On the Sign of Fermion-Mediated Interactions
OOn the Sign of Fermion-Mediated Interactions
Qing-Dong Jiang
Department of Physics, Stockholm University, Stockholm SE-106 91 Sweden
We develop a unified framework for understanding the sign of fermion-mediated interactions byexploiting the symmetry classification of Green’s functions. In particular, we establish a theoremregarding the sign of fermion-mediated interactions in systems with chiral symmetry. The strengthof the theorem is demonstrated within multiple examples with an emphasis on electron-mediatedinteractions in materials.
Introduction.—
That the exchange of a particle can pro-duce a force is one of the most remarkable conceptualadvances in physics. Each of the fundamental interac-tions has an associate bosonic force carrier: for exam-ple, photons mediate Coulomb interactions and gravitonsmediate gravitational interactions. Given these boson-mediated interactions, one may naturally ask the follow-ing innocuous question: “Can fermions also mediate in-teractions?”While the answer is ‘yes’, there is an essential differ-ence between boson-mediated interactions and fermion-mediated interactions. That is, due to the conserva-tion of fermionic parity, fermions need to be exchangedat least twice to produce a force whereas bosons onlyneed to be exchanged once (see the Feynman diagram inFig. 1). As this Feynman diagram resembles that of theCasimir effect, fermion-mediated interactions are occa-sionally called fermionic Casimir effects in the literature .In the case of one-boson-mediated interactions, the signis uniquely determined by the spin of the exchanged par-ticles: Exchanging a scalar, or a tensor particle producesa universally attractive force, while exchanging a vec-tor particle can produce a repulsive force between likecharges . However, unlike the bosonic case, a universalunderstanding of the sign of fermion-mediated interac-tions is currently lacking.In this work, we study the sign of various fermion-mediated interactions. In condensed-matter physics,electron-mediated interactions were initially proposed toexplain the ordering of adsorbates at surfaces , andwere recently considered to be crucial for engineering theproperties of novel materials such as graphene . As animportant mechanism for magnetic ordering, the Ruder-man–Kittel–Kasuya–Yosida (RKKY) interaction is an-other example of electron-mediated interactions . Inthe cold-atom area, fermion-mediated interactions havereceived extensive theoretical investigations and wererecently observed in experiments consisting of a mix-ture of bosonic and fermionic quantum gases . In high-energy physics, fermion-mediated interactions are of par-ticular relevance to the physics of neutron stars andquark-gluon plasma . Given the ubiquitous presenceof fermion-mediated interactions, their sign is of boththeoretical interest and practical significance . Forexample, attractive fermion-mediated interactions couldlead to new phases of matter such as supersolids .By exploiting the symmetry classification of Green’s X X Fermionic environment
Figure 1. Feynman diagram for fermion-mediated interac-tions. The red lines represent fermionic Green’s functions(propagators) G and G that connect X and X . Sincetwo Green’s functions are involved in the Feynman diagram,we say that objects X and X interact with each other byexchanging fermions twice. The blue bubbles represent thescattering matrices of X and X , the form of which will begiven in the text. functions , we present a unified framework for under-standing the sign of fermion-mediated interactions, elim-inating some loopholes and resolving some controversiesin the literature. Specifically, in systems with chiral sym-metry, we show that the sign of fermion-mediated inter-actions U between objects X and X is given by thesimple rule: sign (U ) = ( − η χ χ . (1)Here, χ = ± ; η = 0or 1 depending on the strength of scattering potentialsbeing strong or weak. Between objects with the same chirality, a strong scattering potential leads to repulsion,whereas a weak scattering potential leads to attraction.By sharp contrast, between objects with opposite chiral-ities, a strong scattering potential leads to attraction,whereas a weak scattering potential leads to repulsion.The rest of the paper is organized as follows. We firstderive a nonperturbative expression of fermion-mediatedinteractions involving Matsubara Green’s functions andscattering matrices. Then, we show how nonspatial (lo-cal) symmetries constrain the Green’s functions and scat-tering matrices of a general interacting system. Withthe above preparation, we establish a theorem regardingthe sign of fermion-mediated interactions. Finally, wedemonstrate the power of our theorem by giving a num-ber of examples with an emphasis on electron-mediatedinteractions in materials. a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Field theory of fermion-mediated interactions.—
