University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 12-30-2008 Casimir Effect for Massless Fermions in One Dimension: A Force- Operator Approach Dina Zhabinskaya University of Pennsylvania Jesse M. Kinder University of Pennsylvania Eugene J. Mele University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Physics Commons Recommended Citation Zhabinskaya, D., Kinder, J. M., & Mele, E. J. (2008). Casimir Effect for Massless Fermions in One Dimension: A Force-Operator Approach. Retrieved from https://repository.upenn.edu/physics_papers/95 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 paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/95 For more information, please contact [email protected]. Casimir Effect for Massless Fermions in One Dimension: A Force-Operator Approach Abstract 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. 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: https://repository.upenn.edu/physics_papers/95 RAPID COMMUNICATIONS PHYSICAL REVIEW A 78, 060103͑R͒͑2008͒ 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 2008͒ 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 magni- tude 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 number͑s͒: 03.70.ϩk, 05.30.Fk, 11.80.Ϫm, 68.65.Ϫk Boundaries modify the spectrum of zero-point fluctua- ously tunable from attractive to repulsive by variation of an tions of a quantum field, resulting in fluctuation-induced internal control parameter, realizing the known bounds for forces and pressures on the boundaries that are known gen- the one-dimensional Casimir interaction as two limiting erally as Casimir effects ͓1͔. When sharp boundary condi- cases. The results may be relevant for indirect interactions tions are used to model the Casimir effect, they yield perfect between defects and adsorbed species on carbon nanotubes. reflection of the incident propagating quantum field at all The fermions in our model are massless one-dimensional ͓ ͔ energies 1 . However, in many physical applications this Dirac fermions described by the Hamiltonian hard-wall limit is not appropriate; of special interest in the ͒ ͑ ͒ ͑ ⌿͔ ͒ ͑ ˆ ץ present work are interactions between localized scatterers in ͓ ␴ − i x x + V x − E k x =0, 1 one dimension that have energy-dependent scattering proper- ties controlled by the strength, range, and shape of the po- where we set ប=c=1. In graphene and carbon nanotubes the tential. Along this line, previous work has recognized that the spinor polarizations describe the internal degrees of freedom finite reflectance of partially transmitting mirrors provides a generated by the two-sublattice structure in its primitive cell. ˆ ͑ ͒ H natural high-energy regularization scheme for computing the When V x =0, the eigenstates of 0 are plane waves multi- ⌿ ͑ ͒ ⌽ ikx/ͱ ␲ effect of sharp reflecting boundaries on the zero-point energy plying two-dimensional spinors, k x = ke 2 . When of the electromagnetic field ͓2,3͔. In more recent work, the chemical potential is fixed at ␮=0, the filled Dirac sea ͉ ͉ ⌽T ͑ ϯ ͒/ͱ Sundberg and Jaffe approached the problem of computing has E=− k with Ϯk = 1, 1 2. the effect of confining boundary conditions on a degenerate The general form of the potential entering ͑1͒ is Vˆ ͑x͒ gas of fermions in one dimension as the limiting behavior for ជ =V ͑x͒Iˆ+V͑x͒·␴ជ . The ␴ part of the potential can be elimi- rectangular barriers of finite width and height. Interestingly, 0 x nated by a gauge transformation ͓4͔, and a scalar potential they encounter a divergence of the Casimir energy in the proportional to the identity matrix produces no backscatter- zero-width limit ͑a