Electric conductivity in finite-density SU(2) lattice gauge theory with dynamical fermions
aa r X i v : . [ h e p - l a t ] J u l Electric conductivity in finite-density SU (2) lattice gauge theory with dynamicalfermions P. V. Buividovich, ∗ D. Smith,
2, 3, 4, † and L. von Smekal
2, 3, ‡ Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK Institut f¨ur Theoretische Physik, Justus-Liebig-Universit¨at, 35392 Giessen, Germany Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Giessen, 35392 Giessen, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung, 64291 Darmstadt, Germany (Dated: July 10th, 2020)We study the dependence of the electric conductivity on chemical potential in finite-density SU (2)gauge theory with N f = 2 flavours of rooted staggered sea quarks, in combination with Wilson-Diracand Domain Wall valence quarks. The pion mass is reasonably small with m π /m ρ ≈ .
4. We con-centrate in particular on the vicinity of the chiral crossover, where we find the low-frequency electricconductivity to be most sensitive to small changes in fermion density. Working in the low-densityQCD-like regime with spontaneously broken chiral symmetry, we obtain an estimate of the first non-trivial coefficient of the expansion of conductivity σ ( T, µ ) = σ ( T, (cid:0) c ( T ) ( µ/T ) + O (cid:0) µ (cid:1)(cid:1) inpowers of µ , which takes its maximal value c ( T ) ≈ . ± .
05 around the critical temperature. Atlarger densities and lower temperatures, where the diquark condensation takes place, the conduc-tivity quickly grows with chemical potential, and also becomes closer to the free quark result. As aby-product of our study we confirm the conclusions of previous studies with heavier pion that for SU (2) gauge theory the ratio of crossover temperature to pion mass T c /m π is significantly smallerthan in real QCD. I. INTRODUCTION
Since quarks in QCD have finite electric charge, a hotQCD medium is characterized by some finite electric con-ductivity. It can be directly accessed in heavy-ion colli-sion experiments via the dilepton emission rate [1, 2],and is also of direct importance for the lifetime of strongmagnetic fields generated in off-central heavy-ion colli-sions [3].The temperature dependence of the electric conductiv-ity in QCD and QCD-like theories has been extensivelystudied by now. A lot of first-principle results are avail-able from lattice gauge theory simulations [4–11]. Theelectric conductivity was also calculated using a varietyof approximation methods which complement lattice sim-ulations, for instance, based on Boltzmann or Schwinger-Dyson equations [12–14], or hadron gas models [15, 16].However, there are practically no first-principle resultsregarding the dependence of the electric conductivity onbaryon chemical potential, apart from AdS/CFT calcu-lations [17, 18] which are not directly applicable to non-supersymmetric QCD. Symmetries of the QCD actionsuggest that the electric conductivity should be an evenfunction of the chemical potential µ , and thus can beexpanded in powers of µ as σ ( T, µ ) T = σ ( T, T (cid:18) c ( T ) (cid:16) µT (cid:17) + O (cid:0) µ (cid:1)(cid:19) . (1)The functional form (1) at small µ/T also agrees with ∗ [email protected] † [email protected] ‡ [email protected] the µ dependence of the electric conductivity obtained inAdS/CFT calculations [17, 18].In a calculation based on the off-shell Parton-Hadron-String Dynamics (PHSD) transport approach [19] thecoefficient c ( T ) in (1) was estimated as c ( T ) ≈ . T near the deconfinement transition [19]. A studybased on the Boltzmann equation within the quasiparti-cle approach also gives a result consistent with this es-timate [20], although only for a single non-zero value of µ . Within the dynamical quasiparticle model the depen-dence of the electric conductivity on the baryon chemicalpotential was found to be rather weak [21], which is con-sistent with results obtained using the Functional Renor-malization Group [22]. On the other hand, a kinetic the-ory calculation based on the hadron resonance gas model[23] suggests a strong dependence of σ on µ in the low-temperature hadronic phase, with σ/T changing almostby an order of magnitude as the chemical potential variesfrom µ = 0 . µ = 0 . σ reveals even anon-monotonic dependence of electric conductivity on µ [24].These estimates imply that a finite chemical potentialcan significantly change the electric conductivity in thephysically interesting part of the QCD phase diagramwith µ & T , where the QCD critical point is believed tobe located. This region of the phase diagram is in the fo-cus of ongoing heavy-ion collision experiments at RHICand LHC. Planned experiments at NICA and FAIR fa-cilities will achieve even larger baryon densities at lowertemperatures, hence even larger values of the ratio µ/T .Thus it is important to study the density dependence of σ in non-Abelian gauge theory from first principles in orderto correctly interpret the experimental data on dileptonemission rates.As is well known, due to the notorious fermionic signproblem, first-principle lattice QCD simulations can onlybe performed at small values of µ/T . Moreover, even atsmall µ/T the measurement of many physical observablesin lattice simulations is much more technically challeng-ing than in the case of µ = 0. If one is interested in ob-taining qualitative estimates rather than high-precisionresults, it is often helpful to consider QCD-like theorieswhich behave similarly to QCD in some regions of theirphase diagram, but have no fermionic sign problem. Ex-amples include gauge theories with SU (2) [25, 26] and G [27] gauge groups, as well as QCD at finite isospinchemical potential [28–30].In this work we perform a numerical study of thedependence of the electric conductivity on the fermionchemical potential in SU (2) gauge theory with dynam-ical fermions. “Baryons” in SU (2) gauge theory arediquarks, bound states of two quarks, which have thesame mass m π as the pion and are thus much lighterthan baryons in real QCD. Diquarks hence condense for µ & m π / m n ≫ m π happens at µ ≈ m n / ≫ m π /
2. Atnot very large values of the chemical potential outsideof the diquark condensation phase, however, the proper-ties of finite-density SU (2) gauge theory are expected tobe similar to those of real QCD. In particular, this sim-ilarity makes our estimate of the coefficient c ( T ) in theexpansion (1) relevant for real QCD, in a way analogousto orbifold equivalence, see, e.g. [31].Our main finding is that the electric conductivity ismost strongly sensitive to the quark density in the vicin-ity of the chiral crossover, where our estimate for the co-efficient c ( T ) in (1) is c ( T c ) ≈ . ± .
05, around threetimes larger than the corresponding free-quark result. This estimate implies that the chemical potential shouldbe at least several times larger than the temperature inorder to significantly affect the electric conductivity.Another part of the phase diagram where finite-density SU (2) gauge theory is expected to behave similarly toQCD is the conjectured quarkyonic phase at very lowtemperatures and high densities µ ≫ m n [32, 33]. Cal-culation of the electric conductivity in this part of thephase diagram could shed more light on the propertiesof the quarkyonic/color-superconducting phase. As wewill see, however, measurements of the electric conduc-tivity in this low- T , large- µ regime are numerically verychallenging, and we will leave a detailed study of this forfurther work.We also present data on the phase diagram of finite-density SU (2) gauge theory with N f = 2 fermion Of course, for a free quark gas the exact zero-frequency limitof the electric conductivity is ill-defined. However, it gets somefinite value within numerical analytic continuation methods usedto extract conductivity from Euclidean correlators, such as theBackus-Gilbert method used in this work. flavours which complements previous results [33–42] ob-tained either on smaller and coarser lattices, or forsmaller temperatures and larger densities, or with differ-ent lattice actions. We confirm the findings of [36] thatthe chiral crossover temperature in SU (2) gauge theoryis 3 to 5 times smaller than the pion mass, dependingon the chemical potential, in contrast to real QCD where T c & m π at µ = 0.The outline of the paper is the following: in Section IIwe present the details of our lattice setup and discussthe mixed fermionic action used to calculate the electricconductivity. In Section III we study the phase diagramof SU (2) gauge theory with N f = 2 rooted staggeredfermion flavours in the µ − T plane. In Section IV wediscuss our numerical approach to extract the electricconductivity from current-current correlators. In Sec-tion V we present our numerical results for the electricconductivity, estimated using both the simple “correlatormidpoint” estimate as well as using the more advancedBackus-Gilbert method. We briefly summarize our find-ings in the concluding Section VI. Some technical detailsof our calculations and analytic expressions for spectralfunctions of a free quark gas and a free pion gas at finitedensity are relegated to Appendices. II. LATTICE SETUP
Gauge field configurations were generated using thestandard Hybrid Monte-Carlo algorithm with N f = 2mass-degenerate rooted staggered fermions and a tree-level improved Symanzik gauge action. We acceler-ate both the HMC algorithm and the measurements ofcurrent-current correlators on GPUs. HMC is imple-mented with single-precision arithmetics within the CUDA framework, and measurements use double precision andare implemented using
OpenCL . The same algorithmicand lattice setup has been also used recently in [43].We use lattices with spatial size L s = 24 and tempo-ral sizes L t = 4 . . .
30, changing in steps of two. In thispaper we use a fixed-scale approach, choosing a singlevalue β = 1 . aµ = 0 . , . , . , . , . a . Our largest value of thechemical potential, aµ = 0 .
