Equivalence of Topological Insulators and Superconductors
EEquivalence of topological insulators and superconductors
Emilio Cobanera and Gerardo Ortiz Institute for Theoretical Physics, Utrecht University, 3584 CE Utrecht, The Netherlands ∗ Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
Systems of free fermions are classified by symmetry, space dimensionality, and topological prop-erties described by K-homology. Those systems belonging to different classes are inequivalent. Incontrast, we show that by taking a many-body/Fock space viewpoint it becomes possible to estab-lish equivalences of topological insulators and superconductors in terms of duality transformations.These mappings connect topologically inequivalent systems of fermions, jumping across entries inexistent classification tables, because of the phenomenon of symmetry transmutation by which asymmetry and its dual partner have identical algebraic properties but very different physical inter-pretations. To constrain our study to established classification tables, we define and characterizemathematically Gaussian dualities as dualities mapping free fermions to free fermions (and inter-acting to interacting). By introducing a large, flexible class of Gaussian dualities we show that anyinsulator is dual to a superconductor, and that fermionic edge modes are dual to Majorana edgemodes, that is, the Gaussian dualities of this paper preserve the bulk-boundary correspondence.Transmutation of relevant symmetries, particle number, translation, and time reversal is also inves-tigated in detail. As illustrative examples, we show the duality equivalence of the dimerized Peierlschain and the Majorana chain of Kitaev, and a two-dimensional Kekul´e-type topological insulator,including graphene as a special instance in coupling space, dual to a p-wave superconductor. SinceGaussian duality transformations are also valid for interacting systems we briefly discuss some suchapplications.
I. INTRODUCTION
In this paper we establish equivalences of topolog-ically non-trivial insulators and superconductors.
By means of duality transformations, we show that anyinsulator has a dual superconducting partner, and thepartners are either both topologically trivial or non-trivial. We will focus on non-interacting dual partners,since general classification schemes exist for free-fermionsystems.
As it turns out, the duality transformationsof this paper connect systems that are inequivalent fromthe point of view of these topological classifications. Thisis only possible because of the phenomenon of symmetrytransmutation , by which a duality transformation mapsa symmetry of one system with one physical interpreta-tion, say particle number or time reversal, to a symmetryof the dual system with a different interpretation.From an electromagnetic response viewpoint, insulat-ing and superconducting phases of electron systems aredramatically different. While the insulating phase ischaracterized by a vanishing current-carrying state atzero temperature, the superconducting phase supportsa supercurrent and displays a perfect diamagnetic re-sponse, the Meissner effect. Yet there is a basic sense inwhich both states of matter are equivalent, since manyof their defining properties stem from a common factor,that is, the existence of a gap in the bulk energy spec-trum of fermionic quasiparticles. The additional presenceof gapless, symmetry-protected, extended surface excita-tions defines operationally their topologically non-trivialcharacter. One of the objectives of topological band the-ory is to classify, based on a few preferred (discrete) sym-metries and space dimensionality, topologically distinctnon-interacting (single-particle) Hamiltonians and their concomitant gapless edge excitations.For systems without gauge symmetries, duality trans-formations are implemented by unitary mappings, and so they preserve symmetries; the symmetries of asystem are in one-to-one correspondence with the sym-metries of its dual partner. However, the physical in-terpretation of a symmetry and its dual image can bemarkedly different. Holographic symmetries constitutea most extreme example. For some pairs of dual part-ners, one of the systems displays boundary symmetries,mapped to global symmetries of its dual partner. In thiscase we call the boundary symmetry holographic. Thisphenomenon is remarkable because a symmetry that isformally lost in the thermodynamic limit, the holographicsymmetry, is mapped by duality onto a symmetry thatmay become spontaneously broken in that limit. Becauseof this, not uncommon, example of symmetry transmu-tation, it is conceivable that a duality may map a parti-cle conserving system to a non-conserving one, simply bymapping the U (1) symmetry of particle number to a dual U (1) symmetry that does not have that interpretation.While these arguments are encouraging in the searchfor equivalences of insulators and superconductors,there are at least two other obstacles besides parti-cle (non)conservation. First, in general, dualities forfermions will not preserve the quadratic (or Gaussian)character of a model system, often mapping free-fermionsystems to interacting ones. Second, for topologicallynon-trivial systems, even if one were to find dualitiesmatching non-interacting dual partners, there is in gen-eral no reason to expect that these dualities shouldalso preserve the locality properties of the quasiparticlemodes. For example, at zero energy, modes localized ata boundary may also be interpreted as boundary sym- a r X i v : . [ c ond - m a t . m e s - h a ll ] J un metries. Extrapolating from the experience with holo-graphic symmetries, one would expect these modes tobecome delocalized after a duality transformation.As it turns out, both obstacles may be overcome, and,as a consequence, there is no fundamental obstructionto the construction of equivalences of topological insula-tors and superconductors in terms of dualities. SectionII introduces the special class of duality maps that es-tablishes those equivalences. The starting point is thecharacterization of duality transformations that preservethe quadratic fermionic nature of a given model system;we will call this transformation Gaussian duality . Nextwe will construct a large class of such dualities in anynumber of spatial dimensions, in order to create a toolkitfor generating a topological superconductor from anygiven topological insulator in a systematic fashion. Inother words, given a topological insulator one can alwaysfind at least one dual topological superconductor asso-ciated to it. This process is, of course, reversible sincethe duality transformation is an isometry. Hence our re-sults strongly suggest that there may exist equivalencesof topological insulators and superconductors across allentries of the topological classification table, at least forconstant space dimension. Dimensional reduction by du-alities is possible, but we will not obtain any Gaussianinstance of this phenomenon in this paper. Section IIends with the fundamental concept of symmetry trans-mutation as applied to fermion parity, translation andtime-reversal symmetries. In particular, we will find aquantitative connection between changes in translationsymmetry and breaking of particle conservation.Particular and emblematic examples