Lattice-fermionic Casimir effect and topological insulators
KKEK-TH-2286
Lattice-fermionic Casimir effect and topological insulators
Tsutomu Ishikawa,
1, 2, ∗ Katsumasa Nakayama, † and Kei Suzuki ‡ Graduate University for Advanced Studies (SOKENDAI), Tsukuba, 305-0801, Japan KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195, Japan (Dated: December 22, 2020)The Casimir effect arises from the zero-point energy of particles in momentum space deformedby the existence of two parallel plates. For degrees of freedom on the lattice, its energy-momentumdispersion is determined so as to keep a periodicity within the Brillouin zone, so that its Casimireffect is modified. We study the properties of Casimir effect for lattice fermions, such as the naivefermion, Wilson fermion, and overlap fermion based on the M¨obius domain-wall fermion formulation,in the 1 + 1, 2 + 1, and 3 + 1 dimensional spacetime with the periodic or antiperiodic boundarycondition. An oscillatory behavior of Casimir energy between odd and even lattice size is inducedby the contribution of ultraviolet-momentum (doubler) modes, which realizes in the naive fermion,Wilson fermion in a negative mass, and overlap fermions with a large domain-wall height. Ourfindings can be experimentally observed in condensed matter systems such as topological insulatorsand also numerically measured in lattice simulations.
I. INTRODUCTION
The Casimir effect [1–4] is one of the important phys-ical phenomena especially for microscopic systems withspatial boundary conditions. Although the Casimir effectwas originally predicted as early as 1948 [1], the first suc-cessful experiment was reported fifty years later [5]. Theoriginal Casimir effect was discussed for the photon field,which is described by quantum electrodynamics (QED),but similar concepts can be extended to any field includ-ing scalar, fermion [6, 7], and other gauge fields, whichhave been actively studied.Lattice field theories have been broadly used not onlyas models to study lattice systems realized in solid-statephysics but also as tools to simulate more general (quan-tum) field theories. A lattice formulation can allow usto investigate physical systems without loss of the non-perturbative effects by using numerical methods such asMonte-Carlo simulations. Lattice simulations of quan-tum chromodynamics (QCD), which is the fundamentaltheory to describe the dynamics of quarks and gluons,are successful examples.So far, Casimir(-like) effects on the lattice were nu-merically studied for scalar field theories [8] and U (1)gauge field theory [9–11] as a simple system. As non-perturbative field theories, the compact U (1) gauge the-ory [12–15] and non-Abelian gauge theories such as SU (2) [16, 17] and SU (3) [18] gauge fields are also stud-ied. More complicated and interesting examples are sys- ∗ [email protected] † [email protected] ‡ [email protected] tems with an interaction between different fields. For ex-ample, QCD includes a strong coupling between quarksand gluons, which leads to various nonperturbative phe-nomena such as the confinement, chiral symmetry break-ing, and instantons. Therefore, the roles of the Casimireffects in such interacting fermionic systems will be in-teresting (e.g., see Refs. [19–59]).We should mention the other important context ofthe Casimir effect on the lattice. The various latticefermions such as the staggered fermion [60, 61], Wilsonfermion [62, 63], and domain-wall (DW) fermion [64–66]also appear in condensed matter physics such as Dirac orWeyl semimetals, topological insulators and ultra-coldatom systems. For example, the low-energy band struc-ture in Dirac semimetals [67–69] is Dirac-like (or linear-like), so that it can be regarded as a dispersion rela-tion of relativistic lattice fermion. Also, the mechanismof gapless surface modes induced from the gapped bulkfermions of topological insulators is formally the same asthe chiral fermions realized in the domain-wall fermionformulation. Our motivation is not limited to theoreticalinterests and is devoted to future experiments for thesecondensed-matter materials. If we can experimentallyprepare sufficiently small materials, the Casimir effectfor Dirac-like lattice fermions should influence the ther-modynamic and transport observables.In Ref. [70], we proposed an analytical definition of theCasimir energy for lattice fermions for the first time, andthe various phenomena based on the Casimir energy wereinvestigated. As a sequential investigation, in this paper, As early works about the Casimir effect for lattice scalar fields,see Ref. [71, 72]. a r X i v : . [ h e p - l a t ] D ec we focus on the following new subjects not studied inRef. [70]:1. Relationship between Casimir energy and disper-sion relation —We propose that the structure ofenergy-momentum dispersion relation for a parti-cle gives us a clear and intuitive interpretation ofthe Casimir effect on the lattice. In particular, wewill emphasize novel phenomena induced by contri-butions from the doubler modes2.
Dependence on spatial dimensions —We investigatethe Casimir effects in the 2 + 1 and 3 + 1 dimen-sional spacetime while we studied only the 1 + 1dimensions in Ref. [70].3.
Other types of lattice fermions —For example, Wil-son fermions with a negative mass are interesting inthe sense that they are closely related to the bandstructures of topological insulators.4.
Derivation from the Abel-Plana formulas —TheAbel-Plana formula is used as a mathematical tech-nique to derive the Casimir effect in the continuumtheory. We give a derivation of the Casimir effecton the lattice by using similar formulas.The theoretical construction of lattice fermions isclosely related to the Nielsen-Ninomiya no-go theo-rem [73, 74]. This theorem states that when we naivelydiscretize the space, the additional degrees of freedomwhich are the so-called doubler particles appear [75]. Toeliminate the contribution from doublers in the contin-uum limit, we can use several fermion formulations. TheWilson fermion [62, 63], overlap fermion [76, 77], anddomain-wall fermoin [64–66] are typical examples of thefermion formulation without doublers in the continuumlimit. The Wilson fermion introduces a small but explicitbreaking term of the chiral symmetry to evade the no-gotheorem. The Wilson fermion has difficulty in physicswhich strongly relates to the chiral symmetry but is use-ful for other physics, and then it is broadly applied fornumerical simulations of the QCD. The overlap fermionis a more sophisticated formulation that can define thechiral symmetry on the lattice. It also introduces smallbreaking terms of the chiral symmetry, but we can discussthe details of the chiral symmetry based on the Ginsparg-Wilson relation [78], which represent the chiral symmetryon the lattice. The domain-wall fermion produces a rep-resentation of the overlap fermion. The M¨obius domain-wall (MDW) fermion [79–81] is an improved formalismof the domain-wall fermion.This paper is organized as follows. In Sec. II, we for-mulate the Casimir energy of lattice fermions, based onthe definition given by Ref. [70]. In Sec. III, we see theresults for the naive lattice fermions in the 1 + 1, 2 + 1,and 3 + 1 dimensional spacetime. In Sec. IV, we investi-gate the Wilson fermion with a positive or negative mass.The latter corresponds to the Casimir effect for the bulkmodes of topological insulators. Section V is devoted 𝑥 𝑡 𝑦, 𝑧 (i) Continuous time (ii) Discretized time 𝑥 𝑡 𝑦, 𝑧 Spatial boundary
FIG. 1. Two types of setup for latticized spacetime, whereone spatial direction has a boundary condition. (i) The timeis not latticized. (ii) The time is latticized. to the overlap fermion with the MDW kernel operator,which corresponds to the Casimir effect for surface modesof topological insulators. In Sec. VI, we summarize ourconclusion. In Appendix A, we give a derivation of theCasimir energies for the free massless fermion in the con-tinuum limit with the periodic and antiperiodic bound-ary conditions. In Appendices B–E, we give derivationsof the Casimir energy for the lattice fermions from theAbel-Plana formulas.
II. DEFINITION OF CASIMIR ENERGY ONTHE LATTICE
Before defining the Casimir energy for fermions on lat-tices, we clarify our setup. In this paper, we consider ageometry where only one spatial dimension is compact-ified by a boundary condition. Then one of the spatialmomentum components p is discretized. The other mo-menta stay continuous. This is the simplest geometryinducing a Casimir energy. Besides, we have two choicesto latticize the temporal direction of the geometry or not,which are painted in (i) and (ii) in Fig. 1. (i) When thetime is continuous, the temporal component of momen-tum is not affected by the lattice, which corresponds toa situation realized in a condensed matter system witha small size. (ii) In contrast, latticized time appears inlattice QCD simulations. Then the temporal momentumis discretized in a similar manner to the discretization ofthe spatial momenta. These two setups lead to similarCasimir effects, but the detail is slightly different. In thispaper, we focus on the geometry (i).To define the Casimir energy, we need the energy-momentum dispersion relation for lattice fermions. Thedispersion relation is obtained from the Dirac opera-tor defined in a relativistic fermion action. With non-latticized temporal components, the (dimensionless) en-ergy is defined as aE ( ap ) = a √ D † D, (1)where a is the lattice spacing, and D is the Dirac operatorthat includes the spatial momenta and a few parameterssuch as the mass but does not include the temporal mo-mentum. This expression correctly reflects the positionsof poles in the fermion propagator in momentum space.In the space where one spatial direction is compact-ified, the corresponding momentum is discretized. Thediscretized momentum under the periodic and antiperi-odic boundary conditions is ap → ap P1 ( n ) = 2 nπN , (2) ap → ap AP1 ( n ) = (2 n + 1) πN , (3)respectively, and N is the lattice size. The label n is aninteger bounded by the (first) Brillouin zone: 0 ≤ ap k < π (BZ1) or − π < ap k ≤ π (BZ2). Then the range of n in both the boundary conditions is simply0 ≤ n P , AP < N (BZ1) , (4) − N In this section, we demonstrate the Casimir effect