5, thereby represents a kindof compromise between probing the diquark condensa-tion phase (see Fig. 5) while still staying reasonably wellbelow half-filling and eventual saturation of the quarkdensity as obvious lattice artifacts. As discussed below,to facilitate diquark condensation in a finite volume, for L t ≥
10 we also generate gauge field configurations witha small diquark source term aλ = 5 · − .The numbers of gauge field configurations used in thiswork are summarized in Table I. To obtain these ensem-bles, we have saved gauge field configurations after every λ = 0 aλ = 5 · − L t \ aµ L s = 30 and aµ = 0 .
2, with N conf = 202 gauge fieldconfigurations for L t = 4 , , ,
10 and N conf = 123 gaugefield configurations for L t = 12.The measurements of current-current correlators areperformed mainly using Wilson-Dirac (WD) fermions.One of the technical advantage of using WD fermionsfor measuring the electric conductivity is that all datapoints in the current-current correlator in Green-Kuborelations (3) can be treated uniformly, whereas for stag-gered fermions even and odd time slices are typicallytreated separately [45] in order to filter out the contri-butions from non-taste-singlet states, which effectivelydecreases the signal-to-noise ratio.In addition, some of the measurements are also madewith Domain-Wall (DW) fermions [46], providing a cross-check for the Wilson-Dirac data. We use the distancebetween domain walls (lattice size along the fifth dimen-sion) L = 16, which is typically sufficient to suppressadditive mass renormalization [47, 48] for DW fermions.Such a mixed lattice action with staggered sea fermionsand DW valence fermions has already been used in anumber of studies of the nucleon axial charge [47–49].However, our primary motivation for using DW valencequarks is that we re-use fermion propagators enteringthe current-current correlators (8) to calculate also cor-relators of axial and vector currents. Those correlatorsare related to so-called anomalous transport coefficients[50, 51], which will be the subject of another forthcomingwork. Since the axial anomaly is very subtle for staggeredfermions, the use of DW valence fermions with good chi-ral properties is a big advantage for this kind of calcula-tions.To improve the chiral properties of DW and WDfermions without using much finer and larger lattices,we follow [47] and use HYP smearing [52] for gauge linksin the DW and WD Dirac operators. As in [47–49], bare quark masses in the WD andDW Dirac operators are tuned to match the pion mass m stagπ = 0 . ± .
002 obtained with staggered valencequarks with m stagq = 0 . m π on the bare quark masses of theWD and DW fermions on 24 ×
48 lattice with β = 1 . m π depends on m q as m π ∼ m q + ∆ m in accordancewith the Gell-Mann-Oakes-Renner relation, where ∆ m accounts for additive mass renormalization. From thesedata we have estimated that the bare quark mass shouldbe m W Dq = − .
21 for WD fermions and m DWq = 0 .
01 forDW fermions in order to match m stagπ = 0 . ± . m q = − . ± .
002 to obtain m π = 0 . ρ -meson obtained from the same en-semble on 24 ×
48 lattice with β = 1 . m ρ = 0 . ± . m ρ = 0 . ± .
05 for DW fermions,thus the ratio of pion and ρ -meson masses m π /m ρ ≈ . SU (2) gauge theory. Thepion Compton wavelength is almost four times smallerthan the lattice size, m π L s ≈ .
7, hence we expect finite-size artifacts to be reasonably small.
III. PHASE DIAGRAM OF FINITE-DENSITY SU (2) GAUGE THEORY
In addition to the chiral condensate h ¯ ψψ i and its sus-ceptibility which are conventionally used to map out thechiral crossover on the QCD phase diagram, the orderparameters of SU (2) gauge theory also include the di-quark condensate h ψψ i . In this Section we study theseorder parameters within our lattice setup and map outthe boundaries of regimes which favour chiral or diquarkcondensates.We first discuss the chiral and diquark condensates andthe corresponding susceptibilities. There are two sub-tleties which have to be taken into account when inter-preting the raw lattice data for these observables. First,the chiral condensate contains a UV divergent additivepart which might also depend on temperature and chem-ical potential. As discussed in [43, 53], this UV divergentpart can be removed by subtracting the first-order termof the Taylor expansion of the chiral condensate in pow-ers of the bare quark mass m q : h ¯ ψψ i sub = h ¯ ψψ i − ∂ h ¯ ψψ i ∂m q m q . (2)The effect of this subtraction on the L t dependence ofthe chiral condensate at µ = 0 is illustrated in Fig. 2.One can see that only after subtraction one can observean expected temperature dependence of the chiral con-densate and identify an inflection point which indicates a m p m qWD Wilson−DiracLinear fitStaggered 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 m p m qDW Domain WallLinear fitStaggered
FIG. 1. Squared pion mass m π calculated with Wilson-Dirac (on the left) and Domain Wall (on the right) valence fermions asa function of bare valence quark mass m q . −0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 5 10 15 20 25 30 C h i r a l c onden s a t e L t BareSubtracted x 5
FIG. 2. The effect of subtraction (2) on the temperature-dependent chiral condensate. crossover between the high- and low-temperature regimeswith (approximately) restored and spontaneously brokenchiral symmetry.The second subtlety is that in the chiral limit and at µ = 0 the ground states with nonzero chiral and diquarkcondensates have equal energies. An introduction of aDirac mass term, which is inevitable in Hybrid Monte-Carlo (HMC) simulations, breaks this degeneracy andbiases the system towards the phase with nonzero chi-ral condensate, which makes it difficult to observe thesignatures of the diquark condensation phase. In or-der to counteract this bias, for simulations at sufficientlylow temperatures ( L t >
10) we introduce a small di-quark source term in the action of the form λψψ with aλ = 5 · − (in lattice units) which makes the di-quark condensation more energetically favourable. Asillustrated in Fig. 3, the presence of this small sourceterm has little effect outside of the diquark condensationphase. As illustrated in Fig. 10 below, it also has practi-cally no effect on the electric conductivity. On the otherhand, in the diquark condensation phase it acts to rotate the chiral condensate into a diquark condensate and alsoproduces a clear peak in the diquark susceptibility, thusindicating the crossover temperature.In Fig. 4 we illustrate how the quark density, the sub-tracted chiral condensate, the diquark condensate andtheir corresponding disconnected susceptibilities dependon the temporal lattice size L t at three different valuesof the chemical potential aµ = 0, aµ = 0 . aµ = 0 . aµ = 0 we expect the conventional chiral symme-try breaking pattern, aµ = 0 . aµ = 0 . L t . For aµ ≤ . aµ ≥ . L t obtainedin this way are also in good agreement with the observedpeaks in the corresponding susceptibilities. Note thatsince the thermodynamic singularity in the chiral suscep-tibility is typically associated with disconnected fermiondiagrams, here we follow the conventional way and onlyplot this disconnected contribution. Since susceptibili-ties are much noisier observables than the condensates,we have decided to use the condensates instead of sus-ceptibility peaks to identify the crossover position.The resulting estimates for the boundaries of the chi-rally broken phase and the diquark condensation phaseare shown in Fig. 5. Blue and red points correspond to in-flection points of L t dependence of the chiral and diquarkcondensates, respectively. In this plot, the two pointswith the lowest temperature (corresponding to L t = 30)were obtained in a different way: here, we have fixed L t = 30 and looked for an inflection point in the µ depen-dence of the both the chiral (blue point) and the diquark(red point) condensates. The results coincide within sta-tistical and fitting uncertainties, which suggests that thetwo phases coexist in this region of the phase diagram. −0.001 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 12 14 16 18 20 22 24 26 28 30 C onden s a t e s L t m =0.2 l =0, chiral cond. sub. l =0, diquark cond. l =0.0005,chiral cond. sub. l =0.0005,diquark cond. −2−1.5−1−0.5 0 0.5 1 1.5 2 12 14 16 18 20 22 24 26 28 30 c d i s L t m =0.2 l =0.0, chiral l =0.0, diquark l =0.0005,chiral l =0.0005,diquark FIG. 3. Chiral and diquark condensates (left) and susceptibilities (right) at aµ = 0 . L t with and without a small diquark source. The resulting phase diagram is sketched in Fig. 5 andagrees well with previous results obtained in lattice sim-ulations on sufficiently fine lattices [36, 37, 42], as well aswithin the functional renormalization group approach ineffective low-energy theories [54, 55]. The chiral crossovermoves towards lower temperatures as µ is increased to-wards the diquark condensation threshold. In particular,above this threshold, the critical temperature of the su-perfluid diquark condensation phase only rather weaklydepends on µ as observed previously in two-color QCD[33, 36, 56] and analogously for the pion condensationphase in QCD at finite isospin density as well [30].An interesting feature of the phase diagrams obtainedfrom lattice simulations both in this work and in [36, 42]is that the chiral crossover happens at temperatureswhich are several times lower than the pion mass. Fromour data we estimate T c /m π ≈ .