Toset the stage, we now present a unified expressionfor fermion-mediated interactions. While various ex-pressions for fermion-mediated interactions exist inthe literature , a universal, and nonperturbativederivation is not entirely trivial. A related path-integralapproach has been used to study the Casimir effect inRefs. 18 and 19.Consider two objects X i ( i = 1 ,
2) embedded at thepositions x i in a fermionic environment, and introducelocalized potential operators ˆ V i to describe the scatter-ing effect of X i . We consider bilinear coupling ¯ ψ ˆ V i ψ be-tween the operators ˆ V i and the mediating fermionic fields( ¯ ψ, ψ ); thus the effective Euclidean action reads (we useunits (cid:126) = c = 1) S = − (cid:88) n (cid:90) d x ¯ ψ n ( x ) (cid:16) ˆ G − − ˆ V − ˆ V (cid:17) ψ n ( x ) , (2)where ˆ G = ( iω n + ˆ H ) − represents the Matsubara Green’sfunction of the fermionic host. The partition function ofthe whole system is Z = (cid:82) D ¯ ψ D ψ exp ( −S ). In theabsence of X i , the partition function Z can be obtainedfrom Z by setting ˆ V i = 0. Consequently, the change ofthe energy due to the introduction of X i can be formallyobtained from the reduced partition function E = − β ln ZZ = − β ln det [1 − G ( x , x (cid:48) ) V − G ( x , x (cid:48) ) V ] , (3)where G ( x , x (cid:48) ) = (cid:104) x | ˆ G| x (cid:48) (cid:105) , V i = (cid:104) x (cid:48) | ˆ V i | x (cid:48) (cid:105) δ x (cid:48) , x i , and β is the inverse temperature. Note that the total energy E contains three parts: the self energies of X and X and the mutual interaction between them. Therefore, toobtain the interaction energy, we need to subtract theself-energy contribution that does not depend on relativepositions of X and X . For this purpose, it is convenientto put the energy in a matrix form: E = − β ln det (cid:18) − G V −G V −G V − G V (cid:19) . (4)Here, G ≡ G ( iω n , r ) and G ≡ G ( iω n , − r ) are theMatsubara Green’s functions linking X with X ; G = G ≡ G ( iω n ,
0) are local Matsubara Green’s functions; r = x − x denotes the relative distance between X and X ; and ω n = (2 n + 1) πβ − are fermionic Matsub-ara frequencies. After subtracting the self-energies, i.e.,the diagonal contribution of the matrix, we obtain theuniversal formula for two-particle exchange interactionsU = − β (cid:88) n ln det (1 − G T G T ) , (5)where T = V (1 − G V ) − and T = V (1 − G V ) − represent the scattering matrices for X and X . Notethat Eq. (5) is derived without using any perturbative expansion, and thus it applies to strong scattering poten-tials. If the fermionic host is weakly correlated, G and G represent the renormalized Matsubara Green’s func-tions with interactions encoded in their self-energy parts.To the lowest order expansion, U = β (cid:80) n G T G T represents a two-fermion-exchange interaction, as illus-trated in Fig. 1. Symmetry classification of Green’s functions.—
Wenow present one more ingredient - the symmetry clas-sification of Green’s functions - aiming at a universaltheorem for the sign of fermion-mediated interactions.In recent years, classification of Green’s functions undernonspatial symmetries, namely time-reversal symmetry,charge-conjugation (particle-hole) symmetry and chiralsymmetry, has been used to classify topological phases ofcorrelated fermions . For clarity, we start with the sym-metry classification of non-interacting Hamiltonians and then derive the symmetry classification of Matsub-ara Green’s functions which turns out to be particularlyuseful when interactions are present.Consider a general system of non-interacting fermionsdescribed by the Hamiltonian ˆ H = (cid:80) αβ ψ † α H αβ ψ β withthe fermion creation and annihilation operators satisfyingcanonical anti-commutation relations { ψ α , ψ † β } = δ αβ .Here, the indices α, β refer to relevant degrees of free-dom, such as lattice sites, spins, layers, etc. ; and H , thefirst quantized Hamiltonian, is a complex matrix. Underthe constraints of time-reversal symmetry and charge-conjugation symmetry, the Hamiltonian matrix H satis-fies the following conditions: U † T H ∗ ( k ) U T = H ( − k ) (6a) U † C H ∗ ( k ) U C = −H ( − k ) (6b)where U T and U C are the corresponding unitary matrices.It is crucial to consider an additional discrete symmetry,the chiral symmetry, being the product of time-reversalsymmetry and charge-conjugation symmetry. The chiralsymmetry imposes an additional condition on the Hamil-tonian matrix U † S H ( k ) U S = −H ( k ) . (7)with the chiral matrix U S = U ∗ T U C . Notice thatthe chiral symmetry can be present in cases whereneither time-reversal nor charge-conjugation symmetryis present. The above symmetry constraints lead tothe ten-fold classification of topological insulators andsuperconductors .Equipped with the above Hamiltonian formalism, wenow proceed to present the symmetry classification of theGreen’s functions. We are interested in the MatsubaraGreen’s functions defined as G ( iω n , k ) = [ iω n − H ( k )] − .According to Eqs. (6a), (6b), and (7), time-reversal sym-metry and charge-conjugation symmetry set the follow-ing conditions for Matsubara Green’s functions, i.e., U T G ( iω n , k ) U † T = 1 iω n − H ∗ ( − k ) = G ∗ ( − iω n , − k ) , (8a) U C G ( iω n , k ) U † C = 1 iω n + H ∗ ( − k ) = −G ∗ ( iω n , − k ) . (8b)Furthermore, chiral symmetry implies that the Matsub-ara Green’s function should fulfill the condition U S G ( iω n , k ) U † S = −G ( − iω n , k ) (9)consistent with Eqs. (8a) and (8b). Finally, regardless ofwhether interactions are present or not, the Hermicity ofthe Hamiltonian ensures that G ( iω n , k ) = G † ( − iω n , k ) . (10)As a result, the combination of chiral symmetry and Her-micity leads to the momentum-space condition U S G ( iω n , k ) U † S = −G † ( iω n , k ) , (11)which, after Fourier transform, yields the real-space ex-pression U S G ( iω n , r ) U † S = −G † ( iω n , − r ) (12)which is an essential ingredient in the proof of our the-orem. While the above Eqs. (8a), (8b), and (9) areobtained from non-interacting Hamiltonians, a sophis-ticated field-operator approach shows that the aboveGreen’s function formalism also holds for interacting sys-tems. Interested readers could also find the proof fromthe excellent works Ref.15. The theorem.—
In systems with symmetry, it is oftenconvenient to choose a chiral basis such that the chiraloperator U S is diagonal, i.e., U S = diag ( n , − m ), where n and m are n × n and m × m identity matrices. Theeigenvalue +1 ( −
1) denotes the chirality of the corre-sponding basis. One can then use the Pauli matrix τ z torepresent U S for each pair of basis states with oppositechiralities. (We use τ x,y,z to represent Pauli matrices forgeneral degrees of freedom, while reserving the notation σ x,y,z for real spin.) The Matsubara Green’s functionmatrix in the pair chiral basis ( | χ (cid:105) , | ¯ χ (cid:105) ) reads G ( iω n , r ) = (cid:18) G χχ ( iω n , r ) G χ ¯ χ ( iω n , r ) G ¯ χχ ( iω n , r ) G ¯ χ ¯ χ ( iω n , r ) (cid:19) , (13)where the indices χ = − ¯ χ = ± U S = τ z intoEq.(12), we obtain the crucial conditions required by the chiral symmetry: G χχ ( iω n , r ) = −G χχ ∗ ( iω n , − r ) , (14a) G χ ¯ χ ( iω n , r ) = G ¯ χχ ∗ ( iω n , − r ) . (14b)If two objects have the same chirality, the connectingMatsubara Green’s functions satisfy G ≡ G χχ ( iω n , r ) = −G ∗ ≡ −G χχ ∗ ( iω n , − r ). By contrast, if two ob-jects have opposite chiralities, the connecting Matsub-ara Green’s functions satisfy G ≡ G χ ¯ χ ( iω n , r ) = G ∗ ≡G ¯ χχ ∗ ( iω n , − r ). As indicated by the Eq.(5), the sign ofthe fermion-mediated interaction is identical to the signof the product G T G T . Due to the expression ofthe scattering matrices (below Eq. (5)), two possibilitiescan be distinguished:1) In the limit of a strong potential ( V i → ∞ ), thescattering matrices T = T = G − ( iω n ,
0) arepurely imaginary according to the Eq. (14a), andtherefore T T <
0. It is then straightforward toobtain the sign of fermion-mediated interactions inthe following cases:i. Between two objects with the same chirality,the fermion-mediated interactions are always repulsive becausesign (U ) = sign ( G T G T )= − sign ( G ∗ T G T ) > . (15)ii. Between two objects with opposite chiralities,the fermion-mediated interactions are always attractive due tosign (U ) = sign ( G T G T )= sign ( G ∗ T G T ) < . (16)2) In the limit of a weak potential ( V i → T i = V i are purely real, and T T >
0. Hence, the signs of fermion-mediatedinteractions are opposite to the case of strong po-tential: Between two objects with the same ( op-posite ) chirality, the fermion-mediated interactionsare always attractive ( repulsive ).As the above results can be conveniently summarizedin Eq.(1), we have thus established the theorem for thesign of fermion-mediated interactions. The above analy-sis indicates that one could change the sign of fermion-mediated interactions by tuning the strength of scatter-ing potentials. To demonstrate the power of this the-orem, we now explore a number of examples with anemphasis on electron-mediated interactions in materials. Electron-mediated interactions in non-interacting sys-tems.