include the proofthat the insulating dimerized Peierls and supercon-ducting Kitaev (at vanishing chemical potential) chains are dual partners, and the equivalence of grapheneto a popular example of a weak topological super-conductor in two spatial dimensions. It is in SectionIII that we present these two prototypical equivalences.These dual partners do not simply resemble each other,but are isospectral from a many-body standpoint, for fi-nite lattices and various boundary conditions. No doubt,this fact seems odd at first sight, since the spinfull Peierlschain for example partially breaks translation symmetrybut not (the standard) time reversal or particle conser-vation, while its dual partner, Kitaev’s Majorana chain,breaks time reversal and particle conservation but nottranslation symmetry. (In Appendix A we derive the su-perconducting dual of the m -merized Peierls chain, anddiscuss the differences between m odd and even.) Simi-larly, graphene displays the symmetry of the honeycomblattice while it dual superconducting partner sits on asquare lattice. Remarkably, the Gaussian duality allowsus to qualitatively understand the difference between zig-zag and armchair terminations in graphene. The expla-nation to all these seemingly paradoxical observations issymmetry transmutation. In Appendix B we describe asimple Gaussian duality mapping a ( s -wave) BCS super-conductor to an insulator in any number of dimensions. Another important issue addressed in Section III isthe locality character of our Gaussian dualities, i.e., theproblem of showing that localized zero-energy modesare mapped to dual zero-modes that are also local-ized. It is remarkable to have the possibility to gener-ate localization-preserving Gaussian dualities. In otherwords, there are no holographic symmetries associated tothe Gaussian dualities of this paper: global symmetriesmap to global symmetries, and the localization propertiesof energy modes are also preserved, though edge modesmay be shuffled among boundaries. In particular, thezig-zag boundary of graphene is exactly dual to the “Ki-taev edge” of Refs. 18 and 19. We show how topologicaldefects and edge states map, and also how the nature ofthose excitations transmutes from (canonical) fermionicto Majorana character by duality. Interestingly, we ana-lytically construct exact (as opposed to asymptotic) zero-energy modes for any finite length Kitaev wire when thelength is an odd number of lattice constants.An interesting outcome of our investigation is furtherconfirmation that non-trivial topological quantum orderis a property of a manifold of states interpreted relativeto a given language (a set of preferred observables ); andnot a property of the energy spectrum alone and someHamiltonian singling out those states as energy eigen-states. We also investigate in Section III the interplaybetween dualities and topological invariants of the single-particle Hamiltonian. Indeed, our equivalences are dual-ity mappings and hence necessarily isospectral. How-ever, some of these duality mappings connect systemswith ground states characterized by different topologicalquantum numbers, thus belonging to different topolog-ical classes. For instance, Kitaev wire model belongsto the class D of the Dyson-Altland-Zirnbauer tenfold-way classification, while its dual, the dimerized (spin-less) Peierls chain, belongs to the class AIII. In the past,we have studied dualities mapping systems with topo-logically quantum-ordered ground-state manifolds to sys-tems characterized by local (Landau) orders. In order to take advantage of duality transformations,it is crucial to recognize that the many-body, and not thesingle-particle, representation of the system is the rele-vant one. There is absolutely no doubt that, from a com-putational standpoint, the single-particle representation(e.g., the Bogoliubov-de Gennes equations) is the appro-priate methodology to adopt in the non-interacting ormean-field case. Computationally, it reduces a problemof exponential complexity into one of polynomial com-plexity, thus allowing diagonalization of quite large sys-tem sizes. However, care must be exercised at the mo-ment of analyzing properties like particle conservationthat involve the whole many-body system. In partic-ular, topological classification schemes and counting ofmany-body zero-modes relate directly to the many-bodyground-state manifold. These cautionary remarks areentirely appropriate since Gaussian dualities connectingtopologically non-trivial dual partners often seem at oddswith one form or another of standard wisdom. However,they are entirely natural if one adopts the many-body(Fock-space) language of second quantization, and notthe vector bundle analysis of single-particle Bogoliubov-de Gennes Hamiltonian matrices.Technically, Majorana operators, defined (up to nor-malization) as the real and imaginary parts of the canon-ical fermionic field, generate a complex Clifford alge-bra naturally represented in Fock space, and our Gaus-sian dualities are characterized most naturally as isomor-phisms of these Clifford algebras. The effect in single-particle (mode) space, where a different, exponentiallysmaller Clifford algebra emerges is induced a poste-riori . Crucially, it follows that our Gaussian dualitiescan also be used for investigating interacting many-bodysystems (see for example Section III A 3). Several dif-ferent mathematical simplifications arise when Gaussiandualities are investigated in terms of Majorana opera-tors. These simplify not only the search for equivalences,but also the analysis of symmetries, topological invari-ants and their transformation, and most importantly themapping of boundary excitations.At this point it becomes natural to ask about the ex-tension of our work to bosons, since it is clear that thenotion of Gaussian duality applies to canonical bosonsjust as well. However, the real and imaginary parts ofthe bosonic field satisfy the Heisenberg commutation re-lation, and so the theory of Gaussian dualities for bosonsis bound to be markedly different from that of fermions.Due to this crucial technical difference, we defer the sys-tematic study of bosonic Gaussian dualities to future re-search. Nonetheless, we would still like to illustrate ex-plicitly the point that symmetry transmutation is alsooperative in bosonic systems. Hence, in Appendix C wedescribe a duality mapping of a bosonic Mott insulatorto a quartet superconductor. We also comment brieflyon the relevance of this example for cold atoms.Section IV concludes with a summary and outlook.
II. GAUSSIAN DUALITIES
In statistical mechanics, duality mappings are unitarytransformations that respect the locality structure of themany-body Hamiltonian or transfer matrix.
In whatfollows we will set up the foundations to establish equiv-alences via dualities. Particularly, we will character-ize operator maps relating Hermitian quadratic forms offermions, i.e., Gaussian dualities, including the connec-tion between such many-body dualities and the associ-ated transformation of the single-particle Hamiltonian.Next we will introduce general techniques to decomposea very large class of free fermion models into sums of com-muting Hamiltonians, and finally we will use this tech-nique to construct a general class of Gaussian dualities.