forthe naive lattice fermion which is one of the simplestformulations of lattice fermions, but it contains doublerdegrees of freedom.We define the (dimensionless) Dirac operator of thenaive lattice fermion in momentum space: aD nf ≡ i (cid:88) k γ k sin ap k + am f , (9)where γ k is the gamma matrix with the index k , and m f is the mass of the fermion.From the definition (1), the dispersion relation of thisfermion is written as a E ( ap ) = (cid:88) k sin ap k + ( am f ) . (10)Using the square root of Eq. (10) and the definition (7),the Casimir energy in the 3 + 1 dimensions is aE , nf , PCas ≡ aE , nf , P0 ( N ) − aE , nf , P0 ( N → ∞ ) (11)= c deg (cid:90) d ap ⊥ (2 π ) − (cid:88) n (cid:115) sin nπN + (cid:88) k =2 , sin ap k + ( am f ) + N (cid:90) BZ dap π (cid:115) (cid:88) k =1 , , sin ap k + ( am f ) ,aE , nf , APCas ≡ aE , nf , AP0 ( N ) − aE , nf , AP0 ( N → ∞ ) (12)= c deg (cid:90) d ap ⊥ (2 π ) − (cid:88) n (cid:118)(cid:117)(cid:117)(cid:116) sin (2 n + 1) πN + (cid:88) k =2 , sin ap k + ( am f ) + N (cid:90) BZ dap π (cid:115) (cid:88) k =1 , , sin ap k + ( am f ) , where the integration with respect to ap ⊥ is in the rangeof the first Brillouin zone. Then we can define theCasimir energy without divergence.By using mathematical techniques such as the Abel-Plana formulas (which are shown in Appendix B), onemay get simpler analytic formulas for the Casimir energy.For example, for the naive fermion with m f = 0 in the1 + 1 dimensional spacetime, we can derive the exactformulas for dimensionless Casimir energies [70] (for aderivation, see Appendix C): aE , nf , PCas = 2 Nπ − cot π N ( N = odd) , (13) aE , nf , PCas = 2 Nπ − πN ( N = even) , (14) aE , nf , APCas = 2 Nπ − cot π N ( N = odd) , (15) aE , nf , APCas = 2 Nπ − πN ( N = even) , (16)Here, we find that the odd lattice ( N = odd) and evenlattice ( N = even) exhibit different Casimir energiesfor both the periodic and antiperiodic boundaries. Inother words, the Casimir energy for the naive fermionis oscillatory between the even and odd lattices. Thisbehavior is induced by the existence of the ultraviolet-momentum zero modes (or massless doublers) in thenaive fermion [70]. By expanding these formulas (13)–(16) by a/L or equivalently 1 /N , we can obtain the for-mulas with the small a expansion [70] E , nf , PCas = π L + π a L + O ( a ) ( N = odd) , (17) E , nf , PCas = 2 π L + 2 π a L + O ( a ) ( N = even) , (18) E , nf , APCas = π L + π a L + O ( a ) ( N = odd) , (19) E , nf , APCas = − π L − π a L + O ( a ) ( N = even) . (20) Thus, the terms depending on the lattice spacing a aredominated by the terms with a .In Fig. 2, we plot the Casimir energies for the masslessor positive-mass naive lattice fermions for the periodicor antiperiodic boundary. In these plots, it is convenientto see two types of dimensionless quantities. The first is aE Cas , which is proportional to 1 /N d in the d + 1 dimen-sions for massless particles in the continuum theory. Thesecond is N d − LE Cas , which is a constant in the contin-uum theory. Also, in Fig. 2, we can find that there areoscillatory behaviors of the Casimir energy in the 1 + 1,2 + 1, and 3 + 1 dimensions.For a better understanding, it is useful to comparethe Casimir energy and the corresponding dispersion re-lation. In Fig. 3, we show the (continuous) dispersionrelations of the massless or positive-mass naive fermionin the 1 + 1 dimensions, where the Brillouin zone is − π < ap ≤ π . For the massless naive fermion, we findthe dispersion relation goes to zero at ap = π that is theultraviolet-momentum zero mode as well as at ap = 0that is the infrared zero mode. When a boundary condi-tion is imposed, the continuous energy level is discretizedinto some levels living on this dispersion relation. TheCasimir energy defined as Eq. (8) is determined by thedifference between the sum part including contributionsfrom these discretized levels and the integral part includ-ing the continuous level.The sign of the Casimir energy is often interesting,where positive and negative Casimir energies correspondto the repulsive and attractive Casimir forces, respec-tively. In particular, this sign is related to the number ofthe zero modes with aE = 0:1. Odd N with periodic boundary—There is one in-frared zero mode ( ap = 0). This zero mode sup-presses the contribution of the negative sum partto the Casimir energy. As a result, the Casimir en-ergy is dominated by the positive integral part, andits sign is positive.2. Even N with periodic boundary—There are twozero modes ( ap = 0 , π ). Both the zero modes sup-press the negative sum part, so that the positiveCasimir energy is enhanced. C a s i m i r ene r g y a E C a s Lattice size N=L/a2 π /3N π /6N am f =0am f =0.2 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 L E C a s C a s i m i r ene r g y a E C a s Lattice size N=L/a4 ζ (3)/ π N ζ (3)/2 π N am f =0am f =0.2 0 0.5 1 1.5 2 2.5 0 4 8 12 16 20 N L E C a s C a s i m i r ene r g y a E C a s Lattice size N=L/a8 π /45N π /90N am f =0am f =0.2 0 0.5 1 1.5 2 2.5 0 4 8 12 16 20 N L E C a s −0.8−0.4 0 0.4 0.8 1.2 0 2 4 6 8 10 12 14 16 18 20D=1+1Naive fermionAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a π /6N− π /3Nam f =0am f =0.2−1.5−1−0.5 0 0.5 0 2 4 6 8 10 L E C a s −0.6−0.4−0.2 0 0.2 0 2 4 6 8 10 12 14 16 18 20D=2+1Naive fermionAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a ζ (3)/2 π N −3 ζ (3)/ π N am f =0am f =0.2−2−1.6−1.2−0.8−0.4 0 0.4 0 4 8 12 16 20 N L E C a s −0.4−0.2 0 0.2 0 2 4 6 8 10 12 14 16 18 20D=3+1Naive fermionAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a π /90N −7 π /45N am f =0am f =0.2−3−2−1 0 0 4 8 12 16 20 N L E C a s FIG. 2. Casimir energy for massless or positive-mass naive fermion in the 1 + 1, 2 + 1, and 3 + 1 dimensional spacetime (thetemporal direction is not latticized). Left: Periodic boundary. Right: Antiperiodic boundary. 3. Odd N with antiperiodic boundary—There is oneultraviolet-momentum zero mode ( ap = π ). Thissituation is equivalent to the odd N with the pe-riodic boundary, although the momentum of thezero mode is different from each other. Such equiv-alence on the odd lattice appears not only in the1 + 1 dimensions [70] but also in the 2 + 1 and 3 + 1dimensions. 4. Even N with antiperiodic boundary—In this case,there is no zero mode. Then, the negative sum partof the Casimir energy is enhanced by the highernonzero modes, and as a result the sign of theCasimir energy becomes negative, which is the onlysituation to induce the attractive Casimir force byusing the free naive fermion. D i s pe r s i on a √ DD ✝ Spatial momentum ap Cont. am f =0Naive, am f =0.2Naive, am f =0Wilson, am f =0.2Wilson, am f =0 FIG. 3. Dispersion relations for naive and Wilson latticefermions in the 1 + 1 dimensional spacetime (the temporaldirection is not latticized). IV. CASIMIR ENERGY FOR WILSONFERMION In this section, we investigate the Casimir effects forWilson fermions. In Sec. IV A, we study the conventionalWilson fermion [62, 63]. The results in the 1 + 1 dimen-sions were studied in Ref. [70], so that, in this paper,we focus on the dependence on the spatial dimension.In Sec. IV B, we examine Wilson fermions with negativemasses. A. Wilson fermion The (dimensionless) Dirac operator of the Wilsonfermion with the Wilson parameter r and the fermionmass m f is defined as aD W ≡ i (cid:88) k γ k sin ap k + r (cid:88) k (1 − cos ap k ) + am f . (21)The term proportional to r is called the Wilson term, andit is interpreted as a momentum dependent mass term toeliminate the doublers. From this Dirac operator, thedispersion relation is a E ( ap ) = (cid:88) k sin ap k + (cid:34) r (cid:88) k (1 − cos ap k ) + am f (cid:35) . (22)From the definitions of the Casimir energy, Eqs. (7) and(8), we can get the Casimir energy.For example, when we set r = 1 and m f = 0 in the 1+1dimensional spacetime, we obtain the exact formulas [70](for a derivation, see Appendix D): aE , W , PCas = 4 Nπ − π N , (23) aE , W , APCas = 4 Nπ − π N . (24) Expanding these formulas by a/L , we obtain [70] E , W , PCas = π L + π a L + O ( a ) (25) E , W , APCas = − π L − π a L + O ( a ) . (26)Here, we find that the terms with π L for the periodicboundary and − π L for the antiperiodic boundary agreewith the Casimir energy for the massless fermion in con-tinuum theory, respectively (for a derivation, see Ap-pendix A): E , cont , PCas = π L , (27) E , cont , APCas = − π L . (28)Thus, by taking the continuum limit a → a andthe higher orders are lattice effects (or “lattice artifacts”in the context of lattice simulations) which is related toproperties in the ultraviolet region of the dispersion re-lation.In Fig. 4, we plot the Casimir energy for the masslessor positive-mass Wilson fermion at r = 1. We find that,even for the 2 + 1 and 3 + 1 dimensional spacetime, theCasimir energies for the massless Wilson fermions areagree well with the continuum limit: E , cont , PCas = ζ (3) πL , (29) E , cont , APCas = − ζ (3)4 πL , (30) E , cont , PCas = π L , (31) E , cont , APCas = − π L . (32)Also, we find that the Casimir energy for the positive-mass Wilson fermions is suppressed compared with thatfor the massless one, which is similar to the behavior ofmassive degrees of freedom in the continuum theory (e.g.,see Refs. [7, 82, 83]).Finally, we discuss the relation between the sign of theCasimir energy and the number of the zero modes:1. Periodic boundary—For the massless case, there isone infrared zero mode ( ap = 0). This zero modesuppresses the negative sum part to the Casimirenergy, so that the Casimir energy becomes posi-tive. When the fermion has a positive mass andthe zero mode disappears, both the negative sumpart and positive integral part are enhanced. As aresult, the negative Casimir energy is suppressed. C a s i m i r ene r g y a E C a s Lattice size N=L/a π /3Nam f =0am f =0.2 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 L E C a s C a s i m i r ene r g y a E C a s Lattice size N=L/a ζ (3)/ π N am f =0am f =0.2 0 0.2 0.4 0.6 0.8 0 4 8 12 16 20 N L E C a s C a s i m i r ene r g y a E C a s Lattice size N=L/a π /45N am f =0am f =0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 4 8 12 16 20 N L E C a s −0.8−0.6−0.4−0.2 0 0 2 4 6 8 10 12 14 16 18 20D=1+1WilsonAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a− π /6Nam f =0am f =0.2−0.8−0.6−0.4−0.2 0 2 4 6 8 10 L E C a s −0.8−0.6−0.4−0.2 0 0 2 4 6 8 10 12 14 16 18 20D=2+1WilsonAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a−(3/4) ζ (3)/ π N am f =0am f =0.2−0.8−0.6−0.4−0.2 0 0 4 8 12 16 20 N L E C a s −0.8−0.6−0.4−0.2 0 0 2 4 6 8 10 12 14 16 18 20D=3+1WilsonAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a−7 π /360N am f =0am f =0.2−1.2−1−0.8−0.6−0.4−0.2 0 0 4 8 12 16 20 N L E C a s FIG. 4. Casimir energy for massless or positive-mass Wilson fermion in the 1 + 1, 2 + 1, and 3 + 1 dimensional spacetime (thetemporal direction is not latticized). Left: Periodic boundary. Right: Antiperiodic boundary. 