UCLA/96/TEP/1 hep-th/9601048 UCONN-96-1 Temperature Expansions for Magnetic Systems Daniel Cangemi ∗ Department of Physics, University of California, Los Angeles, CA 90095-1547 Gerald Dunne † 6 9 Department of Physics, University of Connecticut, Storrs, CT 06269-3046 9 1 n a J Abstract 0 1 1 Wederivefinitetemperatureexpansionsforrelativisticfermionsystemsinthe v 8 4 presence of background magnetic fields, and with nonzero chemical potential. 0 1 We use the imaginary-time formalism for the finite temperature effects, the 0 6 proper-time method for the background field effects, and zeta function regu- 9 / h larizationfordevelopingtheexpansions. Weemphasizetheessentialdifference t - p between even and odd dimensions, focusing on 2+1 and 3+1 dimensions. e h : We concentrate on the high temperature limit, but we also discuss the T = 0 v i X limit with nonzero chemical potential. r a I. INTRODUCTION The study of fermion systems in the presence of external electromagnetic fields has ap- plications in diverse areas of physics, including astrophysics, solid state, condensed matter, plasma and particle physics. Indeed, astrophysical considerations led to the first systematic study of relativistic noninteracting fermion systems [1]. The existence of very high inten- sity magnetic fields in gravitationally collapsed objects motivated further investigations of the energy-momentum tensor and equation of state for a degenerate electron gas in a strong uniform magnetic field [2]. More recently, this problem has been addressed in the framework 1 of finite temperature quantum field theory [3–5], using the fact that many of the thermody- namic properties may be derived from the corresponding finite temperature effective action. In this paper we present a pedagogical discussion of the finite temperature (T) and finite chemical potential (µ) effective action for fermions in an external static magnetic field. Our aim is to review and unify the imaginary-time approach [6–8], developed initially for finite temperature free systems , with the Fock-Schwinger proper-time approach [9–11] developed for systems interacting with external electromagnetic fields but at zero temperature. For- mally, these two approaches fit together beautifully for the case of static external magnetic fields; in fact, the finite T and finite µ effects are completely separate from the computation required to compute the effects of the external static magnetic field. Thus, knowledge of the zero T and zero µ proper-time effective action is completely sufficient to write down a corre- sponding expression at nonzero temperature and nonzero chemical potential. However, this expression for the effective action is formal and the separation between T and µ effects and those of the external field may become blurred when one tries to make various approximate asymptotic expansions, such as for high or low temperature. Such expansions are necessarily complicated due to the proliferation of energy scales: thermal energy kT, fermion mass m, chemical potential µ, cyclotron energy B/m, and also possible momentum scales associated with spatial variations of the external field. We concentrate mainly on the high tempera- ture limit, in which T (we use units in which Boltzmann’s constant k = 1) is the dominant energy scale. However, we also present a simple approach to the other extreme: the T = 0 limit. This complements Refs. [3–5], which have focussed on the low T limit, and which have primarily used the real-time formalism for discussing the finite temperature effects. Throughout our analysis, we treat both 2+1 dimensional and 3+1 dimensional theories. This is motivated by the known profound differences between 2 + 1 and 3 + 1 dimensions (and in general between odd and even dimensional space-times) for