Dynamical chiral symmetry breaking in QED 3 at finite density and impurity potential
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Dynamical chiral symmetry breaking in QED at finite density and impurity potential Wei Li and Guo-Zhu Liu
Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R. China
We study the effects of finite chemical potential and impurity scattering on dynamical fermionmass generation in (2+1)-dimensional quantum electrodynamics. In any realistic systems, theseeffects usually can not be neglected. The longitudinal component of gauge field develops a finitestatic length produced by chemical potential and impurity scattering, while the transverse com-ponent remains long-ranged because of the gauge invariance. Another important consequence ofimpurity scattering is that the fermions have a finite damping rate, which reduces their lifetimestaying in a definite quantum state. By solving the Dyson-Schwinger equation for fermion massfunction, it is found that these effects lead to strong suppression of the critical fermion flavor N c and the dynamical fermion mass in the symmetry broken phase. PACS numbers: 11.30.Qc, 11.30.Rd, 11.10.Wx
I. INTRODUCTION
The dynamical chiral symmetry breaking (DCSB)in (2+1)-dimensional quantum electrodynamics (QED )has been investigated intensively for more than twentyyears [1–14]. On the one hand, these investigations mayhelp to gain deeper understanding of DCSB in QCD. Onthe other hand, this non-perturbative phenomenon givesan elegant field theoretic description for the formationof long-range antiferromagnetic order in two-dimensionalquantum Heisenberg antiferromagnet [15–17].Using the Dyson-Schwinger (DS) equation approach,Appelquist et al. first found that DCSB caused byfermion mass generation takes place only when thefermion flavor N is below some critical value N c [3].When N > N c , the fermions remain massless and thechiral symmetry is preserved. Although there was somecontroversy about the existence and precise value of N c in the literature, most analytical and numerical compu-tations confirmed that N c ≈ . m a on DCSB pre-viously [17], and found that the increasing m a leads to asignificant suppression of DCSB.When the action contains only massless Dirac fermionsand non-compact gauge field, the gauge interaction is al-ways long-ranged since the gauge invariance ensures thatno photon mass term can be generated by the radial cor-rection from fermions. If the gauge field also couples toan additional Abelian Higgs model, then the photon canacquire a finite mass via the Anderson-Higgs mechanismin the superconducting state [17]. In the context of hightemperature superconductor, the suppression of DCSBby photon mass m a can be used to qualitatively under-stand the competition between antiferromagnetism and superconductivity, which is one of the most prominentphenomena in high temperature superconductors [17].Apart from Anderson-Higgs mechanism, the gaugefield may also be screened by other physical effects. Forexample, when the fermion density is nonzero, the chemi-cal potential µ should be explicitly taken into account. Inaddition, the fermions may be scattered by some impu-rity atom (defect or imperfection) in realistic interactingsystem described by QED . Although generally chemicalpotential and impurity scattering have distinct influenceson the physical properties of the interacting fermion sys-tem, there is a common feature that they can both inducea finite density of states at the Fermi level. This screensthe time component of the gauge field, which then be-comes short-ranged. However, the transverse gauge fieldremains long-ranged in the sense that the static screen-ing length still diverges as at zero chemical potential inthe clean limit (”clean” means there is completely no im-purity scattering). In principle, the long-range nature oftransverse gauge field is guaranteed by the gauge invari-ance. However, though the transverse part is unscreened,the effective gauge interaction strength is reduced by thefermion density caused by chemical potential and/or im-purity scattering process, which certainly affects the crit-ical flavor N c as well as the dynamical fermion mass. Be-sides generating finite fermion density, the impurity scat-tering leads to the damping of fermionic quantum statesby producing a finite scattering rate