2. Antiperiodic boundary—Even for the masslesscase, there is no zero mode. Then, the negativesum part of the Casimir energy is enhanced by thehigher nonzero modes, so that the Casimir energyis negative. B. Wilson fermions with a negative mass Next, we study the Casimir energy for the Wilsonfermions with a negative mass, am f < −1−0.5 0 0.5 1 1.5 2 0 2 4 6 8 10 12 14 16 18 20D=1+1WilsonPeriodic C a s i m i r ene r g y a E C a s Lattice size N=L/a π /3N− π /6Nam f =0am f =−0.5 am f =−1am f =−1.5am f =−2am f =−3−1−0.5 0 0.5 1 0 2 4 6 8 10 L E C a s −1.5−1−0.5 0 0.5 1 0 2 4 6 8 10 12 14 16 18 20D=2+1WilsonPeriodic C a s i m i r ene r g y a E C a s Lattice size N=L/a ζ (3)/ π N (1/4) ζ (3)/ π N ζ (3)/ π N −(3/4) ζ (3)/ π N am f =0 am f =−1am f =−2am f =−3am f =−4am f =−5 −1−0.5 0 0.5 1 0 4 8 12 16 20 N L E C a s −1−0.5 0 0.5 1 1.5 2 0 2 4 6 8 10 12 14 16 18 20D=1+1WilsonAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a− π /6N π /3Nam f =0am f =−0.5 am f =−1am f =−1.5am f =−2am f =−3−1−0.5 0 0.5 1 0 2 4 6 8 10 L E C a s −1.5−1−0.5 0 0.5 1 0 2 4 6 8 10 12 14 16 18 20D=2+1WilsonAntiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a −(3/4) ζ (3)/ π N (1/4) ζ (3)/ π N −(2/3) ζ (3)/ π N ζ (3)/ π N am f =0 am f =−1am f =−2am f =−3am f =−4am f =−5 −1−0.5 0 0.5 1 0 4 8 12 16 20 N L E C a s FIG. 5. Casimir energy for Wilson fermions with a negative mass in the 1 + 1 and 2 + 1 dimensional spacetime (the temporaldirection is not latticized). Left: Periodic boundary. Right: Antiperiodic boundary. D i s pe r s i on a √ DD ✝ Spatial momentum ap Cont.am f =0.5am f =0am f =−0.5 am f =−1am f =−1.5am f =−2am f =−3 FIG. 6. Negative mass dependence of dispersion relationsfor Wilson fermions in the 1 + 1 dimensional spacetime (thetemporal direction is not latticized). well understood by their dispersion relations, as shownin Fig. 6. Here, our findings are as follows: 1. − < am f ≤ am f = 0, we observe theCasimir energy of massless Wilson fermions. As am f decreases, both the Casimir energy aE Cas andits coefficient LE Cas is suppressed in larger latticesize, which is similar to that for Wilson fermionswith a positive mass. This similarity is understoodfrom the dispersion relations shown in Fig. 6. Forexample, the dispersion for the negative mass with am f = − . am f = 0 . am f = − am f = − aD = iγ sin ap + r ( − cos ap ) . (33)Then, we find that the dispersion relation is a con-stant, as shown in the black line of Fig. 6: a E ( ap ) = 1 . (34)Using the definition (8), we can show that the cor-responding Casimir energy is zero: aE = 0 . (35)Thus, am f = − − < am f < − LE Cas is suppressed in larger latticesize. As shown in the cyan curve of Fig. 6, thisoscillation is dominated by lower nonzero modesaround ap = π , and the contribution from highermodes around ap = 0 is relatively suppressed.4. am f = − am f = − 1. Here, we find an oscilla-tion of the Casimir energy, and LE Cas approachesto a constant in larger lattice size, which impliesthat the Casimir energy is dominated by masslessdegrees of freedom.The Wilson Dirac operator at am f = − aD = iγ sin ap + r ( − − cos ap ) . (36)Then, we find that the dispersion has a ultraviolet-momentum zero mode (in other words, masslessdoubler) at ap = π , as shown in the magenta curveof Fig. 6. Note that Fig. 6 is just a plot of contin-uous spectrum. Although energy levels discretizedby the existence of boundaries is not necessarily topick up this zero mode, which depends on the formof the boundary condition, at least the Casimir en-ergy is dominated by “light” modes around thiszero mode. The exact formulas of the Casimir en-ergies at am f = − r = 1 are (for a derivation,see Appendix E) aE , PCas = 4 Nπ − π N ( N = odd) , (37) aE , PCas = 4 Nπ − π N ( N = even) , (38) aE , APCas = 4 Nπ − π N ( N = odd) , (39) aE , APCas = 4 Nπ − π N ( N = even) . (40)Here, we find that Eqs. (38) and (39) agree withEq. (23) for the periodic boundary at am f = 0.On the other hand, Eqs. (37) and (40) agree withEq. (24) for the antiperiodic boundary at am f = 0.Thus, we find that, on the even lattice, the Casimirenergies for the am f = 0 and am f = − areequivalent to each other. This is because, for theperiodic boundary, the contribution from the in-frared zero mode at am f = 0 is equivalent tothat from the ultraviolet-momentum zero mode at am f = − 2, which holds also for contribution from nonzero modes. The existence of zero modes leadsto the positive Casimir energy. For the antiperi-odic boundary, the dispersion does not contain zeromodes, so that it induces negative Casimir energy.On the odd lattice, the Casimir energy with theperiodic (antiperiodic) boundary at am f = − isequivalent to that with the antiperiodic (periodic)boundary at am f = 0 . This is because the in-frared zero mode at am f = 0 for the periodicboundary plays an equivalent role of the ultraviolet-momentum zero mode at am f = − am f = 0 for the antiperiodic boundary and at am f = − am f < − LE Cas is suppressed in larger lattice size. Such be-havior is similar to the region of − < am f < − am f > am f < − − < am f < ν = 1 corresponding to themapping S → S from momentum space to spin space.The existence of an open boundary condition (or a finitelength) induces a gapless chiral edge mode at the zero-dimensional edge of the insulator. The Casimir effectfor bulk modes inside the topological insulator will affectthe thermodynamic properties of the bulk modes. Notethat, as shown in Eq. (34) and the black line in Fig. 6, am f = − − < am f ≤ am f = 0, the Casimir energy aE Cas is proportional to 1 /N in larger lattice size,and its coefficient N LE Cas is not suppressed. Sucha behavior is induced by the infrared zero mode,( ap , ap ) = (0 , − < am f < aE Cas is not proportional to 1 /N , and N LE Cas is sup-pressed in larger lattice size. This behavior impliesthat the relevant degrees of freedom may be mas-sive. For example, we show the dispersion with0 am f = − am f = − Here, we find an oscillatory behavior of the Casimirenergy between the odd and even lattices. N LE Cas is not suppressed in larger lattice size, which im-plies that the Casimir energy is dominated by mass-less degrees of freedom. As shown in left mid-dle panel of Fig. 7, there are two zero modes,( ap , ap ) = (0 , π ) and ( π, ap , ap ) = (0 , − π ) and ( − π, 0) areoutside the first Brillouin zone ( − π < ap k ≤ π ), sothat they do not contribute to the Casimir effectwe showed.3. − < am f < − N LE Cas is suppressed in larger lat-tice size. We show am f = − | ap k | ∼ π ). These lowest modescontribute to the oscillation of the Casimir energy.4. am f = − N LE Cas is not suppressed in larger lattice size, whichis a similar behavior to am f = − 2. Asshown in the left lower panel of Fig. 7, we cansee that there are an ultraviolet-momentum zeromode, ( ap , ap ) = ( π, π ). Note that the otherzero modes, ( ap , ap ) = ( π, − π ), ( − π, π ), and( − π, − π ), are outside the first Brillouin zone, sothat they do not contribute to the Casimir effect.5. am f < − N LE Cas is suppressed in largerlattice size, which is similar to the case with − 2. We show am f = − am f = − 4, but all the modes are more than 1,and there is no zero mode.Finally, we comment on the relationship to two-dimensional (2D) materials in condensed matter physics.When we consider the 2 × c deg = 1, the corresponding materials are quan-tum anomalous Hall insulators (or Chern insulators).These materials violate time-reversal invariance and can The Wilson fermion satisfying am f + dr = 0, where d is thedimensions in Euclidean lattice space, is known as the central-branch Wilson fermion [86–91] which is a special lattice fermion.In our setup, the time in the 2 + 1 dimensional spacetime isnot on the lattice, so that the Wilson fermion with am f = − FIG. 7. Negative mass dependence of dispersion relationsfor Wilson fermions in the 2 + 1 dimensional spacetime (thetemporal direction is not latticized). be described by the Qi-Wu-Zhang model [92, 93] as a typ-ical model. Then, the Wilson fermion with either a posi-tive mass am f > am f < − − < am f < − < am f < − C = 1 and C = − 1, respectively.Thus, − < am f < − < am f < − × c deg = 2 (the case withKramers doublet such as spin degrees of freedom), thecorresponding materials are time-reversal invariant 2Dtopological insulators, namely quantum spin Hall in-sulators, which are two copies of quantum anomalousHall insulators. Such materials can be described by theBernevig-Hughes-Zhang model [96] and were experimen-tally observed in CdTe/HgTe/CdTe quantum well [97].Even in this case, within our setup, the correspondingCasimir energy for the bulk fermion is qualitatively the1same (except for the factor of the number of the degreesof freedom, c deg = 2) as long as the dispersion relationsof the different spin components degenerate. V. CASIMIR ENERGY FOR OVERLAPFERMION In this section, we study the Casimir energy for theoverlap fermion with an MDW kernel operator. In theDW fermion formulation [64–66], a “bulk” fermion is de-fined in the D + 1-dimensional Euclidean space, whichbecomes a kernel operator in the D -dimensional space.This bulk fermion is projected into the chiral “surface”fermion in the D -dimensional space. Usually, the surfacefermions include information on the finite length of theextra dimension, but for simplicity we consider the infi-nite length. Then the DW fermion is equivalent to theoverlap fermion [76, 77]. A. MDW kernel operators Here, we define the MDW kernel operator D MDW [79–81], aD MDW ≡ b ( aD W )2 + c ( aD W ) , (41)where b and c are called M¨obius parameters . The oper-ator at