View metadata, citationNonrelativistic and similar papers Fermions at core.ac.uk in Magnetic Fields: a Quantum Field Theory Approachbrought to you by CORE provided by CERN Document Server 1 2 2,3 2 O. Espinosa ∗,J.Gamboa †,S.Lepe ‡ and F. M´endez § 1Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile. 2Departamento de F´ısica, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile. 3Instituto de F´ısica, Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile. II. EFFECTIVE ACTION AT LOW ENERGIES The statistical mechanics of nonrelativistic fermions in a constant magnetic field is considered from the quantum field theory point of view. The fermionic determinant is computed In reference [8], a method was proposed to compute using a general procedure that contains all possible regular- effective actions at low energies, based in a path inte- izations. The nonrelativistic grand-potential can be expressed gral derivation of the Foldy-Wouthuysen transformation. in terms polylogarithm functions, whereas the partition func- This method can be used for nonrelativistic fermions as tion in 2+1 dimensions and vanishing chemical potential can well. Let us consider the lagrangian be compactly written in terms of the Dedekind eta function. = ψ¯ iD/ m ψ, (1) The strong and weak magnetic fields limits are easily stud- L − ied in the latter case by using the duality properties of the where D/ = /∂ + igA/ . Dedekind function. Following the procedures developed in [8], we redefine the origin of the energy by the rescaling imt ψ(x)=e− φ(x), (2) to write the lagrangian as I. INTRODUCTION ¯ = φ( iD/ m(1 γ0))φ. (3) The motion of fermions in magnetic fields is an old L − − problem of quantum mechanics that plays a role in a Now we decompose the spinor φ into ‘large’ (ϕ)and wide variety of physical problems, ranging from neutron ‘small’ (χ) components, in whose terms stars [1] to the quantum Hall effect [2]. From the point of = ϕ† iD ϕ + χ† iD +2m χ + ϕ† i~σ Dχ view of quantum field theory, several studies have been L 0 0 · performed, related to the chiral anomaly [3], effective ac- +χ† i~σ Dϕ, (4) tions [4–6] and applications to anyons systems [7]. · In this paper we would like to study another aspect of where the Dirac representation for the γ-matrices has this problem, namely, following the approach proposed in been used. [8], we shall discuss the statistical mechanics of nonrela- The next step is to diagonalize the lagrangian in (4) tivistic fermions embedded in a constant magnetic field. by means of the change of variables Our approach requires the explicit computation of a 0 fermionic determinant, and this is carried out in terms ϕ = ϕ, 0 of polylogarithm and Dedekind functions. Additionally ϕ † = ϕ†, our calculation, in a particular case, allows us to study 0 1 χ = χ +[iD +2m]− i~σ Dϕ, (5) the strong and weak magnetic field limits quite easily, 0 · 0 1 due the modular invariance of the Dedekind function. χ † = χ† + ϕ† i~σ D [ iD0 +2m]− . · The paper is organized as follows: in section 2 we review the approach proposed in [8], in section 3 we formulate This change of variables has a Jacobian equal to unity the problem in the context of the path integral method, and the effective lagrangian, under this transformations, in section 4 we present an explicit computation of the becomes (omitting the primes) nonrelativistic grand potential, and in section 5 we study 1 = ϕ† iD + ~σ D ( iD +2m)− ~σ D ϕ, the partition function in the case of 2+1 dimensions. Our L 0 · 0 · conclusions are given in section 6. + χ† iD0 +2m χ. (6) This lagrangian describes the non-local dynamics of the fermions in terms of two-components spinors. One should note that ϕ and χ decouple and, after expanding ∗E-mail: espinosa@fis.utfsm.cl 1 ( iD +2m)− in powers of 1/m, the partition function E-mail: [email protected] 0 y becomes zE-mail: [email protected] xE-mail: [email protected] iS iS Z = χ† χe χ ϕ† ϕe ' , (7) D D D D Z Z 1 where 1 2 g [ D B µ]f ± =(Ω λ±)f ±, (15) −2m ∓ 2m 0 − n − n n 4 S = d xχ† iD +2m χ, iT˙ (t) ΩT (t)=0, (16) χ 0 − Z 1 2 g 4 1 2 g 2 since the operator D B0 is time-independent. Sϕ = d xϕ† iD + D + ~σ B ϕ + O(1/m ). 