free fermion systems at finite T and µ [12,13], and for zero T fermions in external fields [11]. We find that a consistent treatment of both cases requires careful use of zeta function regularization, which has been used previously for free systems [12,13] (and for systems with constant external A [14]). For the magnetic backgrounds we also find that the 2+1 and 3+1 cases involve 0 2 very different expansions at high temperature. The high temperature behavior of QED 2+1 is also of interest for studying questions of spontaneous symmetry breaking [15–19]. InSectionIIwereviewthestructureofthezerotemperatureeffectiveactionwithnonzero chemical potential for 2 + 1 and 3 + 1 dimensional fermions in external static magnetic fields. Finite temperature is introduced in Section III using the imaginary-time formalism. In Section IV this is combined with the proper-time formalism to provide a general formal expression for the finite T and µ effective action. In Section V we apply this to the high temperature limit of the free fermionic theories, and in Section VI to the high temperature limit of fermions in a static magnetic field. The zero temperature limit is examined in Section VII, andwe conclude in Section VIII with some comments regarding possible further extensions of this approach. II. EFFECTIVE ACTION FOR MAGNETIC SYSTEMS The basic object of interest in this paper is the effective action iS = LogDet iD/ m µγ0 (1) eff − − (cid:16) (cid:17) where D/ = D γν = (∂ +ieA )γν, and we choose Minkowski gamma matrices γν satisfying ν ν ν γν,γσ = 2gνσ = 2diag(1, 1, 1,..., 1) (2) { } − − − − The term µγ0 in the Dirac operator in (1) reflects the presence of a chemical potential µ, corresponding to a term µψ ψ in the Lagrangian. † − In 2+1 dimensions, the irreducible gamma matrices may be chosen to be γ0 = σ3 γ1 = iσ1 γ2 = iσ2 (3) where the σi are the 2 2 Pauli matrices. Note that an alternative choice, γ0 = σ3, × − γ1 = iσ1, γ2 = iσ2, corresponds to changing the sign of the mass, m m, in the Dirac − − → − operator appearing in the effective action (1). The system with effective action (1) is not parityinvariantsinceafermionmasstermbreaksparityin2+1dimensions[15–18]. However, a parity invariant model may be constructed by considering two species of fermions, one of 3 mass m and the other of mass m (see Footnote 11 in Ref. [15]). This may be achieved by − choosing a reducible set of 4 4 gamma matrices × γν 0 Γν = (4)  0 γν  −   in which case the Dirac operator is block diagonal: iD/ m µγ0 0 iD Γν m µΓ0 = − − (5) ν − −  0 iD/ m+µγ0 − −   Thus, the effective action for this parity invariant system may be written as iS = LogDet iD Γν m µΓ0 eff 2+1 ν − − (cid:16) (cid:17) = LogDet (iD µ)2 +m2 + D~ ~γ 2 (6) 2+1 0 − − · (cid:18) (cid:16) (cid:17) (cid:19) In 3+1 dimensions parity symmetry is not an issue, but the effective action (1), which involves the first-order Dirac operator, may still be written in the same form as (6), which involves a second order operator. We use the fact that there exists an additional gamma matrix γ5 satisfying γν,γ5 = 0 and (γ5)2 = 1. Then { } Det iD/ m µγ0 = Det γ5 iD/ m µγ0 γ5 = Det iD/ m+µγ0 (7) 3+1 3+1 3+1 − − − − − − (cid:16) (cid:17) (cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:17) Therefore, the effective action (1) may be expressed as 1 iS = LogDet iD/ m µγ0 iD/ m+µγ0 eff 3+1 2 − − − − = 1LogDet h(cid:16) (iD µ)2 +(cid:17)m(cid:16)2 + D~ ~γ 2 (cid:17)i (8) 3+1 0 2 − − · (cid:18) (cid:16) (cid:17) (cid:19) Note that the spatial operator D~ ~γ 2 which appears in (6) and in (8) is a positive · (cid:16) (cid:17) operator. In 2+1 dimensions it reduces to D D 0 D~ ~γ 2 = D~2 +eBγ0 = − + (9) · − −  0 D D (cid:16) (cid:17) + −   where the magnetic field is B = F = ∂ A ∂ A , and 12 1 2 2 1 − D = D iD (10) 1 2 ± ± In 3+1 