]). The homogeneouslybroken phase, the symmetric phase and theinhomogeneous phase are colored in red, green and blue,respectively. For comparison also the N f → ∞ phaseboundaries are shown as gray lines.The GN model can be solved analytically in the semi-classical approximation or, equivalently, in the limit N f → ∞ (see e.g. Refs. [20, 27, 46, 47]) and it is knownthat extrema of the effective action S eff = 12 g (cid:90) d x σ − ln det D , (14)which one obtains by using L σ from Eq. (2) and integrat-ing over the fermions in the partition function, are notonly given by σ = const. For example in Ref. [20] it wasshown that at large chemical potential and small temper-ature a spatially oscillating function σ ( x ) minimizes thefree energy. For each cycle of the oscillation N f fermionsor antifermions, which can be interpreted as baryons,are located in the region of minimal σ , i.e. where thesign of σ changes. This implies breaking of translationalsymmetry and a crystal of baryons (as shown in Fig. 3).In the present work we investigate, whether tracesof such a baryonic crystal are also present in the GNmodel with a finite number of fermion flavors. We usethe same lattice setup as in our preceding paper [1]. Forall plots shown in the following we performed compu-tations with N f = 8 flavors of SLAC fermions, latticespacing a ≈ . /σ and N s = 63 lattice sites in spatialdirection, corresponding to a periodic spatial directionof extent L = N s a ≈ . /σ . From the extensive set ofsimulations we carried out in Ref. [1] for different a and L , we expect that both lattice discretization errors andfinite volume corrections are small. Note that in Ref. [1]we also performed computations with N f = 8 flavors ofnaive fermions, to check and to confirm our numericalresults. x n B / n B / Figure 3: N f → ∞ results from Ref. [47] for thecondensate σ ( x ) and the baryon density n B ( x ) at( µ/σ , T /σ ) = (0 . , A. Correlation of the baryon density and thecondensate
We start by investigating the location of the fermionsrelative to the spatially oscillating condensate inside theinhomogeneous phase. It is important to note that theeffective action (14) is invariant under spatial transla-tions. Therefore, field configurations, which are spatiallyshifted relative to each other, i.e. σ ( t, x ) and σ ( t, x + δx ),contribute with the same weight e − S eff to the partitionfunction and, thus, should be generated with the sameprobability by the HMC algorithm. Consequently, simpleobservables like (cid:104) σ ( x ) (cid:105) or (cid:104) n B ( x ) (cid:105) , where n B = i ¯ ψγ ψN f , (15)are not suited to detect an inhomogeneous condensateor baryon density in a lattice simulation, because de-structive interference should lead to (cid:104) σ ( x ) (cid:105) = 0 and (cid:104) n B ( x ) (cid:105) = const, even in cases, where all field config-urations exhibit spatial oscillations with the same wave-length. An observable, which does not suffer from de-structive interference and is able to exhibit informationabout possibly present inhomogeneous structures, is thespatial correlation function of σ ( x ), as defined in Eq. (11)(for a more detailed discussion see section 4.3 of Ref. [1]).Similarly, the spatial correlation function of the baryondensity and the squared condensate, C n B σ ( x ) = (cid:28) N t N s (cid:88) t,y n B ( t, y + x ) σ ( t, y ) (cid:29) , (16)can provide insights on the location of the fermionsrelative to the extrema of the condensate inside an in-homogeneous phase. For N f → ∞ the maxima of σ ( x )coincide with the minima of n B ( x ) and vice versa, asone can read off from Fig. 3. Thus, for N f → ∞ thecorrelator C n B σ ( x ) has minima at nλ/ C n B / x C / (a) Analytical results for N f → ∞ and( µ/σ , T /σ ) = (0 . , . C n B / x C / (b) Lattice field theory results for N f = 8 and( µ/σ , T /σ ) = (0 . , . Figure 4: The spatial correlation functions C n B σ ( x ) (top) and C ( x ) (bottom) as defined in Eqs. (16) and (11). Thevertical gray lines indicate the roots of C ( x ).( n + 1 / λ/