A. What constitutes a Gaussian duality ?
We are interested in establishing the conditions underwhich a duality transformation becomes Gaussian. Letus focus for simplicity on systems of free fermions de-fined on a lattice. Then, the most general free fermionHamiltonian is of the form H = L (cid:88) i,j =1 (cid:104) K ij c † i c j + 12 ∆ ij c † i c † j + 12 ∆ ∗ ij c j c i (cid:105) , (1)with one-body and pairing interaction matrices K † = K, ∆ T = − ∆ , (2)where † is the adjoint, T the transpose, and ∗ complexconjugation of a matrix. The creation (annihilation) op-erator of a fermion c † j ( c j ) in the single-particle orbital φ j , satisfying { c i , c † j } = δ ij , is labelled by the genericsubindex j encoding arbitrary quantum numbers, includ-ing position, spin, orbital/band, angular momentum, etc.The total number of single-particle orbitals is L .Equivalently, one can re-write H in Nambu form H = 12 α † h BdG α + 12 Tr K, (3)where the column vector of fermion operators is given by α = (cid:18) cc † (cid:19) , with α j = c j , α L + j = c † j , j = 1 , · · · , L, (4)and the Bogoliubov-de Gennes single-particle Hamilto-nian (2 L × L matrix) h BdG = (cid:18) K ∆ − ∆ ∗ − K ∗ (cid:19) = (5) i ⊗ (cid:61) ( K ) + i τ x ⊗ (cid:61) (∆) + i τ y ⊗ (cid:60) (∆) + τ z ⊗ (cid:60) ( K ) , where τ ν , ν = x, y, z , are Pauli matrices, and (cid:60) ( · )( (cid:61) ( · ))denotes the real (imaginary) part of the matrix. No mat-ter what the specific matrices K and ∆ are, the single-particle Hamiltonian h BdG always anticommutes with theantiunitary (particle-hole) operator C = K τ x ⊗ , C = , (6)i.e., { h BdG , C} = 0, where K denotes complex conjuga-tion. That means that the single particle energy spec-trum is antisymmetric with respect to its zero value, i.e.,a particle-hole symmetric spectrum. By contrast, a chi-ral symmetry U chiral is a unitary transformation that anti-commutes with h BdG . For example, if (cid:61) ( K ) = 0 = (cid:61) (∆),then U chiral = τ x ⊗ .Suppose now that the unitary transformation U d im-plements a duality transformation, H D = U d H U † d , (7)meaning that it transforms a local non-interacting Hamil-tonian H into another, dual H D , that preserves the prop-erty of being also local. The map, however, could gen-erate fermionic density-density interactions for instance.What are the general conditions under which H D is alsoan Hermitian quadratic form of fermions?To answer this question we will recast Hamiltonian H of Eq. (1) in terms of Majorana operators γ j − = c j + c † j , i γ j = c j − c † j , (8)such that γ r = 1, r = 1 , · · · , L . (The notation a j = c j + c † j , i b j = c j − c † j . (9)will be favored in later sections). Then H = i L (cid:88) r,s =1 h rs γ r γ s + 12 Tr K (10)becomes a quadratic form of Majorana fermions, with a2 L × L matrix h that is real and antisymmetric.We can now investigate the dual Hamiltonian. Re-member that we want the duality map to be Gaussian,i.e., H D should also be a quadratic form of Majoranafermions. A naive first, and trivial, attempt would be tokeep the localization properties identical , i.e. H D = i L (cid:88) r,s =1 h rs γ Dr γ Ds + 12 Tr K (11)where the dual operators are related to the originals as γ Dr = U d γ r U † d , (12)and γ Dr is a Majorana fermion operator. This extremelocal map, although Gaussian, is very restrictive and willnot allow us to establish interesting equivalences betweeninsulators and superconductors. We would like to relaxthe extreme locality constraint and allow for changes inthe range of the dual matrix. In other words, we wouldlike to realize a more general Gaussian duality H D = i L (cid:88) r,s =1 h Drs ˜ γ r ˜ γ s + 12 Tr K, (13)for some new Majorana operator ˜ γ r significantly differentfrom the dual Majorana γ Dr and the original one γ r .The argument above suggests setting up the relation U d γ r U † d = L (cid:88) s =1 O d sr ˜ γ s , (14)so that the matrix O d may be computed explicitly as O d sr = 12 L tr (˜ γ s U d γ r U † d ) . (15)Therefore, the duality map U d is Gaussian if, and onlyif, the matrix O d is invertible, in which case it is alsoorthogonal. In the absence of a more educated choice,one may always set ˜ γ s = γ s in Eq. (14). The association U d (cid:55)→ O d shows that Gaussian duali-ties, though many-body in nature, also induce a posteri-ori a duality of the single-particle Hamiltonian h by therelation h Drs = L (cid:88) r (cid:48) ,s (cid:48) =1 h r (cid:48) s (cid:48) O d rr (cid:48) O d ss (cid:48) . (16)How much of the locality of the original system one pre-serves in the dual model will depend on the range of thematrix O d .Finally, let us contrast Gaussian dualities to othertype of dualities that do not preserve the quadraticfermionic nature of the original theory. Consider the non-interacting Hamiltonian H = i L − (cid:88) j =1 [ t x γ j γ j +1 − t y γ j − γ j +2 ] , (17)which can be also described as a spin-1/2 Hamiltonian, H = − L − (cid:88) j =1 [ t x σ xj σ xj +1 + t y σ yj σ yj +1 ] , (18)after the Jordan-Wigner map of Majorana operators γ j − = σ xj j − (cid:89) l =1 σ zl , γ j = σ yj j − (cid:89) l =1 σ zl , (19)in terms of Pauli matrices σ νj , ν = x, y, z . A simple localrotation around the spin y axis, σ xj (cid:55)→ σ zj , σ zj (cid:55)→ − σ xj ( j = 1 , · · · , L ) , (20)induces a non-trivial change in the dual fermionic Hamil-tonian. Although local, it is an interacting Hamiltonian H D = L − (cid:88) j =1 [ t x γ j − γ j γ j +1 γ j +2 − i t y γ j − γ j +2 ] . (21)Therefore, the matrix O d of Eq. (14) should fail to beinvertible, and one can check that this is indeed the case.The dual Hamiltonian of Eq. (21) has an interestingphysical interpretation. It describes the competition be-tween a p -wave superconducting chain in its topologicalphase and a density-density interaction. In the limit inwhich t y vanishes, its ground state is number conserv-ing, a Mott insulating state, otherwise its ground state issuperconducting. For t x = 0, H D has two exact zero-energy modes, γ and γ L − . The evolution of thesemodes with t x may be computed exactly by exploitingthe duality transformation connecting H D to the free-fermion Hamiltonian H . B. Decoupling transformations