37 at µ = 0, and T c /m π ≈ . aµ = 0 . m π /m ρ ), the ratios T c /m π at both µ = 0 and µ = m π / T c /m π are in sharpcontrast with real QCD, where T c ≈
155 MeV [57], m π ≈
135 MeV and hence T c /m π = 1 . >
1. Thisdifference might be explained by the fact that there are5 Goldstone bosons in SU (2) gauge theory with N f = 2flavours [26], in contrast to the 3 pions in N f = 2 QCD. IV. NUMERICAL MEASUREMENTS OFELECTRIC CONDUCTIVITY
By virtue of Green-Kubo relations [58], within thelinear response approximation the electric conductivity σ ( ω ) is related to correlators of same-direction vector currents: 1 V X ~x h j i ( τ, ~x ) j i (cid:16) ,~ (cid:17) i ≡ G ( τ ) == ∞ Z dω K ( τ, ω ) σ ( ω ) ,K ( τ, ω ) = ωπ cosh (cid:0) ω (cid:0) τ − T (cid:1)(cid:1) sinh (cid:0) ω T (cid:1) (3)where j i ( τ, ~x ) is the vector current density in some fixedspatial direction i = 1 , , P ~x denotes summation overspatial lattice coordinates, V = a L s is the spatial latticevolume and τ ∈ [0 . . . aL t ].While the inversion of the relation (3) is a numericallyill-posed problem, a number of practical inversion meth-ods have been developed which either take into accountsome prior knowledge of σ ( ω ) or return σ ( ω ) smearedover a certain frequency range of order of temperature[58]. In this work we use the Backus-Gilbert method [58]with Tikhonov regularization [59, 60] as implemented in[61].Within the Backus-Gilbert method we construct thelinear estimator of the conductivity based on the Eu-clidean current-current correlator (3): σ BG ( ω ) = X τ q τ ( ω ) G ( τ ) == + ∞ Z δ BG ( ω, ω ′ ) σ ( ω ′ ) ,δ BG ( ω, ω ′ ) = X τ q τ ( ω ) K ( τ, ω ′ ) . (4)where the resolution functions q τ ( ω ) are chosen in such away that, combined with the Green-Kubo kernel K ( τ, ω )in (3), they yield a smearing function δ BG ( ω, ω ′ ) whichapproximates the δ -function as closely as possible. In −0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0 5 10 15 20 25 30L tpc = 15.89 +/− 0.74 L t m =0.0, l =0.0000chiral cond. sub.diquark cond.density −0.05 0 0.05 0.1 0.15 0.2 5 10 15 20 25 c d i s L t m =0.0, l =0.0000chiraldiquark ( · tpc = 24.52 +/− 0.76 L t m =0.1, l =0.0005chiral cond. sub.diquark cond.density −0.2−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 30 c d i s L t m =0.1, l =0.0005chiraldiquark−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 4 6 8 10 12 14 16 18 20L tpc = 17.30 +/− 0.38 L t m =0.5, l =0.0005chiral cond. sub.diquark cond.density −8−6−4−2 0 2 4 6 8 10 4 6 8 10 12 14 16 18 20 c d i s L t m =0.5, l =0.0005chiraldiquark FIG. 4. On the left: Quark density, subtracted chiral condensate and diquark condensate as functions of the temporal latticesize L t . On the right: chiral susceptibility and diquark susceptibility as functions of L t . the Backus-Gilbert method, we minimize the “disper-sion” ∞ R dω ′ δ BG ( ω, ω ′ ) ( ω − ω ′ ) . This minimization re-quires an inversion of a certain ill-conditioned matrix con-structed from the kernel K ( τ, ω ). With Tikhonov regu-larization this inversion is regularized by replacing the in-verse singular values 1 /x i of this matrix by x i / (cid:0) x i + ∆ (cid:1) with some small ∆. This effectively cuts off the singularvalues x i which are smaller than ∆ and thus makes thematrix inversion well-defined.In contrast to other regularization schemes which use the covariance matrix for the Euclidean correlator in (3),with Tikhonov regularization the resolution functions donot depend on the data and thus neither on the chemicalpotential µ , which allows for a more meaningful compar-ison of data obtained at different values of µ , and withthe error-free data for free quarks as well. We calculatestatistical errors for the smeared conductivity using databinning [59].Since the smeared conductivity σ BG ( ω ) in practicequite strongly depends on the regularization of the ma-trix inversion and the value of regularization parameters, T / m p L t m / m p m=0.005, b =1.7a m diquarkchiral FIG. 5. Numerical estimate of the phase diagram of finite-density SU (2) gauge theory with N f = 2 rooted staggeredfermions. Blue and red points correspond to inflection pointsof L t dependence of the chiral and diquark condensates, re-spectively. Empty black circles correspond to parameter val-ues for which the electric conductivity was analyzed. the Backus-Gilbert method to some extent still suffersfrom the inherent ambiguity which is typical for numer-ically ill-defined analytic continuation problems [58]. Toassess any residual ambiguity, in addition, we also con-sider an alternative simple estimator of the low-frequencyconductivity [11]. Namely, according to the Green-Kuborelation in (3), the current-current correlator on the l.h.s.of Eq. (3), at the maximal Euclidean time separation τ = aL t /
2, is related to the spectral function via G ( aL t /
2) = ∞ Z dω K ( aL t / , ω ) σ ( ω ) . (5)The function K ( aL t / , ω ) = ω/π (cid:0) sinh (cid:0) ω T (cid:1)(cid:1) − is local-ized within the region of small frequencies ω ∼ T and canbe also considered as a “smeared” δ -function similar tothe one used in the Backus-Gilbert method. The normand width of this function are: N ≡ ∞ Z dω K ( aL t / , ω ) = πT , ∆ ω = vuuut N − ∞ Z dω ω K ( aL t / , ω ) = √ πT. (6)We can thus use the value of the Euclidean correlatorat midpoint as an estimator σ MP of electric conductivity σ ( ω ) smeared over frequencies in the range ω . √ πT ≈ . T : σ MP = 1 πT G ( aL t / . (7)In Fig. 6 we compare the resolution function N − K ( aL t / , ω ) for the midpoint estimator (7)with resolution functions in the Backus-Gilbert method.The contribution of connected fermionic diagrams tothe current-current correlator in (3) for a single gauge N − K ( b / , w ) , d B G ( , w ) w/T MidpointBG, L t =12, D =1.0E−10BG, L t =16, D =2.0E−9BG, L t =20, D =2.0E−7BG, L t =20, D =1.0E−12BG, L t =12, D =1.0E−12 FIG. 6. Comparison of resolution functions δ BG (0 , ω ) and (cid:0) πT (cid:1) − K ( aL t / , ω ) for Backus-Gilbert and midpoint esti-mates of the low-frequency limit of the electric conductivity. field configuration can be written as a single trace overfermionic indices (spin, color and lattice coordinates): h j x,µ j y,ν i conn == C em Tr (cid:18) ∂D∂θ x,µ D − ∂D∂θ y,ν D − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 , (8)where x , y and µ , ν label the sites and the direc-tions on the four-dimensional (for WD fermions) or five-dimensional (for DW fermions) lattice, and D is eitherthe WD or DW Dirac operator in the background of thenon-Abelian gauge fields and an Abelian lattice gaugefield θ x,µ , with link factors e iθ x,µ . The electric chargefactor C em = X f = u,d q f = 5 / u - and d -quarks. In this work we follow most of the previous lat-tice QCD studies of the electric conductivity [4–11] andpresent all the results with C em factored out.At x = y there is also an additional contact term con-tribution Tr (cid:16) ∂ D∂θ x,µ D − (cid:17)(cid:12)(cid:12)(cid:12) θ =0 to the correlator (8). Thiscontact term affects only the high-frequency behavior ofthe electric conductivity, and we disregard it in the fol-lowing. The time slice τ = 0 in the current-current cor-relator (3), for which this contact term is relevant, isdiscarded within the Backus-Gilbert method.In the presence of a nonzero diquark source λ the ex-pression for the connected contribution (8) is somewhatmore complicated, and is given in Appendix C.The contributions from disconnected fermionic dia-grams h j x,µ j y,ν i disc = C disc ×× Tr (cid:18) ∂D∂θ x,µ D − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 Tr (cid:18) ∂D∂θ y,ν D − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 , (10)to the current-current correlator (3) are typically smalland noisy, and in addition weighted by the charge factor C disc = P f = u,d q f ! = 1 /
9, which is five times smallerthan the charge factor C em = 5 / x = 0 . . . L − y = 0 . . . L − L , thus making calculations with DW fermionssignificantly more expensive than with WD fermions.In Appendix B we discuss a small trick which allowsto halve this numerical cost. Further, to obtain phys-ical results with DW fermions it is crucial to subtractthe contribution of five-dimensional bulk Dirac modes,which becomes quite significant at high temperaturesand/or at small values of τ in (3). As discussed in [46],this contribution can be compensated by the contribu-tion of bosonic Pauli-Villars fields which live on the five-dimensional lattice with two times smaller size L in thefifth dimension. This contribution is equal to minus twicethe correlator (8) calculated with L P V = L / m P V = 1 in the DW Dirac operator.
V. NUMERICAL RESULTSA. Euclidean correlators and midpointconductivity estimates
We start the discussion of our numerical results byconsidering Euclidean current-current correlators whichenter the Green-Kubo relations (3). In Fig. 7 we plot con-nected current-current correlators obtained with Wilson-Dirac fermions at three different temperatures corre-sponding to L t = 12 , ,
20 and compare them withthe corresponding disconnected contributions, as well aswith the corresponding correlators for free quarks on thesame lattice (plots on the right). Both connected anddisconnected contributions were calculated for all config-urations in the ensembles listed in Table I, and in addi-tion averaged over 10 . . .