2, where λ is the wavelength of both σ ( x ) and n B ( x ) (see Fig. 4a, where C n B σ ( x ) and C ( x ) are shownfor ( µ/σ , T /σ ) = (0 . , . N f = 8 at the same chemical potentialand temperature exhibit an almost identical behavior (seeFig. 4b). We interpret this as clear signal that baryonsare centered at the roots of the condensate σ , where σ isa periodically oscillating function (we have investigatedthe latter in detail in our preceding work [1]). Thusseparations between neighboring baryons should all besimilar, which is reminiscent to the baryonic crystal foundat N f → ∞ . B. Baryon number and its relation to thecondensate
In this section we study the baryon number B = (cid:28)(cid:90) d x n B (cid:29) (17)with the baryon density n B as defined in Eq. (15) andinvestigate its relation to the average number of cyclesof the oscillating condensate σ . Inside the finite periodiclattice with extent L we have defined and computed ν max = L (cid:104)| k max |(cid:105) π . (18)By k max we denote the dominant momentum of c ( x ) (seeEq. (11)), i.e. that k , which maximizes the absolute valueof the Fourier transform ˜ c ( k ).The central result of this subsection is that B and ν max are almost identical on each field configuration, i.e. evenwhen omitting the average (cid:104) . . . (cid:105) over all generated fieldconfigurations in the definitions (17) and (18). In partic-ular at small T there is almost perfect agreement. This isillustrated in Fig. 5, where we show Monte Carlo histories
550 575 600 625 650 675Monte Carlo sweep8.008.258.508.759.009.25 B , m a x B max Figure 5: Monte Carlo histories of B and of ν max at( µ/σ , T /σ ) = (1 . , . B and of ν max at ( µ/σ , T /σ ) = (1 . , . N f = 8 behaves very similar tothe GN model in the limit N f → ∞ , where B = ν max .In the following we elaborate on this further by present-ing and discussing results obtained at two different valuesfor the temperature, T /σ ≈ .
076 and
T /σ ≈ .
1. Temperature
T /σ ≈ . For temperature
T /σ ≈ .
076 we show both B andΣ = (cid:104) σ (cid:105) σ (19)with σ and σ as defined in Eq. (9) and Eq. (10) as func-tions of the chemical potential µ in Fig. 6. At µ/σ ≈ . B B n B | N f , L L free fermions Figure 6: Baryon number B and Σ as functions of thechemical potential µ at temperature T /σ ≈ . n B | N f ,L →∞ L . The dashed verticalline indicates the phase transition at µ/σ ≈ . B ≈ µ , B suddenlystarts to increase at µ/σ ≈ .
51. At roughly the same µ value Σ rapidly drops from around 1 to 0. Note that B is quite similar to n B | N f ,L →∞ L (the green solid line inFig. 6), where n B | N f ,L →∞ is the analytical infinite volumeresult for the N f → ∞ baryon density according to Ref.[20]. However, the phase transition for N f = 8 takes placeat smaller µ/σ ≈ .
51 compared to µ/σ ≈ /π ≈ . N f → ∞ , where (at T = 0) a baryon corresponds toa kink-antikink field configuration σ with energy 2 σ /π per fermion (see e.g. Refs. [46, 49]). This can also be seenin the phase diagram shown in Fig. 2 and implies thatthe baryon mass (per fermion flavor) is somewhat smallercompared to N f → ∞ . Note that at finite N f a smallerhomogeneously broken phase is expected, because of fluc-tuations in σ , which increase disorder (see the detaileddiscussion in our preceding work [1]).The careful reader might already note the remnants ofthe stair-like low temperature behavior discussed later onin section III B 2. Moreover, at large µ the N f = 8 resultand the N f → ∞ result for the baryon number B are verysimilar, where the latter approaches the correspondingresult for free fermions, as noted in Ref. [27].In Fig. 7 we show B and ν max as functions of the chem-ical potential µ . The two curves are quite similar, whichis a strong indication that the finite- N f theory is qualita-tively well-described by the semi-classical N f → ∞ picture.Note that ν max is slightly below B for µ/σ > ∼ .
51. Thereason can be seen in Fig. 8, where thermalized MonteCarlo histories of B and ν max are shown for µ/σ = 1 . µ/σ > ∼ .
51 are simi-lar). On the majority of generated field configurations B and ν max agree rather well. However, B is an extremely B , m a x B max n B | N f , L L Figure 7: B and ν max as functions of the chemicalpotential µ at temperature T /σ ≈ . n B | N f ,L →∞ L .