dimensions the operator D~ ~γ 2 reduces to · (cid:16) (cid:17) 4 D~ ~γ 2 = D~2 +ie1[γi,γj]F (11) ij · − 4 (cid:16) (cid:17) If we choose the external magnetic field to be directed along the x3 direction and to be independent of x3, then (with a suitable choice of gamma matrices) this may be simplified further to D D 0 0 0 + D~ ·~γ 2 = −∂32 − −00 D+0D− D 0D+ 00  (12) (cid:16) (cid:17)  0 0 −0 D+D   −    where D are as defined for the 2+1 dimensional system in (10). ± Thus, in each case, the spectrum of the spatial operator D~ ~γ 2 is determined by the · spectrum of the 2-dimensional Schr¨odinger-like operators D (cid:16)D . T(cid:17)he operator D~ ~γ 2 is ± ∓ · (cid:16) (cid:17) effectively diagonal and we may write m2 + D~ ~γ 2 2 (13) · ≡ E (cid:16) (cid:17) For static magnetic backgrounds A = 0 and we can replace (iD µ) in (6) or (8) by ω µ, 0 0 − − where ω is an energy eigenvalue. Therefore dω iS = TrLog (ω µ)2 + 2 (14) eff 2π − − E Z h i Here, the trace operation Tr is understood to mean a summation over the eigenvalues 2 E of both the positive operators m2 D D and m2 D D in 2 + 1 dimensions, resp. + + − − − − m2 ∂2 D D and m2 ∂2 D D in 3+1 dimensions. − 3 − + − − 3 − − + III. FINITE TEMPERATURE FORMULATION It is clear from (14) that the effects of the external static magnetic field are contained solely within 2, and are clearly separated from the chemical potential µ and the energy E trace over ω. We can therefore pass to a finite temperature formulation just as in the free case [6–8], by replacing the energy integration with a discrete summation: dω i ∞ 2π → β Z n= X−∞ 2πi 1 ω ω = n+ (15) n → β 2 (cid:18) (cid:19) 5 Here β = 1/T, where T is the temperature and Boltzmann’s constant k has been absorbed into T. The transition to finite temperature would not be so straightforward if there were external electric fields, but here we consider only external static magnetic fields. Also note that in the zero temperature expression (14) it looks as though the dependence on the chemical potential µ may be formally eliminated through a naive shift of the integration variable ω. However, such a shift would violate the boundary conditions used to compute the trace, and a proper treatment at zero temperature leads to the appearance of non- analytic behavior in µ, corresponding to sharp cut-offs in the energy spectrum [7,20]. At finite temperature, these sharp cut-offs are smoothed out, and the dependence on µ is correspondingly smooth, as we shall see below. The effective action (14) may now be expressed as 1 ∞ S = TrLog ω +( µ) ω ( +µ) eff n n β − |E|− − |E| n=X−∞ h (cid:16) (cid:17)(cid:16) (cid:17)i 1 = TrLog ∞ ( µ)2 +(π(2n+1)/β)2 ( +µ)2 +(π(2n+1)/β)2 β |E|− |E| nY=0h(cid:16) (cid:17)(cid:16) (cid:17)i 1 β β = TrLog cosh ( µ)cosh ( +µ) (16) β " 2 |E|− 2 |E| # where in the last step we have used the infinite product representation of the cosh function 4x2 ∞ cosh(x) = 1+ (17) π2(2n+1)2! n=0 Y and we have dropped an infinite contribution that is independent of 2. E It is a simple matter to re-write (16) as (dropping again an irrelevant constant) 1 1 S = Tr + Log 1+e β( µ) + Log 1+e β( +µ) (18) eff − |E|− − |E| "|E| β β # (cid:16) (cid:17) (cid:16) (cid:17) This expression for the effective action generalizes an analogous expression (with µ = 0) derived in [6] for the free case. The only effect of the chemical potential is to shift the ‘energy’ eigenvalue by µ, which corresponds to a shift in the threshold energies for |E| ∓ particles and antiparticles. The only effect of the external static magnetic field is to modify the spectrum of the ‘energy’ eigenvalue from the free spectrum to a spectrum involving |E| dependence on the external B field. Therefore, at least in principle, we can now compute 6 the effective action for any external static