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

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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]

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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.

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.

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 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 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

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 pressures 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␾ . 9 aries of a barrier, respectively; T is calculated by integrating Re 2 T Eq. ͑1͒, The total reflection and transmission coefficients shown in

x2 regions I and III of Fig. 1 are given by T P ͩi͵ dx ␴ ͓E Vˆ ͑x͔͒ͪ ͑ ͒ = x exp x − , 5 2 2 i͑2ka+␯͒ t ͑ ͒ t e x1 T = , R = re−ik 2a+d ͩ1+ ͪ, ͑10͒ 1−r2ei␯ 1−r2ei␯ where Px is a spatial ordering operator. For the square poten- ͑ ͒ ␯ ␦␾ ␦␾ϵ␾ ␾ tial of width a defined in Eq. 2 , the transfer matrix for where =2kd+ and 2 − 1. T1 and R1 in region II negative energy states is of Fig. 1 are given by ͑ ␾ ͒ ␴ ␴ជ ͑ ϫ ជ ͒ t rtei kd+ 2 i xk − · xˆ V ͑ ͒ T = cos͑qa͒ − sin͑qa͒, ͑6͒ T1 = ␯ , R1 = ␯ . 11 q 1−r2ei 1−r2ei ជ ͑ where V=V͑0,sin ␾,cos ␾͒ defines a potential in the yz The coefficients for the waves incoming from the left R1 and ͒ ͑ Ј Ј͒ ͱ 2 2 Ͼ T1 and the ones incoming from the right R1 and T1 are plane, q= k −V , and k 0. ͑␾ ␾ ͒ Ј −i 1+ 2 Ј From the transfer matrix we calculate the scattering ma- related by R1 =R1e and T1 =T1. trix S, which gives the transmitted ͑t͒ and reflected ͑r͒ am- To calculate the force in the two-barrier problem we fix plitudes for wave incident on the barrier from the right and the position of the left barrier in Fig. 1 and differentiate the left. The unitary S matrix for a single square barrier is Hamiltonian with respect to d. To obtain the total force, we sum over the occupied states of the filled Dirac sea at fixed ͑ ␾͒ te−ika re−i 2kx2+ ͩ ͪ ͑ ͒ chemical potential. We find that the force between two S1 = ͑ ␾͒ . 7 rei 2kx1+ te−ika square barriers of finite height and width is ϱ The transmission and reflection coefficients can then be pa- dk ik͑d+2a͒ −i͑kd+␾ ͒ i␯ ␩ ␩ F =−2V͵ Re͓Re − R T*e 2 ͑1+e ͔͒. rametrized t=␶ei and r=iͱ1−␶2ei , where ␲ 1 1 0 2 ␭ k tanh ␭a ͑12͒ ␶ = , ␩ = tan−1ͩ ͪ, ͑8͒ ͑V2 cosh2 ␭a − k2͒1/2 ␭ The first term in the integrand arises from the exterior modes with ␭=−iq=ͱV2 −k2. To obtain the hard-wall limit, we fix pushing the two barriers together. The second term accounts ⌫=Va and take ⌫→ϱ. In this limit, ͉r͉2 →1 and ͉t͉2 →0 at all for the confined modes in between the barriers pushing them energies. apart. Since incoming waves are fully transmitted at high For a single barrier, the contributions to the force from the energies for barriers of finite height and width, the integral in particles incoming from the right and left cancel, resulting in Eq. ͑12͒ converges even in the case of sharp barriers no net force. A nonzero force arises from the multiple reflec- ͑a→0͒, with ⌫=Va fixed. Thus, the reflection coefficient tion of electron waves between two barriers. An illustration provides a natural cutoff for the computation of the force of a scattering process for two square potentials with differ- ͑though not the energy ͓4͔͒ even in the limit of infinitely high ␾ ␾ ent spinor polarizations 1 and 2 separated by distance d is barriers. shown in Fig. 1. The contributions from waves incoming The Casimir force for hard-wall boundary conditions re- from the right are also included in the calculation. quires the limits of infinite barrier strength ⌫→ϱ and zero The S matrix for the two-barrier system ͓6͔ in Fig. 1 is width a→0. This limit enforces a vanishing current at the

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CASIMIR EFFECT FOR MASSLESS FERMIONS IN ONE… PHYSICAL REVIEW A 78, 060103͑R͒͑2008͒