350 400 450Monte Carlo sweep0.02.55.07.510.012.515.0 B , m a x B max max probabilityB probability Figure 8: Left plot: Monte Carlo histories of B and of ν max at ( µ/σ , T /σ ) = (1 . , . B and of ν max approximatingtheir probability distribution.stable quantity, while ν max exhibits sizable fluctuations onaround 25% of the generated field configurations, mostlyfluctuations towards small values, significantly below themedian of ν max . Such fluctuations are more common forlarger values of µ . This is expected from the Fourier trans-formed spatial correlation function, which we investigatedin Ref. [1] in detail, and also reflected by the correspond-ing histogram for ν max shown in the right part of Fig. 8.To summarize, Fig. 7 indicates that the baryon number B is quite similar to the number of cycles of the spatialoscillation of the condensate σ , as for N f → ∞ . Thisobservation supports our conclusions above that, also atfinite N f , baryons are the relevant excitations of the GNmodel and that their number is closely related to theshape of the condensate σ . B cold start hot start Figure 9: Monte Carlo histories of B at( µ/σ , T /σ ) = (0 . , . th measurement is shown.
2. Temperature
T /σ ≈ . Autocorrelatios at
T /σ ≈ .
038 turned out to be ratherlarge in our simulations, in particular near the boundary ofthe homogeneously broken phase and the inhomogeneousphase, in the region 0 . < ∼ µ/σ < ∼ .
65. This is illustratedin Fig. 9, where we compare Monte Carlo histories of B at µ/σ = 0 .
65 for a cold start (each field variable σ ( t, x ) =1) and a hot start (each field variable drawn randomlyfrom a Gaussian distribution with mean 0). After around1500 Monte Carlo sweeps the two Monte Carlo historieseventually converge and the simulations seem to havethermalized. Nevertheless the autocorrelation time isquite large, of the order of the average number of MonteCarlo sweeps needed to create or annihilate a baryon, i.e. > ∼ µ/σ ≈ .
65 the errors we showfor our results might be somewhat underestimated.For larger µ/σ , i.e. farther away from the phase bound-ary, autocorrelation times become smaller. For example,in Fig. 5 we show the Monte Carlo histories of B andof ν max at µ/σ = 1 .
10 after thermalization. For almostall field configurations B ≈ ν max and their values areeither close to 8 or close to 9. Even though we onlyshow 150 Monte Carlo sweeps, there are many transtionsbetween B ≈ ν max = 8 and B ≈ ν max = 9, indicatingthat the HMC algorithm is able to frequently increaseor decrease the number of cycles of the spatial oscilla-tion of the condensate σ . In Fig. 10 we show B and ν max as functions of µ in the range 0 . ≤ µ/σ ≤ . ν max (which are more pronounced at low temperature dueto less thermal fluctuations) are explained by the semi-classical commensurability constraint, namely that thewave length of the periodic condensate must divide the B , m a x B max B | N f , L < n B | N f , L L Figure 10: B and ν max as functions of the chemicalpotential µ at temperature T /σ ≈ . N f → ∞ lattice fieldtheory result for B at spatial lattice extent L ≈ . /σ (the same extent used in our simulations), while thepurple curve is the analytically known N f → ∞ continuum result for n B | N f ,L →∞ L .box length L . In the limit N f → ∞ the baryon numberis equal to the cycles of the condensate. Thus, a secondbaryon appears at chemical potential µ > µ c ≈ (2 /π ) σ ,a third baryon at µ > µ , etc. Hence, with increasing µ the mean separations of baryons decreases which leads tomore interaction. That the semi-classical picture explainsthe simulations so well is a further indication that theGN model at N f = 8 is quite similar to the GN model for N f → ∞ , i.e. quantum fluctuations at N f = 8 seem to berather weak.We note that the stair-like behavior is a consequence ofthe finite spatial extent L . To support this we minimizedthe SLAC regularized GN action with a specific ansatzfor the chiral condensate, σ ( x ) = A cos (cid:18) πL qx (cid:19) , q ∈ N , (20)in the variables A and q . This ansatz is a reasonableapproximation for the considered values of µ as the ana-lytically known chiral condensate in the N f , L → ∞ caserapidly reduces to a cos-shape for increasing chemicalpotential. This enables us to calculate B for N f → ∞ and finite L . The result for a ≈ . /σ and N s = 63corresponding to L = N s a ≈ . /σ (i.e. the same latticespacing and extent used in our simulations) exhibits clearsteps as shown in Fig. 10. The steps disappear for L → ∞ as shown by n B | N f ,L →∞ L , which is also plotted in Fig. 10.For µ/σ < ∼ .