Γ . Such dampingeffect reduces the lifetime of a fermion staying at a defi-nite quantum state, specified by such quantum numbersas momentum and/or energy. This will also have impor-tant effects on DCSB.While there appeared several papers discussing the ef-fect of chemical potential on DCSB in QED [18], toour knowledge the effect of impurity scattering has neverbeen considered in previous work. Indeed, in any real-istic applications of QED to condensed matter physics,such as high temperature superconductor, the chemicalpotential and impurity scattering are usually importantand thus should not been ignored. The purpose of thispaper is to study the dependence of DCSB on finite den-sity and impurity scattering.The paper is organized as follows. In Sec. II, we set upthe Lagrangian and then discuss the screening effect ofgauge interaction by including chemical potential and im-purity scattering rate into the vacuum polarization func-tion. We then solve the Dyson-Schwinger equation andpresent the critical flavor and fermion mass function inSec. III. The discussion is presented in Sec. IV. II. DYSON-SCHWINGER EQUATION IN THEPRESENCE OF µ AND Γ The Lagrangian for (2+1)-dimensional QED with N flavors of massless fermions is given by L = − F µν F µν + N X a =1 ψ a ( i∂/ µ − eA/ µ ) ψ a . (1)In (2+1)-dimensional space-time, the lowest rank spino-rial representation is two-component spinor whose 2 × γ µ = ( σ , iσ , iσ ). However, it is impossible to definea 2 × ψ , whose conjugate spinor fieldbeing defined as ¯ ψ = ψ † γ [2]. The 4 × γ -matricescan be defined as γ µ = ( σ , iσ , iσ ) ⊗ σ , satisfying thestandard Clifford algebra { γ µ , γ ν } = 2 g µν with metric g µν = diag(1 , − , − × γ = i (cid:18) II (cid:19) , γ = i (cid:18) I − I (cid:19) , which anticommute with all γ µ . The massless Lagrangian(Eq. (1)) preserves a continuous U(2N) chiral symmetry ψ → e iαγ , ψ . The mass term generated by fermion-anti-fermion pairing will break this global chiral sym-metry dynamically to subgroup U ( N ) × U ( N ). In thefollowing we consider a general large N and perform the1 /N expansion. For convenience, we work in units where ~ = k B = 1.In the Euclidian space, the full gauge field propagator D µν ( q ) is given by the equation D − µν ( q ) = D (0) − µν ( q ) + Π µν ( q ) , (2)with the free photon propagator being D (0) µν ( q ) = 1 q ( g µν − q µ q ν q ) , (3)in the Landau gauge. To the leading order of 1 /N expan-sion, the one-loop contribution to vacuum polarizationtensor Π µν isΠ µν ( q ) = − α Z d k (2 π ) Tr[ γ µ k/γ ν ( q/ + k/ )] k ( q + k ) , (4) where α = N e . The QED can be treated using the 1 /N expansion, with the product α = N e being fixed when N → ∞ and e →
0. This tensor can also be writtenas Π µν ( q ) = ( g µν − q µ q ν q )Π( q ) according to the gaugeinvariance. It is easy to find the polarization function:Π( q ) = αq . Now the propagator of gauge field has theform D − µν ( q ) = 1 q + Π( q ) ( g µν − q µ q ν q ) . (5)In order to study the dynamical fermion mass genera-tion, one can write the following DS equation for fermionspropagator S − F ( p ) = S (0) − F ( p ) − e Z d k (2 π ) γ µ S F ( k ) D µν ( p − k )Γ ν ( k, p ) . (6)To the leading order in 1 /N expansion, the vertex func-tion Γ ν is replaced by the bare matrix γ ν . The inversefull propagator of fermion is S F ( p ) − = ip/ + Σ( p ) , (7)where the wave-function renormalization is neglected.Taking trace on both sides of the DS equation, we getan integral equationΣ( p ) = e Z d k (2 π ) Σ( k ) k + Σ ( k ) Tr[ γ µ D µν ( p − k ) γ ν ] . (8)Using the gauge field propagator Eq. (5), we haveΣ( p ) = 2 αN Z d k (2 π ) Σ( k ) k + Σ ( k ) 1( p − k ) + α ( p − k )8 , (9)which was first obtained by Appelquist et al. [3]. Theyfound that DCSB takes place only when the fermion fla-vor is less than a critical value N c = 32 /π . Motivated bythis interesting prediction, a great many attentions havebeen paid on this issue. After taking into account thehigh order corrections to wave function renormalization,the critical flavor was found to change to N c = 128 / π [4]. Pennington et al [8] used a more careful truncation ofthe fermion DS equation and found that DCSB can occurfor all values of N . When the DS equations of fermionself-energy is coupled self-consistently to those of gaugefield propagator, Maris [9] showed that N c ≃ .