π boundaries, the so-called bag boundary conditions. Since the F/(24d2 ) force in Eq. ͑12͒ is multiplied by V, we keep terms to O͑k/V͒ 2 in the integrand. The first term in Eq. ͑12͒ becomes proportional to k, thus implying a continuous spectrum of modes scattering off the barriers from the outside. The second term 1 exhibits resonances that arise from the quantized modes between the boundaries. These resonances, similar to ones seen 0 in Fabry-Perot cavities, are represented by Dirac ␦ functions df ͓3͔ to constrain the k integration: −3π −2π −π π 2π 3p ϱ -1 ␶2 ␲ ␦͑ ͒ ͑ ͒ lim 2 i͑␯+2␩͒ 2 = ͚ k − kn , 13 ␶→0 ͉1+͑1−␶ ͒e ͉ 2d n=0 FIG. 2. ͑Color online͒ Force between two barriers as a function ␦␾ where k =␲͓n+͑1−␦␾/␲͒/2͔/d and ␩→0 in the limit of of their relative spinor polarization . The solid and dashed lines n ͑ ͒ ͑ ͒ infinite potential strength. Here ␦␾ is the difference in the represent the forces in Eqs. 16 and 17 , respectively. The magni- ␦␾ tude of the force in the ⌫Ӷ1 limit, the dashed curve, is rescaled to spinor polarizations of the two scattering potentials and ⌫ / =2␲n denotes the situation for identical scatterers. An in- =1 2, so the two curves can be compared. coming wave vector satisfying the resonance condition in ϳa/d as seen in Eq. ͑17͒, analogous to a multipole expan- Eq. ͑13͒ gets fully transmitted through the two-barrier sys- sion of an electrostatic interaction. tem. The modes in between the barriers, on the other hand, The relative orientation can be expressed as ␦␾ are fully reflected, yielding the appropriate quantization con- −1͓ ជ ជ /͉͑ ជ ͉ ͉ ជ ͉͔͒ =cos V1 ·V2 V1 · V2 . When the two potentials are dition. Combining these results, we obtain aligned at ␦␾=2␲n, we have F=−␲/24d2. This yields the ϱ ϱ ͓ ͔ dk ␲ 1 attractive fermionic Casimir force as found in Ref. 4 . When F =2͵ kͫ1− ͚ ␦͑k − k ͒ͬ + Oͩ ͪ. ͑14͒ ␦␾=͑2n+1͒␲, the relative polarization of the defect poten- ␲ n 0 2 d n=0 V tials is antiparallel and F=␲/12d2, i.e., a repulsive Casimir The Casimir force in Eq. ͑14͒ can be calculated by apply- force is obtained. An analog of our result for a one- ing the generalized Abel-Plana formula dimensional bosonic field is obtained by imposing mixed Dirichlet and Neumann boundary conditions where attractive ϱ ϱ and repulsive Casimir forces are found for like and unlike ͵ tdt− ͚ ͑n + ␤͒ boundary conditions, respectively ͓9͔. A Casimir force that 0 n=0 oscillates as a function of defect separation d is known to ϱ ͑ sinh͑2␲t͒ arise from large momentum backscattering Friedel oscilla- =−͵ tdtͩ +1ͪ, ͑15͒ tions͒ of the Fermi gas ͓8͔. However, the interaction we cal- cos͑2␲␤͒ − cosh͑2␲t͒ 0 culate here is monotonic as a function of distance. In our which is valid for 0ഛ␤Ͻ1. Due to the rapid convergence of calculation, the magnitude and sign of the force vary as a the integral in Eq. ͑15͒, the result does not require the intro- function of the relative polarization of two scatterers at a duction of an explicit ultraviolet cutoff function ͓7͔. More fixed distance. As shown in Fig. 2, this behavior occurs for generally, since the reflection coefficient vanishes at high both finite barriers and hard-wall boundaries. energy, it will regularize the calculation of the force. Using The cusps seen in Fig. 2 at the odd multiples of n result Eq. ͑15͒, we obtain the force for two barriers satisfying bag from a sum over the discrete number of energy levels ͑␦␾͒ ͑ ͒ boundary conditions, En . The energy bands found in Eq. 13 cross zero energy at ␦␾=͑2n+1͒␲ as shown in Fig. 3. At fixed chemical 2 ␲ ␦␾ potential, with negative energy states of the Dirac sea occu- F =− ͫ1−3ͩ ͪ ͬ, ͑16͒ 24d2 ␲ pied, the number of states changes by 1 in each 2␲-periodic region indicated by dotted vertical lines in Fig. 3. Conse- ␲ഛ␦␾Ͻ␲ for − , beyond which it is periodic. We also ex- quently, the force exhibits a discontinuity in slope in Fig. 2 plore the force between two scatterers of finite height and exactly at the values of ␾ at which there is a jump in the width. In the small barrier strength limit, the force becomes number of occupied energy levels. When the barrier strength ⌫2 cos͑␦␾͒ a is finite, the cusps in the force disappear. The resonance con- F =− ͫ1+Oͩ ͪͬ. ͑17͒ dition resulting in quantized states between the barriers is 2␲d2 d only valid for hard-wall boundaries since the quasibound The force in the limits of ⌫→ϱ and ⌫Ӷ1 for a→0 is plot- states between finite barriers exhibit a continuous spectrum. ted for three periods in ␦␾ in Fig. 2. The interaction, Eq. ͑16͒, is likely to be important for The scaling of the force with distance as 1/d2 and the defect interactions on carbon nanotubes and possibly for ratio of 1/2 between the repulsive and attractive forces are other one-dimensional systems as well. Reinserting dimen- universal results for massless one-dimensional fluctuating sional factors this force corresponds to an interaction energy ␲ប / ប fields in the limit dӷa. When the range of the potentials Ec =− vF 24d for two identical scatterers. With vF becomes comparable to their separation, the first-order cor- ϳ5.4 eV Å this gives an energy of 1.4 meV at a range d rection due to the shape of the scatterer scales with ␦F/F =50 nm. Note that its spatial form follows the same scaling

