Casimir effect for lattice fermions
KKEK-TH-2220
Casimir e ff ect for lattice fermions Tsutomu Ishikawa a,b , Katsumasa Nakayama c , Kei Suzuki d a Graduate University for Advanced Studies (SOKENDAI), Tsukuba, 305-0801, Japan b Studies, High Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan c NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany d Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai, 319-1195, Japan
Abstract
We propose a definition of the Casimir energy for free lattice fermions. From this definition, we study the Casimir e ff ects for themassless or massive naive fermion, Wilson fermion, and (M¨obius) domain-wall fermion in 1 + ff erence between odd and even lattice sizes. For the Wilson fermion, in the small lattice size of N ≥
3, theCasimir energy agrees very well with that of the continuum theory, which suggests that we can control the discretization artifactsfor the Casimir e ff ect measured in lattice simulations. We also investigate the dependence on the parameters tunable in M¨obiusdomain-wall fermions. Our findings will be observed both in condensed matter systems and in lattice simulations with a small size.
1. Introduction
The Casimir e ff ect [1–4] is known as negative energy andattractive force caused by a zero-point energy shift of photonfields between two parallel plates. It was first predicted in1948 [1] and was experimentally discovered after fifty years [5].The original Casimir e ff ect is physics related to photon fields,which is perturbatively described in quantum electrodynamics(QED), but a similar concept can be applied to any field such asscalar fields, fermion fields, and other gauge fields.Nowadays, lattice simulations are utilized as a tool for con-trollably studying various quantum phenomena, and Casimir(-like) e ff ects on the lattice for not only the U (1) gauge theory [6–8] but also scalar field theories [9], and nonperturbative theo-ries such as the compact U (1) gauge theory [10–13] and Yang-Mills theories with S U (2) [14, 15] and
S U (3) [16] gauge fieldswere measured. Furthermore, in the future, the Casimir e ff ectin lattice gauge theories coupled to dynamical fermions, suchas QED and quantum chromodynamics (QCD) will draw at-tention. In particular, the QCD Casimir e ff ect contains com-plicated contributions from not only perturbative quarks andgluons but also nonperturbative phenomena such as the sponta-neous chiral-symmetry breaking, confinement, instantons, ande ff ects from hadron degrees of freedoms (for related worksfrom fermionic e ff ective models, see Refs. [17–54]). How-ever, a discretized lattice-field theory is di ff erent from the orig-inal continuum theory, so that, to precisely interpret physicsmeasured on the lattice, we also need the understandings of theCasimir e ff ects for lattice field theories. Email addresses: [email protected] (Tsutomu Ishikawa), [email protected] (Katsumasa Nakayama), [email protected] (Kei Suzuki) The first example of the Casimir e ff ects for massless quarks and gluonswas introduced in the context of the MIT bag model by Johnson [55]. The purpose of this Letter is to define the Casimir energy forfree lattice fermions for the first time. The Casimir energy forcontinuous degrees of freedom can be derived by dealing withdivergent zero-point energy. On the other hand, the zero-pointenergy for lattice degrees of freedom is not divergent becauseof the lattice regularization. In this sense, the definition of theCasimir energy on the lattice is not trivial. Furthermore, ourmotivations for this study are as follows:1. It will be important for deeply understanding the elemen-tal properties of lattice fermion actions. This is becausethe Casimir e ff ect might be sensitive to the properties oflattice fields in the ultraviolet (UV) region. In the UV re-gion, the properties of lattice fermions are deformed by anonzero lattice spacing a , so that the physics would dif-ferent from that in the continuum theory. In this sense,the comprehensive examination of the phase structure of afermion action in finite (especially, small) volume will beimportant, which is similar to theoretical studies at finitetemperature and / or density.2. It will be useful for the estimate or