The results of the previous section, especially the ex-ample at the end of the section, show that generic du-alities do not preserve the non-interacting character of atheory. Hence, to systematically establish equivalences oftopological superconductors and insulators it is necessaryto determine all possible Gaussian dualities. Recall thatthe key difficulty in searching for dualities is to identifyunitary transformations that respect the locality struc-ture of the Hamiltonian, meaning that Eq. (14) is notthe full answer to our problem. We still need to addressthe issue of locality for the specific purpose of relating topological insulators to topological superconductors .Let us denote by r the sites of a lattice Λ, i.e., r ∈ Λ ,defined in arbitrary space dimensions. A generic (second-quantized) electron system where the number of electrons N is conserved is described by the Hamiltonian H = − (cid:88) r , r (cid:48) ,σ (cid:104) t r , r (cid:48) c † r ,σ c r (cid:48) ,σ + t r (cid:48) , r c † r (cid:48) ,σ c r ,σ (cid:105) , (22)where c † r ,σ represents a canonical fermion creation op-erator at site r and spin σ = ↑ , ↓ . The hopping am-plitude t r , r (cid:48) is related to t r (cid:48) , r by complex conjugation, t r (cid:48) , r = t ∗ r , r (cid:48) .Generically, the Hamiltonian above displays a brokentime reversal symmetry unless the hopping amplitudesare purely real or purely imaginary. For purely imag-inary amplitudes, an internal decoupling occurs in thesystem that splits the particle conserving Hamiltonian H into four identical, independent and decoupled super-conductors. The proof of this assertion relies on rewritingthe particle conserving Hamiltonian H above in terms ofMajorana fermions a r ,σ and b r ,σ , such that a r ,σ = c r ,σ + c † r ,σ , i b r ,σ = c r ,σ − c † r ,σ , (23)with the result H = − (cid:88) r , r (cid:48) ,σ (cid:20)(cid:18) t r , r (cid:48) − t r (cid:48) , r (cid:19) ( a r ,σ a r (cid:48) ,σ + b r ,σ b r (cid:48) ,σ )+ i (cid:18) t r , r (cid:48) + t r (cid:48) , r (cid:19) ( a r ,σ b r (cid:48) ,σ − b r ,σ a r (cid:48) ,σ ) (cid:21) . (24)Hence, if t r , r (cid:48) is purely imaginary, H = (cid:80) σ ( ˜ H ,σ + ˜ H ,σ )where the particle non-conserving Hamiltonians˜ H ,σ = − (cid:88) r , r (cid:48) t r , r (cid:48) a r ,σ a r (cid:48) ,σ , (25)˜ H ,σ = − (cid:88) r , r (cid:48) t r , r (cid:48) b r ,σ b r (cid:48) ,σ , (26)commute, [ ˜ H ,σ , ˜ H ,σ (cid:48) ] = 0.The four decoupled superconductors˜ H , ↑ , ˜ H , ↓ , ˜ H , ↑ , ˜ H , ↓ are isospectral: Any one of themcan be mapped into any other one by a local unitarytransformation. On one hand, ˜ H , ↑ ( ˜ H , ↑ ) is mapped to˜ H , ↓ ( ˜ H , ↓ ) by a rotation in spin space. On the otherhand, the unitary transformation U σ = (cid:81) r (cid:16) + a r ,σ b r ,σ √ (cid:17) maps ˜ H ,σ to ˜ H ,σ . Hence it is possible to unequivocallyassociate H with a new Hamiltonian H reduced H (cid:55)→ H reduced = − (cid:88) r , r (cid:48) t r , r (cid:48) γ r γ r (cid:48) , (27) of spinless Majorana fermions γ r that includes only onefourth of the original number of fermionic degrees of free-dom, and represents any of the four Hamiltonians ob-tained by decoupling H .The equivalence of Eq. (27) states that the spectrumof H can be reconstructed from that of H reduced . In par-ticular, the zero-energy modes of the particle conservingHamiltonian H are explained by the zero-energy modes ofthe spinless superconductor H reduced . If H reduced has pre-cisely s zero modes, then H has precisely s zero modes.It is also possible to add spin interaction terms andretain some level of decoupling. For example, spin-orbitterms of the Rashba, Dimmock, or Dresselhaus type arelinear in momentum and hence purely imaginary. Thusthey couple ˜ H , ↑ to ˜ H , ↓ (and ˜ H , ↑ to ˜ H , ↓ ), but theydo not couple a to b Majoranas. In the presence ofthese types of spin terms, the decoupling transforma-tion decomposes the particle conserving system into tworotationally-invariant superconductors with non-trivialspin dynamics.Many of the Majorana lattice models investigated inthe literature can be interpreted as the H reduced associ-ated to a particle-conserving Hamiltonian. The Majo-rana chain of Kitaev is obtained in one dimension froma chain with only nearest-neighbor hoppings. In twodimensions it is possible to obtain Kitaev’s Honeycombmodel (see Ref. 28 for the immediate connection) as thereduced Hamiltonian associated to the simplest modelof graphene (see below), and the triangular Majoranalattice from a corresponding particle-conserving modelon the triangular lattice. Decoupling on the square lat-tice obtains variations of Majorana arrays investigati-gated in Ref. 18.An apparently less general but more often useful ver-sion of the decoupling transformation exits for purely realhopping amplitudes on a bipartite lattices Λ = A ∪ B ,with generic lattice sites r ∈ Λ. Let us denote by x ∈ A and y ∈ B the sites of each sublattice. The generic bi-partite Hamiltonian H = − (cid:88) x , y ,σ (cid:2) t x , y c † x ,σ c y ,σ + t y , x c † y ,σ c x ,σ (cid:3) , (28)allows only for hopping from sublattice A to B or vicev-ersa. If t x , y is purely real-valued, that is, t x , y = t y , x ,then H = (cid:88) σ [ H ,σ − H ,σ ] , (29)where the superconducting Hamiltonians H ,σ = − i (cid:88) x , y t x , y a x ,σ b y ,σ , (30) H ,σ = − i (cid:88) x , y t x , y b x ,σ a y ,σ , (31)commute, [ H ,σ , H ,σ (cid:48) ] = 0. The spectral equivalence of H ,σ and H ,σ is established by the unitary transforma-tion U σ = (cid:89) x , y (cid:18) + b x ,σ a x ,σ √ (cid:19) (cid:18) + a y ,σ b y ,σ √ (cid:19) , (32)that maps H ,σ ↔ H ,σ . An interesting corollary to Eqs.(29) and (32) is that the unitary transformation C = U ↑ U ↓ anticommutes with the Hamiltonian, C H = − H C ,and so it defines a chiral symmetry in the sense that thespectrum of H is particle-hole symmetric, i.e., for eachpositive eigenvalue E α there exists a negative − E α .Just as before it is possible to associate a reducedHamiltonian to H .Consider, as an example, the chain of spinless fermions H = − M (cid:88) j =1 (cid:104) t j ( c † j c j +1 + c † j +1 c j ) + (cid:15) j ( n j − / (cid:105) , (33)with quenched disorder in the real-valued hopping am-plitudes t j and on-site atomic energy (cid:15) j . The chain has L = 2 M lattice sites and periodic boundary conditionsare assumed ( c † L +1 = c † ). Since the lattice is bipartite, H is the difference of two identical, independent super-conductors, coupled by the on-site atomic energies. Letus associate a pair of Majorana fermions a j , b j to eachsite j , as in Eq. (23), and rewrite the Hamiltonian abovein terms of Majorana degrees of freedom H = − i M (cid:88) j =1 [ t j ( a j +1 b j − b j +1 a j ) + (cid:15) j a j b j ]= H − H + H (cid:15) , (34)with commuting Hamiltonians H = − i M (cid:88) j =1 [ t j − a j b j − − t j b j +1 a j ] ,H = − i M (cid:88) j =1 [ t j − b j a j − − t j a j +1 b j ] , (35)and H (cid:15) = − ( i / M (cid:88) j =1 [ (cid:15) j − a j − b j − + (cid:15) j a j b j ] . (36) C. A class of Gaussian duality transformations