b = 1 and c = 1 corresponds to the conventionalShamir-type formulation [65], and that at b = 2 and c = 0is Bori¸ci-type (or Wilson-type) [98, 99]. D W is the Wil-son Dirac operator with r = 1, as defined in Eq. (21),where the original fermion mass m f is replaced by the domain-wall height am f → − M that plays a role as thenegative mass of bulk fermions. B. Overlap fermion with MDW kernel Using the MDW kernel operator D MDW , we define theoverlap Dirac operator D OV with a fermion mass m f anda Pauli-Villars mass m PV , aD OV ≡ (2 − cM ) M am PV × (1 + am f ) + (1 − am f ) V (1 + am PV ) + (1 − am PV ) V , (42)with V ≡ γ sign( γ aD MDW ) = D MDW (cid:113) D † MDW D MDW . (43)The Pauli-Villars mass m PV was introduced so as tosatisfy the Ginsparg-Wilson relation. The scaling fac-tor (2 − cM ) M m PV with a constraint 2 − cM > a → D † OV D OV = p for m f = 0. Note that if we consider a finite length ofthe extra dimension, then the sign function in Eq. (43)is replaced by an approximate functional form dependingon the finite length.As a result, the dispersion relation of the overlapfermion is written as a E = [(2 − cM ) M m PV ] × am f ) ] + [1 − ( am f ) ]( V † + V )2[1 + m ] + [1 − m ]( V † + V ) . (44)where we used V † V = 1 and the commutation relationbetween V † + V and V . Using D W , V † + V is V † + V = (cid:16) D † MDW + D MDW (cid:17) (cid:113) D † MDW D MDW , = 2( D W + D † W + cD † W D W ) 1 (cid:113) D † W D W × (cid:113) c ( D † W + D W ) + c D † W D W , (45)where we used D † W D W > c ( D † W + D W ) + c D † W D W > b . The b dependencecan appear when we consider a finite length of the extradimension. By substituting the square root of the dis-persion relation (44) into the definitions, (7) and (8), wecan calculate the Casimir energy for the overlap fermion. C. Numerical results In Fig. 8, we show the dependence of the Casimir en-ergy for the overlap fermion with the MDW kernel op-erator on the domain-wall height ( M = 0 . 5, 1 . 0, and1 . c = 0 and m PV = 1 . M . Note that M = 1 . am f = 0.First, we summarize the properties of the Casimir en-ergy in the three regions (namely, small N , intermediate N , and large N ):1. Suppression of Casimir energy in small N —InFig. 8, for M (cid:46) . 0, we find that the Casimir en-ergy for the overlap fermion is suppressed as thelattice size N decreases. Such suppression of theCasimir energy at a small lattice size is intuitivelyunderstood by considering the case at N = 1.2 PV =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/a π /3NM =0.5M =1.0M =1.5 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 L E C a s PV =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/a ζ (3)/ π N M =0.5M =1.0M =1.5 0.2 0.4 0.6 0.8 0 5 10 15 20 N L E C a s PV =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/a π /45N M =0.5M =1.0M =1.5M =1.8−1−0.5 0 0.5 1 0 5 10 15 20 N L E C a s −1.5−1−0.5 0 0 2 4 6 8 10 12 14 16 18 20D=1+1MDWc=0,m PV =1Antiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a− π /6NM =0.5M =1.0M =1.5−1.5−1−0.5 0 0 2 4 6 8 10 L E C a s −0.6−0.4−0.2 0 0 2 4 6 8 10 12 14 16 18 20D=2+1MDWc=0,m PV =1Antiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a−(3/4) ζ (3)/ π N M =0.5M =1.0M =1.5−0.8−0.6−0.4−0.2 0 0 5 10 15 20 N L E C a s −0.4−0.3−0.2−0.1 0 0 2 4 6 8 10 12 14 16 18 20D=3+1MDWc=0,m PV =1Antiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/a−7 π /360N M =0.5M =1.0M =1.5M =1.8−1−0.5 0 0.5 1 1.5 0 5 10 15 20 N L E C a s FIG. 8. Casimir energy for overlap fermions with MDW kernel operator in the 1 + 1, 2 + 1, and 3 + 1 dimensional spacetime(the temporal direction is not latticized). M¨obius parameter c = 0 and Pauli-Villars mass m PV = 1 are fixed. Left: Periodicboundary. Right: Antiperiodic boundary. For example, for the periodic boundary in the 1 + 1dimensions, the possible momentum and energy at N = 1 are only ap = 0 and aE = 0, respectively.Then the sum part of aE Cas defined as Eq. (8) iszero, so that aE Cas is determined by only the pos-itive integral part. The integral part at a smaller M is smaller than that at a larger M , so thatthe corresponding (positive) Casimir energy at the smaller M is also suppressed compared with thelarger M .The case with the antiperiodic boundary is morecomplicated than that with the periodic boundary.In the 1 + 1 dimensions, the possible momentumat N = 1 is only ap = π which is the maxi-mum energy level within the Brillouin zone, and3 PV =1 D i s pe r s i on a √ DD ✝ Spatial momentum ap Cont.M =0.5M =1.0 M =1.5M =1.8 FIG. 9. Domain-wall height M dependence of dispersionrelations for overlap fermions with MDW kernel operator inthe 1+1 dimensional spacetime (the temporal direction is notlatticized). this energy level dominates the negative sum partof aE Cas . The difference between the negative sumpart and positive integral part determines the neg-ative aE Cas . With decreasing M , both the partsare suppressed, where the negative aE Cas is deter-mined by a balance between the two parts. As aresult, aE Cas is also smaller. Thus a small M canlead to the suppression of the Casimir energy.2. Enhancement of Casimir energy in intermediate N —Next we focus on the intermediate N region.For M (cid:46) . 0, we find that the Casimir energyfor the overlap fermion is enhanced, compared withthat expected in the continuum theory. Such en-hancement is similar to that for the Wilson fermionat am f = 0 in the small N region. For the periodicboundary, this enhancement is induced by contri-butions from nonzero modes ( ap > 0) which ap-pears at N ≥ 2. For the antiperiodic boundary,this is caused by contributions from lower modes( ap < π ) except for the maximum mode ( ap = π ).3. Good agreement with continuum theory in large N —In the large N region, the Casimir energy forthe overlap fermion agrees well with that in thecontinuum theory, which is similar to the Wilsonfermion at am f = 0. Thus, the Casimir energy inthe large N region is determined by the infraredpart of the dispersion relations.Next, we summarize the domain-wall height M de-pendence:1. No oscillation at M < . M = 0, theoverlap fermion is not defined. In the region with0 < M ≤ . 0, we find that there is no oscilla-tion. This is because the doublers are sufficientlymassive, so that their contributions are irrelevant. When M (cid:54) = 0 is small enough, both the suppres-sion of Casimir energy at small N and the enhance-ment at intermediate N are visible.2. Good agreement with continuum theory at M =1 . M = 1 . Oscillation at M > . M > . 0, we find an oscillatory behavior of the Casimirenergy, where we find the enhancement on the oddlattice and suppression on the even lattice. This os-cillation is induced by the contributions from mas-sive doublers which is irrelevant at M ≤ . 0. Asshown in Fig. 9, the dispersion relation at M = 1 . N regionis quite different. Therefore, if we try to repro-duce the Casimir energy for the continuum Diracfermion by using the overlap fermion, M ∼ . M .In lattice simulations for fermions interacting withgauge fields, a setup with M > . M > . c and the Pauli-Villars mass m PV as additionaltuning parameters. The results in the 1 + 1 dimensionsare shown in Ref. [70]. While the M¨obius parameter c does not induce oscillation of the Casimir energy, a largevalue of the Pauli-Villars mass m PV leads to an oscilla-tion of the Casimir energy, which is a similar mechanismto the massive doublers in M > . VI. CONCLUSION AND OUTLOOK In this paper, based on the formulation given byRef. [70], we investigated the Casimir energies for the free In the domain-wall fermion formulation, the doublers with aheavy mass still survive. We call such ultraviolet-momentummodes “massive doublers”. and carbon nanotubes, quan-tum anomalous Hall insulators described by the Haldanemodel [124], and quantum spin Hall insulators describedby the Kane-Mele model [125, 126] will be experimen-tally measured in small-size materials. In particular,when we consider the edge of the small-size direction ofhoneycomb lattice ribbons, there are two types of struc-tures: the armchair edge and the zigzag edge. These twotypes of edges are known to lead to different band struc-tures of bulk modes [127], and the size dependence ofobservables is related to the Casimir effect for the bulkmodes. Moreover, a rolled-up graphene sheet is a carbonnanotube, which is nothing but a 2D honeycomb latticewith the periodic boundary condition, and electrons liv-ing on this lattice are affected by the Casimir effect withthe periodic boundary. For carbon nanotubes, there arethree types of structures: the armchair, zigzag, and chi-ral configurations are known to exhibit different bandgap energies [128, 129]. As a result, in a small-size tube,such three structures lead to different Casimir effects, andthen it should influence transport/thermodynamic prop-erties such as electric/thermal conductivity and specificheat. In addition, Casimir effects on “curved” lattice de-formed by insertion of structural defects such as latticekirigami [130, 131] are also interesting, which is analogousto the Casimir effect in curved spacetime in continuumtheory. Thus a wide range of lattice Casimir physics isleft for future work. ACKNOWLEDGMENTS The authors are grateful to Yasufumi Araki for care-fully reading our manuscript and Daiki Suenaga for giv-ing us helpful comments about the Abel-Plana formu-las. This work was supported by Japan Society for thePromotion of Science (JSPS) KAKENHI (Grant Nos.JP17K14277 and JP20K14476). Appendix A: Derivation of Casimir energy for massless fermions in continuum limit We quickly review the derivation of the Casimir energy for a massless fermion in D = d + 1 dimensional spacetime.The dispersion relation of the fermion is E = (cid:112) p , (A1) For the electromagnetic Casimir effects induced betweengraphene another material (or between two graphene sheets),there are already many works (e.g., see Refs. [113–117]). Carbon nanotubes in the continuum limit can be approximatedas the cylindrically compactified space with the spatial topology R × S or the toroidally compactified space with the spatialtopology S × S . The Casimir effect for Dirac fermions in suchcontinuous spacetime was investigated in Refs. [118–123]. If onetries to investigate more realistic carbon nanotubes, systems