As our first example, let us consider electron-mediated interactions between two adatoms embedded inmonolayer graphene. The physics of single layer grapheneis captured by the continuum low-energy Hamiltonian H SLG = v F ( k x τ x + k y τ y ), where τ x,y,z represent thePauli matrices in the sublattice basis, and v F is the Fermivelocity . One can easily identify the chiral transfor-mation matrix U S = τ z . Therefore, without further de-tailed calculation, we can apply the theorem to graphene.In the strong-impurity limit , adatoms residing on thesame (different) sublattices repel (attract), whereas, inthe weak-impurity limit, impurities reside on the same(different) sublattices attract (repel). This is consis-tent with detailed calculations carried out in Refs. One can consider more general Hamiltonians such as H = v F (cid:0) k nx τ x + k my τ y (cid:1) , where m and n can be arbitraryodd numbers. As the chiral matrix for this Hamiltonianis still U S = τ z , our theorem also applies to this model.Our second example deals with the electron-mediatedinteractions in the Bernal-stacked bilayer graphene, theunit cell of which includes A and B atoms on layer 1and A and B atoms on layer 2. Expressed in the basis( ψ A , ψ B , ψ A , ψ B ) (here subindices denote the latticetype), the low-energy effective Hamiltonian reads H BLG = γ v F k − γ v F k + v F k − v F k + , (17)where k ± = k x ± ik y and γ denotes the interlayer cou-pling. It can be verified that this Hamiltonian preserveschiral symmetry with the chiral matrix U S = (cid:18) τ z τ z (cid:19) , (18)which indicates that sublattices (A , A ) have positivechirality while sublattices (B , B ) have negative chiral-ity. Accordingly, regardless the layer index, the electron-mediated interactions are always repulsive (attractive)between impurities on the same type of sublattices (i.e.,AA or BB) in the strong (weak) impurity limit. In con-trast, when impurities reside on different sublattices (i.e.,AB), the electron-mediated interactions are always at-tractive (repulsive) in the strong (weak) impurity limit. Electron-mediated interaction in correlated systems.—
Our theorem is applicable to weakly interacting systems.We examine electron-mediated interactions in the Hub-bard model defined on a bipartite latticeˆ H = (cid:88) (cid:104) i,j (cid:105) ,σ t ij ˆ c † i,σ ˆ c j,σ − µ (cid:88) i ˆ n iσ + U (cid:88) i ˆ n i ↑ ˆ n i ↓ , (19)where ˆ c i,σ and ˆ c † i,σ is the electron annihilation and cre-ation operators at site i with spin σ = ↑ or ↓ andˆ n iσ = ˆ c † iσ ˆ c iσ . Here, the nearest-neighbor hopping ma-trix elements t ij = t ji need to be real; µ and U representthe chemical potential and on-site interaction strength,respectively. One can verify that the Hamiltonian is in-variant at half-filling ( µ = U/
2) under the following chi-ral symmetry transformation: ˆ S ˆ c i ↑ / ↓ ˆ S − = κ ( i ) ˆ c † i ↓ ( ↑ ) ,ˆ S ˆ c † i ↑ ( ↓ ) ˆ S − = κ ( i ) ˆ c i ↓ ( ↑ ) with κ ( i ) = 1 for one of thesublattices and κ ( i ) = − U is small enough for the ground stateto preserve chiral symmetry so that our result Eq.(5) stillholds. Without performing any perturbative calculation,we can apply our theorem to the correlated bipartite lat-tices: fermion-mediated interactions are repulsive or at-tractive depending on the impurities located on the sameor different sublattices. Our theorem can be applied toall weakly coupled models with chiral symmetry, such asBdG systems with time-reversal symmetry and QCDat high density . RKKY interactions in systems with chiralsymmetry.—