We are now ready to introduce a large class of Gaussianduality transformations. Consider the cases for which thesublattices A and B of previous section are equivalent,meaning that there is a shortest, typically non-uniquetranslation δ ( δ ) mapping sublattice A ( B ) to sublat-tice B ( A ). In set notation, A + δ = B, B + δ = A. (37)For the purpose of the duality transformation that we areabout to introduce it is often convenient to choose δ , δ to be as parallel and short as possible. This conditionguarantees that the range of the hoppings in the dualHamiltonian deviates as little as possible from that of theoriginal Hamiltonian. A hypercubic lattice is simplest inthat one may choose δ = δ .The mapping a y ,σ → b y + δ ,σ , a x ,σ → a x ,σ ,b y ,σ → b y ,σ , b x ,σ → − a x + δ ,σ , (38)induces a unitary transformation that leaves H ,σ un-changed and transforms H ,σ as H ,σ → H D ,σ = − i (cid:88) x , y t y − δ , x − δ b x ,σ a y ,σ . (39)In rearranging the sum over sites, we have assumed pe-riodic boundary conditions or that the system is infinite.The dual superconducting Hamiltonian, H D = (cid:88) σ (cid:2) H ,σ − H D ,σ (cid:3) = (40) − i (cid:88) x , y ,σ [ t x , y a x ,σ b y ,σ − t y − δ , x − δ b x ,σ a y ,σ ] , can be rewritten in terms of creation and annihilationoperators, H D = − (cid:88) x , y ,σ (cid:104) t av x , y ( c † x ,σ c y ,σ + c † y ,σ c x ,σ ) +∆ x , y ( c † y ,σ c † x ,σ + c x ,σ c y ,σ ) (cid:105) , (41)where t av x , y = t x , y + t y − δ , x − δ , ∆ x , y = t x , y − t y − δ , x − δ . Even though H + H (cid:15) is roughly as general as possiblefor a band electronic system of independent fermions, H D + H D(cid:15) remains a superconductor at vanishing chem-ical potential µ . It is possible to include spin terms in H and still obtain a dual superconductor featuring only lo-cal interactions. Just as the duality breaks particle con-servation in general, we expect it to modify rotationalproperties since it has a highly non-trivial action on theoperators of total spin. Since, and when, spin does notplay any decisive role in the studied physical phenomenonwe will drop it from the discussion in order to avoid con-fusing notation and obscure explanations.It is now straightforward to apply the general dual-ity transformation of bipartite models to the disorderedchain of the previous section. For this one-dimensionalsystem, the mapping defined in Eqs. (38) reduces to a j − → b j , a j → a j b j − → b j − , b j → − a j +1 , (42)always identifying the index L + 1 with 1, and 0 with L −
1. Thus, while H = H D remains invariant, H transforms as H D = − i M (cid:88) j =1 [ t j − b j a j − − t j − a j +1 b j ] (43)(the case M = 1 is special in that the duality map keeps H D also invariant). The on-site atomic energy termtransforms like H D(cid:15) = − i M (cid:88) j =1 [ (cid:15) j − b j b j − − (cid:15) j a j a j +1 ] (44)= i M (cid:88) j =1 (cid:104) (cid:15) j ( c † j c j +1 − c † j +1 c j ) + ( − j (cid:15) j ( c † j c † j +1 − c j +1 c j ) (cid:105) . Combining all of these results, we obtain the dual super-conductor H D = H − H D + H D(cid:15) H D = − M (cid:88) j =1 (cid:104) (cid:18) t j + t j − − i (cid:15) j (cid:19) c † j c j +1 + (45)( − j (cid:18) t j − − t j − i (cid:15) j (cid:19) c † j c † j +1 + H . c . (cid:105) .
1. Symmetry transmutation: particle number and fermionicparity
The duality transformation Eqs. (38), breaks parti-cle conservation in general because the particle number(charge) operatorˆ N = (cid:88) r ,σ [ n r ,σ − /
2] = i (cid:88) r ,σ a r ,σ b r ,σ . (46)associated to, and a symmetry of, H is drastically mod-ified by the duality. Since a x ,σ b x ,σ → − a x ,σ a x + δ ,σ ,a y ,σ b y ,σ → − b y ,σ b y + δ ,σ , (47)the duality transformation maps ˆ N to a symmetry ˆ N D of H D that does not have the interpretation of a chargeoperator,ˆ N D = − i (cid:88) σ (cid:34)(cid:88) x a x ,σ a x + δ ,σ + (cid:88) y b y ,σ b y + δ ,σ (cid:35) , (48)while it is still true that [ ˆ N D , H D ] = 0.There is, however, a quantum number important fromthe point of view of superconductivity that is almost pre-served by duality: fermionic parity. The operator offermionic parity( − F = e i π (cid:80) r ,σ n r ,σ = (cid:89) r ,σ ( − i a r ,σ b r ,σ ) (49)measures the parity of the total number of fermions. TheBCS mean field approximation breaks the symmetry ofparticle conservation down to conservation of fermionicparity. The duality transformation maps( − F → (50) (cid:89) x ,σ ( i a x ,σ a x + δ ,σ ) (cid:89) y ,σ ( i b y ,σ b y + δ ,σ ) = ( − σ ( − F , where ( − σ is the sign accumulated after permutationsof the Majorana fermions to establish the original order( − F .Incidentally, Eq. (47) shows that the duality H → H D is more general in scope than the decoupling transforma-tion that motivated it. For example, adding an on-siteenergy term H (cid:15) = − (cid:88) r ,σ (cid:15) r ( n r ,σ − /
2) (51)( r ∈ A ∪ B ) to H , couples the reduced superconductors H ,σ and H ,σ . It transforms as H D(cid:15) = i (cid:88) x ,σ (cid:15) x a x ,σ a x + δ ,σ + i (cid:88) y ,σ (cid:15) y b y ,σ b y + δ ,σ . (52)Hence the effect of the on-site atomic energy term (cid:15) r is torenormalize (by a purely imaginary amount) the hoppingand pairing amplitudes of the dual superconductor in thedirections δ , and δ .
