Nonrelativistic Fermions in Magnetic Fields

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provided by CERN Document Server 1 2 2,3 2 O. Espinosa ∗,J.Gamboa †,S.Lepe ‡ and F. Méndez § 1Departamento de F´ısica, Universidad Técnica 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 Católica 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 † becomes ‡E-mail: [email protected]

§E-mail: [email protected] iS iS Z = χ† χe χ ϕ† ϕe ϕ , (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 iSϕ 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]. − β 3/2 − k=0 βE± Z−∞ r The second contribution, log(1+e− n ), which is finite X and independent of the constants c1±,2, coincides (up to where we have introduced the polylogarithm function, afactorβ) with the grand potential in nonrelativistic defined as the analytic continuation to the whole complex statistical mechanics. z-plane of the series

∞ zn Lis(z) , (36) IV. GRAND-POTENTIAL ≡ ns n=1 X In this section we compute the nonrelativistic grand defined for z < 1ands C. Therefore, potential (Ω) directly from its thermodynamical defini- | | ∈ ∞ tion, rather than deriving it form the partition function + gB0 √2πm βµ (k+1)βω Ω (T,µ)=V Li ( e e− ), (Z). 4π2 β3/2 3/2 − k The nonrelativistic grand potential is given by X=0 (37) g˜ ∞ ∞ βE` Ω(β,µ)= dpz log(1 + e− n ), (27) −β and `=+, n=0 X− Z−∞ X + gB0 √2πm βµ + =Ω (β,µ)+Ω−(β,µ), (28) Ω−(T,µ)=V Li ( e )+Ω (T,µ). (38) 4π2 β3/2 3/2 − 2 whereg ˜ is the degeneracy factorg ˜ =(gB0/4π )V . Clearly, we need to consider the following object It is instructive to check that our result for Ω = + Ω +Ω− reduces to the correct result in the limit of zero temperature (β ). Depending on the sign of ∞ bn S(A, b) log 1+Ae− . (29) µ kω, the argument→∞ of the polylog function will tend ≡ n=0 either− to zero (in which case Li (0) = 0 and there is X 3/2 In terms of S we have no contribution) or infinity. To get the corresponding contribution in the latter case we need the asymptotic g˜ ∞ behavior of Li3/2(z)asz . This can be obtained Ω±(T,µ)= dpz S(A±,βω) (30) →∞ −β from Joncquière’s relation [15], Z−∞

3 s isπ (2π) isπ/2 log z where Lis(z)+e Lis(1/z)= e ζ(1 s, ). (39) Γ(s) − 2πi βω θ = , In particular, for x>0wehave 2 + β (2π)3/2 1 1 βx δ = (ω µ), Li ( eβx) e3iπ/4ζ( , + ) (40) 2 − 3/2 − −→ Γ(3/2) −2 2 2πi βµ δ− = . (47) 2 β3/2 − 2 x3/2, (41) + −→ − 3 Γ(3/2) 2θ 2δ 2δ− Defining q = e− , z = e− andz ˜ = e− ,then(46) becomes because of the asymptotic behavior ζ( 1 ,q) 2 q3/2 − 2 →−3 of the Hurwitz zeta function. With Γ(3/2) = √π/2, we ∞ + 2nθ δ +δ− n n obtain the correct nonrelativistic result Z0 = e e (1 + zq )(1+˜zq ) . (48) n=0 1 Y Ω(T =0,µ)= V This is most general expression for the (reduced) par- µ/ω tition function in 2+1-dimensions. 2gB0 b c 1 A particular situation arises when the chemical poten- √2m (µ kω)3/2 µ3/2 . (42) − 3π2  − − 2  tial is zero. In such a case (48) becomes k X=0   η(2iθ/π) 2 Here x is the floor of the real number x. Z = , (49) b c 0 η(iθ/π) A similar calculation can be performed in 2+1- dimensions, but then the result one obtains is nothing where more than the starting expression (28), with the corre- 2 2 sponding 2+1-degeneracy factorg ˜2 =(gB0/2π)L , L ∞ η(τ)=q1/24 (1 qn), (50) being the area of the 2-dimensional quantization box: − n=1 Y g˜2 ∞ βµ (k+1)βω 2iπτ Ω (T,µ)= 2 log 1+e e− is the Dedekind eta function [12], with q = e . 2+1 − β " k=0 It is not difficult to see that formula (49) reproduces X result (43) in the case of vanishing chemical potential. +log 1+eβµ . (43) Indeed, from the definition of the eta function and writing # 1 q2n =(1 qn)(1 + qn) we find − − ∞ 1/12 n 2 Z0 = q (1 + q ) , (51) n=1 V. PARTITION FUNCTION IN Y 2+1-DIMENSIONS βω which, with q = e− leads to In section we compute the partition function in 2+1- 1 ∞ nβω dimensions. In order to do that, one can proceed as fol- log Zϕ = βω +2 log 1+e− . (52) −12 lows: the product (14) is first written as n=1 X