30 random source positions inorder to reduce statistical errors.One can see that as the chemical potential graduallyincreases, the connected current-current correlators (8)around mid-point become larger and more flat, whichagrees qualitatively with the expected growth of the elec-tric conductivity in a finite-density system. On the otherhand, the behavior at small Euclidean time separations,which is most sensitive to the high-frequency part of thespectrum, is practically unaffected by finite density.We’ve also invested a significant amount of CPU/GPUtime into measuring disconnected contributions (10), and were not able to detect any statistically significant devia-tion from zero. For small values of the chemical potentialwe were able to reduce statistical errors of disconnectedcontributions such that they are at least 3 − µ and smaller T we were not ableto reduce statistical errors of disconnected contributionsbelow 10 . . .
20% of the connected ones. Thus we cannotrule out that disconnected contributions might becomeimportant at very large densities and low temperatures,for instance, in the quarkyonic phase.We also note that for lower temperatures and largervalues of the chemical potential the current-current cor-relators become significantly noisier. In addition, theirstatistical distribution seems to develop heavy tails, sothat contributions from outlier configurations becomemore and more important. These outlier configura-tions present a major challenge for calculating connectedcurrent-current correlators at low temperatures and highdensities.A comparison with the free quark results is shown onplots on the right in Fig. 7). For this as well as for allother calculations with free quarks we use the bare quarkmass am = 0 .
01, which corresponds to the optimal valueof quark mass in the DW Dirac operator which, for in-teracting theory, reproduces the pion mass obtained withstaggered fermions (see Fig. 1). This choice is dictatedby the expected smallness of mass renormalization effectsfor DW fermions, which makes the bare and renormal-ized masses close to each other. Clearly, for free WDfermions mass renormalization is absent, and we can usethe same value of bare mass as for free DW fermions. Inany case, current-current correlators depend only weaklyon the bare quark mass, and changing its value by ± aµ = 0 . µ the current-current correlator around midpoint becomes smaller thanthe corresponding free-quark result.In order to check how the chiral properties of lat-tice fermions might affect the electric conductivity,in Fig. 8 we plot the ratios of connected contribu-tions (8) to current-current correlators in (3) calculatedwith Wilson-Dirac (WD) and with Domain-Wall (DW)quarks. Around mid-point, the results obtained withboth WD and DW Dirac operators agree within statis-tical errors. For DW Dirac operators, the latter are no-ticeably larger due to smaller statistics (as calculationswith DW quarks are more than an order of magnitudemore expensive than with WD quarks). A salient fea- −4 −3 −2 −1
0 2 4 6 8 10 12 G W D ( t ) / C e m t aL t = 12a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 12 G W D ( t ) / G W D ( t ) t aL t = 12a m =0.00a m =0.05a m =0.10 a m =0.20a m =0.5010 −4 −3 −2 −1
0 2 4 6 8 10 12 14 16 G W D ( t ) / C e m t aL t = 16a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 2 4 6 8 10 12 14 16 G W D ( t ) / G W D ( t ) t aL t = 16a m =0.00a m =0.05a m =0.10 a m =0.20a m =0.5010 −4 −3 −2 −1
0 5 10 15 20 G W D ( t ) / C e m t aL t = 20a m =0.00a m =0.05a m =0.10a m =0.20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 G W D ( t ) / G W D ( t ) t aL t = 20a m =0.00a m =0.05a m =0.10a m =0.20 FIG. 7.
On the left: current-current correlators G WD ( τ ) obtained with Wilson-Dirac fermions. On the right: the ratio ofcurrent-current correlators G WD ( τ ) in SU (2) theory to the current-current correlators G WD ( τ ) calculated for non-interactingWilson-Dirac quarks. On both sides, empty symbols show the corresponding disconnected contribution, multiplied by the ratioof charge factors C disc /C em = 1 / ture of the current-current correlators for DW fermionsis that they strongly deviate from the Wilson-Dirac resultat short Euclidean time separations, where the contribu-tion of five-dimensional bulk modes becomes importantand is not completely cancelled by Pauli-Villars regulatorfields.In order to estimate possible finite-volume artifactsin our study, in Fig. 9 we compare finite-density con-nected current-current correlators calculated on latticeswith L s = 24 and L s = 30 using Wilson-Dirac fermions with aµ = 0 . SU (2) lattice gauge theoryand for free quarks. The deviations clearly grow towardslower temperatures, but do not exceed 2%. An importantobservation is that in the full gauge theory deviations dueto finite-volume effects appear to have opposite sign tothose in the free quark case.We now turn to the estimates of the low-frequencyelectric conductivity σ MP based on the mid-point val-ues of current-current correlators, as defined in (7). Asdiscussed in Section IV above, these estimates are com-0 G D W ( t ) / G W D ( t ) t aL t = 12a m =0.00, freea m =0.50, freea m =0.00a m =0.10a m =0.20a m =0.50 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 G D W ( t ) / G W D ( t ) t aL t = 16a m =0.00, freea m =0.50, freea m =0.00a m =0.10a m =0.20a m =0.50 FIG. 8. The ratio of current-current correlators G WD ( τ ) and G DW ( τ ) obtained with Wilson-Dirac and Domain Wall fermions,respectively. For comparison, we also plot the corresponding ratio G WD ( τ ) /G DW ( τ ) for free Wilson-Dirac and Domain Wallquarks. −0.06−0.04−0.02 0 0.02 0.04 0.06 0 2 4 6 8 10 12 G W D ( t ) / G W D ( t ) − t aa m =0.2 L t =4L t =6L t =8L t =10L t =12 FIG. 9. Relative difference G WD ( τ, L s = 30) /G WD ( τ, L s = 24) − G WD ( τ ) calculated on lattices withspatial sizes L s = 24 and L s = 30. Solid lines show the samerelative difference calculated for free quarks. pletely model-independent and do not depend in any wayon the method of performing numerical analytic contin-uation of Euclidean data. An analysis of current-currentcorrelators for free quarks (see Appendix D) suggests thatthe midpoint estimator is also somewhat less affected byfinite-volume effects.In Fig. 10 we show the dependence of the ratio σ MP / ( C em T ) on the inverse temperature 1 / ( aT ) ≡ L t in lattice units, calculated using Wilson-Dirac fermions.We combine the data points obtained with zero diquarksource λ = 0 at L t <
10 and with aλ = 5 · − at L t ≥
10. As we demonstrate in Fig. 11 below, introduc-ing a small diquark mass term has no noticeable effecton current-current correlators for all temperatures andchemical potentials which we consider. Points with errorbars correspond to lattice data in the full gauge theory,and solid lines are free quark results on the same lattices s M P / ( C e m T ) L t T/T c a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 FIG. 10. Midpoint estimate σ MP of the low-frequency elec-tric conductivity obtained with Wilson-Dirac fermions on thelattice with L s = 24 as a function of temperature at differ-ent values of the chemical potential. The data with λ = 0and L t <
10 is combined with data with aλ = 5 · − and L t ≥
10. Solid lines are the results obtained with free quarksat the same temperature and chemical potential. as well as in the infinite-volume and continuum limit.For small values of chemical potential the ratio σ MP / ( C em T ) in full gauge theory appears to be slowlydecreasing towards lower temperatures. On the otherhand, the corresponding free quark result grows ratherquickly towards lower temperatures. As a result, at lowtemperatures σ MP / ( C em T ) in the full gauge theory is2 . σ MP /T towards low temperatures for free quarks is afinite-volume artifact, and in the infinite-volume, con-tinuum and massless limits σ MP /T is constant for freefermions: σ MP /T = N c π = 0 . T /T c ∼ . s M P / ( C e m T ) L t a m =0.00continuum,free,inf.vol.WD, l =0,freeWD,a l =0.0005WD, l =0DW, l =0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 s M P / ( C e m T ) L t a m =0.20continuum,free,inf.vol.WD, l =0,freeWD,a l =0.0005WD, l =0DW, l =0 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 s M P / ( C e m T ) L t a m =0.50continuum,free,inf.vol.WD, l =0,freeWD,a l =0.0005WD, l =0DW, l =0 FIG. 11. Midpoint estimates of the low-frequency conductivity for different lattice ensembles and different valence quarkactions: Wilson-Dirac quarks with λ = 0 and aλ = 5 · − , Domain-Wall quarks with λ = 0, free Wilson-Dirac quarks with λ = 0, and free quarks in the continuum and infinite-volume limits. For all lattice data L s = 24. Clearly, simulations at larger lattices are necessary toprecisely determine the electric conductivity at low tem-peratures. Especially at low temperatures and at finitedensity, such simulations with dynamical fermions arecomputationally very challenging. We note, however,that since at low temperatures the correlation length ismuch larger for free quarks than for the full gauge theory,finite-volume effects can be expected to be smaller in thegauge theory.It is also interesting to note that for our largest twovalues of the chemical potential, aµ = 0 . aµ = 0 .
5, the mid-point estimate σ MP /T appears to becloser to the free quark result than for the lower densities.This is probably due to the fact that large densities movethe system closer to the high-energy regime of asymptoticfreedom.According to Fig. 5, for aµ = 0 . L t = 20 ,
22 weshould already be in the superconducting diquark con-densation phase. Interestingly, superconductivity doesnot show up as a sharp increase in conductivity, here. In-stead, the conductivity even falls slightly below the freequark result as we reach the diquark condensation phasewith lowering the temperature at aµ = 0 .