3. In arecent publication, Fischer et al. [11] found that N c ≃ claim that there isno DCSB for N > et al. foundthat the absence of DCSB can be attributed to the largeinfrared cutoff used in lattice studies and the smallness ofthe generated mass scale [13]. In summary, most analyt-ical and numerical computations seem to agree that thecritical fermion flavor should be N c ≈ . et al. within the lowest order of 1 /N expansion.The above investigation and result are valid at zerochemical potential in clean fermion system. Till now, lit-tle attention has been paid to the case with finite fermiondensity and finite impurity potential. Since the chemi-cal potential and impurity scattering are generally veryimportant in realistic applications, it is important to ex-amine their effects on DCSB. The aim of this paper is tostudy this problem.First of all, at nonzero chemical potential the fermionshave a finite density at the Fermi level, which screensthe temporal component of gauge field and weakens thegauge interaction. The influence of chemical potentialon DCSB was previously discussed by Feng et al. [18].They derived an explicit equation to include the chemicalpotential into the DS equation. One crucial assumptionin their work is the neglecting of chemical potential in thevacuum polarization function. The numerical results ofFeng et al. show that the chemical potential leads to onlyinsignificant change of N c and fermion mass. We thinkthat the screening effect caused by chemical potential isvery important and hence will pay special attention to thescreening of gauge field induced by chemical potential bystudying the vacuum polarization. As will be shown inthe context, the chemical potential suppresses both N c and fermion mass strongly.The effect of impurity scattering is more complicatedthan chemical potential. Generally, the scattering offermions by impurity potential has two important effects.First, it generates a finite density of states at low energy,which also screens the gauge field. This effect is expectedto weaken the gauge interaction, analogous to the roleplayed by the chemical potential. Second, the impurityscattering produces a finite damping of fermion quan-tum states and thus reduces the time for massless Diracfermions to interact with their anti-particles. These twoeffects are both very important.The behavior of massless Dirac fermions in a randompotential is of great interest in the context of condensedmatter physics since the low-energy properties of somemany-body systems, such as d -wave high temperaturesuperconductor and graphene, are largely controlled bythe interaction of Dirac fermion with impurities. Unfor-tunately, at present the problem of random Dirac fermionhas not been fully understood even when there is no di-rect interaction between fermions. When the impurityscattering and gauge interaction are both important, theproblem is basically out of theoretical control. If we con-sider a single impurity atom, then the impurity scatteringcan be treated by the self-consistent Born approximation.Within this approximation, the retarded fermion self-energy function develops a finite imaginary part, which isusually represented by a constant scattering rate Γ . Tostudy the problem about impurity scattering, it is mostconvenient to work in the Matsubara formalism and writethe fermion propagator as S F ( iω n , p ) = 1 iω n γ − γ · p , (10) where the frequency is iω n = i (2 n +1) πβ with β = T . Oncethe impurity scattering rate is taken into