65 autocorrelation times are extremelylarge and we were not able to reach thermal equilibriumin our simulations. This is shown in Fig. 11, where wepresent results for B obtained from cold starts and fromhot starts. For 0 . < ∼ µ/σ < ∼ .
65 the cold and the hotcurves differ and the largest discrepancy is observed closeto the phase transition around µ/σ ≈ .
51. We expectthe true result for B ( µ ) to be somewhere between the B T / T / T / Figure 11: B as a function of µ for T /σ ≈ . T /σ ≈ . B ( µ ). This is also supportedby our simulation results for B ( µ ) at T /σ ≈ .
076 (seesection III B 1), which is bounded by the cold and the hotcurves obtained at
T /σ ≈ .
038 in this range. Fig. 11as well as the exceedingly long autocorrelation times re-mind us of hysteresis effects near a first order transition.From the good agreement with the semi-classical picture,one could conjecture that in a finite volume the prob-ability distribution e − S eff /Z has two peaks (due to thecommensurability constraint) which leads to the observedhysteresis effects. The problem will probably go away forvery large volumes, but for high-precision simulations onfinite lattices near the transition improved algorithms areneeded, which support the creation and annihilation ofextended baryons.We remark that consequences of the large autocorrela-tions are also visible in the phase diagram shown in Fig. 2.In the problematic region, i.e. for 0 . < ∼ µ/σ < ∼ .
65 and
T /σ < ∼ .
05, the boundary between the homogeneouslybroken phase and the inhomogeneous phase suddenlyturns towards the origin, which amounts to an inhomo-geneous phase larger than expected and qualitativelydifferent from the N f → ∞ boundary. Knowing that allsimulations for this phase diagram were started with hotfield configurations, this behavior can now be understoodas a thermalization problem (see Fig. 11, where the resultfor B ( µ ) obtained with a hot start incorrectly indicatesthe phase boundary at a rather small value for µ . IV. CONCLUSIONS
In this work we investigated the distribution of thebaryon density n B in the N f = 8 GN model enclosed ina finite box of size L at finite chemical potential µ . Thesimulations were performed with chiral SLAC fermions.We compared with recent results on the spatial inhomo-geneities in the 1 + 1-dimensional GN model [1], whichis interpreted as modulation of baryonic matter densityas observed in the N f → ∞ limit of the model [20] andwhich is well-known in solid state physics [25, 26]. Sincetranslation symmetry is inherent to Monte Carlo simu-lations on finite lattices (the phase of a quasi-periodicconfiguration is a collective parameter) we could not mea-sure the baryon density n B ( x ) directly. Instead we founda strong correlation between the dominant wave numberof the spatial inhomogeneities and the baryon number.This is clear evidence for a region in the phase diagramcorresponding to a regime of modulated baryonic matter.Via this detour we explicitly circumvented the questionabout the breaking of translation symmetry. The delicatequestion whether we found a rigid baryon crystal (as seenin the large- N f limit) or rather a baryonic liquid [50],where the baryons have a preferred separation locally,but are disordered on large scales, has been addressed inRef. [1] and needs further investigations with improvedalgorithms.Our results shed further light on what can happenin quantum field theories at large fermion densities. Inparticular, it shows that mean-field and large- N f approxi-mations may contain more (hidden) information on thephysics at finite N f than one would expect. This is re-assuring, since in particle physics and even more so insolid state physics we often rely on these approximations.Our results may also be of relevance in condensed mattersystems, e.g. for large, almost 1-dimensional polymers[51].However, it is still unclear if the above results are rele-vant for QCD. On the one hand, we established that theinterpretation as baryonic matter is not spoiled by takingquantum fluctuations into account. On the other hand,although recent numerical lattice-studies of the GN modelin 2 + 1 dimensions and for N f → ∞ spotted inhomoge-nous condensates, the spatial modulation is related to thecutoff scale and disappears in the continuum limit [52, 53].Clearly, if this happens in the limit N f → ∞ , then wecannot expect a breaking of translation invariance for afinite number of flavours. Thus, extending our numeri-cal studies and simulations to fermion systems in higherdimensions is an important task. Interesting candidatesare for example the Nambu-Jona-Lasinio model or thequark-meson model in 3 + 1 dimensions. ACKNOWLEDGMENTS
