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Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
ON FINITE TEMPERATURE CASIMIR EFFECTFOR DIRAC LATTICES
IRINA PIROZHENKO
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear ResearchandDubna State University, Dubna, 141980, [email protected]
Received (Day Month Year)Revised (Day Month Year)We consider polarizable sheets modeled by a lattice of delta function potentials. TheCasimir interaction of two such lattices is calculated at nonzero temperature. The heatkernel expansion for periodic singular background is discussed in relation with the hightemperature asymptote of the free energy.
Keywords : Vacuum energy; scattering; delta function potential.PACS Nos.: 11.10.Wx, 12.20.Ds
1. The System
In recent years, we have witnessed the discoveries of ever new unusual proper-ties of two-dimensional materials, such as 2d electron gas, graphene, or othermonoatomic layers. In the present note we deal with a system which mimics twoparallel monoatomic layers and discuss van der Waals and Casimir forces betweenthem. Two-dimensional rectangular lattices of delta function potentials separatedby a distance b are considered. Our objective is to evaluate the vacuum energy of ascalar field in the background of these lattices and compute the finite temperaturecorrections.A periodic delta potentials are a well studied in quantum mechanics, the simplestcase being the Kronig-Penney model (‘Dirac comb’). In more than one dimensiona Hamilton operator with a delta function potential is not self adjoint, however aself-adjoint extension may be defined. For a Laplace operator with tree-dimensional δ function, ∆ a = ∆ + aδ ( x ), the self-adjoint extension was analyzed in the QFTcontext. Close relation of self-adjoint extension and zero range potential approachwith regularization and renormalization was traced.
3, 4
The scattering on a single two-dimensional lattice 3D delta functions was consid-ered in Ref. 4 for scalar and electromagnetic field. In Ref. 6 the scattering approachwas used to obtain the separation dependent part of the vacuum energy for a scalar ovember 18, 2019 1:25 WSPC/INSTRUCTION FILE Pirozhenko˙CS Irina Pirozhenko field in the presence of two parallel lattices. In the notations of this paper, the latticesites of two rectangular 2D lattices, (A) and (B), are given by 3D vectors ~a A n = (cid:18) a n + c b (cid:19) , ~a B n = (cid:18) a n (cid:19) , a n = a (cid:18) n n (cid:19) , (1) a n are 2 D vectors in ( x, y )-plane, n and n are integers, a is the lattice spacing, b is the separation, c is the displacement vector in ( x, y )-plane.The wave equation for a scalar field φ ( ~x ) , − ω − ∆ + g X n (cid:16) δ (3) ( ~x − ~a A n ) + δ (3) ( ~x − ~a B n ) (cid:17)! φ ( ~x ) = 0 , (2)endowed with two lattices of three dimensional delta functions is not well defined,and g , having the dimension of length, should be regarded as a bare coupling. In the scattering approach the separation dependent part of the vacuum energyis given by so called ‘TGTG’-formula, see for example, E = 12 π Z ∞ dξ Tr ln(1 − M ( iξ, ~x, ~x ′ )) , M = T A G T B G , (3)where ξ is the imaginary frequency, ω = iξ , and the trace is taken with respect to ~x . The free Green’s function is denoted by G ( ~x − ~x ′ ) = Z d k (2 π ) e i~k ( ~x − ~x ′ )+ i Γ | x − x ′ | i Γ( k ) , Γ( k ) = p ω − k + i , (4)and the T operators describe scattering on the lattices A and BT A,B ( ~x, ~x ′ ) = X n , n ′ δ ( ~x − ~a A , B n )Φ − n , n ′ δ ( ~a A , B n ′ − ~x ′ ) . (5)Here Φ − n , n ′ is the inverse matrix to Φ n , n ′ = g δ n , n ′ − G ( ~a n − ~a n ′ ) with diagonalelements defined after the renormalization of the coupling g so that Φ n , n = 1 /g .Due to translational invariance with respect to the lattice step the momentum k may be split into quasi momentum and integer part, k = q + πa N . The infinitemomentum integration is replaced by R d k = R d q P N with the components of q = ( q , q ) restricted to − π/a ≤ q , < π/a . One can derive Tr ln(1 − M ) in (3)expanding the logarithm in powers of M ( iξ ), which after some transformations appear to be diagonal with respect to q . Consequently the expression (3) can bewritten in a Lifshitz-like form. For the vacuum energy per lattice cell we get E = 12 Z ∞ dξπ a Z d q (2 π ) ln (cid:0) − | h ( iξ, q ) | (cid:1) , (6)with h ( ω, q ) = 1 a X N e i Γ( k ) b + i πa Nc i Γ( k ) ˜ φ ( k ) , ˜ φ ( k ) = 1 g − π X n ′ | a n | e iω | a n | + i ka n (7)ovember 18, 2019 1:25 WSPC/INSTRUCTION FILE Pirozhenko˙CS On Finite Temperature Casimir Effect for Dirac Lattices (in the primed sum the term with n = 0 is dropped). For week coupling g itcorresponds to the vacuum energy of parallel plates with “reflection coefficient” r ( ω, k ) = g/ (2 a )( ω − k ) − / .The general formula (6) obtained in Ref. 6 for the vacuum energy of two latticesat zero temperature will be used to find finite temperature corrections.