interpretation of dis-cretization artifacts contaminating in lattice simulations.In this work, for simplicity, we focus on only the free (non-interacting) fermions, but the studies of qualitative proper-ties would be useful also for lattice simulations with inter-acting fermions such as lattice QCD.3. It can be related to the Casimir e ff ects in similar systems incondensed matter physics. For example, Wilson fermion-like dispersion relations appear in Dirac semimetals [58–60]. Also, the domain-wall fermions are well known asan analogy to zero-mode Dirac fermions realized on the There are a few analytical works about the Casimir e ff ects for lattice scalarfields [56, 57]. a r X i v : . [ h e p - l a t ] M a y urface of topological insulators [61, 62]. In this sense,this study is not limited to theoretical interests, and it canalso provide us motivations for future tabletop experimentsin condensed matter. In such situations, we can exper-imentally observe Casimir e ff ects for (Dirac-like) latticefermions.When we naively formalize fermion fields on the lattice [63],we confront the so-called doubler problem, which is known asthe Nielsen-Ninomiya no-go theorem [64, 65]. To evade fromthe doubler problem, one have to introduce “improved” fermionactions such as the Wilson fermions [66, 67] and domain-wall(DW) fermions [68–70]. Although the Wilson fermion breaksthe chiral symmetry, it has been broadly used in the various sim-ulations of QCD. On the other hand, the DW fermion formu-lation realizes the chiral symmetry on the lattice and has beensuccessfully applied to the investigation of various physics. TheM¨obius domain-wall (MDW) fermion [71–73] is an improve-ment of the DW fermion using a real M¨obius transformation ofthe Dirac operator.The contents of this Letter are organized as follows. InSec. 2, we give a definition of Casimir energy for latticefermions. As an example of applications of this definition, inSec. 3, we investigate the properties of the Casimir energy forthe naive lattice fermion. Here, we will find a characteristicoscillation of Casimir energy. In Secs. 4 and 5, we study theCasimir energies for the Wilson and overlap fermions, respec-tively. Section. 6 is devoted to our conclusion and outlook.
2. Definition of Casimir energy on the lattice
In this section, we give a definition of the Casimir energy forlattice fermions. In this Letter, as a situation with the Casimire ff ect, we consider only the compactification of one spatial di-mension. Then only the first component p of the spatial mo-mentum of a fermion is discretized: p → p ( n ), where n isthe label of discretized levels. The other spatial components(e.g., p and p in three spatial dimensions) are not discretized.Moreover, we can choose two types of definitions for the tem-poral component ( p in Euclidean space or p in Minkowskispacetime): (i) continuous temporal component and (ii) latti-cized temporal component. The definition (i) corresponds tomaterials with a small size in condensed matter systems, wherethe energy p of a fermion is not latticized. On the other hand,the definition (ii) corresponds to numerical lattice simulationswith the temporal direction, where the (Euclidean) temporalcomponent is also artificially latticized. The situations (i) and(ii) share some properties of the Casimir e ff ect, but the detail isslightly di ff erent. In this Letter, we apply only the definition (i).For the case (ii), see our future studies [74].First, we define the energy-momentum dispersion relation offermions on the lattice. This is defined by the combination ofthe Dirac operators. In the 3 + aE ( ap ) = a (cid:113) D † k D k , (1)with the lattice spacing a , where k = , , N = L / a , the spatial momen-tum for this direction with the periodic and antiperiodic bound-ary conditions is discretized as ap → ap P1 ( n ) = n π N , (2) ap → ap AP1 ( n ) = (2 n + π N , (3)respectively. The label n is an integer ( n = , ± , , · · · ). Whenwe choose the Brillouin zone for the three spatial momenta as − π < ap k ≤ π or 0 ≤ ap k < π , and then the lower and up-per bounds of n are determined by the Brillouin zone and theboundary condition: − π < ap k ≤ π → − N < n P ≤ N , (4) − π < ap k ≤ π → − N − < n AP ≤ N − , (5)0 ≤ ap k < π → ≤ n P < N , (6)0 ≤ ap k < π → − ≤ n AP < N − , (7)where N and n should be an integer, so that the range of n AP inEq. (7) is practically 0 ≤ n AP < N . Note that the Casimir energydefined below does not depend on the choice of the Brillouinzone.By discretization of the momentum p , the zero-point energy(per area) is redefined as aE ( N → ∞ ) = − Nc deg (cid:90) d ap (2 π ) aE ( ap ) → aE ( N ) = − c deg (cid:90) d ap ⊥ (2 π ) (cid:88) n aE n ( ap ⊥ , N ) , (8)where c deg is the degeneracy factor, such as spin of fermion,and we set c deg = fromthe zero-point energy and the factor of 2 from the particle andantiparticle cancel out each other.Here, we define the Casimir energy for lattice fermions (withone compactified space in the 3 + ff erence between aE ( N ) and aE ( N → ∞ ): aE ≡ aE ( N ) − aE ( N → ∞ ) = c deg (cid:90) d ap ⊥ (2 π ) − (cid:88) n aE n ( ap ⊥ , N ) + N (cid:90) BZ dap π aE ( ap ) , (9) The original zero-point energy in the three dimensional space is defined as a ˆ E ( N → ∞ ) = − Va c deg (cid:90) d ap (2 π ) aE ( ap ) , where V = L × A with the two-dimensional area A . In this sense, Eq. (8) is thezero-point energy divided by the area. p runs over the defined Bril-louin zone. Application to other dimensions is straightforward.For example, in the 1 + aE ≡ c deg − (cid:88) n aE n ( N ) + N (cid:90) BZ dap π aE ( ap ) . (10)
3. Casimir energy for naive fermion
First, we study the Casimir e ff ect for the naive latticefermion. The (dimensionless) Dirac operator of the naive lat-tice fermion with a mass m f in momentum space is defined as aD nf ≡ i (cid:88) k γ k sin ap k + am f , (11)where γ k is the gamma matrix.From Eq. (1), we can evaluate the dispersion relation: a E ( ap ) = (cid:88) k sin ap k + ( am f ) , (12)From Eq. (9), we can calculate the Casimir energy. Here, theintegration with respect to ap ⊥ is limited to the first Brillouinzone, so that the Casimir energies are determined without anydivergence. Then we can numerically evaluate the Casimir en-ergy. In order to get the analytic formulas of the Casimir en-ergy, one also can utilize a mathematical technique such as theAbel-Plana formulas [74].For the naive fermion with m f = + aE , nf , PCas = N π − cot π N ( N = odd) , (13) aE , nf , PCas = N π − π N ( N = even) , (14) aE , nf , APCas = N π − cot π N ( N = odd) , (15) aE , nf , APCas = N π − π N ( N = even) , (16)where we distinguished the solutions into the odd lattice ( N = odd) and even lattice ( N = even).When we expand these formulas by a small lattice spacing a ,we obtain E , nf , PCas = π L + π a L + O ( a ) ( N = odd) , (17) E , nf , PCas = π L + 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)In Fig. 1, we plot the Casimir energy for the massless andmassive naive lattice fermions in the 1 + aE Cas and LE Cas . Here, for the massless fermion, we compared the ex-act formulas (13)–(16) and the approximate formulas (17)–(20).From this figure, our findings are as follows: 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 −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 Figure 1: Casimir energy for massless or massive naive lattice fermion in the1 + Oscillation of Casimir energy —For both the periodic andantiperiodic boundaries, we find the oscillatory behaviorof the Casimir energy. This behavior is caused by the dif-ference between the properties of the odd lattice and evenlattice. While, for the periodic boundary, the Casimir en-ergy oscillates between two di ff erent repulsive forces, forthe antiperiodic boundary, the Casimir energy oscillatesbetween attractive and repulsive forces.These behaviors can be qualitatively interpreted by thecounting of the momentum zero modes in energy levelsdiscretized by a finite volume. For example, from thethe definition of the Brillouin zone, Eqs. (4)–(7), the dis-cretized levels on the even lattice with the periodic bound-ary has two zero modes ( ap = , π ). Because of the ap-pearance of zero modes, the vacuum energy in finite vol-ume increases, which corresponds to the positive Casimirenergy (and repulsive Casimir force). Furthermore, theodd lattice with the periodic or antiperiodic boundary hasone zero mode ( ap = π ), which also leads to the re-pulsive Casimir energy. On