We acknowledge useful discussions with Martin Ammon,Michael Buballa, Philippe de Forcrand, Holger Gies, FelixKarbstein, Adrian K¨onigstein, Maria Paola Lombardo,Dirk Rischke, Alessandro Sciarra, Lorenz von Smekal,Michael Thies and Marc Winstel on various aspects offermion theories and spacetime symmetries.We thank Philippe de Forcrand for bringing reference[50] to our attention.We also thank Marc Winstel for providing the codebase which was used to obtain the finite Volume N f → ∞ lattice results.Furthermore we thank Michael Thies for providing usthe analytic expression for the spatial correlator in the N f , L → ∞ limit.J.J.L. and A.W. have been supported by theDeutsche Forschungsgemeinschaft (DFG) under GrantNo. 406116891 within the Research Training Group RTG2522/1. L.P. and M.W. acknowledge support by theDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) through the CRC-TR 211 “Strong-interaction matter under extreme conditions” – projectnumber 315477589 – TRR 211. M.W. acknowledgessupport by the Heisenberg Programme of the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) – project number 399217702.Calculations on the GOETHE-HLR high-performancecomputers of the Frankfurt University as well as on theARA cluster of the University of Jena supported in part byDFG grants INST 275/334-1 FUGG and INST 275/363-1FUGG were conducted for this research. We would liketo thank HPC-Hessen, funded by the State Ministry ofHigher Education, Research and the Arts, and Andr´eSternbeck from the Universit¨atsrechenzentrum Jena forprogramming advice. Appendix A: Continuum limit for free fermions withlinearly coupled chemical potential µ In this appendix we calculate the correction term, whichmust be added to the baryon density, when using a linearlycoupled chemical potential and removing the lattice cutoffas it is typically done in lattice field theory.We consider non-interacting fermions linearly coupledto µ enclosed in a ( d − L subject to periodic boundary conditions. Theallowed wave numbers of the corresponding fermion fieldat finite temperature T = 1 /β are k ∈ { ( k , k ) } = (cid:26)(cid:18) ω n , πL n (cid:19)(cid:27) , ( n, n ) ∈ Z d (A1)with Matsubara frequencies ω n = 2 πβ (cid:18) n + 12 (cid:19) . (A2)The corresponding eigenvalues of the free Dirac operatorwith chemical potential are λ ± k = m ± (cid:112) ( µ + i k ) − k , (A3) such that λ + k λ − k = ( k − i µ ) + E k , E k = k + m . (A4)Because of spin, the degeneracy of the eigenvalues is C = 2 [ d/ − , where [ x ] denotes the greatest integer lessthan or equal to x .The logarithm of the grand partition function Z ( µ )divided by βV is the pressure p . The µ -derivative of p isthe baryon density, n B = ∂p∂µ = ∂∂µ ln Z ( µ ) βV = C βV (cid:88) k v ( E k ) , (A5)with v ( E k ) = ∂∂µ (cid:88) ω n ln λ + k λ − k . (A6)The sum defining v ( E k ) is convergent for any fixed k .But when we sum over the spatial momenta, E k (which isthe positive square root of E k ) becomes arbitrarily largeand we will show that removing the cutoffs in frequencyspace and momentum space does not commute.To this end, we regularize the sum over the Matsubarafrequencies (A6) by only admitting N t frequencies. It isconvenient to choose these frequencies symmetric to theorigin (which is only possible for even N t ), i.e. we restrict ω n according to | ω n | ≤ πN (cid:48) t β , N (cid:48) t = N t − . (A7)To calculate the truncated series over the ω n , denoted by v ( E k , N t ), we combine terms with ± ω n and obtain v ( E k , N t ) = (cid:88) | n + |≤ N (cid:48) t µ − E k ω n + ( µ − E k ) + (cid:0) E k → − E k (cid:1) . (A8)where ( E k → − E k ) represents the previous term withopposite sign of the energy. v ( E k , N t ) is finite for N t → ∞ and can be calculated, v ( E k ) = v ( E k , ∞ ) = βe β ( E k − µ ) + 1 − (cid:0) µ → − µ (cid:1) . (A9)Inserting (A9) into (A5) yields the baryon density n B = C V (cid:88) k (cid:18)