2m 2m 0 2m 2m · Equation (15) is just Schr¨odinger’s± equation for the Lan- Z (8) dau problem, whose eigenvalues are known, 2 Sϕ is the nonrelativistic action for fermions interacting 1 1 pz En± =Ω λn± =(n + )ω + µ, (17) with a magnetic field. Sχ is the action associated to the − 2 ± 2 2m − lower contribution of the spinor and can be neglected in the nonrelativistic limit. with ω = gB0/m. The equation for T (t) has a solution only if π III. NONRELATIVISTIC FERMIONS IN Ω=Ωm = (2m +1), (18) MAGNETIC FIELDS T where is the period and m an integer, in virtue of the T Using (7) and (8) one can explicitly study the nonrel- antiperiodic boundary condition on T (t). ativistic quantum field theory of fermions at finite tem- Thus, the eigenvalues in (14) are given by perature. π The partition function associated to this problem is λ± = (2m +1) E±. (19) m,n − n T iS' Zϕ = ϕ ϕ† e , The statistical mechanics of the fermionic system un- D D Z der consideration is described by the grand potential – 1 2 g basically the logarithm of the partition function after go- =det[iD0 + D + ~σ B + µ], (9) 2m 2m · ing to Euclidean space, i.e. replacing by iβ,where T = λn, (10) β =1/T is the inverse temperature. The logarithm of n the partition function is Y where g stands for the electric charge of the fermions, µ ∞ ∞ + is the chemical potential, and the infinite product runs log Zϕ = dpz[ log(λm,n) + log(λm,n− )], n=0 m= over all the eigenvalues λn of the operator that appears X X−∞ Z in (9), ∞ + dpz[L + L−], (20) ≡ n n 1 2 g n=0 Z [iD + D + ~σ B + µ]φn = λnφn, (11) X 0 2m 2m · where Ln± are infinite sums defined as subject to the usual antiperiodic boundary conditions in the imaginary time direction. ∞ For a constant magnetic field one can choose the gauge Ln± = log (λm,n± ). (21) m= X−∞ A =( B0y,0, 0), − Although the series in (21) are divergent, they can be A =0, 0 computed by using a definite regularization prescription. in which case the fermionic determinant can be computed Let us start considering the divergent series from ∞ L(a, b)= log ( am+ b). (22) 1 2 g [i∂t + D B0 + µ]φn± = λn±φn±, (12) m= 2m ± 2m X−∞ and thus the partition function (9) becomes where a and b are constants. The second derivative of L(a, b)is + Zϕ =det[i∂t H + µ]det[ i∂t H− + µ], (13) − − 2 + d L(a, b) ∞ 1 = λn λn−, (14) = , db2 − (am + b)2 n n m= Y Y X−∞ π2 csc2(bπ/a) where the H± can be read off from (12). = , (23) Each determinant in (13) is evaluated by explicitly − a2 solving the eigenvalue equation (12) by means of the so that upon integration one finds Ansatz φn±(x,t)=fn±(x)T (t), which yields 2 c bc πb with L(a, b)=log e 1+ 2 sin , (24) a + β(ω µ) βp2 /2m βµ βp2 /2m A = e− − e− z ,A− = e e− z , (31) where c1 and c2 are two arbitrary integration constants. In our case, a and b can be read from (19) and (21). The function S(A, b) has the following Taylor series in Therefore the variable A: n+1 π E± ∞ ( 1) c1±+c2±[ i β En±] n n Ln± =log e − − cosh β (25) S(A, b) = log(1 + A)+ − A , (32) 2 n(enb 1) n=1 X − c±π β βE n+1 c i 2 +( c + )E +log(1+e n± ). (26) ∞ ( 1) 1± β 2± 2 n± − = − An. (33) ∼ − − n(1 e nb) n=1 − X − Clearly, the constants c1± and c2± parametrize the arbi- n n nγp2 trariness of the regularization. Since the Euclidean space Since A has the form a e− z ,thepz integral can be effective action must be real, we choose c1±,2 in such a way performed explicitly to yield that (26) has no imaginary part. n+1 At this point we can compare our result (26) with the ∞ 2πm ∞ ( 1) n dp S(A±,b)= − (a±) . nonrelativistic limit of the general result given in [4] for z β n3/2(1 e nb) r n=1 − the effective action. The contribution Z−∞ X − (34) c2±π β c± i +( c± + )E± 1 − β − 2 2 n Expanding now indeed corresponds – after adding the analogous positron nb 1 ∞ nb k (1 e− )− = (e− ) , contribution– to the first term, Tr , in equation (18) of − |E| k=0 reference [4], and the dependence on the arbitrary con- X stants c1±,2 reflects the regularization that needs to be we find performed in order to define the trace. For instance the ∞ choice c1± =0=c2± is consistent with ζ-function regular- ∞ 2πm kb dpzS(A±,b)= Li ( a±e− ), (35) ization [10].