g˜2 This coincides with the result for βΩ2+1 from (43) Zϕ =(Z0) , (44) at µ = 0, up to terms due to the regularization− used. The high temperature limit (or equivalently the weak in view of the degeneracyg ˜ of each Landau eigen- 2 magnetic ﬁeld limit) corresponds to q 1, where the value. Using ζ-function regularization the reduced parti- Dedekind function can be approximated→ by [13] tion function Z0 can be put into the form

π2 + 1/24 1/2 6(1 q) ∞ E β E β η(τ) q (1 q)− e− − , (53) Z = cosh n cosh n− . (45) ∼ − 0 2 2 n=0 and, therefore, the partition function in the high temper- Y ature limit becomes Using the explicit form of the eigenvalues for the Lan- 2 5 βω+ 2π dau problem in 2+1 dimensions (pz = 0) from (17), (45) e 12 3sinhβω can be written as Z0 . (54) βω 1 ∼ βω cosh 2 ∞ + Z0 = cosh(θn + δ )cosh(θn + δ−). (46) Moreover, as the Dedekind function satisﬁes the mod- n=0 ular property Y

4 i 1/2 1 η(τ)= η , (55) τ − τ one can compute the partition function in the strong magnetic ﬁeld limit (or low temperature limit) directly [1] J. Shapiro and S. Teukolsky, Black Holes, White Dwarfs from (54). Indeed, by using (54) and (55) we ﬁnd and Neutron Stars: The Physics of Compact Objects,J.

5π2 2π2 Wiley (1983). 12βω 2 2 − − 3sinh 2π π [2] R. B. Laughlin in The Quantum Hall Effect,R.R.Prange Z0 e βω cosh . (56) βω 1 ∼ βω and S. M Girvin (Eds.). [3] K. Ishikawa, N. Maeda, cond-mat/ 0102347. [4] D. Cangemi and G. Dunne, Annals Phys. 249, 582 (1996). VI. CONCLUSIONS [5] S. K. Blau, M. Visser and A. Wipf, Int. J. Mod. Phys. A6, 5409 (1991). In this paper we have studied the motion of nonrela- [6] G. Dunne and T. M. Hall, Phys. Lett. B419, 322 (1998). tivistic fermions in a constant magnetic field, following a [7] See e.g. F. Wilczek, Fractional Statistics and Anyon Su- quantum field theory approach. The main point is the perconductivity, World Scientific ( 1987). calculation of the fermionic determinant by using a gen- [8] J. Gamboa, S. Lepe and L. Vergara. Mod. Phys. Lett. A eral method that contains all the possible regularizations (in press), hep-ph/0007089. of the infinite product, in the form of particular choices [9] L. Landau and E. Lifshitz, Non-Relativistic Quantum of the coefficients c . Mechanics, Pergamon Press (1975). 1±,2 [10] J. L. Cortés, J. Gamboa, I. Schmidt and J. Zanelli, Phys. In addition, we obtained explicit expressions for the Lett. B444, 451 (1998); J. L. Cortés and J. Gamboa, nonrelativistic grand-potential, in terms of a series of Phys. Rev. D59 105016 (1999). polylogarithms, and for the partition function in 2+1 di- [11] C. Dib and O. Espinosa, The magnetized electron gas mensions and zero chemical potential, in terms of the in terms of Hurwitz zeta functions, math-ph/0012010, to Dedekind eta function. In the latter case, the modular appear in Nucl. Phys B. properties of the Dedekind function allow us to relate [12] M. Abramowitz and I. Stegun Editors, Handbook of the strong and weak magnetic field limits. This relation mathematical functions, Dover Publications, 1970. could be useful in the dynamics of the quantum Hall ef- [13] See for example J. Polchinski, String Theory, Vol 1, Cam- fect. bridge University Press, 1998 or M. Green, J. Schwartz and E. Witten, Superstring Theory , Vol 1, Cambridge University Press, 1987. ACKNOWLEDGMENTS [14] L. D. Landau and E. M. Lifshitz, Statistical Physics, Pergamon Press (1975). We would like to thank G. V. Dunne by pointed out [15] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tri- several important references. This work was partially comi, Higher transcendental functions,Vol.1,McGraw- supported by grants Fondecyt 8000017 (P.L.C.), 1010596, Hill, 1953. 2990037, 3000005 and Dicyt-USACH.