A similar line of reasoning enables usto predict the sign of RKKY interactions in the presenceof chiral symmetry: magnetic moments with the same ( opposite ) chirality favor a ferromagnetic ( antiferro-magnetic ) state. We demonstrate this statement byconsidering two magnetic moments ( S and S ) em-bedded in a fermionic system described by the effectiveaction S = − (cid:88) n (cid:90) d x ¯ ψ n ( x ) (cid:20) G − − J S · σ δ ( x − x ) − J S · σ δ ( x − x ) (cid:21) ψ n ( x ) . (20)with G being the Green’s function of the host system and J as the coupling constant between the impurity mag-netic moment and the spin of electrons σ µ ( µ = x, y, z )located at x and x , respectively. Note that this ac-tion has the same form as Eq. (2) given the substitutionˆ V i = S i · σ δ ( x − x i ). According to Eq. (5), we thenobtain the interaction energy of the nuclear spins (to thelowest order of J ):U = J β (cid:88) µ,ν,n tr [ S µ σ µ G ( iω n , r ) S ν σ ν G ( iω n , − r )]= − J α ( r ) S · S , (21)where the susceptibility is defined as α ( r ) = − β (cid:80) n G ( iω n , r ) G ( iω n , − r ) with r = x − x as the rel-ative distance between the localized spins. Note thatwe have assumed that the Green’s function is spin-independent in the above derivation. Additional termsuch as Dzyaloshinskii-Moriya type of interaction canemerge in spin-orbital coupled systems where the Green’sfunctions depend on spin .The two possibilities, α ( r ) > α ( r ) <
0) correspondsto a ferromagnetic (antiferromagnetic) alignment of mag-netic moments. Consider two magnetic moments (spins)that are embedded in a system with chiral symmetry,and have definite chiralities χ and χ , respectively. Thesusceptibility can then be written as α χ χ = − β (cid:88) n G χ χ ( iω n , r ) G χ χ ( iω n , − r ) . (22)Based on the symmetry properties of Green’s functions(see Eqs. (14a) and (14b)), we can verify that α χχ > α χ ¯ χ <
0. As a result, magnetic moments with thesame (opposite) chirality favor a ferromagnetic (antifer-romagnetic) alignment. This applies to all the examplesdiscussed in the previous sections. Note that our theoremexhibits two major differences compared to the theoremconcerning the sign of RKKY interactions on noninter-acting bipartite lattices . First, our theorem has abroader scope of application, as the bipartite model isonly a specific system preserving chiral symmetry. Sec-ond, our theorem can be applied to interacting modelsas long as chiral symmetry is present. This indicates thepossibility of chiral symmetry breaking in the previouscalculations, which showed that the sign of the RKKYinteractions can be modified by electronic interactions,edge states, strains, or flat bands . Concluding Remarks.—