the other hand, the even latticewith the antiperiodic boundary has no zero mode, so thatthe vacuum energy relatively decreases. This understand-3ng is based on the simple energy level structures of thenaive fermion, and for more complicated dispersion rela-tions, a detailed discussion should be required.2. Equivalence between periodic and antiperiodic bound-aries on odd lattice —As seen in Eqs. (13) and (15), wefind that, on the odd lattice, the Casimir energy for the pe-riodic and antiperiodic boundaries is equivalent. This isbecause these dispersion relations are e ff ectively equiva-lent to each other.3. Comparison with continuous Dirac fermions —In the con-tinuum theory with the one spatial dimension, the Casimirenergy for free massless Dirac fermions is proportional tothe inverse of the interval: E Cas ∝ / L . The Casimir ener-gies for the periodic and antiperiodic boundary conditionsare given by [75–78] E , cont , PCas = π L , (21) E , cont , APCas = − π L , (22)respectively. Therefore, the Casimir energy for naive lat-tice fermions is completely di ff erent from that for the con-tinuous Dirac fermion. This disagreement remains evenafter the a → Dependence on lattice spacing —In Fig. 1, we comparethe approximated formulas with the small a expansion,Eqs. (17)–(20), and the exact formulas (13)–(16). Theseformulas are in good agreement with each other in the re-gion of N ≥
3, which indicates that the approximated for-mulas will be useful for estimating the Casimir energy asfar as we focus on N ≥ Suppression of massive Casimir energy —In Fig. 1, wecompare the results of the massless fermion and massiveone with am f = .
2. When the fermion has a nonzeromass, the Casimir energy is suppressed. This tendency issimilar to the usual Casimir energy with massive degreesof freedom. We emphasize that although the Casimir en-ergy is suppressed, the oscillatory behavior survives.
4. Casimir energy for Wilson fermion
Next, we investigate the properties of Casimir e ff ects for Wil-son fermions. We define the Wilson Dirac operator D W with theWilson coe ffi cient r : aD W ≡ i (cid:88) k γ k sin ap k + r (cid:88) k (1 − cos ap k ) + am f , (23)The dispersion relation is a E ( ap ) = (cid:88) k sin ap k + r (cid:88) k (1 − cos ap k ) + am f , (24)From Eq. (9), we can calculate the Casimir energy. For theWilson fermion with r = am f = + C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.am 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 −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/aCont.am f =0am f =0.2−0.8−0.6−0.4−0.2 0 2 4 6 8 10 L E C a s Figure 2: Casimir energy for massless or massive Wilson fermion in the 1 + spacetime, we can get the exact formulas: aE , W , PCas = N π − π N , (25) aE , W , APCas = N π − π N . (26)When we expand these formulas by a small lattice spacing a ,we obtain E , W , PCas = π L + π a L + O ( a ) (27) E , W , APCas = − π L − π a L + O ( a ) . (28)In Fig. 2, we show the results of the Wilson fermion at r = + Agreement with continuum theory in large size —In thelarger lattice size with N ≥
3, the Casimir energy forthe Wilson fermion agrees very well the Casimir energies,(21) and (22), in the continuum theory. This result indi-cates that when one tries to measure the fermionic Casimir4nergy in this region by using numerical simulations, onecan obtain the results consistent with the continuum theorywithin a small discretization error. In particular, the oscil-lation of the Casimir energy found for the naive fermion isremoved by the Wilson term. In other words, the contin-uum limit can be easily taken.Moreover, we emphasize that our procedure with the a → Overestimate in small size —Furthermore, we find that, inthe smaller lattice size of N ≤
3, the Casimir energy for theWilson fermion is overestimated by the discretization ef-fects. This is because the energy-momentum dispersionrelation of the Wilson fermions is underestimated com-pared to that of the Dirac fermion in the continuum theory.Thus, the Casimir energy for the Wilson fermion is largerthan that of the Dirac fermion. Therefore, in order to ob-serve the Casimir energy for lattice fermions, the materialwith the Wilson-fermion-like band structure will be morepreferable.3.