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ZHABINSKAYA, KINDER, AND MELE PHYSICAL REVIEW A 78, 060103͑R͒͑2008͒

En In order to fully understand the Casimir effect between defects on carbon nanotubes, one needs to consider the sym- metry and range of the potentials produced by localized de- −7π −5π −3π −π π 3π 5π fects. The spinor polarization discussed in this paper is de- δφ ␴ ␴ termined by the form of the impurity potential: z and y potentials define a sublattice-asymmetric and bond-centered defects, respectively. In addition, the electronic spectrum contains two distinct Fermi points at inequivalent corners of the two-dimensional Brillouin zone. Short-range potentials couple the two Fermi points, resulting in intervalley scattering ͓10͔. Therefore, both the structure of the defects and the effect of intervalley scattering determine the sign and magnitude of the Casimir interaction. In the context of our FIG. 3. ͑Color online͒ Quantized energy bands for massless model, a sharp potential is one with a range on the order of Dirac fermions due to hard-wall boundary conditions as a function the tube radius R for which the effects of intervalley scatter- / of the relative polarization of the two potentials ␦␾. Solid lines ing are suppressed by a factor of ac R, where ac is the width denote the energy levels of the filled Dirac sea. Vertical dashed lines of the graphene primitive cell. Atomically sharp scatterers, define 2␲-periodic states where the number of occupied states on the other hand, will usually require a treatment of the changes by 1. effects of intervalley as well as intravalley scattering. To summarize, we introduced a force operator approach law as the Coulomb interaction between uncompensated for calculating the Casimir effect and obtained the ␲ប / 2 ϳ charges, but it is reduced by a factor vF 24e 0.05. fluctuation-induced force between two finite square barriers Thus, for charge-neutral dipoles p=es whose electrostatic mediated by massless Dirac fermions in one dimension. In ϳ 2 / 3 ͑ 2 / ͒ϫ͑ / ͒2 interactions scale as Ed −p d =− e d s d , they are taking the limit of sharp barriers of infinite strength, we ob- տ dominated by the Casimir interaction in the far field d 5s. tained a Casimir force that scales as 1/d2 and is tunable from Similarly, this one-dimensional Casimir interaction com- attractive to repulsive form as a function of the relative pletely dominates the familiar van der Waals interactions be- spinor polarizations of the two scattering potentials. tween charge neutral species that are mediated by the fluc- tuations of the exterior three-dimensional electromagnetic This work was supported by the Department of Energy fields. under Grant No. DE-FG02-ER45118.

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