2. Translation Symmetry
For the Gaussian dualities of Eq. (38), the non-conservation of particle number for the dual partner H D is precisely related to (partial) breaking of translationsymmetry for H , since the pairing potential ∆ x , y van-ishes if t y − δ , x − δ = t x , y (and H D = H in this case).What is less obvious is that the translation symmetry of H D may be be higher than that of H , in which case weare enlarging one group of symmetries (translations) atthe expense of breaking another symmetry, particle con-servation. This is explained by the transmutation of thetranslation operation under duality.Let us focus for simplicity on a closed chain of spin-less fermions. The extension to more general settings isstraightforward but notationally cumbersome. The mapof Eq. (42) leads to c j − (cid:55)→ c D j − = 12 ( b j + i b j − ) ,c j (cid:55)→ c D j = 12 ( a j − i a j +1 ) , (53)or, more explicitly, c D j − = 12 ( c j − − i c j ) −
12 ( c † j − − i c † j ) ,c D j = 12 ( c j − i c j +1 ) + 12 ( c † j − i c † j +1 ) . (54)These expressions show already transmutation of thetranslation operation, let us call it ˆ T . On one hand,ˆ T c j ˆ T † = c j +1 ( j = j + L ) , (55)and consequently ( ˆ T D = U d ˆ T U † d ),ˆ T D c Dj ( ˆ T D ) † = c Dj +1 ( j = j + L ) . (56)On the other hand, ˆ T c Dj ˆ T † (cid:54) = c Dj +1 , (57)and so ˆ T (cid:54) = ˆ T D . (58)This is the point to notice. If ˆ T happens to be a sym-metry of H , then ˆ T D is necessarily a symmetry of H D .However, ˆ T D cannot possibly have the interpretation of atranslation by one site, since that physical interpretationcontinues to be attached to ˆ T ! (The action of ˆ T D onthe c j may be computed by inverting Eqs. (54).) Notice,however, by the same reasoning thatˆ T = e i α d ( ˆ T D ) , (59)where the (possibly trivial) phase on the right-hand sideis determined by the actual duality transformation.The concrete significance of this result will become ap-parent in the next section when we investigate the equiv-alence of the dimerized Peierls chain and the Majoranachain of Kitaev. What happens in that case actually isthe following. The dimerized Peirls chain commutes withˆ T and a very non-evident symmetry U † d ˆ T U d (not to beconfused with ˆ T D = U d ˆ T U † d ). As a consequence, its dualpartner (the Majorana chain) commutes with ˆ T . This il-lustrates how Gaussian dualities may increase translationsymmetry at the expense of breaking (transmuting) othersymmetries, e.g., particle conservation.It is revealing to to rewrite the Gaussian duality of Eq.(42) as a map of fermions in crystal momentum space.The even-odd structure of the duality mapping evincedby Eqs. (54) for example suggests that we should take aunit cell with two sites. To keep the notation simple, wewill assume that L = 2 M , with M odd. Then c j − = M − (cid:88) l = − M − e − i kj √ M ˆ c ,k , c j = M − (cid:88) l = − M − e − i kj √ M ˆ c ,k , (60)where k = 2 πl/M . Let us emphasize that we are notmaking any assumption about the symmetries of any par-ticular Hamiltonian. We are just going to recast our dual-ity transformation in a new light. With these definitions,the dual fermions in momentum space areˆ c D ,k = 12 (ˆ c ,k − i ˆ c ,k ) −
12 (ˆ c † , − k − i ˆ c † , − k ) , (61)ˆ c D ,k = 12 (ˆ c ,k − i e − i k ˆ c ,k ) + 12 (ˆ c † , − k − i e − i k ˆ c † , − k ) . The key point is that the dual fermions of momentum k are combinations of the original fermions of momentum k and − k . It follows that the induced single-particle dual-ity O d is not block diagonal with respect to momentum.
3. Time-reversal Symmetry
The standard antiunitary operation T of motion re-versal may be specified by its action on the creation andannihilation operators, T c j, ↑ T − = c j, ↓ , T c † j, ↑ T − = c † j, ↓ , T c j, ↓ T − = − c j, ↑ , T c † j, ↓ T − = − c † j, ↑ , (62)with the important consequence that T = ( − ˆ N . (63)The duality transformation Eq. (38) is spin diagonal.Hence the dual fermions c D j − ,σ = 12 ( c j − ,σ − i c j,σ ) −
12 ( c † j − ,σ − i c † j,σ ) ,c D j,σ = 12 ( c j,σ − i c j +1 ,σ ) + 12 ( c † j,σ − i c † j +1 ,σ ) , (64)are just as before, except for the additional spin label.By construction, the dual antiunitary operation T D = U d T U † d , with ( T D ) = ( − ˆ N D , (65)acts as standard time-reversal on the dual fermions (fora discussion of ˆ N D , see Section II C 1). One may checkthat T (cid:54) = T D , (66)and so there is transmutation of time-reversal symmetry.To put this result in perspective, suppose that both H and H D commute with T . As we will see, this is the casefor example in polyacetylene and its dual superconduct-ing partner (class DIII). Then, since [ H D , T D ] = 0, wehave uncovered a (unitary) symmetry T D T of H D .An example of transmutation of time-reversal forcanonical bosons (phonons) can be found in Ref. 11,page 730. III. EQUIVALENCES OF TOPOLOGICALINSULATORS AND SUPERCONDUCTORS
Can Gaussian dualities in general, and in particular,the ones of this paper, establish equivalences betweentopological insulators and topological superconductors?The discussion of the previous section shows that thereis no obstruction for this to be the case: many-body du-alities can jump across entries in single-particle classifi-cation schemes simply by transmuting key symmetries.But symmetry transmutation is a necessary, not a suf-ficient condition. For example, the duality of Eq. (42)maps the clearly trivial insulator, H = − (cid:15) M (cid:88) j =1 ( n j − /
2) (67)to the equally trivial superconductor, H D = i (cid:15) M (cid:88) j =1 (cid:104) ( c † j c j +1 − c † j +1 c j ) (68)+( − j ( c † j c † j +1 − c j +1 c j ) (cid:105) , in spite of the symmetry rearrangements it causes.For the Gaussian dualities of the previous section inparticular, it is not hard to convince oneself that it mustbe the case that they map topologically (non)trivial sys-tems to equally non(trivial) dual partners. In this sectionwe will study some paradigmatic examples of non-trivialpartners. In one dimension, we find that the dimerizedPeirls chain and the Majorana chain of Kitaev are dualpartners, and we also investigate the mapping of topolog-ical defects under duality. The more general m − merizedPeierls chain is investigated in the Appendix A. In twodimensions, we study a topological insulator based on aKekul´e-like pattern of hopping matrix elements that in-cludes graphene as a special case. It dual partner is ap-wave superconductor. In the limit where the insulatorbecomes graphene (a semi-metal), the dual superconduc-tor reduces to a stack of Kitaev chains interconnected bypure kinetic hopping in the direction perpendicular to thechains. This superconducting realization of Dirac conesseems to be new in the literature. Appendix B shows anequivalence of a trivial BCS superconductor to a trivialinsulator in any number of space dimensions. A. The Peierls chain is dual to the Kitaev chain