We have established a univer-sal theorem regarding the sign of fermion-mediated in-teractions in systems with chiral symmetry. We havedemonstrated the strength of the theorem by consider- ing multiple examples ranging from noninteracting mod-els to weakly correlated systems. Without involving anyspatial symmetry, the theorem is robust to disorders anddefects. Furthermore, our theorem suggests a new routeto probe chiral symmetry breaking via fermion-mediatedinteractions: When additional parameters break chiralsymmetry, the sign of fermion-mediated interactions willnot be definite, and will oscillate with respect to theseparameters. In the future, it will be interesting to in-vestigate how other symmetries ( e.g. crystalline symme-tries) constrain fermion-mediated interactions. It wouldalso be desirable to study the sign of two-boson-mediatedinteractions in systems with chiral symmetry . Acknowledgements.—
We thank F. Wilczek for his en-couragement and support, and E. Bergholtz, L. Liang,Y. Kedem, and especially T. H. Hansson for indispens-able clarifying discussions. We thank K. Dunnett forher proofreading of the manuscript. This work was sup-ported by the Swedish Research Council under ContractNo. 335-2014-7424. A. Bulgac and A. Wirzba, Phys. Rev. Lett. , 120404(2001); P. Sundberg and R. L. Jaffe, Ann. Phys. (N.Y.) , 442 (2004); D. Zhabinskaya, J. M. Kinder, and E. J.Mele, Phys. Rev. A , 060103(R) (2008). An excellent introduction can be found in A. Zee,
QuantumField Theory in a Nutshell (Orient Longman, Princeton,2005). T. B. Grimley, Proc. Phys. Soc. , 751 (1967); ibid . ,776 (1967); T. L. Einstein and J. R. Schrieffer, Phys. Rev.B , 3629 (1973); N. R. Burke, Surf. Sci. , 349 (1976);K. H. Lau and W. Kohn, ibid . , 69 (1978). For recent experiments, see Jascha Repp, et al. , Phys. Rev.Lett. , 2981(2000); Fabien Silly, et al. , ibid . , 016101(2004); Markus Ternes, et al. , ibid . , 146805 (2004). V. V. Cheianov and V. I. Fal’ko, Phys. Rev. Lett. ,226801 (2006); V. V. Cheianov, O. Sylju˚asen, B. L. Alt-shuler, and Vladimir Fal’ko, Phys. Rev. B , 233409(2009); V. V. Cheianov, O. Sylju˚asen, B. L. Altshuler,and V. I. Fal’ko, Europhys. Lett. , 56003 (2010); D.A. Abanin, A. V. Shytov, and L. S. Levitov, Phys. Rev.Lett. , 086802 (2010). M. A. Ruderman and C. Kittel, Phys. Rev. , 99 (1954);T. Kasuya, Progress of Theoretical Physics , 45 (1956);K. Yosida, Phys. Rev. , 893 (1957). M. M. Parish, B. Mihaila, B. D. Simons, and P. B. Little-wood, Phys. Rev. Lett. , 240402 (2005); J. M. Acton,M. M. Parish, and B. D. Simons, Phys. Rev. A , 063606(2005); S. T. Chui and V. N. Ryzhov, ibid . , 043607(2004); D. H. Santamore and E. Timmermans, ibid . ,013619 (2008); S. De and I. B. Spielman, Appl. Phys. B , 527 (2014); J. J. Kinnunen and G. M. Bruun, Phys.Rev. A , 041605(R) (2015); D. Suchet, Z. Wu, F. Chevy,and G. M. Bruun, ibid . , 043643 (2017). B. DeSalvo, K. Patel, G. Cai, and C. Chin, Nature (Lon-don), , 61 (2019); H. Edri, B. Raz, N. Matzliah, N.Davidson, and R. Ozeri, Phys. Rev. Lett. , 163401(2020). A. Bulgac and P. Magierski, Nucl. Phys. A , 695 (2001),and references therein; Y. Yu et al. , Phys. Rev. Lett. ,412 (2000). A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, and V. F.Weisskopf, Phys. Rev. D , 3471 (1974); G. Neergaard andJ. Madsen, ibid . , 034005 (2000), and earlier referencestherein; K. Hattori, K. Itakura, S. Ozaki, and S. Yasui, ibid . , 065003 (2015). A. V. Shytov, D. A. Abanin, and L. S. Levitov, Phys. Rev.Lett. , 016806 (2009). S. LeBohec, J. Talbot, and E. G. Mishchenko, Phys. Rev.B , 045433 (2014); M. Agarwal and E. G. Mishchenko,Phys. Rev. B , 075411 (2017). A. Recati, J. N. Fuchs, C. S. Pe¸ca, and W. Zwerger, Phys.Rev. A , 023616 (2005); E. B. Kolomeisky, J. P. Straley,and M. Timmins, ibid . , 022104 (2008); A. Mering andM. Fleischhauer, ibid . , 011603(R) (2010); A. Flachi,Phys. Rev. Lett. , 060401 (2013); A. Flachi, M. Nitta,S. Takada, and R. Yoshii, ibid . , 031601 (2017); H. P. B¨uchler and G. Blatter, Phys. Rev. Lett. , 130404(2003); P. P. Orth, D. L. Bergman, and K. Le Hur, Phys.Rev. A , 023624 (2009). V. Gurarie, Phys. Rev. B , 085426 (2011); Z. Wang andS.-C. Zhang, ibid . , 165116 (2012). S. Ryu, A. Schnyder, A. Furusaki, and A. Ludwig, New J.Phys. , 065010 (2010); C.-K. Chiu, J. C. Y. Teo, A. P.Schnyder, and S. Ryu, Rev. Mod. Phys. C. D. Fosco and E. L. Losada, Phys. Rev. D , 025017(2008); S. Bellucci and A. A. Saharian, Phys. Rev. D ,105003 (2009). O. Kenneth and I. Klich, Phys. Rev. Lett. 97, 160401(2006); Phys. Rev. B , 014103 (2008); T. Emig, N. Gra-ham, R. L. Jaffe, and M. Kardar, Phys. Rev. Lett. ,170403 (2007); S. J. Rahi, T. Emig, N. Graham, R. L.Jaffe, and M. Kardar, Phys. Rev. D , 085021 (2009). Q.-D. Jiang and F. Wilczek, Phys. Rev. B , 125403(2019). J. M. Byers, M. E. Flatt´e, and D. J. Scalapino, Phys. Rev.Lett. , 3363 (1993); M. I. Salkola, A. V. Balatsky, andD. J. Scalapino, ibid . , 1841 (1996); W. Ziegler, D. Poil-blanc, R. Preuss, W. Hanke, and D. J. Scalapino, Phys.Rev. B , 8704 (1996); C. Bena and S. A. Kivelson, ibid . , 125432 (2005). Due to the properties of the log function log(1 − x ) < x >
0, and log(1 − x ) > x < A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009). T. O. Wehling, S. Yuan, A. I. Lichtenstein, A. K. Geim,and M. I. Katsnelson, Phys. Rev. Lett. , 056802 (2010);T. O. Wehling, M. I. Katsnelson, and A. I. Lichtenstein,Chem. Phys. Lett. , 125 (2009). E. McCann and Vladimir I. Fal’ko, Phys. Rev. Lett. ,086805 (2006); J. Nilsson, A. H. Castro Neto, N. M. R.Peres, and F. Guinea, Phys. Rev. B , 214418 (2006);M. Koshino, T. Nakanishi, and T. Ando, ibid . , 205436(2010). M. S. Foster and A. W. W. Ludwig, Phys. Rev. B ,165108 (2008). J. Verbaarschot, Phys. Rev. Lett. , 2531 (1994); S. Yasuiand K. Sudoh, Phys. Rev. C , 015201 (2013); K. Hat- tori, K. Itakura, S. Ozaki, and S. Yasui, Phys. Rev. D ,065003 (2015). F. Ye, G. H. Ding, H. Zhai, and Z. B. Su, Europhys. Lett. , 47001 (2010); J.-J. Zhu, D.-X. Yao, S.-C. Zhang, andK. Chang, Phys. Rev. Lett. , 097201 (2011); A. A.Zyuzin and D. Loss, ibid. , 125443 (2014); D. K. Efimkin and V. Galitski, Phys. Rev. B , 115431(2014); H.-R. Chang, J. Zhou, S.-X. Wang, W.-Y. Shan,and D. Xiao, ibid. , 241103(R) (2015); V. Kaladzhyan,A. A. Zyuzin, and P. Simon, ibid. , 165302 (2019). S. Saremi, Phys. Rev. B , 184430 (2007); L. Brey, H.A. Fertig, and S. Das Sarma, Phys. Rev. Lett. , 116802(2007); E. Kogan, Phys. Rev. B , 115119 (2011). A. M. Black-Schaffer, Phys. Rev. B , 073409 (2010); ibid . , 205416 (2010); J. E. Bunder and H.-H. Lin, ibid . ,153414 (2009); P. D. Gorman, J. M. Duffy, M. S. Ferreira,and S. R. Power, ibid. , 085405 (2013); D. O. Oriekhovand V. P. Gusynin, ibid. , 235162 (2020). For example, we have shown that anomalous Casimir in-teractions can be induced by a chiral-symmetry-breakingmedium or a time-reversal-symmetry-breaking medium(see Q.-D. Jiang and F. Wilczek, Phys. Rev. B ,201104(R) (2019); ibid.99