withnonzero lattice spacing should be taken into account. p = p + p + · · · + p d . The thermodynamic potential Ω at zero temperature and zero chemical potential isthe integral of the dispersion relation: Ω V = − (cid:90) d d p (2 π ) d (cid:112) p , (A2)where V is the volume of the system. It can be considered as the sum of the zero-point energies for all momenta. Thecoefficient 1 / c deg = 2 has to be multiplied, but we set c deg = 1 and neglect the spin degrees of freedom for simplicity. Thethermodynamic potential is still divergent, and usually one neglects it. In the compactified spacetime, however, wecan get a finite value by a subtraction scheme, and it modifies thermodynamic properties.Let us consider the spatial geometry where one spatial dimension, x , is compactified. Then the boundary condition isnecessary. Here we limit ourselves to the periodic boundary condition (PBC) and the antiperiodic boundary condition(ABC). The momentum along the x direction is discretized and depends on the boundary condition: p = 2 πL n (PBC) , (A3) p = 2 πL (cid:18) n + 12 (cid:19) (ABC) , (A4)where L is the length of the x direction, and n is an integer. We replace the integral by the sum: (cid:90) dp π → L ∞ (cid:88) n = −∞ . (A5)The momenta along the other direction are continuous and integral variables. We can compute the thermodynamicpotential in the anisotropic system, the so-called Casimir energy.The derivation of the Casimir energy is based on the analyticity of the dispersion relation. The potential densityΩ( L ) V = − L ∞ (cid:88) n = −∞ (cid:90) d d − p ⊥ (2 π ) d − (cid:113) p ⊥ + p (A6)is still divergent, where p ⊥ = p + · · · + p d . We have to extract a finite part by removing the divergence in the infinitevolume from Eq. (A6). In the massless case, it can be done by the zeta function regularization and the analyticcontinuation.Let us move on to the derivation of the Casimir energy. By using the well-known formula for a d dimensionalintegral with respect to k , for parameters l and ∆, (cid:90) d d k (2 π ) d (cid:0) k + ∆ (cid:1) − l = 1(4 π ) d/ Γ( l − d )Γ( l ) ∆ d − l , (A7)and with the replacement (cid:112) p ⊥ + p → (cid:0) p ⊥ + p (cid:1) − s with a new parameter s , we can integrate the potential density:Ω( L ; s ) V = − L ∞ (cid:88) n = −∞ (cid:90) d d − p ⊥ (2 π ) d − (cid:0) p ⊥ + p (cid:1) − s (A8)= − L ∞ (cid:88) n = −∞ π ) ( d − / Γ( s − d − )Γ( s ) | p | d − − s . (A9)For the PBC, the sum of p can be replaced by the zeta function through the analytic continuation:1 L ∞ (cid:88) n = −∞ | p | d − − s = 2 L ∞ (cid:88) n =1 (cid:18) πL n (cid:19) d − − s = 2 L (cid:18) πL (cid:19) d − − s ζ (2 s − d + 1) . (A10)Then the potential can be the simpler formΩ( L ; s ) V = − L π ) ( d − / Γ( s − d − )Γ( s ) (cid:18) πL (cid:19) d − − s ζ (2 s − d + 1) (A11)= − L π ) ( d − / π − / Γ (cid:0) d − s (cid:1) ζ ( d − s )Γ( s ) 2 d − − s L d − − s , (A12)6where, in order to avoid the divergence of Γ( − 1) at d = 2 in the limit s → − / 2, we have applied the reflectionformula Γ( z/ ζ ( z ) = π z − / Γ (cid:18) − z (cid:19) ζ (1 − z ) . (A13)Eq. (A12) is already finite even in the limit s → − / 2, and the final form isΩ( L ) V = 2( L √ π ) d +1 Γ (cid:18) d + 12 (cid:19) ζ ( d + 1) . (A14)We show the results in d = 1 , , 3: Ω( L ) V = π L ( d = 1) , (A15)Ω( L ) V = ζ (3) πL ( d = 2) , (A16)Ω( L ) V = π L ( d = 3) . (A17)By multiplying these formulas by L , we can obtain E Cas discussed in the main text.For the ABC, we just have to replace the zeta function ζ (2 s − d + 1) by the Hurwitz zeta function ζ (2 s − d + 1 , / ζ ( z, / 2) = (2 z − ζ ( z ).Then, we can take the limit s → − / L ) V = − d − d L √ π ) d +1 Γ (cid:18) d + 12 (cid:19) ζ ( d + 1) . (A18)We also show the expressions in d = 1 , , 3: Ω( L ) V = − π L ( d = 1) , (A19)Ω( L ) V = − ζ (3)4 πL ( d = 2) , (A20)Ω( L ) V = − π L ( d = 3) . (A21) Appendix B: Abel-Plana formulas in finite range The Abel-Plana formula (APF) is a conventional and powerful tool to study the Casimir effect (for early works, seeRefs. [7, 132–135]). Usually, this formula is used to calculate the finite Casimir energy from an infinite integral andan infinite sum. For lattice fermions, the momentum of a fermion has a periodicity within a Brillouin zone, and thenthe momentum space can be restricted to the first Brillouin zone. Therefore, the APF should also be modified asthat in a finite range . In this appendix, we derive the APF in a finite range. Our derivation is based on the followingformula (see, Refs. [136, 137]): (cid:90) ba f ( x ) dx = R [ f ( z ) , g ( z )] − (cid:90) + i ∞− i ∞ [ g ( u ) + σ ( z ) f ( u )] u = b + zu = a + z dz, σ ( z ) ≡ sgn(Im z ) , (B1)where f ( z ) and g ( z ) with z = x + iy are meromorphic functions for a ≤ x ≤ b and satisfylim h → (cid:90) b ± iha ± ih [ g ( z ) ± f ( z )] dz = 0 . (B2)The residue part is R [ f ( z ) , g ( z )] = πi (cid:34)(cid:88) k Res z = z g,k g ( z ) + (cid:88) k σ ( z f,k ) Res z = z f,k f ( z ) (cid:35) . (B3) z f,k and z g,k stand for the poles of f ( z ) and g ( z ), respectively. In the following, we assume f ( z ) is regular in a < x < b ,and then the second term of Eq. (B3) vanishes.7 1. Integer First, we consider the APF for f ( n ), where n ∈ Z is the index of summation. We set g ( z ) = − i cot( πz ) f ( z ), wherethe residue of g ( z ) at z = n is − if ( n ) /π . Then, Eq. (B1) is (cid:98) b (cid:99) (cid:88) n = (cid:100) a (cid:101) f ( n ) − (cid:90) ba dxf ( x ) − (cid:18) f ( a ) + 12 f ( b ) if a, b ∈ Z (cid:19) = 12 (cid:90) + i ∞− i ∞ [ σ ( z ) − i cot( πu )] f ( u ) | u = b + zu = a + z dz = 12 (cid:90) + i ∞ [+1 − i cot( πu )] f ( u ) | u = b + zu = a + z dz + 12 (cid:90) − i ∞ [ − − i cot( πu )] f ( u ) | u = b + zu = a + z dz, (B4)where (cid:100) x (cid:101) and (cid:98) x (cid:99) are the ceiling function and the floor function, respectively. If z = a and/or z = b are poles of g ( z ), we have to avoid the poles on the integral path. When we avoid the poles along a small semicircle, we obtain − f ( a ) and/or − f ( b ). By using the exponential form of ± − i cot( πu ), ± − i cot( πu ) = ± e iπu − e − iπu e iπu − e − iπu + e iπu + e − iπu e iπu − e − iπu = 2 e ± iπu e iπu − e − iπu = ∓ e ∓ iπu − , (B5)the right-hand side of Eq. (B4) is written as+ 12 (cid:90) + i ∞ dz (cid:20) − f ( u ) e − iπu − (cid:21) u = b + zu = a + z + 12 (cid:90) − i ∞ dz (cid:20) f ( u ) e iπu − (cid:21) u = a + zu = b + z = + i (cid:90) ∞ dy (cid:20) − f ( u ) e − iπu − (cid:21) u = b + iyu = a + iy − i (cid:90) ∞ dy (cid:20) f ( u ) e iπu − (cid:21) u = b − iyu = a − iy = i (cid:90) ∞ dy f ( a + iy ) e π ( y − ia ) − − i (cid:90) ∞ dy f ( a − iy ) e π ( y + ia ) − − i (cid:90) ∞ dy f ( b + iy ) e π ( y − ib ) − i (cid:90) ∞ dy f ( b − iy ) e π ( y + ib ) − . (B6)Finally, we obtain the APF for f ( n ): (cid:98) b (cid:99) (cid:88) n = (cid:100) a (cid:101) f ( n ) − (cid:90) ba dxf ( x ) − (cid:18) f ( a ) + 12 f ( b ) if a, b ∈ Z (cid:19) = i (cid:90) ∞ dy f ( a + iy ) e π ( y − ia ) − − i (cid:90) ∞ dy f ( a − iy ) e π ( y + ia ) − − i (cid:90) ∞ dy f ( b + iy ) e π ( y − ib ) − i (cid:90) ∞ dy f ( b − iy ) e π ( y + ib ) − . (B7) 2. Half-integer Next, we consider the APF for f ( n + 1 / n + 1 / n ∈ Z ). We set g ( z ) = i tan( πz ) f ( z ), where the residueof g ( z ) at z = n + 1 / − if ( n + 1 / /π . Then, Eq. (B1) is (cid:98) b − / (cid:99) (cid:88) n = (cid:100) a − / (cid:101) f ( n + 1 / − (cid:90) ba dxf ( x ) − (cid:18) f ( a ) + 12 f ( b ) if a − , b − ∈ Z (cid:19) = 12 (cid:90) + i ∞ [+1 + i tan( πu )] f ( u ) | u = b + zu = a + z dz + 12 (cid:90) − i ∞ [ − i tan( πu )] f ( u ) | u = b + zu = a + z dz. (B8)8By using the exponential forms of ± i tan( πu ), ± i tan( πu ) = ± e iπu + e − iπu e iπu + e − iπu + e iπu − e − iπu e iπu + e − iπu = ± e ± iπu e iπu + e − iπu = ± e ∓ iπu + 1 , (B9)the right-hand side of Eq. (B8) is12 (cid:90) + i ∞ [+1 + i tan( πu )] f ( u ) | u = b + zu = a + z dz + 12 (cid:90) − i ∞ [ − i tan( πu )] f ( u ) | u = b + zu = a + z dz = (cid:90) + i ∞ (cid:20) f ( u ) e − iπu + 1 (cid:21) u = b + zu = a + z dz − (cid:90) − i ∞ (cid:20) f ( u ) e +2 iπu + 1 (cid:21) u = b + zu = a + z dz = i (cid:90) ∞ (cid:20) f ( u ) e − iπu + 1 (cid:21) u = b + iyu = a + iy dy − i (cid:90) ∞ (cid:20) f ( u ) e +2 iπu + 1 (cid:21) u = b − iyu = a − iy dy = − i (cid:90) ∞ f ( a + iy ) e π ( y − ia ) + 1 dy + i (cid:90) ∞ f ( a − iy ) e π ( y + ia ) + 1 dy + i (cid:90) ∞ f ( b + iy ) e π ( y − ib ) + 1 dy − i (cid:90) ∞ f ( b − iy ) e π ( y + ib ) + 1 dy. (B10)Finally, we obtain the APF for f ( n + 1 / (cid:98) b − / (cid:99) (cid:88) n = (cid:100) a − / (cid:101) f ( n + 1 / − (cid:90) ba dxf ( x ) − (cid:18) f ( a ) + 12 f ( b ) if a − , b − ∈ Z (cid:19) = − i (cid:90) ∞ f ( a + iy ) e π ( y − ia ) + 1 dy + i (cid:90) ∞ f ( a − iy ) e π ( y + ia ) + 1 dy + i (cid:90) ∞ f ( b + iy ) e π ( y − ib ) + 1 dy − i (cid:90) ∞ f ( b − iy ) e π ( y + ib ) + 1 dy. (B11) Appendix C: Derivation for naive lattice fermion In this appendix, we derive the Casimir energy for massless naive lattice fermions from the APFs in finite rangesuch as Eqs. (B7) and (B11). From the Dirac operator (9) at am f = 0, the dispersion relation in the 1 + 1 dimensionalspacetime is a (cid:113) D † nf D nf = (cid:113) sin ap ( n ) , (C1)where the momenta for the periodic and antiperiodic boundaries are ap ( n ) = 2 πn/N and ap ( n ) = 2 π ( n + 1 / /N ,respectively.After the analytic continuation ( n → z for the periodic boundary and n + 1 / → z for the antiperiodic boundary),the resulting complex function f ( z = x + iy ) = (cid:113) sin (2 πz/N ) has branch cuts along the y direction at x = 0, x = N/ x = N . Therefore, to avoid the branch cut, we separately apply the APF to the two regions, 0 ≤ x < N/ N/ < x < N (when the Brillouin zone is defined as 0 ≤ ap < π ). For example, we consider a path along thebranch cut at x = N/ 2, where the path is shifted by an infinitesimal parameter (cid:15) → +0 from the cut. Then f ( z ) onthis path is (if y > 0) sin (cid:18) πN (cid:18) N (cid:15) ± iy (cid:19)(cid:19) (cid:39) (cid:18) ∓ i sinh (cid:18) πyN (cid:19) − (cid:15) (cid:19) (cid:39) − sinh (cid:18) πyN (cid:19) ± i(cid:15), (C2) (cid:115) sin (cid:18) πN (cid:18) N (cid:15) ± iy (cid:19)(cid:19) (cid:39) e ± iπ/ sinh (cid:18) πyN (cid:19) = ± i sinh (cid:18) πyN (cid:19) . (C3)9Similarly, f ( z ) on the paths at x = N/ − (cid:15) , x = 0 + (cid:15) , and x = N − (cid:15) are (cid:115) sin (cid:18) πN (cid:18) N − (cid:15) ± iy (cid:19)(cid:19) (cid:39) ∓ i sinh (cid:18) πyN (cid:19) , (C4) (cid:115) sin (cid:18) πN (0 + (cid:15) ± iy ) (cid:19) (cid:39) ± i sinh (cid:18) πyN (cid:19) , (C5) (cid:115) sin (cid:18) πN ( N − (cid:15) ± iy ) (cid:19) (cid:39) ∓ i sinh (cid:18) πyN (cid:19) . (C6)In the following, these expressions will be used for evaluating integrals. 