account, thefermion frequency should be replaced by iω n → iω n + i Γ sgn( ω n ) . (11)The fermion damping effect can be intuitively understoodas follows. After analytical continuation, iω n → ω + iδ, (12)the retarded fermion propagator has the form S ret F ( ω, p ) = 1( ω + i Γ ) γ − γ · p . (13)After Fourier transformation, it becomes S F ( t, r ) = e iωt − Γ t e i p · r (14)in real space. Apparently, the parameter Γ measuresthe decaying rate of the fermionic state characterized bysuch quantum numbers as ( ω, p ), which is known as theLandau damping effect. Starting from the propagatorEq. (13), various physical quantities can be calculatedand compared with experiments. Intuitively, the damp-ing effect is at variance with DCSB since a fermion maybe scattered into another state before it combines withits anti-particle to form a stable pair.Before we set up the DS equation for fermion self-energy, we need first to calculate the photon propaga-tor and discuss the screening effect caused by chemicalpotential and impurity scattering within the Matsubaraformalism. With finite chemical potential µ and finitedamping rate Γ , the fermion propagator reads S F ( iω n , p ) = 1( iω n + i Γ sgn( ω n ) − µ ) γ − γ · p . (15)The energy shift caused by chemical potential and thedamping effect caused by impurity scattering are bothreflected in this propagator. The screening of gauge in-teraction induced by them can be seen from the corre-sponding vacuum polarization functions. We will use thispropagator to calculate the vacuum polarization func-tions and to construct the DS equation for fermion mass.The inverse photon propagator for frequency q = mπβ and spatial momentum | q | is given by∆ − µν ( q , q , β ) = ∆ (0) − µν ( q , q , β ) + Π µν ( q , q , β ) , (16)where the ∆ (0) µν is the free photon propagator. Here, weuse different symbols D µν and ∆ µν to denote the gaugeboson propagator at zero temperature and finite temper-ature, respectively. Taking advantage of the transversecondition, q µ Π µν ( q ) = 0, the vacuum polarization tensordefined by Eq. (4) can be decomposed in terms of twoindependent transverse tensors [19],Π µν ( q , q , β ) = Π A ( q , q , β ) A µν + Π B ( q , q , β ) B µν , (17)where A µν = (cid:16) δ µ − q µ q q (cid:17) q q (cid:16) δ ν − q q ν q (cid:17) , (18) B µν = δ µi (cid:16) δ ij − q i q j q (cid:17) δ jν . (19)They are orthogonal and related by the relationship A µν + B µν = δ µν − q µ q ν q . (20)The functions Π A ( q , q , β ) and Π B ( q , q , β ) are relatedto the temporal and spatial components of vacuum po- larization tensor Π µν by the following expressionsΠ A = q q Π , Π B = Π ii − q q Π . (21)Now the full finite temperature photon propagator∆ µν ( q , q , β ) can be written as∆ µν ( q , q , β ) = A µν q + Π A ( q , q , β ) + B µν q + Π B ( q , q , β ) . (22)At finite temperature, the fermion contribution to vac-uum polarization functions should beΠ µν ( q, β ) = − αβ ∞ X n = −∞ Z d K (2 π ) Tr[ γ µ k/γ ν ( q/ + k/ )] k ( q + k ) = − αβ Z dx ∞ X n = −∞ Z d L (2 π ) l µ l ν + (1 − x )( l µ q ν + q µ l ν − q · lδ µν ) + 2 x (1 − x )( q δ µν − q µ q ν )[ l + x (1 − x ) q ] − αβ Z dx ∞ X n = −∞ Z d L (2 π ) − δ µν l + x (1 − x ) q , (23)where q = ( q , q ), Q = | q | , q = mπβ for gauge boson and k = ( k , k ), K = | k | , k = (2 n +1) πβ for fermion. Here, anew momentum variable is defined by l = k + xq with l = ( l , l ) , L = | l | , l = 2 πβ ( n + xm + 12 ) . (24)In the presence of impurity scattering rate Γ and chem-ical