sharp boundary͒ even for finite potential ͓ ͔ ing in the massless Dirac equation. Therefore, we consider strength 4 . ជ In this Rapid Communication we address the problem of potentials for which V lies in the yz plane. In this paper, we Casimir interactions between scatterers mediated by a one- consider the effects of the orientation of the potential deter- ␾ dimensional Fermi gas. The fermions in our calculation are mined by the angle . Thus, a square barrier potential lo- massless Dirac fermions appropriate to describe, for ex- cated between points x1 and x2 is written as ͑ ͒ ample, the single-valley electronic spectrum of a metallic ˆ ͑ ␾͒ ˆ ͑␾͒␪͑ ͒␪͑ ͒ ͑ ͒ carbon nanotube. We employ the Hellmann-Feynman theo- V x, = V x − x1 x2 − x , 2 ␴ ␾/ ␴ ␾/ rem to calculate the force, rather than energy, of the interac- ˆ ͑␾͒ i x 2␴ −i x 2 ␪͑ ͒ where V =Ve ze and x is a step function. tion between two scatterers as a function of their separation To study the force on a square well scatterer, we use the d. This approach renders our calculation free from ultraviolet ͔ ͓ ␭ץ/ ץ ␭͘ץ/Hˆ ͑␭͒ץ͗ divergences even for the limiting case of sharp scatterers. We Hellmann-Feynman theorem = E 5 . Tak- ␭ ͑ ͒/ demonstrate that for the case of identical scatterers, this for- ing the control parameter = x1 +x2 2=¯x, the ground-state malism recovers the well-known attractive 1/d2 Casimir average gives the force acting on a rigid barrier. For a barrier force in one dimension. Furthermore, we find that for Dirac with sharp walls the expectation value becomes ˆHץ -fermions the internal structure of the matrix-valued scatter ing potential admits a long-range Casimir interaction which ͗⌿͑x͉͒ ͉⌿͑x͒͘ = ͗⌿͑¯x + a/2͉͒Vˆ ͉⌿͑¯x + a/2͒͘ x¯ץ can also be repulsive or even compensated. This provides a physical situation where the Casimir interaction is continu- − ͗⌿͑¯x − a/2͉͒Vˆ ͉⌿͑¯x − a/2͒͘, ͑3͒ where Vˆ is the square-barrier potential, ¯x is its center, and a *[email protected] is its width. The total force is the expectation value of this 1050-2947/2008/78͑6͒/060103͑4͒ 060103-1 ©2008 The American Physical Society RAPID COMMUNICATIONS ZHABINSKAYA, KINDER, AND MELE PHYSICAL REVIEW A 78, 060103͑R͒͑2008͒ (x, summed over all the occupied V(x¯ץ/ ˆHץ−= ˆforce operator, F states; Eq. ͑3͒ then gives the difference between the pres- sures exerted on the right and left sides of the barrier. For V potentials of general shape, a similar expression can be de- φ T1 φ veloped in terms of an integral over the scattering region. 1 2 T First, we apply Eq. ͑3͒ to calculate the force on an iso- R R1 lated barrier. The eigenstates are represented as linear com- a a H ⌿͑ ͒ binations of right- and left-moving solutions of 0: x x 1 ͑␣ ⌽ ikx ␤ ⌽ −ikx͒ ␣ ␤ -d/2 d/2 = ͱ2␲ k ke + k −ke , where k and k represent the amplitudes of the counterpropagating waves in each region. Region I Region II Region III The yz-polarized potential defined in Eq. ͑2͒ gives Ϯi␾ Vˆ ͑␾͒⌽Ϯ =Ve ⌽ϯ , so the general expression for the ex- k k FIG. 1. ͑Color online͒ Scattering of massless Dirac fermions pectation values in Eq. ͑3͒ at some position x is ͑incoming from the left͒ between two square barriers of height V, ͑ ͒ V ͑ ␾͒ width a, and separation d. The two potentials defined in Eq. 2 ͗⌿͑x͉͒Vˆ ͑␾͉͒⌿͑x͒͘ = Re͓␣ ␤*ei 2kx+ ͔. ͑4͒ ␾ ␲ k k have a spinor polarization determined by angle . The reflection and transmission coefficients are labeled in each scattering region. ␣ We use a transfer matrix to obtain the coefficients k and ␤ ͑ ͒ ␾ k entering Eq. 4 . The transfer matrix is defined so that TRei 1 ⌿͑ ͒ ⌿͑ ͒ ͩ ͪ ͑ ͒ x2 =T x1 , where x1 and x2 are the left and right bound- S2 = −i␾ .