Massive Casimir energy —In the massive case, we find thatthe Casimir energy is suppressed for N ≥
2, which isconsistent with the usual suppression of the Casimir ef-fect by massive degrees of freedom. On the other hand,for N = N = ap = π , and then the discretized energy level is dominatedby aE = + am f . As a result, such a single “heaviest”mode induces a Casimir energy more attractive than themassless case. This situation is di ff erent from the periodicboundary, where the discretized energy level is dominatedby ap = aE = am f . Thus, we emphasize that theCasimir e ff ect enhanced by a nonzero mass is a rare exam-ple in the long history of the Casimir e ff ect.
5. Casimir energy for overlap fermion
In this section, we investigate the Casimir energy of the over-lap fermion with the MDW kernel operator. In the domain-wallfermion formulation [68–70], a “bulk” fermion defined as ker-nel operators in the D + D -dimensionalEuclidean space. The length of the extra dimension is usuallyfinite, but we assume infinite length for simplicity, which cor-responds to the overlap fermion [79, 80]. We define the MDW kernel operator D MDW using the M¨obiusparameters b and c [71–73], aD MDW ≡ b ( aD W )2 + c ( aD W ) . (29) This kernel operator is a generalization of the conventionalShamir-type ( b = , c =
1) [69] and Boric¸i-type (or Wilson-type) ( b = , c =
0) [81, 82] formulations. D W is the WilsonDirac operator defined as Eq. (23) with r =
1, but the fermionmass m f is replaced by the domain-wall height am f → − M which is a negative mass. Using the MDW kernel operator D MDW , we introduce theoverlap Dirac operator D OV with fermion mass m f , aD OV ≡ (2 − cM ) M m PV × (1 + am f ) + (1 − am f ) V (1 + m PV ) + (1 − m PV ) V , (30)with V ≡ γ sign( γ aD MDW ) = D MDW (cid:113) D † MDW D MDW . (31)The Pauli-Villars mass m PV was introduced so as to satisfy theGinsparg-Wilson relation. The scaling factor (2 − cM ) M m PV with a constraint 2 − cM > a → D † OV D OV = p for m f = a E = [(2 − cM ) M m PV ] × + ( am f ) ] + [1 − ( am f ) ]( V † + V )2[1 + m ] + [1 − m ]( V † + V ) , (32)where we used the V † V = V † + V and V . V † + V are represented by the Wilsonoperator D W , V † + V = (cid:16) D † MDW + D MDW (cid:17) (cid:113) D † MDW D MDW , = (cid:113) D † W D W D W + D † W + cD † W D W ) (cid:113) + c ( D † W + D W ) + c D † W D W , (33)where we used the properties D † W D W > + c ( D † W + D W ) + c D † W D W >
0. Note that, in thisform, the parameter b dependence is completely eliminated. Inthis work, the b dependence is not relevant since we considerthe infinite length of the extra dimension. From the dispersionrelation (32) and the definition (9), we can calculate theCasimir energy of the overlap fermion. In Fig. 3, we show the dependence of the Casimir energyfor the overlap fermion on the domain-wall height ( M = . .
0, and 1 .
5) at fixed c = m PV = .
0. Among them, theCasimir energy at M = . PV =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.M =0.5M =1.0M =1.5 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 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/aCont.M =0.5M =1.0M =1.5−1−0.8−0.6−0.4−0.2 0 2 4 6 8 10 L E C a s Figure 3: Domain-wall height M dependence of Casimir energy for overlapfermions with MDW kernel operator in the 1 + M = .
5, we find the overestimate of the Casimir energy inthe larger lattice size and underestimate in the smaller latticesize. At M = .
5, there is an oscillatory behavior for both theperiodic and antiperiodic boundaries, where we find the overes-timate on the odd lattice and underestimate on the even lattice.This oscillation is induced by the appearance of massive dou-blers, which is absent at M ≤ .