1. Periodic boundary For the periodic boundary, we substitute f ( z ) = (cid:113) sin (2 πz/N ) into the APF for integers, Eq. (B7). We put( a, b ) = (0 + (cid:15), N/ − (cid:15) ) in the first region and ( a, b ) = ( N/ (cid:15), N − (cid:15) ) in the second region, and then aE , nf , PCas = − i (cid:90) ∞ dye πy − (cid:16) f ( z ) | z = (cid:15) + iyz = (cid:15) − iy − f ( z ) | z = N − (cid:15) + iyz = N − (cid:15) − iy (cid:17) + i (cid:90) ∞ dye π ( y + iN/ − (cid:16) f ( z ) | z = N/ − (cid:15) + iyz = N/ − (cid:15) − iy − f ( z ) | z = N/ (cid:15) + iyz = N/ (cid:15) − iy (cid:17) = 4 (cid:90) ∞ dy sinh(2 πy/N ) e πy − (cid:90) ∞ dy sinh(2 πy/N ) e π ( y + iN/ − . (C7)For e π ( y ± iN/ of the denominator in the APF, we used e + iNπ = e − iNπ for a integer N . The contributions from thesemicircles in the APF are zero because f (0) = f ( N/ 2) = f ( N ) = 0. Here, the integrations are performed: (cid:90) ∞ dy sinh(2 πy/N ) e πy − N π − 14 cot (cid:16) πN (cid:17) , (C8) (cid:90) ∞ dy sinh(2 πy/N ) e πy + 1 = − N π + 14 csc (cid:16) πN (cid:17) . (C9)Finally, aE , nf , PCas = (cid:40) Nπ − cot π N ( N = odd) Nπ − πN ( N = even) . (C10)For N = odd, we used cot( x ) + csc( x ) = cot( x/ 2. Antiperiodic boundary For the antiperiodic boundary, we substitute f ( z ) = (cid:113) sin (2 πz/N ) into the APF for half-integers, Eq. (B11): aE , nf , APCas = i (cid:90) ∞ dye πy + 1 (cid:16) f ( z ) | z = (cid:15) + iyz = (cid:15) − iy − f ( z ) | z = N − (cid:15) + iyz = N − (cid:15) − iy (cid:17) − i (cid:90) ∞ dye π ( y + iN/ + 1 (cid:16) f ( z ) | z = N/ − (cid:15) + iyz = N/ − (cid:15) − iy − f ( z ) | z = N/ (cid:15) + iyz = N/ (cid:15) − iy (cid:17) = − (cid:90) ∞ dy sinh(2 πy/N ) e πy + 1 − (cid:90) ∞ dy sinh(2 πy/N ) e π ( y + iN/ + 1= (cid:40) Nπ − cot π N ( N = odd) Nπ − πN ( N = even) . (C11)For the third equality, we used Eqs. (C8) and (C9).0 Appendix D: Derivation for massless Wilson fermion From the Dirac operator (21) at r = 1 and am f = 0, the dispersion relations of massless Wilson fermion in the1 + 1 dimensional spacetime is a (cid:113) D † W D W = (cid:112) − ap ( n ) = 2 (cid:115) sin (cid:18) ap ( n )2 (cid:19) , (D1)where the momenta for the periodic and antiperiodic boundaries are ap ( n ) = 2 πn/N and ap ( n ) = 2 π ( n + 1 / /N ,respectively.In the dispersion relations after the analytic continuation, f ( z = x + iy ) = 2 (cid:113) sin ( πz/N ) has a branch cut alongthe y direction at x = 0 and x = N . f ( z ) on paths along the branch cuts at x = 0 and x = N are (if (cid:15) → +0 , y > (cid:16) πN (0 + (cid:15) ± iy ) (cid:17) (cid:39) − sinh (cid:16) πyN (cid:17) ± i(cid:15), (D2) (cid:114) sin (cid:16) πN (0 + (cid:15) ± iy ) (cid:17) (cid:39) ± i sinh (cid:16) πyN (cid:17) , (D3)sin (cid:16) πN ( N − (cid:15) ± iy ) (cid:17) (cid:39) − sinh (cid:16) πyN (cid:17) ∓ i(cid:15), (D4) (cid:114) sin (cid:16) πN ( N − (cid:15) ± iy ) (cid:17) (cid:39) ∓ i sinh (cid:16) πyN (cid:17) . (D5) 1. Periodic boundary For the periodic boundary, we substitute f ( z ) = 2 (cid:113) sin ( πz/N ) into the APF (B7). We put ( a, b ) = (0 + (cid:15), N − (cid:15) ),and then aE , W , PCas = − i (cid:90) ∞ dye πy − (cid:16) f ( z ) | z = (cid:15) + iyz = (cid:15) − iy − f ( z ) | z = N − (cid:15) + iyz = N − (cid:15) − iy (cid:17) (D6)= 8 (cid:90) ∞ dy sinh( πy/N ) e πy − Nπ − π N , (D7)where the contributions from the semicircles in the APF are zero because f (0) = f ( N ) = 0. 2. Antiperiodic boundary For the antiperiodic boundary, we substitute f ( z ) = 2 (cid:113) sin ( πz/N ) into the APF (B11), aE , W , APCas = i (cid:90) ∞ dye πy + 1 (cid:16) f ( z ) | z = (cid:15) + iyz = (cid:15) − iy − f ( z ) | z = N − (cid:15) + iyz = N − (cid:15) − iy (cid:17) (D8)= − (cid:90) ∞ dy sinh( πy/N ) e πy + 1 = 4 Nπ − π N . (D9) Appendix E: Derivation for Wilson fermion with negative mass am f = − The dispersion relations of the Wilson fermion at r = 1 and am f = − a (cid:113) D † D ( am f = − 2) = (cid:112) ap ( n ) = 2 (cid:115) cos (cid:18) ap ( n )2 (cid:19) , (E1)where the momenta for the periodic and antiperiodic boundaries are ap ( n ) = 2 πn/N and ap ( n ) = 2 π ( n + 1 / /N ,respectively.1In the dispersion relations after the analytic continuation, f ( z = x + iy ) = 2 (cid:112) cos ( πz/N ) has a branch cut alongthe y direction at x = N/ 2. Therefore, to avoid the branch cut, we separately apply the APF to the two regions,0 ≤ x < N/ N/ < x < N (when the Brillouin zone is defined as 0 ≤ ap < π ). For 0 ≤ x < N/ 2, the firstand second terms in the right-hand side of the APF are zero because of f (0 + iy ) − f (0 − iy ) = 0, and the third andfourth terms are nonzero. For N/ < x < N , the first and second terms are nonzero, and the third and fourth termsare zero because of f ( N + iy ) − f ( N − iy ) = 0. Note that these situations are different from the procedures shownfor naive fermion and massless Wilson fermion since these fermions have the branch cuts at x = 0 and x = N . f ( z ) on paths along the branch cut at x = N/ (cid:15) → +0 , y > (cid:18) πN (cid:18) N (cid:15) ± iy (cid:19)(cid:19) (cid:39) − sinh (cid:16) πyN (cid:17) ± i(cid:15), (E2) (cid:115) cos (cid:18) πN (cid:18) N (cid:15) ± iy (cid:19)(cid:19) (cid:39) ± i sinh (cid:16) πyN (cid:17) , (E3)cos (cid:18) πN (cid:18) N − (cid:15) ± iy (cid:19)(cid:19) (cid:39) − sinh (cid:16) πyN (cid:17) ∓ i(cid:15), (E4) (cid:115) cos (cid:18) πN (cid:18) N − (cid:15) ± iy (cid:19)(cid:19) (cid:39) ∓ i sinh (cid:16) πyN (cid:17) . (E5) 1. Periodic boundary For the periodic boundary, we substitute f ( z ) = 2 (cid:112) cos ( πz/N ) into the APF for integers, Eq. (B7). We put( a, b ) = (0 + (cid:15), N/ − (cid:15) ) in the first region and ( a, b ) = ( N/ (cid:15), N − (cid:15) ) in the second region, and then aE , PCas ( am f = − 2) = − i (cid:90) ∞ dye π ( y + iN/ − (cid:16) f ( z ) | z = N/ (cid:15) + iyz = N/ (cid:15) − iy − f ( z ) | z = N/ − (cid:15) + iyz = N/ − (cid:15) − iy (cid:17) (E6)= 8 (cid:90) ∞ dy sinh (cid:0) πyN (cid:1) e π ( y + iN/ − (cid:40) Nπ − π N ( N = odd) Nπ − π N ( N = even) . (E7)The contributions from the semicircles around z = 0 and z = N , namely, the third term in the left-hand side ofthe APF (B7), is f (0) + f ( N ). The Casimir energy is defined as (cid:80) N − n =0 − (cid:82) N dxf ( x ) where the Brillouin zone isdefined as 0 ≤ ap < π , while the APF is now defined as (cid:80) Nn =0 f ( n ) − (cid:82) N dxf ( x ). Therefore, we have to subtract f ( N ) from the APF in order to obtain the Casimir energy. After we subtract f ( N ) from f (0) + f ( N ), we get f (0) − f ( N ) = 0. On the other hand, the contribution from the semicircles around z = N/ N = even) disappearsdue to f ( N/ 2) = 0. 2. Antiperiodic boundary For the antiperiodic boundary, we substitute f ( z ) = 2 (cid:112) cos ( πz/N ) into the APF for half-integers, Eq. (B11), aE , APCas ( am f = − 2) = i (cid:90) ∞ dye π ( y + iN/ + 1 (cid:16) f ( z ) | z = N/ (cid:15) + iyz = N/ (cid:15) − iy − f ( z ) | z = N/ − (cid:15) + iyz = N/ − (cid:15) − iy (cid:17) (E8)= − (cid:90) ∞ dy sinh (cid:0) πyN (cid:1) e π ( y + iN/ + 1 = (cid:40) Nπ − π N ( N = odd) Nπ − π N ( N = even) . (E9)The contributions from the semicircles around z = N/ N = odd) are zero because f ( z = N/ 2) = 0. [1] H. B. G. Casimir, “On the Attraction Between TwoPerfectly Conducting Plates,” Proc. Kon. Ned. Akad. Wetensch. , 793–795 (1948). [2] V. M. Mostepanenko and N. N. Trunov, “The CasimirEffect and Its Applications,” Sov. Phys. Usp. , 965–987 (1988).[3] Michael Bordag, U. Mohideen, and V. M. Mostepa-nenko, “New developments in the Casimir effect,” Phys.Rept. , 1–205 (2001), arXiv:quant-ph/0106045[quant-ph].[4] K. A. Milton, The Casimir effect: Physical manifesta-tions of zero-point energy (2001).[5] S. K. Lamoreaux, “Demonstration of the Casimir Forcein the 0.6 to 6 µm Range,” Phys. Rev. Lett. , 5–8(1997), [Erratum: Phys. Rev. Lett. , 5475 (1998)].[6] K. Johnson, “The M.I.T. Bag Model,” Acta Phys.Polon. B , 865 (1975).[7] S. G. Mamaev and N. N. Trunov, “Vacuum expecta-tion values of the energy-momentum tensor of quantizedfields on manifolds with different topologies and geome-tries. III,” Sov. Phys. J. , 551–554 (1980).[8] M. N. Chernodub, Harold Erbin, I. V. Grishmanovskii,V. A. Goy, and A. V. Molochkov, “Casimir effect withmachine learning,” Phys. Rev. Res. , 033375 (2020),arXiv:1911.07571 [hep-lat].[9] Oleg Pavlovsky and Maxim Ulybyshev, “Casimir en-ergy calculations within the formalism of the noncom-pact lattice QED,” Int. J. Mod. Phys. A , 2457–2473(2010), arXiv:0911.2635 [hep-lat].[10] O. V. Pavlovsky and M. V. Ulybyshev, “Casimir energyin noncompact lattice electrodynamics,” Theor. Math.Phys. , 1051–1063 (2010), [Teor. Mat. Fiz. , 262(2010)].[11] Oleg Pavlovsky and Maxim Ulybyshev, “Monte-Carlocalculation of the lateral Casimir forces between rectan-gular gratings within the formalism of lattice quantumfield theory,” Int. J. Mod. Phys. A , 2743–2756 (2011),arXiv:1105.0544 [quant-ph].[12] Oleg Pavlovsky and Maxim Ulybyshev, “Casimir energyin the compact QED on the lattice,” arXiv:0901.1960[hep-lat].[13] M. N. Chernodub, V. A. Goy, and A. V. Molochkov,“Casimir effect on the lattice: U(1) gauge theory in twospatial dimensions,” Phys. Rev. D , 094504 (2016),arXiv:1609.02323 [hep-lat].[14] M. N. Chernodub, V. A. Goy, and A. V. Molochkov,“Nonperturbative Casimir effect and monopoles: com-pact Abelian gauge theory in two spatial dimensions,”Phys. Rev. D , 074511 (2017), arXiv:1703.03439 [hep-lat].[15] M. N. Chernodub, V. A. Goy, and A. V. Molochkov,“Casimir effect and deconfinement phase transition,”Phys. Rev. D , 094507 (2017), arXiv:1709.02262 [hep-lat].