5. Most likely,the dramatic changes expected in the transport-peak partof the electric conductivity are simply not captured byour frequency-smeared conductivity estimates, and arealso to some extent compensated by the suppression ofthe higher-frequency conductivity at large µ . A moredetailed study of the electric conductivity in the super-conducting phase would certainly be interesting, but isbeyond the scope of this work.In order to further check the independence of our esti-mates on the choice of the fermionic action and diquarksource term, in Fig. 11 we compare the temperature de-pendence of the midpoint estimator σ MP /T for DomainWall fermions and for Wilson-Dirac fermions with andwithout the diquark source term. One can see that allestimates agree within statistical errors.Finally, we use our results for σ MP /T to estimate theexpansion coefficient c ( T ) in (1), which characterizes thesensitivity of low-frequency electric conductivity to chem-ical potential. To this end we use the difference between σ MP at µ = 0 and aµ = 0 .
05. This value of aµ is a c L t T/T c WD,MPWD,BGFree WD quarks,MPFree WD quarks,BGFree cont.quarks,MP
FIG. 12. Numerical estimate of the coefficient c in (1) fromthe finite difference between midpoint estimates σ MP of thelow-frequency electric conductivity at µ = 0 and aµ = 0 . compromise between having a finite difference which isconsiderably larger than statistical errors, and being stillin the small- µ QCD-like regime far from the diquark con-densation phase. We thus approximate c ( T ) as c ( T ) ≈ σ MP ( T, aµ = 0 . − σ MP ( T, µ = 0)( aµL t ) σ MP ( T, µ = 0) . (11)The resulting temperature dependence of c ( T ) forWilson-Dirac fermions is illustrated in Fig. 12, along withreference results for free fermions on the lattice and in thecontinuum. While in the infinite-volume, continuum andmassless limits the coefficient c ( T ) based on the mid-point estimate for free quarks is c ( T ) = − π π = 0 . c ( T ) be-comes negative. As discussed in Appendix D, for freequarks this behavior is a finite-volume artifact, and inthe large-volume limit the lattice free-quark estimate of c ( T ) becomes closer to the continuum value and dependsweaker on the temperature.A noticeable feature is that the temperature depen-2dence of c ( T ) in the full gauge theory is quite differentfrom the free quark result, especially in the vicinity of thechiral crossover, where c ( T ) is almost three times largerthan the free quark result. This suggests that the electricconductivity should be most sensitive to finite density inthe crossover regime. Since this statement is based onthe data obtained in the low-density QCD-like regime of SU (2) gauge theory, it should be also qualitatively cor-rect for the full QCD. B. Estimates of spectral functions from theBackus-Gilbert method
In this Section we turn to the estimates of the elec-tric conductivity based on the Backus-Gilbert methodoutlined in Section IV. We implement the resolutionfunctions in the Backus-Gilbert transformation (4) onthe discrete grid of frequency values ω = j T with j = 0 , , . . . L t . For each L t we tune the value ofthe Tikhonov regularization parameter ∆ in the Backus-Gilbert method as follows: we perform the analysis with∆ = 1 · − , · − , · − , · − , . . . , · − ,starting from ∆ = 1 · − , and choose the least value of∆ for which the Backus-Gilbert estimate of the spectralfunction is positive and its maximal relative error is lessthan 10% for all values of chemical potential. Such tuningyielded the following values: ∆ = 1 · − for L t = 12,∆ = 1 · − for L t = 14, ∆ = 2 · − for L t = 16and ∆ = 2 · − for L t = 18 , ,
22. As illustrated inFig. 6, with these values of ∆ the resolution functions δ BG (0 , ω ) are still very close to resolution functions cal-culated with a very small reference value ∆ = 10 − .They are noticeably narrower than the resolution func-tion N − K ( aL t / , ω ) for the midpoint estimate (7).In Fig. 13 we show the Backus-Gilbert estimates of thespectral function σ BG ( ω ) for Wilson-Dirac fermions atdifferent values of chemical potential and different tem-peratures, and compare them with corresponding esti-mates for free quarks. To understand how the inherentsmearing within the Backus-Gilbert method as well aslattice artifacts and finite-volume effects affect the spec-tral functions, in the plots on the right in Fig. 13 we alsocompare the Backus-Gilbert estimates σ BG ( ω ) for freequarks with analytically calculated spectral functions inthe continuum theory. Analytic expressions for spectralfunctions are summarized in Appendix A. For illustrativepurposes, in Fig. 13 we have replaced the infinitely nar-row transport peaks of free continuum quarks (the termproportional to the δ -function in (A2)) by Breit-Wignerdistributions ( α/T ) / (cid:16) ω/T ) (cid:17) of unit width, where α is the δ -function prefactor in (A2).Finite chemical potential affects the spectral functionsof free continuum quarks in two competing ways, whichbecome especially evident in the zero-temperature limit(see equation (A8)). On the one hand, the height ofthe transport peak at ω = 0 grows approximately as µ (neglecting the small quark mass), i.e. in proportionto the area of the Fermi surface. On the other hand,chemical potential makes the finite-frequency part of thespectral function vanish for w < { m q , µ } . In thecondensed-matter physics language, the transport peakand the finite-frequency part of the spectral function orig-inate from intraband and interband transitions, respec-tively.For low frequencies the Backus-Gilbert estimator σ BG ( ω ) receives contributions from both the transportpeak and the finite-frequency spectral functions. As a re-sult, the strength of the transport peak is rather stronglyover-estimated for low densities, as one can also see fromthe plots on the right in Fig. 13. For larger densities( aµ = 0 . aµ = 0 .
5) the finite-frequency part of thespectral function is separated from the transport peak bya rather wide gap, wider than the width of the resolutionfunctions in the Backus-Gilbert method. As a result, forlarge densities the Backus-Gilbert method captures thestrength of the transport peak more precisely. As couldbe expected, the Backus-Gilbert estimates of the spec-tral functions most strongly deviate from the continuumresults in the vicinity of the gap between the transportpeak and the finite-frequency part of the spectral func-tion. This is a direct consequence of the smearing whichremoves sharp threshold effects and also smears out thetransport peak. The deviation of lattice and continuumresults in the high-frequency tails of the spectral func-tions is most likely a lattice artifact.Comparing now the Backus-Gilbert estimates for thefull gauge theory and for free quarks, we see that, asthe temperature is decreased, the low-frequency spec-tral function becomes significantly smaller for the fullgauge theory at all values of the chemical potential.On the other hand, for ω/T ∼ aω ≈ . . . . . ρ -meson resonance, which becomes very wide due tosmearing. At the largest value of the chemical potential aµ = 0 . µ and as func-tions of µ at fixed temperature. Data points with errorbars correspond to the full gauge theory, and solid linescorrespond to Backus-Gilbert estimates for free quarks3 s B G ( w ) / ( C e m T ) w/TL t = 12 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0.2 0.5 2 0.1 1 0 5 10 15 20 s B G ( w ) / ( C e m T ) w/TL t = 12 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0.2 0.5 2 0.1 1 0 5 10 15 20 25 30 s B G ( w ) / ( C e m T ) w/TL t = 16 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0.2 0.5 2 0.1 1 0 5 10 15 20 25 30 s B G ( w ) / ( C e m T ) w/TL t = 16 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0.2 0.5 2 0.1 1 0 5 10 15 20 25 30 35 40 s B G ( w ) / ( C e m T ) w/TL t = 20 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 0.2 0.5 2 0.1 1 0 5 10 15 20 25 30 35 40 s B G ( w ) / ( C e m T ) w/TL t = 20 a m =0.00a m =0.05a m =0.10a m =0.20a m =0.50 FIG. 13.
On the left:
Frequency-smeared electric conductivity σ BG ( ω ) extracted from Euclidean correlation functions usingthe Backus-Gilbert method (points with error bars). Solid lines are the smeared spectral functions obtained for free quarks usingthe same procedure. On the right we compare these free-fermion smeared spectral functions (points) with analytic expression(A2) for the electric conductivity of continuum Dirac fermions (solid lines). To illustrate the magnitude of the “transport peak”term in (A2), the delta-function δ ( ω ) was replaced by a fixed-width Breit-Wigner distribution ( α/T ) / (cid:0) ω/T ) (cid:1) , where α is the δ -function prefactor in (A2). on the same lattices. The overall picture is consistentwith the results from the midpoint estimators - for thefull gauge theory the temperature dependence of the con-ductivity is much weaker than it is for free quarks. Whileat high temperatures the conductivity is close to the freequark result, it differs by a factor of 2 − aµ = 0 . L t = 20 , c ( T )in the expansion (1) of electric conductivity in powersof µ/T . We again use the finite-difference approxima-4 s B G ( w = ) / ( C e m T ) L t a m =0.00a m =0.05a m =0.10 a m =0.20a m =0.50 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 s B G ( w = ) / ( C e m T ) a m L t =12L t =14L t =16 L t =18L t =20L t =22 FIG. 14. On the left: Backus-Gilbert estimate of the ω → σ BG ( ω = 0) /T of the electric conductivity as a function ofthe inverse temperature L t = 1 / ( aT ). On the right: σ BG ( ω = 0) /T as a function of chemical potential at different L t . tion (11), replacing σ MP with σ BG ( ω = 0). The resultis shown in Fig. 12 together with the result based on themidpoint estimate. Both results appear to be consistentwith each other, within statistical errors, and they ex-hibit the largest deviations from the free quark result inthe vicinity of the chiral crossover. VI. CONCLUSIONS AND DISCUSSION
We have studied the low-frequency electric conductiv-ity in finite-density SU (2) gauge theory with dynam-ical fermions at various temperatures across the chi-ral crossover, both within the phase with spontaneouslybroken chiral symmetry and around the transition tothe diquark condensation phase. As a by-product ofour study, we have also obtained new estimates of thephase boundaries of SU (2) gauge theory, as summarizedin Fig. 5. An interesting observation, which confirmsthe findings of [36, 42], is that in SU (2) gauge theorythe chiral crossover happens at rather low temperatures, T c /m π ≈ .