11 + e β ( E k − µ ) −
11 + e β ( E k + µ ) (cid:19) . (A10)An integration with respect to the chemical potential givesthe pressure of the Fermi gas. The integration constantis the divergent contribution of the quantum fluctuationsat zero temperature and zero chemical potential, pV C = (cid:88) k (cid:18) E k + 1 β ln (cid:0) e − β ( E k − µ ) (cid:1) + ( µ → − µ ) (cid:17) . (A11)0To summarize: if we first perform the continuum limit inEuclidean time direction, which means sum over all n ∈ Z in Eq. (A6), then we get the sum (cid:80) E k at µ = T = 0plus the finite sum known from quantum statistics.Since the sum over the Matsubara frequencies is onlyconditionally convergent and the sum over the k is di-vergent, we get a different result, when we remove thecutoff in the Matsubara frequencies together with thecutoff in the spatial momenta, as one does in lattice fieldtheory. To show that, we consider the difference betweenthe series (A9) and the finite sum (A8),∆ v ( E k , N t ) = v ( E k , ∞ ) − v ( E k , N t )= βπ F Nt +12 (cid:18) β ( µ − E k )2 π (cid:19) − βπ F Nt +12 (cid:18) β ( µ + E k )2 π (cid:19) , (A12)where we introduced the function F κ ( z ) = ∞ (cid:88) n =0 zz + ( n + κ ) = ∞ (cid:88) n,m =0 ( − m z m +1 ( n + κ ) m +2 (A13)with κ = ( N t + 1) /
2. For large κ the sum over n isapproximately given by ∞ (cid:88) n =0 n + κ ) s +1 = 1 sκ s − κ s +1 + . . . (A14)In the following we focus on free fermions in 1 + 1 dimen-sions, where only the first term gives a finite contributionto the error. The other terms are suppressed by inversepowers of N t . Thus, keeping the relevant term we arriveat F κ ( z ) = ∞ (cid:88) m =0 ( − m m + 1 (cid:18) zκ (cid:19) m +1 . (A15)For large spatial momenta we have µ (cid:28) E k and( µ + E k ) m +1 + ( µ − E k ) m +1 ∼ m + 1) µE m k , (A16)such that∆ v ( E k , N t ) = 2 βµπ ∞ (cid:88) m =0 ( − m (cid:18) β πκ (cid:19) m +1 E m k . (A17) To study the error for the baryon density∆ n B = 1 βL (cid:88) k ∆ v ( E k , N t ) (A18)(in 2 spacetime dimensions C = 1) as a function of the tem-poral and spatial cutoffs, we cut off the spatial momentaas | k | ≤ πN (cid:48) s L , N (cid:48) s = N s − . (A19)For convenience we choose the spatial momenta symmetricto the origin. Inserting Eq. (A17) into the regularizedsum (A18) we can calculate the leading term with thehelp of N (cid:48) s (cid:88) n = − N (cid:48) s E m k ∼ m + 1 (cid:18) πL (cid:19) m (cid:18) N s (cid:19) m +1 . (A20)This way we end up with∆ n B = 2 µπ ∞ (cid:88) m =0 ( − m m + 1 (cid:18) N s βN t L (cid:19) m +1 = 2 µπ atan (cid:18) N s βN t L (cid:19) . (A21)If we regularize the system on a lattice with the samelattice constant a in temporal and spatial direction thenthe argument of atan is equal to 1 and we obtain∆ n B = µ π . (A22)To summarize: If we use a linearly coupled chemical poten-tial for free fermions on the lattice, then we must subtractfrom the resulting baryon density n B a term linear in µ ,in order to recover the result obtained in a conventionalcontinuum calculation. In spacetime dimensions d > κ ∝ N t does not anymore balance the sum overthe spatial momenta.Finally, we note that higher order terms in this cor-rection (vanishing for N t → ∞ ) are now straightforwardto compute. For future reference, we just show the O ( a )correction on a hypercubic lattice with lattice spacing a ( β = aN t , L = aN s ):∆ n B = µ π (cid:18) − aπβ (cid:19) + O (cid:0) a (cid:1) . (A23)In higher orders in a , also terms ∼ µ n +1 , n ∈ N , appear. [1] Lenz, Julian and Pannullo, Laurin and Wagner, Marc andWellegehausen, Bj¨orn and Wipf, Andreas, Inhomogeneousphases in the Gross-Neveu model in 1+1 dimensions atfinite number of flavors , Phys. Rev. D (2020), Nr. 9 094512, arXiv:2004.00295.[2] B. Friman, C. H¨ohne, J. Knoll, S. Leupold, J. Randrup,R. Rapp und P. Senger,
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