[16] M. N. Chernodub, V. A. Goy, A. V. Molochkov, andHa Huu Nguyen, “Casimir Effect in Yang-Mills Theoryin D = 2 + 1,” Phys. Rev. Lett. , 191601 (2018),arXiv:1805.11887 [hep-lat].[17] M. N. Chernodub, V. A. Goy, and A. V. Molochkov,“Phase structure of lattice Yang-Mills theory on T × R ,” Phys. Rev. D , 074021 (2019), arXiv:1811.01550[hep-lat].[18] Masakiyo Kitazawa, Sylvain Mogliacci, Isobel Kolb´e,and W. A. Horowitz, “Anisotropic pressure induced byfinite-size effects in SU(3) Yang-Mills theory,” Phys.Rev. D , 094507 (2019), arXiv:1904.00241 [hep-lat].[19] S. K. Kim, W. Namgung, K. S. Soh, and J. H. Yee, “Dynamical symmetry breaking and space-time topol-ogy,” Phys. Rev. D , 3172 (1987).[20] D. Y. Song and J. K. Kim, “Dynamical symmetry break-ings on a nontrivial topology,” Phys. Rev. D , 3165(1990).[21] D. Y. Song, “Four-fermion interaction model on R × S :A dynamical dimensional reduction,” Phys. Rev. D ,3925 (1993).[22] D. K. Kim, Y. D. Han, and I. G. Koh, “Chiral sym-metry breaking in a finite volume,” Phys. Rev. D ,6943–6946 (1994).[23] A. S. Vshivtsev, B. V. Magnitsky, and K. G. Kli-menko, “Tricritical point in the Gross-Neveu model witha chemical potential and a nontrivial topology of thespace,” JETP Lett. , 871–874[Pisma Zh. Eksp. Teor.Fiz. , 847 (1995)].[24] M. A. Vdovichenko and A. K. Klimenko, “Oscillationphenomena in polyacetylene: R × S Gross-Neveumodel with a chemical potential,” JETP Lett. , 460–466 (1998), [Pisma Zh. Eksp. Teor. Fiz. , 431 (1998)].[25] A. S. Vshivtsev, M. A. Vdovichenko, and K. G. Kli-menko, “Oscillatory phenomena in cold matter withfour-fermion interaction,” J. Exp. Theor. Phys. , 229–238 (1998), [Zh. Eksp. Teor. Fiz. , 418 (1998)].[26] J. Braun, B. Klein, and H. J. Pirner, “Volume de-pendence of the pion mass in the quark-meson-model,”Phys. Rev. D , 014032 (2005), arXiv:hep-ph/0408116[hep-ph].[27] J. Braun, B. Klein, and H. J. Pirner, “Influence of quarkboundary conditions on the pion mass in finite volume,”Phys. Rev. D , 034017 (2005), arXiv:hep-ph/0504127[hep-ph].[28] J. Braun, B. Klein, H. J. Pirner, and A. H. Rezaeian,“Volume and quark mass dependence of the chiral phasetransition,” Phys. Rev. D , 074010 (2006), arXiv:hep-ph/0512274 [hep-ph].[29] L. M. Abreu, M. Gomes, and A. J. da Silva, “Finite-size effects on the phase structure of the Nambu-Jona-Lasinio model,” Phys. Lett. B , 551–562 (2006),arXiv:hep-th/0610111 [hep-th].[30] D. Ebert, K. G. Klimenko, A. V. Tyukov, andV. Ch. Zhukovsky, “Finite size effects in the Gross-Neveu model with isospin chemical potential,” Phys.Rev. D , 045008 (2008), arXiv:0804.4826 [hep-ph].[31] L. F. Palhares, E. S. Fraga, and T. Kodama, “Chiraltransition in a finite system and possible use of finitesize scaling in relativistic heavy ion collisions,” J. Phys.G , 085101 (2011), arXiv:0904.4830 [nucl-th].[32] L. M. Abreu, A. P. C. Malbouisson, J. M. C. Mal-bouisson, and A. E. Santana, “Finite-size effects onthe chiral phase diagram of four-fermion models infour dimensions,” Nucl. Phys. B , 127–138 (2009),arXiv:0909.5105 [hep-th].[33] L. M. Abreu, A. P. C. Malbouisson, and J. M. C.Malbouisson, “Phase structure of difermion condensatesin the Nambu–Jona-Lasinio model: the size-dependentproperties,” EPL , 11001 (2010).[34] Masako Hayashi and Tomohiro Inagaki, “Curvature andtopological effects on dynamical symmetry breaking in afour- and eight-fermion interaction model,” Int. J. Mod.Phys. A , 3353–3374 (2010), arXiv:1003.3163 [hep-ph].[35] D. Ebert and K. G. Klimenko, “Cooper pairing andfinite-size effects in a NJL-type four-fermion model,” Phys. Rev. D , 025018 (2010), arXiv:1005.0699 [hep-ph].[36] Jens Braun, Bertram Klein, and Piotr Piasecki, “Onthe scaling behavior of the chiral phase transition inQCD in finite and infinite volume,” Eur. Phys. J. C ,1576 (2011), arXiv:1008.2155 [hep-ph].[37] L. M. Abreu, A. P. C. Malbouisson, and J. M. C. Mal-bouisson, “Nambu-Jona-Lasinio model in a magneticbackground: Size-dependent effects,” Phys. Rev. D ,065036 (2011).[38] D. Ebert, T. G. Khunjua, K. G. Klimenko, and V. Ch.Zhukovsky, “Charged pion condensation phenomenon ofdense baryonic matter induced by finite volume: TheNJL model consideration,” Int. J. Mod. Phys. A ,1250162 (2012), arXiv:1106.2928 [hep-ph].[39] Jens Braun, Bertram Klein, and Bernd-Jochen Schae-fer, “On the phase structure of QCD in a finite volume,”Phys. Lett. B , 216–223 (2012), arXiv:1110.0849[hep-ph].[40] Antonino Flachi, “Interacting fermions, boundaries, andfinite size effects,” Phys. Rev. D , 104047 (2012),arXiv:1209.4754 [hep-th].[41] Antonino Flachi, “Strongly Interacting Fermions andPhases of the Casimir Effect,” Phys. Rev. Lett. ,060401 (2013), arXiv:1301.1193 [hep-th].[42] B. C. Tiburzi, “Chiral Symmetry Restoration froma Boundary,” Phys. Rev. D , 034027 (2013),arXiv:1302.6645 [hep-lat].[43] Ralf-Arno Tripolt, Jens Braun, Bertram Klein, andBernd-Jochen Schaefer, “Effect of fluctuations on theQCD critical point in a finite volume,” Phys. Rev. D , 054012 (2014), arXiv:1308.0164 [hep-ph].[44] Tran Huu Phat and Nguyen Van Thu, “Finite-size ef-fects of linear sigma model in compactified space-time,”Int. J. Mod. Phys. A , 1450078 (2014).[45] D. Ebert, T. G. Khunjua, K. G. Klimenko, and V. Ch.Zhukovsky, “Interplay between superconductivity andchiral symmetry breaking in a (2+1)-dimensional modelwith a compactified spatial coordinate,” Phys. Rev. D , 105024 (2015), arXiv:1504.02867 [hep-th].[46] Gabor Almasi, Robert Pisarski, and Vladimir Skokov,“Volume dependence of baryon number cumulantsand their ratios,” Phys. Rev. D , 056015 (2017),arXiv:1612.04416 [hep-ph].[47] Antonino Flachi, Muneto Nitta, Satoshi Takada, andRyosuke Yoshii, “Sign Flip in the Casimir Force forInteracting Fermion Systems,” Phys. Rev. Lett. ,031601 (2017), arXiv:1704.04918 [hep-th].[48] Muneto Nitta and Ryosuke Yoshii, “Self-consistentlarge-N analytical solutions of inhomogeneous conden-sates in quantum C P N − model,” JHEP , 145 (2017),arXiv:1707.03207 [hep-th].[49] L. M. Abreu and E. S. Nery, “Finite-size effects on thephase structure of the Walecka model,” Phys. Rev. C , 055204 (2017), arXiv:1711.07934 [nucl-th].[50] Qing-Wu Wang, Yonghui Xia, and Hong-Shi. Zong, “Fi-nite volume effects with stationary wave solution fromNambu–Jona-Lasinio model,” (2018), arXiv:1802.00258[hep-ph].[51] Qingwu Wang, Yonghui Xiq, and Hongshi Zong,“Nambu–Jona-Lasinio model with proper time regular-ization in a finite volume,” Mod. Phys. Lett. A ,1850232 (2018), arXiv:1806.05315 [hep-ph].[52] Tsutomu Ishikawa, Katsumasa Nakayama, and Kei Suzuki, “Casimir effect for nucleon parity doublets,”Phys. Rev. D , 054010 (2019), arXiv:1812.10964 [hep-ph].[53] Tomohiro Inagaki, Yamato Matsuo, and Hiromu Shi-moji, “Four-Fermion Interaction Model on M D − ⊗ S ,”Symmetry , 451 (2019), arXiv:1903.04244 [hep-th].[54] Kun Xu and Mei Huang, “Zero-mode contributionand quantized first-order apparent phase transition ina droplet quark matter,” Phys. Rev. D , 074001(2020), arXiv:1903.08416 [hep-ph].[55] L. M. Abreu, Emerson B. S. Corrˆea, Cesar A. Linhares,and Adolfo P. C. Malbouisson, “Finite-volume and mag-netic effects on the phase structure of the three-flavorNambuJona-Lasinio model,” Phys. Rev. D , 076001(2019), arXiv:1903.09249 [hep-ph].[56] Tsutomu Ishikawa, Katsumasa Nakayama, Daiki Sue-naga, and Kei Suzuki, “ D mesons as a probe of Casimireffect for chiral symmetry breaking,” Phys. Rev. D ,034016 (2019), arXiv:1905.11164 [hep-ph].[57] L. M. Abreu and E. S. Nery, “Critical behaviour of aneffective relativistic mean field model in the presence ofmagnetic background and boundaries,” Eur. Phys. J. A , 108 (2019), arXiv:1907.04486 [hep-th].[58] L. M. Abreu, E. B. S. Corrˆea, and E. S. Nery, “Bound-ary effects on constituent quark masses and on chi-ral susceptibility in a four-fermion interaction model,”(2020), arXiv:2004.11237 [hep-ph].[59] Shen-Song Wan, Daize Li, Bonan Zhang, and MarcoRuggieri, “Finite Size Effects on the Chiral PhaseTransition of Quantum Chromodynamics,” (2020),arXiv:2012.05734 [hep-ph].[60] John Kogut and Leonard Susskind, “Hamiltonian for-mulation of Wilson’s lattice gauge theories,” Phys. Rev.D , 395 (1975).[61] Leonard Susskind, “Lattice fermions,” Phys. Rev. D ,3031 (1977).[62] Kenneth G. Wilson, Gauge Theories and Modern FieldTheory (MIT Press, Cambridge, 1975).[63] Kenneth G. Wilson, New Phenomena in SubnuclearPhysics, Part A (Plenum Press, New York, 1977) p. 69.[64] David B. Kaplan, “A method for simulating chiralfermions on the lattice,” Phys. Lett. B , 342–347(1992), arXiv:hep-lat/9206013 [hep-lat].[65] Yigal Shamir, “Chiral fermions from lattice bound-aries,” Nucl. Phys. B , 90–106 (1993), arXiv:hep-lat/9303005 [hep-lat].[66] Vadim Furman and Yigal Shamir, “Axial symmetries inlattice QCD with Kaplan fermions,” Nucl. Phys. B ,54–78 (1995), arXiv:hep-lat/9405004 [hep-lat].[67] Shuichi Murakami, “Phase transition between the quan-tum spin Hall and insulator phases in 3D: emergence ofa topological gapless phase,” New J. Phys. , 356 (2007),arXiv:0710.0930 [cond-mat.mes-hall].[68] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane,E. J. Mele, and A. M. Rappe, “Dirac Semimetalin Three Dimensions,” Phys. Rev. Lett. , 140405(2012), arXiv:1111.6483 [cond-mat.mes-hall].[69] N. P. Armitage, E. J. Mele, and Ashvin Vish-wanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. , 015001 (2018),arXiv:1705.01111 [cond-mat.mes-hall].[70] Tsutomu Ishikawa, Katsumasa Nakayama, and KeiSuzuki, “Casimir effect for lattice fermions,” Phys. Lett.B , 135713 (2020), arXiv:2005.10758 [hep-lat]. [71] A. Actor, I. Bender, and J. Reingruber, “Casimir effecton a finite lattice,” Fortsch. Phys. , 303–359 (2000),arXiv:quant-ph/9908058 [quant-ph].[72] Michael Pawellek, “Finite-sites corrections to theCasimir energy on a periodic lattice,” arXiv:1303.4708[hep-th].[73] H. B. Nielsen and M. Ninomiya, “Absence of neutri-nos on a lattice: (I). Proof by homotopy theory,” Nucl.Phys. B , 20 (1981), [Erratum: Nucl. Phys. B195 ,541 (1982)].[74] H. B. Nielsen and M. Ninomiya, “Absence of neutri-nos on a lattice: (II). Intuitive topological proof,” Nucl.Phys. B , 173–194 (1981).[75] Kenneth G. Wilson, “Confinement of quarks,” Phys.Rev. D , 2445 (1974).[76] Herbert Neuberger, “Exactly massless quarks on the lat-tice,” Phys. Lett. B , 141–144 (1998), arXiv:hep-lat/9707022.