37. In contrast, in real QCD T c /m π & aµ = 0 . σ (0) /T ≈ . ± .
02 at tem-peratures around T c is in agreement with the results ob-tained in full lattice QCD [7–9]. The decrease of theabsolute value of σ (0) /T across the crossover in SU (2)gauge theory turns out to be not as significant as in fullQCD. This is expectable, since for smaller N c the differ-ence in the number of degrees of freedom between confine-ment ( O (1)) and deconfinement O (cid:0) N c (cid:1) regimes is also smaller. However, in comparison with the free quark re-sult the conductivity in the full gauge theory drops by afactor of 2 − T /T c ∼ .
8, which is again in agreementwith [7–9]. It is interesting that for all temperatures anddensities which we have considered the conductivity isstill much larger than the conductivity of a free pion gas,calculated in Appendix A.The result which should be most relevant for real QCDis our estimate of the first nontrivial coefficient c ( T ) inthe expansion (1) of low-frequency electric conductivityin powers of chemical potential over temperature µ/T .This result is obtained within the low-density QCD-likephase with spontaneously broken chiral symmetry andno diquark condensation. The maximal value of c ( T ) is c ( T ) ≈ . ± .
05 in the vicinity of the chiral crossover,which is almost three times larger than the free quarkresult obtained within the same regularization. This es-timate suggests that even for T ≈ T c and µ/T ∼ − c ( T ) becomes closer to thefree quark result. However, especially at low tempera-tures c ( T ) is plagued by large finite-volume artifacts (seeAppendix A), which calls for computationally expensivesimulations on larger lattices. For free quarks the originof the finite-volume artifacts can be traced back to theinterplay between the discrete spectrum of states and theFermi surface: at low temperatures only the states in thevicinity of the Fermi surface contribute to conductivity.However, for small µ and small lattices the number ofdiscrete states which are inside the Fermi surface tendsto become very small and changes in discrete steps aseither L s or µ change, thus leading to dramatic finite-volume artifacts. Clearly, for full SU (2) gauge theory inthe confinement regime, where excitations are bosonic,the Fermi surface is not defined. Thus one can expect5the finite-volume artifacts to be of different nature in theconfinement regime.The maximum of c ( T ) in the vicinity of the chiralcrossover can be also explained by the following qualita-tive argument. As shown in Appendix A, c ( T ) decreasestowards higher temperatures for free quarks, but growsfor free pion gas. Thus a maximum can be expectedfor intermediate temperatures between the regimes whereeach of these two approximations are valid.We have also observed that, as the quark density in-creases and the temperature decreases, the contributionof disconnected fermionic diagrams to the current-currentcorrelators becomes more significant. We cannot rule outthat it can be as large as ∼ . . .
20% of the connectedcontributions for the largest value of the chemical poten-tial aµ = 0 . ACKNOWLEDGMENTS [1] S. Campbell, Nucl. Phys. A , 177 (2017),ArXiv:1704.06307.[2] L. D. McLerran and T. Toimela,Phys. Rev. D , 545 (1985).[3] L. McLerran and V. Skokov,Nucl. Phys. A , 184 190 (929), ArXiv:1305.0774.[4] B. B. Brandt, A. Francis, T. Harris, H. B. Meyer,and A. Steinberg, EPJ Web Conf. , 07044 (2018),ArXiv:1710.07050.[5] H. Ding, O. Kaczmarek, and F. Meyer,Phys. Rev. D , 034504 (2016), ArXiv:1604.06712.[6] J. Ghiglieri, O. Kaczmarek, M. Laine, and F. Meyer,Phys. Rev. D , 016005 (2016), ArXiv:1604.07544.[7] B. B. Brandt, A. Francis, B. Jaeger, and H. B. Meyer,Phys. Rev. D , 054510 (2016), ArXiv:1512.07249.[8] G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands, andJ. Skullerud, JHEP , 186 (2015), ArXiv:1412.6411.[9] A. Amato, G. Aarts, C. Allton, P. Giudice, S. Hands,and J. Skullerud, Phys. Rev. Lett. , 172001 (2013),ArXiv:1307.6763.[10] P. V. Buividovich, M. N. Chernodub, D. E. Kharzeev,T. Kalaydzhyan, E. V. Luschevskaya, and M. I.Polikarpov, Phys. Rev. Lett. , 132001 (2010),ArXiv:1003.2180.[11] H. Ding, A. Francis, O. Kaczmarek,F. Karsch, E. Laermann, and W. Soeldner,Phys. Rev. D , 034504 (2011), ArXiv:1012.4963.[12] M. Greif, I. Bouras, Z. Xu, and C. Greiner,Phys. Rev. D , 094014 (2014), ArXiv:1408.7049.[13] A. Puglisi, S. Plumari, and V. Greco,Phys. Rev. D , 114009 (2014), ArXiv:1408.7043.[14] S. Qin, Phys. Lett. B , 358 (2015),ArXiv:1307.4587. [15] M. Greif, C. Greiner, and G. S. Denicol,Phys. Rev. D , 059902 (2017), ArXiv:1602.05085.[16] D. Fernandez-Fraile and A. GomezNicola, Phys. Rev. D , 045025 (2006),ArXiv:hep-ph/0512283.[17] A. Karch and A. O’Bannon, JHEP , 024 (2007),ArXiv:0705.3870.[18] K.-Y. Kim, S.-J. Sin, and I. Zahed,JHEP , 096 (2008), ArXiv:0803.0318.[19] T. Steinert and W. Cassing,Phys. Rev. C , 035203 (2014), ArXiv:1312.3189.[20] P. K. Srivastava, L. Thakur, and B. K. Patra,Phys. Rev. C , 044903 (2015), ArXiv:1501.03576.[21] O. Soloveva, P. Moreau, and E. Bratkovskaya,Phys. Rev. C , 045203 (2020), ArXiv:1911.08547.[22] R.-A. Tripolt, C. Jung, N. Tanji, L. von Smekal,and J. Wambach, Nucl. Phys. A , 775 (2019),ArXiv:1807.04952.[23] G. Kadam, H. Mishra, and L. Thakur,Phys. Rev. D , 114001 (2018), ArXiv:1712.03805.[24] S. Ghosh, Phys. Rev. D , 036018 (2017),ArXiv:1607.01340.[25] J. B. Kogut, D. K. Sinclair, S. J. Hands, andS. E. Morrison, Phys. Rev. D , 094505 (2001),ArXiv:hep-lat/0105026.[26] J. B. Kogut, M. A. Stephanov, D. Toublan,J. J. M. Verbaarschot, and A. Zhitnitsky,Nucl. Phys. B , 477 (2000), ArXiv:hep-ph/0001171.[27] A. Maas, L. von Smekal, B. Wellegehausen, and A. Wipf,Phys. Rev. D , 111901 (2012), ArXiv:1203.5653.[28] D. T. Son and M. A. Stephanov,Phys. Rev. Lett. , 592 (2001),ArXiv:hep-ph/0005225. [29] K. Kamikado, N. Strodthoff, L. von Smekal, andJ. Wambach, Phys. Lett. B , 1044 (2013),ArXiv:1207.0400.[30] B. B. Brandt, G. Endrodi, and S. Schmalzbauer,Phys. Rev. D , 054514 (2018), ArXiv:1712.08190.[31] M. Hanada and N. Yamamoto, JHEP , 138 (2012),ArXiv:1103.5480.[32] L. McLerran and R. D. Pisarski,Nucl. Phys. A , 83 (2007), ArXiv:0706.2191.[33] V. V. Braguta, E. Ilgenfritz, A. Y. Ko-tov, A. V. Molochkov, and A. A. Nikolaev,Phys. Rev. D , 114510 (2016), ArXiv:1605.04090.[34] S. Hands, S. Kim, and J. Skullerud,Eur. Phys. J. C , 193 (2006), ArXiv:hep-lat/0604004.