potential µ , the variable l should be replaced by l ( n ) = 2 πβ ( n + xm + 12 + i β π µ + β π Γ sgn( ω n )) . (25)Now the spatial and temporal component of polarizationtensor Π µν can be expressed asΠ ij ( q , q , β ) = 4 αβ Z dx Z d L (2 π ) × [2 x (1 − x )( q δ ij − q i q j ) S − (1 − x ) δ ij q S ∗ ] , (26)Π ( q , q , β ) = 4 αβ Z dx Z d L (2 π ) × (cid:2) S − L + x (1 − x ) q ) S + (1 − x ) q S ∗ (cid:3) . (27)where, following Ref. [19], we defined S i = ∞ X n = −∞ l ( n ) + L + x (1 − x ) q ] i , (28) S ∗ = ∞ X n = −∞ l ( n )[ l ( n ) + L + x (1 − x ) q ] . (29) It is not easy to calculate Eq. (26) and Eq. (27) ana-lytically. Nevertheless, within the widely used instanta-neous approximation q = 0, the integration can be per-formed by the methods presented in Ref. [19] and [20].After tedious but straightforward computation, we ob-tain the following expressions for polarization functions:Π A ( q , β ) = 2 απβ Z dx ln 2 X Γ + 12 × ln (cid:20) (cid:18) cosh (cid:20) πX q ( x )10 ln 2 X Γ + 1 (cid:21) + cosh (cid:20) πX µ ln 2 X Γ + 1 (cid:21)(cid:19)(cid:21) , Π B ( q , β ) = 4 αβ Z dx X q ( x ) × sinh h πX q ( x )10 ln 2 X Γ +1 i cosh h πX q ( x )10 ln 2 X Γ +1 i + cosh h πX µ ln 2 X Γ +1 i , (30)where X q ( x ) = β π p x (1 − x ) q , X Γ = β π Γ , X µ = β π µ .In either of the limits β → ∞ ( T →
0) or q → ∞ , boththe above functions reduce to α q / q →
0, Π A ( q , β ) becomes afunction of temperature, chemical potential, and impu-rity scattering rate, M ( β, X Γ , X µ ). In the zero energyand zero momentum limit, the photon propagator is∆ µν ( q = 0 , q → , β ) = A µν q + M + B µν q + 0 . (31)Apparently, the temporal component of the propagatornow acquires an effective mass and becomes M = s απβ (10 ln 2 X Γ + 1) ln h (cid:2) πX µ ln 2 X Γ + 1 (cid:3)i . (32)The mass appearing in the longitudinal photon impliesthat the electric field acquires a static screening lengthdetermined by the chemical potential and impurity scat-tering rate. The transverse photon, however, remainsmassless and hence the corresponding magnetic field isstill long-ranged, albeit dynamically screened.We now study the problem of DCSB at finite temper-ature. As in the case of zero temperature, this problemcan be studied by the DS equation approach. The DSequation for fermion propagator S F at finite tempera-ture is given by S − F ( p , p , β ) = S (0) − F ( p , p , β ) − e β ∞ X n = −∞ Z d k (2 π ) γ µ S F ( k , k , β )Γ ν ∆ µν ( q , q , β ) , (33)where Γ ν is the fermion-photon vertex and q = p − k .The full fermion propagator can be written as S − F ( p , p , β ) = A ( p , p , β ) S (0) − F + Σ( p , p , β ) , (34)where A ( p , p , β ) is the wave-function renormalization.Substituting Eq. (15) and Eq. (34) into Eq. (33) andthen taking trace on both sides of this equation, we getthe following couple of closed integral equations A ( p , p , β ) = 1 + e β X ω n Z d k (2 π ) Tr h S (0) F γ µ S F Γ ν ∆ µν i , Σ( p , p , β ) = − e β X ω n Z d k (2 π ) Tr [ γ µ S F Γ ν ∆ µν ] . (35)Here, ∆ µν is the photon propagator as defined by Eq.