[77] Herbert Neuberger, “More about exactly masslessquarks on the lattice,” Phys. Lett. B , 353–355(1998), arXiv:hep-lat/9801031.[78] Paul H. Ginsparg and Kenneth G. Wilson, “A remnantof chiral symmetry on the lattice,” Phys. Rev. D ,2649 (1982).[79] Richard C. Brower, Hartmut Neff, and KostasOrginos, “M¨obius fermions: Improved domain wall chi-ral fermions,” Nucl. Phys. Proc. Suppl. , 686–688(2005), arXiv:hep-lat/0409118 [hep-lat].[80] R. C. Brower, H. Neff, and K. Orginos, “M¨obiusfermions,” Nucl. Phys. B Proc. Suppl. , 191–198(2006), arXiv:hep-lat/0511031.[81] Richard C. Brower, Harmut Neff, and Kostas Orginos,“The M¨obius domain wall fermion algorithm,” Com-put. Phys. Commun. , 1–19 (2017), arXiv:1206.5214[hep-lat].[82] P. Hays, “Vacuum fluctuations of a confined massivefield in two dimensions,” Annals Phys. , 32–46(1979).[83] Jan Ambjørn and Stephen Wolfram, “Properties of theVacuum. I. Mechanical and thermodynamic,” AnnalsPhys. , 1 (1983).[84] W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitonsin Polyacetylene,” Phys. Rev. Lett. , 1698 (1979).[85] W. P. Su, J. R. Schrieffer, and A. J. Heeger, “SolitonExcitations in Polyacetylene,” Phys. Rev. B , 20991(1980), [Erratum: Phys. Rev. B , 1138 (1983)].[86] Michael Creutz, Taro Kimura, and Tatsuhiro Misumi,“Aoki phases in the lattice Gross-Neveu model with fla-vored mass terms,” Phys. Rev. D , 094506 (2011),arXiv:1101.4239 [hep-lat].[87] Taro Kimura, Shota Komatsu, Tatsuhiro Misumi,Toshifumi Noumi, Shingo Torii, and Sinya Aoki, “Re-visiting symmetries of lattice fermions via spin-flavorrepresentation,” JHEP , 048 (2012), arXiv:1111.0402[hep-lat].[88] Tatsuhiro Misumi, “New fermion discretizations andtheir applications,” Proceedings, 30th InternationalSymposium on Lattice Field Theory (Lattice 2012):Cairns, Australia, June 24-29, 2012 , PoS LAT-TICE2012 , 005 (2012), arXiv:1211.6999 [hep-lat].[89] Abhishek Chowdhury, A. Harindranath, JyotirmoyMaiti, and Santanu Mondal, “Many avatars of the Wil-son fermion: A perturbative analysis,” JHEP , 037(2013), arXiv:1301.0675 [hep-lat]. [90] Tatsuhiro Misumi and Yuya Tanizaki, “Latticegauge theory for Haldane conjecture and central-branch Wilson fermion,” PTEP , 033B03 (2020),arXiv:1910.09604 [hep-lat].[91] Tatsuhiro Misumi and Jun Yumoto, “Varieties andproperties of central-branch Wilson fermions,” Phys.Rev. D , 034516 (2020), arXiv:2005.08857 [hep-lat].[92] Xiao-Liang Qi, Yong-Shi Wu, and Shou-Cheng Zhang,“Topological quantization of the spin hall effect in two-dimensional paramagnetic semiconductors,” Phys. Rev.B , 085308 (2006), arXiv:cond-mat/0505308 [cond-mat.mes-hall].[93] Xiao-Liang Qi, Taylor L. Hughes, and Shou-ChengZhang, “Topological field theory of time-reversal in-variant insulators,” Phys. Rev. B , 195424 (2008),arXiv:0802.3537 [cond-mat.mes-hall].[94] D.J. Thouless, M. Kohmoto, M.P. Nightingale, andM. den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Phys. Rev. Lett. ,405–408 (1982).[95] M. Kohmoto, “Topological invariant and the quantiza-tion of the Hall conductance,” Ann. Phys. , 343–354(1985).[96] B. Andrei Bernevig, Taylor L. Hughes, and Shou-Cheng Zhang, “Quantum Spin Hall Effect and Topo-logical Phase Transition in HgTe Quantum Wells,” Sci-ence , 1757–1761 (2006), arXiv:cond-mat/0611399[cond-mat.mes-hall].[97] Markus K¨onig, Steffen Wiedmann, Christoph Br¨une,Andreas Roth, Hartmut Buhmann, Laurens W.Molenkamp, Xiao-Liang Qi, and Shou-Cheng Zhang,“Quantum Spin Hall Insulator State in HgTe QuantumWells,” Science , 766–770 (2007), arXiv:0710.0582[cond-mat.mes-hall].[98] A. Bori¸ci, “Truncated overlap fermions,” Nucl. Phys.B Proc. Suppl. , 771–773 (2000), arXiv:hep-lat/9909057.[99] Artan Bori¸ci, “Truncated overlap fermions: the link be-tween overlap and domain wall fermions,” NATO Sci.Ser. C , 41–52 (2000), arXiv:hep-lat/9912040.[100] T. Blum et al. , “Quenched lattice QCD with domainwall fermions and the chiral limit,” Phys. Rev. D ,074502 (2004), arXiv:hep-lat/0007038.[101] Y. Aoki et al. , “Domain wall fermions with improvedgauge actions,” Phys. Rev. D , 074504 (2004),arXiv:hep-lat/0211023.[102] Debasish Banerjee, R. V. Gavai, and Sayantan Sharma,“Thermodynamics of the ideal overlap quarks on the lat-tice,” Phys. Rev. D , 014506 (2008), arXiv:0803.3925[hep-lat].[103] R. V. Gavai and Sayantan Sharma, “Thermodynamicsof free domain wall fermions,” Phys. Rev. D , 074502(2009), arXiv:0811.3026 [hep-lat].[104] J. N. Fuchs, A. Recati, and W. Zwerger, “Oscillat-ing casimir force between impurities in one-dimensionalfermi liquids,” Phys. Rev. A , 043615 (2007),arXiv:cond-mat/0610659 [cond-mat.mes-hall].[105] P. W¨achter, V. Meden, and K. Sch¨onhammer, “Indirectforces between impurities in one-dimensional quantumliquids,” Phys. Rev. B , 045123 (2007), arXiv:cond-mat/0703128 [cond-mat.str-el].[106] Eugene B. Kolomeisky, Joseph P. Straley, and MichaelTimmins, “Casimir effect in a one-dimensional gasof free fermions,” Phys. Rev. A , 022104 (2008), arXiv:0706.2887 [cond-mat.mes-hall].[107] Dina Zhabinskaya and E. J. Mele, “Casimir interac-tions between scatterers in metallic carbon nanotubes,”Phys. Rev. B , 155405 (2009), arXiv:0909.1731 [cond-mat.mes-hall].[108] Sinya Aoki, “New phase structure for lattice QCD withWilson fermions,” Phys. Rev. D , 2653 (1984).[109] A. Bermudez, L. Mazza, M. Rizzi, N. Goldman,M. Lewenstein, and M. A. Martin-Delgado, “Wil-son Fermions and Axion Electrodynamics in Opti-cal Lattices,” Phys. Rev. Lett. , 190404 (2010),arXiv:1004.5101 [cond-mat.quant-gas].[110] Leonardo Mazza, Alejandro Bermudez, Nathan Gold-man, Matteo Rizzi, Miguel Angel Martin-Delgado,and Maciej Lewenstein, “An optical-lattice-based quan-tum simulator for relativistic field theories and topo-logical insulators,” New J. Phys. , 015007 (2012),arXiv:1105.0932 [cond-mat.quant-gas].[111] Yoshihito Kuno, Ikuo Ichinose, and Yoshiro Takahashi,“Generalized lattice Wilson-Dirac fermions in (1+1) di-mensions for atomic quantum simulation and topologi-cal phases,” Sci. Rep. , 10699 (2018), arXiv:1801.00439[cond-mat.quant-gas].[112] T. V. Zache, F. Hebenstreit, F. Jendrzejewski, M. K.Oberthaler, J. Berges, and P. Hauke, “Quantum simu-lation of lattice gauge theories using Wilson fermions,”Sci. Technol. , 034010 (2018), arXiv:1802.06704 [cond-mat.quant-gas].[113] M. Bordag, I.V. Fialkovsky, D.M. Gitman, and D.V.Vassilevich, “Casimir interaction between a perfect con-ductor and graphene described by the Dirac model,”Phys. Rev. B , 245406 (2009), arXiv:0907.3242 [hep-th].[114] G. G´omez-Santos, “Thermal van der Waals interactionbetween graphene layers,” Phys. Rev. B , 245424(2009), arXiv:0908.0821 [cond-mat.mtrl-sci].[115] D. Drosdoff and Lilia M. Woods, “Casimir forces andgraphene sheets,” Phys. Rev. B , 155459 (2010),arXiv:1007.1231 [cond-mat.mes-hall].[116] Ignat V. Fialkovsky, Valery N. Marachevsky, andDmitri V. Vassilevich, “Finite temperature Casimir ef-fect for graphene,” Phys. Rev. B , 035446 (2011),arXiv:1102.1757 [hep-th].[117] B. E. Sernelius, “Casimir interactions in graphene sys-tems,” EPL , 57003 (2011), arXiv:1011.2363 [cond-mat.mes-hall].[118] S. Bellucci and A. A. Saharian, “Fermionic casimir den-sities in toroidally compactified spacetimes with appli-cations to nanotubes,” Phys. Rev. D , 085019 (2009),arXiv:0902.3726 [hep-th].[119] S. Bellucci and A. A. Saharian, “Fermionic casimir ef-fect for parallel plates in the presence of compact di-mensions with applications to nanotubes,” Phys. Rev.D , 105003 (2009), arXiv:0907.4942 [hep-th].[120] S. Bellucci, A. A. Saharian, and V. M. Bardeghyan,“Induced fermionic current in toroidally compacti-fied spacetimes with applications to cylindrical andtoroidal nanotubes,” Phys. Rev. D , 065011 (2010),arXiv:1002.1391 [hep-th].[121] E. Elizalde, S.D. Odintsov, and A. A. Sahar-ian, “Fermionic condensate and Casimir densities inthe presence of compact dimensions with applica- tions to nanotubes,” Phys. Rev. D , 105023 (2011),arXiv:1102.2202 [hep-th].[122] S. Bellucci and A. A. Saharian, “Fermionic current fromtopology and boundaries with applications to higher-dimensional models and nanophysics,” Phys. Rev. D ,025005 (2013), arXiv:1207.5046 [hep-th].[123] S. Bellucci, E. R. Bezerra de Mello, and A. A. Saharian,“Finite temperature fermionic condensate and currentsin topologically nontrivial spaces,” Phys. Rev. D ,085002 (2014), arXiv:1312.1686 [hep-th].[124] F. D. M. Haldane, “Model for a Quantum Hall Effectwithout Landau Levels: Condensed-Matter Realizationof the ’Parity Anomaly’,” Phys. Rev. Lett. , 2015–2018 (1988).[125] C. L. Kane and E. J. Mele, “Quantum Spin Hall Ef-fect in Graphene,” Phys. Rev. Lett. , 226801 (2005),arXiv:cond-mat/0411737.[126] C. L. Kane and E. J. Mele, “ Z Topological Order andthe Quantum Spin Hall Effect,” Phys. Rev. Lett. ,146802 (2005), arXiv:cond-mat/0506581.[127] Mitsutaka Fujita, Katsunori Wakabayashi, KyokoNakada, and Koichi Kusakabe, “Peculiar LocalizedState at Zigzag Graphite Edge,” J. Phys. Soc. Jpn. ,1920–1923 (1996).[128] Noriaki Hamada, Shin-ichi Sawada, and Atsushi Os-hiyama, “New one-dimensional conductors: Graphiticmicrotubules,” Phys. Rev. Lett. , 1579 (1992).[129] R. Saito, M. Fujita, G. Dresselhaus, and M. S Dressel-haus, “Electronic structure of chiral graphene tubules,”Appl. Phys. Lett. , 2204 (1992).[130] Eduardo V. Castro, Antonino Flachi, Pedro Ribeiro,and Vincenzo Vitagliano, “Symmetry Breaking and Lat-tice Kirigami,” Phys. Rev. Lett. , 221601 (2018),arXiv:1803.09495 [hep-th].[131] Antonino Flachi and Vincenzo Vitagliano, “Symme-try breaking and lattice kirigami: Finite temper-ature effects,” Phys. Rev. D , 125010 (2019),arXiv:1904.06912 [hep-th].[132] S. G. Mamaev, V. M. Mostepanenko, and Alexei A.Starobinsky, “Particle Creation from Vacuum Near anHomogeneous Isotropic Singularity,” JETP , 823(1976), [Zh. Eksp. Teor. Fiz. , 1577 (1976)].[133] S. G. Mamaev and N. N. Trunov, “Dependence of thevacuum expectation values of the energy-momentumtensor on the geometry and topology of the manifold,”Theor. Math. Phys. , 228–234 (1979).[134] S. G. Mamaev and N. N. Trunov, “Vacuum means ofenergy-momentum tensor of quantized fields on mani-folds of different topology and geometry. I,” Sov. Phys.J. , 766–770 (1979).[135] S. G. Mamaev and N. N. Trunov, “Vacuum averagesof the energy-momentum tensor of quantized fields onmanifolds of various topology and geometry. II,” Sov.Phys. J.22