In one dimension, the dimerized Peierls chain at half-filling, proposed by Su, Schrieffer and Heeger (SSH) tomodel polyacetylene , is the prototype of a topologi-cally non-trivial band insulator, while the Kitaev chainis the prototype of a topologically non-trivial supercon-ductor. In spite of their physical differences, the Kitaevand Peierls chains are isospectral in second quantization ,that is, as many-fermion systems. The reason is thatthere exists a Gaussian duality connecting both models.As we will see below, the mapping is different in detail forperiodic and open boundary conditions. The Gaussianduality for open boundary conditions is crucial to under-stand the way boundary excitations are related, i.e., howthe Majorana charge-neutral zero-energy edge modes ofthe Kitaev chain map into charged (canonical fermion)zero-modes of the Peierls chain.For simplicity, and pedagogical reasons, in the follow-ing we will consider the spinless case. (The original modelfor polyacetylene involves spin-1/2 electrons and thisfact is relevant for topological classification purposes butit is not from the standpoint of Gaussian dualities thatpreserve spin, such as the ones defined in this paper.)Since the unit cell of the dimerized Peierls chain consistsof two sites, the length (number of sites) L of the closedchain must be even, L = 2 M . There are only two in-dependent, periodically repeated hopping terms t and t t t t · · · j = t t t t M + 1 L = 2 M j = M + 1 L = 2 M FIG. 1. Peierls chain with periodic boundary conditions onthe left and its dual Kitaev chain superconductor on the right.Here, L = 2 M . t , see Fig. 1. It follows that the Peierls chain is a spe-cial case of the generic one-dimensional Hamiltonian ofEq. (33), with t j − = t , t j = t , (cid:15) j = 0 , j = 1 , · · · , L. (69)The dual of the Peirls chain, H D = − i L (cid:88) j =1 (cid:104) t a j +1 b j − t b j +1 a j (cid:105) (70)= − L (cid:88) j =1 (cid:104) t c † j c j +1 + ∆ c † j c † j +1 + H . c . (cid:105) , is obtained by the same specialization of Eq. (45), with t = t + t , ∆ = t − t . (71)The Hamiltonian H D is precisely the Majorana chain ofKitaev, at vanishing chemical potential µ . The interplayof symmetries is noteworthy. While the dimerized Peierlschain shows reduced translation symmetry and conserva-tion particle number, its dual the Kitaev chain has fulltranslation symmetry and particle-number conservationis broken.Next, we would like to comment on polyacetylene(class AII), that is the case with real electrons (spinfullfermions). In this case, we obtain two copies of a Peierlschain, one for each spin component. Time-reversal sym-metry is preserved and our Gaussian duality maps theSSH model to two copies of Kitaev’s chain (class DIII),one for each spin component.In Appendix A we analyze the superconducting equiv-alent of an m -merized Peierls chain with m ≥
3. Thereis a clear distinction between distortions with m evenand odd. In the latter case, the periodicity of the dualsuperconductor is doubled, and there are no zero-energymodes. For m ≥ m even there are zero-energy modes.0
1. Mapping of topological invariants
A main goal of any topological classification of matteris to divide all possible quantum states of matter intoequivalence classes. There is some degree of arbitrari-ness in the criteria used to define those classes. Once thecriteria is established, e.g., by symmetry/dimension andreduced K-homology of bundles, two states in the sameclass are connected through a continuous map whose in-verse is also continuous, i.e., a homeomorphism. To es-tablish a characterization of the class one uses topologicalinvariants, i.e., quantities that are preserved under thehomeomorphism. That a particle-conserving system suchas a Peierls insulator may be dual to a superconductorraises some conceptual issues for the topological classifi-cation of systems of free fermions. What is the relationbetween the topological invariants characterizing thesedifferent but dual states of matter? The fermionic parityof the ground state of a superconductor such as Kitaev’s,with L = 2 M even, constitutes a good quantum num-ber whose value depends on the boundary conditions.For periodic boundary conditions, the topologically non-trivial ground state of the Kitaev chain is non-degenerateand fermionic parity is odd, while it is even in the triv-ial phase. These facts hold independently of M = L/ M = sgn( µ + 2 t ) sgn( µ − t ) , (72)defined as the product of signs of Pfaffians of an anti-symmetric matrix at momenta 0 and - π . This quantityidentifies the topologically non-trivial phase as the onewith M = −
1. On the other hand, we have seen thatthe dimerized Peierls chain maps into a Kitaev’s chain at µ = 0. The Peierls chain is always in a topologically non-trivial insulating phase, as long as t (cid:54) = t . However, thefermion parity of the Peierls’ insulating non-degeneratemany-body ground state is given by ( − M , and thereforeit is defined by the parity of M , i.e., it can be odd or evendepending on M . It is instructive to express the Peierls’chain in terms of Majorana fermions and compute theMajorana number, now with a doubled unit cell, to real-ize that indeed M = ( − M . The point is that fermionparity is not the good topological quantum number tocharacterize the Peierls’ insulating phase despite the factthat it is exactly dual to Kitaev’s chain model.From the point of view of the many-body duality trans-formation, the mismatch is explained by the (very mild)transmutation of fermionic parity, Eq. (50).
2. Mapping of topological defects and boundary modes
Consider for simplicity a dimerized, spinless, chainwith periodic boundary conditions, L = 2 M with M ∈ odd, and two defects symmetrically located at positions j = 1 and j = M . This corresponds to the Hamiltonian H = − M − (cid:88) j =2 t j − mod ( c † j c j +1 + c † j +1 c j ) (73) − L − (cid:88) j = M +2 t j − ( M +1)( mod ( c † j c j +1 + c † j +1 c j ) − t ( c † c + c † L c + c † M c M +1 + c † M +1 c M +2 + H . c . ) , where the first two terms represent two identical dimer-ized Peierls chains each of length M −
1, and the lastterm represents the pair of defects, see Fig. 2. t t t t · · · j = t t t t M + 1 L = 2 M j = M + 1 L = 2 M t t t t O r d e r e d C h a i n O r d e r e d C h a i n O r d e r e d C h a i n t t t t FIG. 2. Peierls chain (periodic boundary conditions) with acouple of defects on the left and its dual Josephson junctionsof Kitaev chain superconductors on the right. Here, L = 2 M . Applied to the Hamiltonian (73), the duality trans-formation of the periodic chain, Eq. (42), produces twodual superconductors coupled by two particle-conservingsegments, H D = − M − (cid:88) j =2 ( t c † j c j +1 + ∆ c † j c † j +1 + H . c . ) (74) − L − (cid:88) j = M +2 ( t c † j c j +1 − ∆ c † j c † j +1 + H . c . ) − t ( c † c + c † M +1 c M +2 + H . c . ) . (75)Notice the change in sign of the superconducting orderparameter across the links. Because of this phase dif-ference, a Majorana zero mode is trapped at each weaklink.For an infinite chain with a single defect located atthe origin, one may use the same duality map, Eq. (42)in order to obtain a dual Majorana zero-mode localizedat the origin. The defect that famously traps fractionalcharge ± e/