(16) and the Γ ν is the full vertex function. To treat thisequation, a number of approximations should be made.At present, there is no well-controlled way to choosethe vertex function Γ ν . In order to satisfy the Ward-Takahashi identity, Maris et al. [9] studied several dif-ferent Ans ¨ a tze for the full vertex function and comparedthe results. For the sake of simplicity, they assumed thatΓ ν = f ( A ( p ) , A ( k ) , A ( p − k )) γ ν , with A ( p ) ≡ f -function andthe bare vertex is actually a good approximation. OnceΓ ν = γ ν is assumed, then the Ward-Takahashi identityrequires that the wave function renormalization shouldbe A ( p ) ≡ T .To simplify the theoretical and numerical analysis, wekeep only the leading order of 1 /N expansion by replac-ing the vertex function Γ ν by γ ν and taking A ( p ) = 1.The DS equation of fermion Eq. (34) can now be simpli-fied as S − F ( p , p , β ) = [( iω n + i Γ sgn( ω n ) − µ ) γ − γ · p ]+Σ( p , p , β ) . (36)The fermion mass function Σ( p , p , β ) satisfies the fol-lowing integral equationΣ( p , p , β ) = e β X ω n Z d k (2 π ) Tr[ γ µ γ ν ∆ µν ( q , q , β )] × Σ( k , k , β )4 [( ω n + Γ sgn ( ω n ) + iµ ) + k + Σ ( k , k , β )]= αN β π ∞ X n = −∞ Z d k (2 π ) ∆ µµ ( q , q , β ) × Σ( k , k , β ) (cid:2) (( n + ) + X Γ sgn ( ω n ) + iX µ ) + M (cid:3) , (37)where X T = β π , M = X T p k + Σ ( k , β ) . (38)Note that chemical potential and impurity scattering rateboth appear in two places: in the occupation numberand in the polarization function. The energy shift in-duced by chemical potential and the fermion damping ef-fect induced by impurity scattering are both representedin the former place. The screening of gauge interactioncaused by chemical potential and impurity scattering isreflected in the vacuum polarization function. In orderto perform the frequency summation in Eq. (37), we uti-lize the approximation Σ( p , p , β ) ≃ Σ( p = 0 , p , β ). At ω m = q = 0, the summation over ω n in Eq. (37) yieldsthe equationΣ( p , β ) = βα N π Z d k (2 π ) Σ( k , β ) M ∆ µµ ( q = 0 , q , β ) ×ℑ m (cid:20) ψ (cid:2)
12 + X Γ ± iX µ + iM (cid:3)(cid:21) . (39)Substituting the full photon propagator Eq. (22) into themass equation and use the same approximative methodas that presented in Ref. [20], we haveΣ( p , β )= βα N π Z d k (2 π ) (cid:20) q + Π A ( q , β ) + 1 q + Π B ( q , β ) (cid:21) × (cid:26) Σ( k , β ) M ℑ m (cid:20) ψ (cid:2)
12 + X Γ ± iX µ + iM (cid:3)(cid:21)(cid:27) ≈ βα N π Z d k (2 π ) (cid:20) q + Π A ( q , β ) + 1 q + Π B ( q , β ) (cid:21) × (cid:26) Σ( k , β ) M π (cid:20) π ( M ± X µ )10 ln 2 X Γ + 1 (cid:21)(cid:27) . (40)If this nonlinear integral equation develops a nontrivialsolution, then the massless fermion acquires a finite mass. III. SOLUTION OF THE DYSON-SCHWINGEREQUATION
The nonlinear integral equation can be solved by thebifurcation theory and parameter imbedding method[21, 22]. The basic idea and detailed computation pro-cedures are presented in previous papers [17, 21, 22]. Todetermine the bifurcation point that separates the chiralsymmetric phase and symmetry broken phase, we need tofind the eigenvalues of the associated linearized equation.Taking the Frˆechet derivative of the nonlinear integralequation Eq. (40), we haveΣ( p , β )= βα N π Z d k (2 π ) (cid:20) q + Π A ( q , β ) + 1 q + Π B ( q , β ) (cid:21) × Σ( k , β ) ∂∂ Σ (cid:26) Σ( k , β ) M π (cid:20) π ( M ± X µ )10 ln 2 X Γ + 1 (cid:21)(cid:27) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Σ=0 = β N π Z d k (2 π ) (cid:20) q + Π A ( q , β ) + 1 q + Π B ( q , β ) (cid:21) × Σ( k , β ) X T | k | π (cid:20) π ( X T | k | ± X µ )10 ln 2 X Γ + 1 (cid:21) . (41)To facilitate numerical calculation, we divide the mo-menta, temperature, fermion mass, chemical potentialand impurity scattering rate by parameter α and makethem dimensionless. The UV cutoff for momentum now isreplaced by Λ /α , which should be properly chosen to en-sure the results insensitive to UV cutoff. As pointed outby Appelquist et al. [2, 3], the integral in the zero tem-perature DS equation damps rapidly for momenta greaterthan α , it is thus natural to set Λ /α ≃
