Casimir Effect for Massless Fermions in One Dimension: A Force-Operator Approach

We calculate the Casimir interaction between two short-range scatterers embedded in a background of onedimensional massless Dirac fermions using a force-operator approach. We obtain the force between two finitewidth square barriers and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers, we recover the conventional attractive onedimensional Casimir force. For the general problem with inequivalent scatterers, we find that the magnitude and sign of this force depend on the relative spinor polarizations of the two scattering potentials, which can be tuned to give an attractive, a repulsive, or a compensated null Casimir interaction. Disciplines Physical Sciences and Mathematics | Physics Comments Suggested Citation: D. Zhabinskaya, J.M. Kinder and E.J. Mele. (2008). "Casimir effect for massless fermions in one dimension: A force-operator approach." Physical Review A. 78, 060103. © 2008 The American Physical Society http://dx.doi.org/10.1103/PhysRevA.78.060103 This journal article is available at ScholarlyCommons: http://repository.upenn.edu/physics_papers/95 Casimir effect for massless fermions in one dimension: A force-operator approach Dina Zhabinskaya,* Jesse M. Kinder, and E. J. Mele Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Received 31 July 2008; published 30 December 2008 We calculate the Casimir interaction between two short-range scatterers embedded in a background of one-dimensional massless Dirac fermions using a force-operator approach. We obtain the force between two finite-width square barriers and take the limit of zero width and infinite potential strength to study the Casimir force mediated by the fermions. For the case of identical scatterers, we recover the conventional attractive one-dimensional Casimir force. For the general problem with inequivalent scatterers, we find that the magnitude and sign of this force depend on the relative spinor polarizations of the two scattering potentials, which can be tuned to give an attractive, a repulsive, or a compensated null Casimir interaction. DOI: 10.1103/PhysRevA.78.060103 PACS number s : 03.70. k, 05.30.Fk, 11.80. m, 68.65. k Boundaries modify the spectrum of zero-point fluctuations of a quantum field, resulting in fluctuation-induced forces and pressures on the boundaries that are known generally as Casimir effects 1 . When sharp boundary conditions are used to model the Casimir effect, they yield perfect reflection of the incident propagating quantum field at all energies 1 . However, in many physical applications this hard-wall limit is not appropriate; of special interest in the present work are interactions between localized scatterers in one dimension that have energy-dependent scattering properties controlled by the strength, range, and shape of the potential. Along this line, previous work has recognized that the finite reflectance of partially transmitting mirrors provides a natural high-energy regularization scheme for computing the effect of sharp reflecting boundaries on the zero-point energy of the electromagnetic field 2,3 . In more recent work, Sundberg and Jaffe approached the problem of computing the effect of confining boundary conditions on a degenerate gas of fermions in one dimension as the limiting behavior for rectangular barriers of finite width and height. Interestingly, they encounter a divergence of the Casimir energy in the zero-width limit a sharp boundary even for finite potential strength 4 . In this Rapid Communication we address the problem of Casimir interactions between scatterers mediated by a onedimensional Fermi gas. The fermions in our calculation are massless Dirac fermions appropriate to describe, for example, the single-valley electronic spectrum of a metallic carbon nanotube. We employ the Hellmann-Feynman theorem to calculate the force, rather than energy, of the interaction between two scatterers as a function of their separation d. This approach renders our calculation free from ultraviolet divergences even for the limiting case of sharp scatterers. We demonstrate that for the case of identical scatterers, this formalism recovers the well-known attractive 1 /d2 Casimir force in one dimension. Furthermore, we find that for Dirac fermions the internal structure of the matrix-valued scattering potential admits a long-range Casimir interaction which can also be repulsive or even compensated. This provides a physical situation where the Casimir interaction is continuously tunable from attractive to repulsive by variation of an internal control parameter, realizing the known bounds for the one-dimensional Casimir interaction as two limiting cases. The results may be relevant for indirect interactions between defects and adsorbed species on carbon nanotubes. The fermions in our model are massless one-dimensional Dirac fermions described by the Hamiltonian − i x x + V̂ x − E k x = 0, 1 where we set =c=1. In graphene and carbon nanotubes the spinor polarizations describe the internal degrees of freedom generated by the two-sublattice structure in its primitive cell. When V̂ x =0, the eigenstates of H0 are plane waves multiplying two-dimensional spinors, k x = ke / 2 . When the chemical potential is fixed at =0, the filled Dirac sea has E=− k with k T = 1, 1 / 2. The general form of the potential entering 1 is V̂ x =V0 x Î+V x · . The x part of the potential can be eliminated by a gauge transformation 4 , and a scalar potential proportional to the identity matrix produces no backscattering in the massless Dirac equation. Therefore, we consider potentials for which V lies in the yz plane. In this paper, we consider the effects of the orientation of the potential determined by the angle . Thus, a square barrier potential located between points x1 and x2 is written as V̂ x, = V̂ x − x1 x2 − x , 2 where V̂ =Vei x /2 ze x /2 and x is a step function. To study the force on a square well scatterer, we use the Hellmann-Feynman theorem Ĥ / = E / 5 . Taking the control parameter = x1+x2 /2= x̄, the ground-state average gives the force acting on a rigid barrier. For a barrier with sharp walls the expectation value becomes