[35] S. Hands, S. Kim, and J. Skullerud,Phys. Rev. D , 091502 (2010), ArXiv:1001.1682.[36] S. Cotter, P. Giudice, S. Hands, and J. Skullerud,Phys. Rev. D , 034507 (2013), ArXiv:1210.4496.[37] T. Boz, P. Giudice, S. Hands, J. Skullerud, andA. G. Williams, AIP Conf. Proc. , 060019 (2016),ArXiv:1502.01219.[38] V. G. Bornyakov, V. V. Braguta, E. Ilgenfritz, A. Y.Kotov, A. V. Molochkov, and A. A. Nikolaev,JHEP , 161 (2018), ArXiv:1711.01869.[39] L. Holicki, J. Wilhelm, D. Smith, B. Wellegehausen,and L. von Smekal, PoS LATTICE2016 , 052 (2017),ArXiv:1701.04664.[40] K. Iida, E. Itou, and T. Lee, JHEP , 181 (2020),ArXiv:1910.07872.[41] R. Contant and M. Q. Huber,Phys. Rev. D , 014016 (2020), ArXiv:1909.12796.[42] T. Boz, P. Giudice, S. Hands, and J. Skullerud,Phys. Rev. D , 074506 (2020), ArXiv:1912.10975.[43] J. Wilhelm, L. Holicki, D. Smith, B. Wellegehausen,and L. von Smekal, Phys. Rev. D , 114507 (2019),ArXiv:1910.04495.[44] D. Scheffler, C. Schmidt, D. Smith, and L. von Smekal,PoS LATTICE2013 , 191 (2013), ArXiv:1311.4324.[45] G. Aarts, C. Allton, J. Foley, S. Hands, andS. Kim, Phys. Rev. Lett. , 022002 (2007),ArXiv:hep-lat/0703008.[46] V. Furman and Y. Shamir,Nucl. Phys. B , 54 (1995), ArXiv:hep-lat/9405004.[47] R. G. Edwards, G. T. Fleming, P. Hagler, J. W. Negele,K. Orginos, A. Pochinsky, D. B. Renner, D. G. Richards,and W. Schroers, Phys. Rev. Lett. , 052001 (2006),ArXiv:hep-lat/0510062.[48] E. Berkowitz, D. Brantley, C. Bouchard, C. Chang,M. A. Clark, N. Garron, B. Joo, T. Kurth, C. Monahan,H. Monge-Camacho, A. Nicholson, K. Orginos, E. Ri-naldi, P. Vranas, and A. Walker-Loud, “An accurate cal-culation of the nucleon axial charge with lattice QCD,”(2017), ArXiv:1704.01114.[49] D. B. Renner, W. Schroers, R. Edwards,G. T. Fleming, P. Hagler, J. W. Negele,K. Orginos, A. V. Pochinski, and D. Richards,Nucl. Phys. Proc. Suppl. , 255 (2005),ArXiv:hep-lat/0409130.[50] I. Amado, K. Landsteiner, and F. Pena-Benitez,JHEP , 081 (2011), ArXiv:1102.4577.[51] P. V. Buividovich, Nucl. Phys. A , 218 (2014),ArXiv:1312.1843.[52] A. Hasenfratz and F. Knechtli,Phys. Rev. D , 034504 (2001), ArXiv:hep-lat/0103029.[53] W. Unger, The chiral phase transition of QCD with 2+1flavors : a lattice study on Goldstone modes and uni-versal scaling , Ph.D. thesis, Bielefeld University, PhysicsDepartment (2010).[54] N. Strodthoff, B. Schaefer, and L. von Smekal,Phys. Rev. D , 074007 (2012), ArXiv:1112.5401.[55] N. Strodthoff and L. von Smekal,Phys. Lett. B , 350 (2014), ArXiv:1306.2897.[56] N. Y. Astrakhantsev, V. G. Bornyakov, V. V.Braguta, E. Ilgenfritz, A. Y. Kotov, A. A. Niko-laev, and A. Rothkopf, JHEP , 171 (2019),ArXiv:1808.06466.[57] A. Bazavov, T. Bhattacharya, M. Cheng, C. DeTar,H. Ding, S. Gottlieb, R. Gupta, P. Hegde, U. M. Heller,F. Karsch, E. Laermann, L. Levkova, S. Mukherjee,P. Petreczky, C. Schmidt, R. A. Soltz, W. Soeldner,R. Sugar, D. Toussaint, W. Unger, and P. Vranas,Phys. Rev. D , 054503 (2012), ArXiv:1111.1710.[58] H. B. Meyer, Eur. Phys. J. A , 86 (2011),ArXiv:1104.3708.[59] M. Ulybyshev, C. Winterowd, and S. Zafeiropoulos,Phys. Rev. B , 205115 (2017), ArXiv:1707.04212.[60] R. Tripolt, P. Gubler, M. Ulybyshev, and L. vonSmekal, Comput.Phys.Commun. , 129 (2019),ArXiv:1801.10348.[61] M. V. Ulybyshev, “Green-Kubo solver,” GitHub reposi-tory (2017).[62] S. Ghosh, S. Mitra, and S. Sarkar,Nucl. Phys. A , 237 (2018), ArXiv:1711.08257. Appendix A: Current-current correlators andspectral functions for free quarks and free pions
Using the standard tools of finite-temperature fieldtheory, we obtain the following expression for the Eu-clidean current-current correlator of free quarks with N c colors: G qE ( τ ) = N c π ∞ Z m dǫ (cid:0) ǫ − m (cid:1) ǫ ×× (cid:0) ǫ − µ T (cid:1) + 1cosh (cid:0) ǫ + µ T (cid:1) ! ++ N c π ∞ Z m dǫ ǫ p ǫ − m (cid:18) m ǫ (cid:19) ×× cosh (cid:0) ǫ (cid:0) τ − T (cid:1)(cid:1) cosh (cid:0) ǫ − µ T (cid:1) cosh (cid:0) ǫ + µ T (cid:1) (A1)The corresponding spectral function in the Green-Kuborelation (3) is σ q ( ω ) = αN c π T δ ( ω ) ++ N c π Re (cid:0) ω − m (cid:1) (cid:18) m ω (cid:19) ×× sinh (cid:0) ω T (cid:1) cosh (cid:0) ω − µ T (cid:1) cosh (cid:0) ω +2 µ T (cid:1) , (A2)7where α = ∞ Z m dǫ (cid:0) ǫ − m (cid:1) ǫ ×× (cid:0) ǫ − µ T (cid:1) + 1cosh (cid:0) ǫ + µ T (cid:1) ! . (A3)In the phase with spontaneously broken chiral sym-metry, the conductivity is expected to be dominated bycharged pions contributions. The leading-order contri-bution is just the conductivity of free massive chargedscalar fields at finite chemical potential [16], with elec-tric current defined as j µ = i (cid:0) ¯ φ∂ µ φ − (cid:0) ∂ µ ¯ φ (cid:1) φ (cid:1) . (A4)A straightforward calculation yields the following expres-sion for the Euclidean correlator of spatial currents of acharged scalar field: G πE ( τ ) = 16 π ∞ Z m dǫ (cid:0) ǫ − m (cid:1) ǫ ××
14 sinh (cid:0) ǫ + µ T (cid:1) + 14 sinh (cid:0) ǫ − µ T (cid:1) ++ 2 e ǫ/T cosh (cid:0) ǫ (cid:0) τ − T (cid:1)(cid:1)(cid:0) e ( ǫ + µ ) /T − (cid:1) (cid:0) e ( ǫ − µ ) /T − (cid:1) ! . (A5)The corresponding AC conductivity is σ π ( ω ) = α π π T δ ( ω ) ++ 148 π Re (cid:0) ω − m (cid:1) ω ×× (cid:0) e ω/T − (cid:1)(cid:16) e ω +2 µ T − (cid:17) (cid:16) e ω − µ T − (cid:17) , (A6)where α π = ∞ Z m dǫ (cid:0) ǫ − m (cid:1) ǫ ×× (cid:0) ǫ − µ T (cid:1) + 1sinh (cid:0) ǫ + µ T (cid:1) ! . (A7)It is instructive also to consider the low-temperature limitof these expressions. In the limit T →
0, free quarkconductivity takes the formlim T → σ q ( ω ) = N c √ π Re (cid:0) µ − m (cid:1) µ δ ( ω ) ++ N c π Re (cid:0) ω − m (cid:1) (cid:18) m ω (cid:19) θ ( ω − µ ) , (A8) where θ ( x ) is the Heaviside unit step function. In thesame limit, the free pion conductivity takes the formlim T → σ π ( ω ) = 148 π Re (cid:0) ω − m (cid:1) ω θ ( ω − µ ) . (A9)In contrast to the free fermion case, the term with the δ -function vanishes in the limit of zero temperatures as √ πm T π δ ( ω ) e ( µ − m ) /T . Of course, for free bosons theconductivity is only defined for µ ≤ m .It is instructive to compare the midpoint estimatesof the low-frequency electric conductivity, which we usein Subsection V A, for free quarks and free pions. Tothis end we use the bare quark mass m = 0 .
01, andthe pion mass m π = 0 . aµ = 0 .
05, which is below thepion/diquark condensation threshold, and temperaturesin the range 1 / ( aT ) = 16 . . .
22, we find that the mid-point conductivity estimate is 5 . . .