1. This is alsotrue for DS equation at finite temperature [7]. In thefollowing we take the advantage of this fact and assumethat Λ /α = 1.The dependence of critical flavor number N c on chem-ical potential µ and impurity scattering rate Γ areshown in Fig. 1 for a list of values of temperature T = 10 − , − , − . It is easy to see that the criti-cal flavor N c is a decreasing function of T , µ , and Γ ,implying a strong suppression of DCSB by thermal fluc-tuation, fermion density, and impurity scattering.The fermion mass function Σ( p , β ) can be obtained af-ter solving the DS equation Eq. (40) using the straight-forward iteration method. The results are presented inFig. 2 and Fig. 3. In Fig. 2, we show the re-scaled zero-momentum fermion mass Σ(0) for several different values -10 -8 -6 -4 -2 T = 10 -8 = 0 = 10 -8 = 10 -6 = 10 -4 = 10 -2 N c 0 -10 -8 -6 -4 -2 T = 10 -6 = 0 = 10 -8 = 10 -6 = 10 -4 = 10 -2 N c 0 -10 -8 -6 -4 -2 T = 10 -4 = 0 = 10 -8 = 10 -6 = 10 -4 = 10 -2 N c 0 FIG. 1: Dependence of N c on µ and Γ for different temper-atures T = 10 − , − and 10 − . of fermion flavor N at a fixed temperature T = 10 − ,with µ = Γ = 0. Apparently, the fermion mass Σ(0)decreases rapidly as the fermion flavor N increases. Thisis easy to understand since the inverse of fermion flavor1 /N serves as the effective coupling strength of gaugeinteraction.The dynamical fermion mass as a function of momen-tum is shown in Fig. 3 for N = 2 at different values oftemperatures, chemical potential and impurity scatteringrate. We notice that the dynamical fermion mass is sig-nificantly suppressed by the increasing chemical potentialand impurity scattering rate. It also decreases with theincreasing fermion momentum.We next would like to compare the results presentedabove with those in QED with finite photon mass gen- -26 -22 -18 -14 -10 -6 -2 (0) T = 10 -100 = 0 = 0 N FIG. 2: Zero-momentum mass Σ(0) for different values of Nat T = 10 − and µ = Γ = 0. erated by Anderson-Higgs mechanism. Different fromthe partial screening of gauge interaction, the Anderson-Higgs mechanism induces a complete static screening. Inthis case, the gauge boson acquires a physical mass byeating the massless Goldstone bosons, so that the prop-agator (Eq. (22)) becomes∆ µν ( q , q , β ) = A µν q + Π A ( q , q , β ) + m a + B µν q + Π B ( q , q , β ) + m a . (42)If m a = 0, both temporal and spatial components ofgauge field develop a finite screening length, which isgiven by the gauge boson mass. Such screening lengtheliminates the contribution of small momenta to the DSequation Eq. (41). It is expected that a large m a willprevent the DCSB. The dependence of critical flavor N c on m a and Γ is shown in Fig. 4. The gauge boson massand impurity scattering lead to similar suppression effectof DCSB. IV. SUMMARY AND DISCUSSION