magnetic field for which we know the spectrum of the operators D D appearing in (9) and (12). ± ∓ When µ = 0 and we take the zero temperature limit (β ) then the effective action → ∞ in (18) reduces to S = Tr( ) (19) eff µ=0;β | →∞ |E| which is the familiar T = 0, µ = 0 effective action in a static magnetic background [10]. When µ = 0 and β we must distinguish between the cases µ < m and µ > m. 6 → ∞ When µ < m, all low temperature thermal excitations are exponentially suppressed because µ > 0 (since m). Therefore, in this case the effective action reduces just as |E| ± |E| ≥ in (19). However, if µ > m, then in the infinite β limit the first logarithmic term in (18) contributes for the portion of the spectrum for which µ < 0. Thus, we have |E|− S = Tr +(µ )θ(µ ) eff µ=0;β | 6 →∞ |E| −|E| −|E| (cid:20) (cid:21) = Tr µ θ(µ )+ θ( µ) (20) −|E| |E| |E|− (cid:20) (cid:21) where θ is the step function. This is the standard expression for the effective action at zero temperature and with nonzero chemical potential [7,20]. The first equality in (20) emphasizes the correction from the zero T and zero µ expression (19), while the second emphasizes the physical content of the effective action with zero T and nonzero µ as µ times the number of occupied particle states plus the trace of the energy eigenvalues above the threshold µ [20]. In Section VII we examine this T = 0 limit in detail for fermions in a static magnetic background. The form of the effective action (18) is reminiscent of the grand partition function in non-relativistic statistical mechanics. Indeed, the non-relativistic limit corresponds to the situation in which the rest mass energy m is the dominant contribution to , |E| = m2 + D~ ~γ 2 = m+ 1 D~ ~γ 2 +... (21) |E| · 2m · r (cid:16) (cid:17) (cid:16) (cid:17) In this limit, with µ > m and m , the first logarithmic term in (18) dominates over the → ∞ second (i.e antiparticles are suppressed), and we are left with 7 Seff 1TrLog 1+eβµNRe−β(D~·~γ)2/2m (22) → β (cid:18) (cid:19) where we have identified µ = µ m as the non-relativistic chemical potential. The expres- NR − sion (22) is 1/β times the logarithm of the grand partition function for the corresponding non-relativistic fermion system. IV. PROPER TIME FORMULATION The proper time formulation provides an efficient method for computing the effective action at zero temperature [9–11], and furthermore has the virtue that the generalization to finite temperature and nonzero chemical potential naturally separates out the influence of a static background magnetic field. Using an integral representation of the logarithm to define the logarithm of an operator, we may express the finite temperature version of the effective action (14) as 1 (ω µ)2 2 ∞ n S = TrLog − + |E| eff β "− m2 m2 # n= X−∞ 1 ds (ω µ)2 2 = ∞ ∞ Tr exp n − s |E| s (23) −β n= Z0 s " m2 − m2 !# X−∞ where we have chosen to refer all energy scales to m in order to have dimensionless operators in the exponent. For massless theories one must choose a different reference energy scale, as discussed in Section VI. Expression (23) may be re-cast in terms of the second elliptic theta function θ (u τ), 2 | 1 ds 2πµs 4πis ( 2 µ2) S = ∞ θ Tr exp |E| − s (24) eff −β Z0 s 2 βm2 (cid:12)(cid:12)β2m2! " − m2 !# (cid:12) (cid:12) where [21,22] (cid:12) θ (u τ) = 2 ∞ eiπτ(n+1/2)2 cos((2n+1)u) (25) 2 | n=0 X APoissonsummationformulaconverts thesecondthetafunctionintoafourththetafunction according to the identity [21]: u 1 i 1/2 θ = − eiu2/(πτ) θ (u τ) (26) 4 2 τ −τ τ | (cid:18) (cid:12) (cid:19) (cid:18) (cid:19) (cid:12) (cid:12) (cid:12) 8 This converts the expression (24) for the effective action into a form involving θ : 4 m ds iβµ iβ2m2 2 S = ∞ θ Tr exp |E| s (27) eff −2√π Z0 s3/2 4 2 (cid:12)(cid:12) 4πs ! " − m2 !