10 times smaller forthe pion gas than for free quarks. On the other hand, themidpoint estimate for the pion gas conductivity showsmuch stronger dependence on the chemical potential, asalso noticed in [11, 62]. Also, for pion gas the coefficient c ( T ) in (1) grows with temperature, whereas for the freequark gas c ( T ) decreases with temperature. Appendix B: Efficient calculation of correlators ofconserved currents for Wilson-Dirac and DomainWall fermions
Since for either the Wilson-Dirac or Domain Wallfermions the conserved current operator j z,µ =¯ ψ x ∂D xy ∂θ z,µ ψ y with the single-particle current operator ∂D xy ∂θ z,µ == iP + µ U z,µ δ x,z δ y,z +ˆ µ − iP − µ U † z,µ δ x,z +ˆ µ δ y,z . (B1)is localized on two lattice adjacent lattice sites z and z +ˆ µ ,a straightforward calculation of the connected part (8) ofcurrent-current correlators requires2 × × N d × N c (B2)inversions of the Dirac operator. In this expression N c is the number of independent source vector orientationsin color space which is obviously equal to the numberof colors, N d = 4 is the number of independent sourceorientations in spinor space, the first factor of two comesfrom the necessity to have source vectors localized at twolattice sites z and z + ˆ µ , and the second factor of twoaccounts for the inversions of both D and D † .Especially for Domain Wall fermions these inversionsbecome extremely costly due to the summation of five-dimensional vector current over the fifth dimension,which is necessary to obtain the conserved vector cur-rent and the correct form of the axial current.8Here we describe a small trick which was used in thiswork to halve the number of Dirac operator inversions.A straightforward idea is to try to diagonalize the single-particle current operator in the 2 N c N d -dimensional lin-ear space spanned on source vectors localized either at z or z + ˆ µ and having all possible colour and spin orienta-tions. However, a simple check reveals that the matrix ∂D x,y ∂θ z,µ (with x , y and the corresponding implicit spinorand color indices considered as matrix indices, and z and µ as parameters) is nilpotent and cannot be diagonalized.A physical reason for this nilpotency is that the currentoperator j x,µ moves electric charge from lattice site x tosite x + ˆ µ . After the first application of j x,µ to somestate there is no electric charge at site x , thus applying j x,µ second time just produces zero. Instead of diago-nalization, in this situation one should rather use Jordandecomposition. We have found that ∂D x,y ∂θ z,µ admits Jordandecomposition of the following form: ∂D ( x,α,a );( y,β,b ) ∂θ z,µ = X A,γ =1 , N c X c =1 ψ ( A,γ,c ) x,α,a ¯ χ ( A,γ,c ) y,β,b ,ψ (1 ,γ,c ) x,α,a = iδ x,z φ (+ γ ) α κ ( c ) a e iθ c ,χ (1 ,γ,c ) y,β,b = δ y,z +ˆ µ φ (+ γ ) β κ ( c ) b ,ψ (2 ,γ,c ) x,α,a = − iδ x,z +ˆ µ φ ( − γ ) α κ ( c ) a e − iθ c ,χ (2 ,γ,c ) y,β,b = δ y,z φ (+ γ ) β κ ( c ) b . (B3)Here φ ( ± γ ) α are orthonormal eigenspinors of the projectionoperators P ± µ , with (cid:0) P ± µ (cid:1) αβ = X γ =1 , φ ( ± γ ) α ¯ φ ( ± γ ) β . (B4)Similarly, orthonormal color vectors κ ( c ) a and phases e iθ c form an eigensystem of the link matrix U z,µ :( U z,µ ) ab = N c X c =1 κ ( c ) a e iθ c ¯ κ ( c ) b . (B5)Omitting all matrix indices, the Jordan decomposition(B3) can be compactly written as ∂D∂θ z,µ = X A,γ,c ψ ( A,γ,c ) ¯ χ ( A,γ,c ) . (B6) Inserting this decomposition for one of the current oper-ators in the current-current correlator (8), we obtain X ~y Tr (cid:18) ∂D∂θ z,µ D − ∂D∂θ y,ν D − (cid:19) == X A,γ,c ¯ χ ( A,γ,c ) D − X ~y ∂D∂θ y,ν D − φ ( A,γ,c ) , (B7)where Dirac vectors χ and φ are constructed for the link( z, µ ) according to (B3) and P ~y denotes summation overspatial lattice volume with time-like component y fixed.The operator P ~y ∂D∂θ y,ν is obviously a local lattice opera-tor which can be applied to Dirac vectors in CPU timecomparable with the application of the Dirac operatoritself. Expression (B7) suggests that the connected con-tribution to current-current correlators (8) can be cal-culated as follows. For each A = 1 , γ = 1 , c = 1 . . . N c we have to do two Dirac operator inver-sions, one to calculate D − φ ( A,γ,c ) and the other to cal-culate ¯ χ ( A,γ,c ) D − = (cid:0) D † (cid:1) − χ ( A,γ,c ) . This amounts to8 N c Dirac operator inversions in total, to be comparedwith 16 N c inversions which would be required for a morestraightforward calculation.The same trick can be applied to the calculation ofdisconnected current-current correlators of the form X ~y h Tr (cid:18) ∂D∂θ z,µ D − (cid:19) Tr (cid:18) ∂D∂θ y,ν D − (cid:19) i . (B8)In this case one can use the Jordan decomposition trickto calculate the trace Tr (cid:16) ∂D∂θ z,µ D − (cid:17) , and stochastic es-timator techniques for the second trace. Appendix C: Current-current correlators in thepresence of diquark sources
In the presence of diquark sources, the current-currentconnected correlator is different from (8) and takes some-what more complicated form: h j x,µ j y,ν i conn = ∂ ∂θ x,µ ∂θ y,ν Tr ln (cid:0) DD † + λ (cid:1) == Re Tr (cid:18) D † (cid:0) DD † + λ (cid:1) − ∂D∂θ x,µ D † (cid:0) DD † + λ (cid:1) − ∂D∂θ y,ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 ++ λ Re Tr (cid:18)(cid:0) D † D + λ (cid:1) − ∂D∂θ x,µ ∂D † ∂θ y,ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 ++ Re Tr (cid:18) D † (cid:0) DD † + λ (cid:1) − ∂ D∂θ x,µ ∂θ y,ν (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) θ =0 (C1)9In the usual conductivity measurement setup, the lasttwo terms in (C1) are contact terms which only affectthe time slice with τ = 0. This time slice is anywaydiscarded in our analysis. Appendix D: Finite-volume and lattice artifacts forcurrent-current correlators and electric conductivityfor free quarks and free pions
In Section V we have seen that the T - and µ -dependence of the electric conductivity on the lattice isquite different from the one in infinite-volume contin-uum theory. In this Appendix we quantify the finite-volume and lattice artifacts in electric conductivity forfree Wilson-Dirac and Domain Wall quarks on the lat-tice and demonstrate that lattice results agree well withcontinuum theory in the large-volume limit. We considerboth midpoint and Backus-Gilbert estimators. For esti-mates made with the Backus-Gilbert method, we use thesame values of ∆ as for the analysis of the real latticedata.In Fig. 15 we show the temperature dependence ofzero-density electric conductivity, obtained with both themidpoint and the Backus-Gilbert estimators on latticeswith different spatial volumes. As already discussed inSection V, for lattice size L s = 24 the deviations fromthe infinite-volume limit are very large, and for L t = 22the lattice and the continuum results differ by a factor of 2 . L t .
14, where free quark approxi-mation is not unreasonable, finite-volume effects are con-siderably smaller and do not exceed 20%. A comparisonof Wilson-Dirac and Domain Wall fermions suggests thatall deviations are finite-volume rather than discretizationartifacts, and discretization artifacts only become impor-tant for L t .
6. They are much larger for Domain Wallfermions due to large contributions from bulk modes andPauli-Villars regulator fields which compensate them.In Fig. 16 we also illustrate the finite-volume effects inthe estimates of the coefficient c ( T ) in the expansion (1),calculated from the finite difference between aµ = 0 . µ = 0. We use both the midpoint and Backus-Gilbert estimates. Again we see that c ( T ) quickly be-comes negative towards lower temperatures, and becomessufficiently close to the continuum value only for L s & s M P ( w = , m = ) / ( C e m T ) L t DW,L s =24DW,L s =30DW,L s =36WD,L s =24WD,L s =30WD,L s =36WD,L s =48continuum,inf.vol. 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 12 14 16 18 20 22 s B G ( w = , m = ) / ( C e m T ) L t WD,L s =24WD,L s =30WD,L s =36WD,L s =48 FIG. 15. Temperature dependence of the low-frequency electric conductivity of free quarks at zero chemical potential anddifferent lattice volumes.
On the left: estimated from the correlator midpoint according to (7) compared with the freecontinuum result at the same bare quark mass.
On the right: estimated using the Backus-Gilbert method. −0.2−0.15−0.1−0.05 0 0.05 0.1 4 6 8 10 12 14 16 18 20 22 c ( T ) L t DW,L s =24DW,L s =30DW,L s =36DW,L s =48WD,L s =24WD,L s =30WD,L s =36WD,L s =48continuum,inf.vol. −0.15−0.1−0.05 0 0.05 0.1 12 14 16 18 20 22 c ( T ) L t WD,L s =24WD,L s =30WD,L s =36WD,L s =48 FIG. 16. Temperature dependence of the second-order expansion coefficient c ( T ) of electric conductivity in powers of µ (1)at different lattice volumes. On the left: estimated from the correlator midpoint according to (7) compared with the freecontinuum result at the same bare quark mass.