In summary, we studied the effects of finite chemi-cal potential and impurity scattering on dynamical massgeneration in QED . In the realistic applications ofQED to condensed matter systems, these effects usu-ally can not be ignored. By solving the DS equationfor fermion mass function, we found that both chemicalpotential and impurity scattering lead to strong reduc-tion of critical fermion flavor N c and dynamical fermionmass. In reality, the chemical potential, impurity scat-tering, and even a finite gauge boson mass may coexist atthe same time. When their effects are all important, theDCSB is completely suppressed. These results impose animportant constraint on the applicability of QED in var-ious physical systems. For a system with large fermiondensity and impurity potential, it seems impossible thatDCSB can indeed take place. -10 -8 -6 -4 -2 -42 -34 -26 -18 -10 (0) N = 2.0, T = 10 -8 = 0 = 10 -8 = 10 -6 = 10 -4 -12 -8 -4 -24 -19 -14 -9 -4 ( )p N = 2.0, T = 10 -6 = = 0 = = 10 -8 = = 10 -6 p FIG. 3: (a) Zero-momentum mass Σ(0) for different valuesof µ and Γ at T = 10 − ; (b) Dependence of mass Σ( p ) onmomentum p for different values of µ and Γ at T = 10 − . -10 -8 -6 -4 -2 T = 10 -8 = 10 -6 , m a = 0 = 10 -6 , m a = 10 -2 N c 0 FIG. 4: Dependence of N c on m a and Γ at T = 10 − . We now would like to briefly discuss the meaning offinite chemical potential and the possible experimentalstudy of its effect on DCSB. In condensed matter physics,the zero temperature ground state (vacuum) of a many-body system is characterized by the presence of a Fermilevel which separates the fully occupied and fully emptystates. In some condensed matter systems, such as hightemperature superconductor and graphene, the valenceband and conduction band touch at discrete Dirac points.When the Fermi level lies exactly at Dirac points, thechemical potential is zero and the low-energy excitationsare massless Dirac fermions. The Fermi level movesupwards (downwards) from the Dirac points once thefermion density increases (decreases). Now the chemi-cal potential is just the quantity that measures how thenew Fermi level is far from the original Dirac points. Atzero chemical potential, the pairs are formed by fermionsslightly above Dirac points and anti-fermions (holes incondensed matter terminology) slightly below the Diracpoints. Once the fermions become massive, they are con-fined. At finite chemical potential, the pairs are formedby fermions slightly above the finite Fermi surface andanti-fermions slightly below Fermi surface. The massivefermions are also confined by the gauge force.In realistic condensed matter systems, the fermion den-sity or chemical potential can be continually turned bychemical doping or gate voltage. We first have a numberof samples, each with different chemical potential, andlet them stay at some finite temperature where DCSB iscompletely prevented by thermal fluctuations. We thencool down these samples and study whether DCSB in-deed take place at very low temperature by measuringsome observable quantities, such as specific heat, suscep-tibility, and electric conductivity. Through this way, inprinciple the effect of finite chemical potential on the fate of DCSB can be experimentally studied.We emphasize that our treatment on the random po-tential applies only to the case of weak impurity poten-tial where the self-consistent Born approximation is valid.When the random potential caused by impurities takesthe form of Gaussian white noise, such treatment mightbe questionable. For this kind of impurities, the randompotential has to be averaged by performing proper func-tional integration, which yields an effective four-fermioninteraction in the whole action. This new effective inter-action can contribute a new term to the DS equation. Atpresent, it is unclear technically how to study the DCSBand the fermion damping effect in a unified framework.This problem is subject to further investigation.
V. ACKNOWLEDGMENTS
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