# (cid:12) (cid:12) where the fourth theta function is (cid:12) θ (u τ) = 1+2 ∞ ( 1)neiπτn2cos(2nu) (28) 4 | − n=1 X This expression (27) for the effective action is particulary useful as it shows clearly the separation between the effects of the static background magnetic field, which appear solely in the trace factor, and the effects of finite temperature and nonzero chemical potential, which appear solely in the θ iβµ iβ2m2 factor. Indeed, when µ 0 and β , the θ 4 2 4πs ≡ → ∞ 4 (cid:16) (cid:12) (cid:17) factor reduces to 1 and we are left(cid:12) with the standard proper time expression for the zero (cid:12) temperature effective action in a static system. This corresponds to keeping just the term “1” in the expansion (28) of the θ function, so the remaining summation over n in (28) 4 represents the nonzero temperature correction. The utility of elliptic theta functions in the computation of finite temperature effective actions has been noted previously in [23] for fermions (without chemical potential) and in [24] for bosonic systems. All information about the static magnetic background is neatly encapsulated in the proper time propagator 2 (D~ ~γ)2 Tr exp |E| s = e s Tr exp · s (29) − " − m2 !#  − m2     which is computed independent ofany reference to temperature or chemical potential. Thus, if one computes the zero temperature effective action using the proper time method it is completely straightforward to then write down an expression for the effective action at finite temperature and at nonzero chemical potential simply by inserting the θ factor as in (27). 4 However, while the expression (27) illustrates the separate roles of finite β, µ and the externalstaticfield, itisnotsostraightforwardtouseittoobtainusefulnumericalestimates. This is because of the wildly oscillatory behavior of the θ -function in (27) for large values 4 of the proper time parameter s. This oscillatory behavior is also sensitive to the magnitude of the dimensionless parameter βµ which appears in the first argument of the θ function. 4 9 These difficulties are further complicated by the proliferation of energy scales - for zero T and µ the only scales are the fermion mass m, the characteristic strength scale B (with dimensions of m2) of the external magnetic field, and possibly also characteristic length scales associated with spatial variations in the magnetic field. The generalization to nonzero temperature and chemical potential introduces two further energy scales: β and µ. In this paper, we concentrate mainly on high temperature expansions (in which T = 1/β is the dominant energy scale), although we also discuss the zero temperature limit (with nonzero chemical potential) in Section VII. To conclude this brief review of the finite temperature formalism for fermionic systems, we show how the general expression (27) relates to the previous expression (18), for which our statistical mechanics intuition is most direct. Using (28) and rescaling the proper time variable s in (27), s/m2 β2s/4, we obtain → 1 ds iβµ i S = ∞ θ Tr e β2 2s/4 eff −β√π Z0 s3/2 4 2 (cid:12)(cid:12)πs! h − E i 1 ds (cid:12) = ∞ 1+2 ∞(cid:12) ( 1)ne n2/scosh(nβµ) Tr e β2 2s/4 (30) − − E −β√π Z0 s3/2 " nX=1 − # h i The integrations over s may now be performed, with those in the summation term requiring the identity (see [25] Eqs. 3.471.9 and 8.469.3) 1 ds n2 β2 2 e βn ∞ exp E s = − |E|, n > 0 (31) √π Z0 s3/2 − s − 4 ! n Thus 2 ( 1)n+1 S = Tr + ∞ − e βn cosh(nβµ) (32) eff − |E| |E| β n n=1 X which is just expression (18). V. FREE THEORIES Before discussing the temperature expansions for fermions in the presence of background magnetic fields, we first describe the high temperature expansions for free theories. This is partly to establish some notation and to introduce some number theoretic functions (the 10