2. Finite Temperature
To find the finite temperature corrections the Matsubara formalism is used. Thefree energy is given by the sum over Matsubara frequencies iξ → ξ n = 2 πT n , F = T ∞ X n = −∞ Tr ln (1 − M ( ξ n )) = T a ∞ X n = −∞ Z d q (2 π ) ln (cid:0) − | h ( ξ n , q ) | (cid:1) . (8)We define three temperature regions with respect to the parameters of the model,namely the lattice spacing a and the separation of two lattices b . Low temperaturecorresponds to T a, T b ≪
1. Medium temperature meets the inequality
T a < 1, the Abel-Plana formula may be used F = E + F T , F T = i Z ∞−∞ dξ π n T ( ξ ) Tr [ln(1 − M ( iξ )) − ln(1 − M ( − iξ ))] . (9)Here n T ( ξ ) = (exp( | ξ | /T ) − − is the Boltzman factor. At low temperatures theintegral in (9) is determined by small ξ . After the development around ξ = 0, eachpower of ξ adds a power of T , F T = 1 π ∞ Z −∞ dξ exp( | ξ | T ) − M ξ + M ξ + . . . ] = M πT M π T 15 + . . . , (10) M = − Tr M ′ − M , M = − Tr ( M ′ − M ) + M ′ M ′′ − M ) + M ′′′ − M ) ) . . . . At high temperature the leading contribution is given by the zeroth therm of theMatsubara sum (8) which is proportional to the derivative of the spectral zeta func-tion, ζ ′ (0), of considered system. The subleading terms are expressed through the co-efficients of the heat kernel expansion K ( t ) = P j e − λ j t ∼ (4 πt ) − / P ∞ n =0 t n/ a n/ , t → 0. Thus the entire high temperature expansion aquires the form, F ( T ) ≃ − T ζ ′ (0) + a T ~ π − a / π / T ~ ζ R (3) − a T ~ + . . . . (11)The heat kernel in the spectral problem (2) with two parallel delta lattices A and B is given by an integral equation, K ( x, y ; t ) = K ( x, y ; t ) + g Z t ds X i = A,B X n K ( x, a i n ; t − s ) K ( a i n , y ; s ) , (12)ovember 18, 2019 1:25 WSPC/INSTRUCTION FILE Pirozhenko˙CS Irina Pirozhenko with the free heat kernel K ( x, y ; t ) = (4 πt ) − / e − ( x − y )24 t . The integral equation (12)can be iterated. 7, 8 After taking the trace one arrives at K ( t ) = K ( t ) + K (1) ( t ) + K (2) ( t ) + K (3) ( t ) ... , where K ( t ) = 1(4 πt ) / V, K (1) ( t ) = 2 N g (4 π ) / t / , . . . . (13)Here N is the number of the lattice points. Unfortunately, starting from g term wecome across divergent integrals which may be regularized by point-splitting. Similarproblem was discussed in Ref. 8 . It is worth comparing (13) with the exact solutionfor the heat kernel trace of the operator with a single three-dimensional δ -function,obtained in Ref. 2 K ( t ) = 1(4 πt ) / + 12 e π g t (cid:20) − Φ (cid:18) πg √ t (cid:19)(cid:21) , Φ( z ) = 2 √ π Z z dxe − x . (14)The small g expansion of (14) coincides in its leading terms with K ( t ) and K (1) ( t ),up to volume and arear factors. Thus, from (13) the leading heat kernel coefficients a and a / may be extracted. 3. Conclusion We considered the Casimir effect for two-dimensional lattices of delta functions atzero and finite temperature. Our approach is based on the T GT G formula, with T -operators derived in terms of lattice sums. The generalization to finite temperatureis tricky but straightforward. Here scaling properties of the system and relationsbetween various limiting cased proved to be useful at low and high temperatures.The heat kernel of the Laplace operator with double delta lattice potential was an-alyzed in relation with the high temperature asymptote of the free energy. Mediumtemperature region is left for numerical study. Acknowledgments The author is thankful to the Organizers of 4th Symposium on the Casimir effect.She also acknowledges fruitful discussions with Michael Bordag. References 1. S. Albeverio et al. , Solvable Models in Quantum Mechanics (Springer, 1988).2. S. N. Solodukhin, Nucl. Phys. B , 461 (1999).3. R. W. Jackiw, in Diverse Topics in Theoretical and Mathematical Physics , Section I.3(World Scientific, 1995).4. M. Bordag and J. M. Munoz-Castaneda, Phys. Rev. D , 065027 (2015).5. I. Klich and O. Kenneth, Journal of Physics: Conference Series , 012020 (2009)6. M. Bordag and I. G. Pirozhenko, Phys. Rev. D , 056017 (2017).7. B. Gaveau and L. Schulman, J. Phys. A: Math. Gen. , 1833 (1986).8. M. Bordag and D. V. Vassilevich, J. Phys. A: Math. Gen.32