0. The setup with M > . M may usefulfor correcting the fermion dispersion relations e ff ectively mod-ified by gauge fields. A similar oscillation is also found at finitetemperature [85, 86].In Fig. 4, we show the dependence of the Casimir energy onM¨obius parameter ( c =
0, 0 .
5, 1 .
0, and 1 .
5) at fixed M = . m PV = .
0. We find that the Casimir energy at c =
0, whichis equivalent to the Wilson fermion, best reproduces that in thecontinuum theory. For a nonzero c , we find the overestimate ofthe Casimir energy in the larger lattice size and underestimate inthe smaller lattice size. As c is larger, the region with deviationfrom the continuum theory becomes broader. In particular, at c = .
5, this deviation can be observed even on the lattice with N = =1,m PV =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.c=0c=0.5 c=1.0c=1.5 0.6 0.8 1 1.2 0 5 10 15 20 25 30 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+1MDWM =1,m PV =1Antiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.c=0c=0.5 c=1.0c=1.5−0.8−0.6−0.4−0.2 0 0 5 10 15 20 L E C a s Figure 4: M¨obius parameter c dependence of Casimir energy for overlapfermions with MDW kernel operator in the 1 + In Fig. 5, we show the dependence of the Casimir energy onthe Pauli Villars mass m PV dependence ( m PV = . . m PV = .
0, we find an oscillatory behavior. This oscillation isalso induced by the appearance of massive doublers, which issimilar to M > .
0. A heavy Pauli Villars mass, m PV > .
6. Conclusion and outlook
In this Letter, we defined the Casimir energy for latticefermions, Eq. (9), for the first time. From this definition,we investigated the properties of the Casimir energy for themassless / massive naive fermion, Wilson fermion, and overlapfermion with the MDW kernel operator. In particular, for sometypes of fermions, we found a characteristic oscillatory behav-ior between odd and even lattices.For some simple cases, we analytically obtained the exactformulas for the Casimir energies, and for more complicatedcases, we numerically calculated the Casimir energies. In orderto carefully examine our formulas, the confirmation from othermathematical derivation is left for future works.In this Letter, we focused on only free fermions, but the6 =1Periodic C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.m PV =1.0m PV =3.0−2 0 2 4 0 2 4 6 8 10 L E C a s −1.5−1−0.5 0 0 2 4 6 8 10 12 14 16 18 20D=1+1MDWc=1,M =1Antiperiodic C a s i m i r ene r g y a E C a s Lattice size N=L/aCont.m PV =1.0m PV =3.0−4−2 0 2 0 2 4 6 8 10 L E C a s Figure 5: Pauli Villars mass m PV dependence of Casimir energy for overlapfermions with MDW kernel operator in the 1 + Casimir energy for interacting lattice fermions would be alsoimportant. In particular, one might be interested in the rela-tion between the Casimir e ff ect and the parity-broken (Aoki)phase [88] realized in the strong-coupling regime of the Wil-son fermion and domain-wall fermion. Furthermore, otherfermion actions not studied in this Letter, such as staggeredfermions [89, 90], will be interesting. Also, we showed onlythe results in the 1 + + + ff ects for lattice fermions will be observed in table-top experiments with Dirac electron systems when an extremelysmall lattice size is realized . For example, a (very short) 1D ringmade of a Dirac material can induce the Casimir energy withthe periodic boundary condition in the one-dimensional space.Similarly, a (very small) cylinder made of 2D thin films leadsto the Casimir energy in the two-dimensional space (for worksrelated to carbon nanotubes, see Refs. [91, 92]). In this sense,to investigate fermionic Casimir e ff ects in honeycomb latticeswill be important. Another possible candidate for studying theCasimir e ff ects for lattice fermions would be cold-atom simu-lations [93–96] with a small size. Acknowledgment
The authors are grateful to Yasufumi Araki and Daiki Sue-naga for giving us helpful comments. This work was supportedby Japan Society for the Promotion of Science (JSPS) KAK-ENHI (Grant Nos. JP17K14277 and JP20K14476).
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