2, per spin direction, in the Peierls chain isdual to a defect that traps a Majorana zero-mode!For open boundary conditions, it is necessary to takethe number of sites L = 2 M + 1 to be odd, see Fig. 3.1The duality transformation a j − → a L − j − , a j → a j b j − → b j − , b j → b L +1 − j , (76)leaves H = H D invariant, and transforms H as follows H D = − i M (cid:88) j =1 ( t j a L − j b L +1 − j − t j − b L +1 − j a L − j − ) , = − i M (cid:88) j =1 ( t L +1 − j a j − b j − t L − j b j a j +1 ) . (77) { t t t L = 2 M + 1 t t j = · · · t t t t t FIG. 3. Peierls chain with open boundary conditions on thetop and its dual Kitaev chain superconductor on the bottom.Here, L = 2 M + 1. In the dimerized case there are only two different al-ternating hopping terms which satisfy t j − = t L − j = t , t j = t L +1 − j = t , (78)with the end result that the dual total Hamiltonian rep-resents a spinless superconductor H D = − i L − (cid:88) j =1 ( t a j +1 b j − t b j +1 a j ) (79)= − L − (cid:88) j =1 (cid:16) t c † j c j +1 + ∆ c † j c † j +1 + H . c . (cid:17) , that is again the Kitaev chain Hamiltonian at vanishing, µ = 0, chemical potential.The Kitaev chain of this section (open boundary condi-tions, odd length, and vanishing chemical potential) hastwo exact zero-energy modes, one per boundary point.The chain is reflection symmetric with respect to the cen-tral site j = M + 1, and the many-body ground state istwo-fold degenerate. The zero-energy mode associatedto the left boundary can be computed from the set ofcommutators [ − i H D , a ] = t b , [ − i H D , a ] = t b + t b , ...[ − i H D , a L − ] = t b L − + t b L − , [ − i H D , a L ] = t b L − . (80) Let η = t /t , and assume without loss of generality | η | <
1. From these commutators it is possible to show thatthe combination γ D left = 1 N M (cid:88) j =0 ( − η ) j a j +1 (81)of a j fermions at odd sites j commutes with the Hamilto-nian H D . It constitutes an exact symmetry for any finite M . The normalization factor is N ( η ) = (cid:115) − η M +1) − η . (82)A similar calculation establishes the right symmetry γ D right = 1 N M (cid:88) j =0 ( − η ) j b L − j . (83)These exact Majorana zero-energy modes are exponen-tially localized. If | η | >
1, the corresponding localizedsymmetries are obtained by rescaling γ Dα → ( − /η ) M γ Dα ,and changing the normalization factor to N (1 /η ). At | η | = 1 the mass gap vanishes in the thermodynamiclimit L → ∞ .Notice the fundamental difference between chains ofeven or odd lengths L . While it is possible to determine exact zero modes when L is odd, this is not the case for L even where the Majorana character of the edge modesis only asymptotically exact in the thermodynamic limit.The reason is simple. For any finite L , H D commuteswith the global symmetries U z = L (cid:89) j =1 (1 − n j ) , U x = L (cid:89) j =1 i ( b † j + b j ) , (84)where b † j = c † j (cid:81) j − l =1 (1 − n l ) is a hard-core boson. How-ever, it is only when L is odd that { U z , U x } = 0. Inturn, this implies that the whole many-body spectrum of H D is, at least, exactly two-fold degenerate. This sym-metry analysis applies just as well to the more generalHamiltonian of Eq. (33) with open boundary conditions,provided the on-site potential (cid:15) j vanishes.The duality transformation of Eq. (76) maps theboundary Majorana zero-modes of the Kitaev chain intocorresponding boundary symmetries of the Peierls chain, γ right , = 1 N M (cid:88) j =0 ( − η ) j a L − j ,γ right , = 1 N M (cid:88) j =0 ( − η ) j b L − j . (85)Unlike for the Kitaev chain, these two zero modes resideon one and the same edge. Hence, it is natural to re-combine them into one exponentially localized fermionicmode, c † right = γ right , − i γ right , . (86)2It is not surprising that both boundary symmetries of thePeierls chain appear on one boundary point, and not bothas in the Kitaev case. The reason is the lack of reflectionsymmetry about j = M + 1 in the mapped Peierls chain.Nonetheless, its many-body ground state is two-fold de-generate, just as its dual Kitaev superconductor, indicat-ing that the Z symmetry of fermionic parity is odd inthe thermodynamic limit also for the Peierls chain. Thisstatement is of course confirmed by the exact solution ofthe Peierls Hamiltonian.
3. Density-density interactions
As mentioned above Gaussian dualities can be used toestablish equivalences in interacting many-body systems.Here, we illustrate this fact in another paradigmatic ex-ample. Consider the case of a dimerized Peierls chain (oflength L = 2 M ) at half-filling where electrons interactthrough a density-density Coulomb repulsion VH = M (cid:88) j =1 (cid:104) − t j ( c † j c j +1 + c † j +1 c j )+ V − n j )(1 − n j +1 ) (cid:105) . (87)The duality transformation of Eq. (42) maps the densityoperators as follows n j − →
12 (1 − i b j − b j ) ,n j →
12 (1 − i a j a j +1 ) , (88)and the resulting dual superconducting equivalent isgiven by H D = − L (cid:88) j =1 (cid:104) t c † j c j +1 + ∆ c † j c † j +1 + H . c . (cid:105) + V L (cid:88) j =1 e i πn j +1 (cid:104) − c † j c j +2 + e i πj c † j c † j +2 + H . c . (cid:105) , (89)which clearly shows the competition and interplay be-tween the band and Mott gaps.A natural question that emerges is how robust is thePeierls phase to the presence of Coulomb interactions?The second, related, question is can interactions alonegenerate a topological Mott phase in the case where thenon-interacting phase is metallic, i.e., ∆ = 0? Sincein the latter case the model is exactly (Bethe ansatz)solvable, we know that for sufficiently large repulsion V ,there exists a Mott phase.In Appendix C we analyze the phenomenon of symme-try transmutation in interacting boson systems. x y t t (cid:15) x (cid:15) y t t (cid:109) Dual Superconductor r r rr t av r , r ∆ r , r ∆ r , r t av r , r + π − π + π − π FIG. 4. (Top) Topological insulator characterized by a par-ticular Kekul´e-type pattern of hopping matrix elements asdepicted in the figure. Double bonds along the horizontal(vertical) direction represent the hopping matrix element t ( t (cid:48) ), while single bonds along the horizontal (vertical) direc-tion represent t ( t (cid:48) ). The (cid:15) x ( y ) are on-site atomic energies,constant on each sublattice. (Bottom) Dual chiral topolog-ical superconductor with π fluxes per square plaquette dis-tributed antiferromagnetically, and a p x + i p y superconduct-ing order parameter. The dual relation between parametersis explained in the main text. B. Graphene is dual to a “weak” TopologicalSuperconductor
In this section we establish a non-trivial equivalences intwo space dimensions. For conciseness, we investigate atopological insulator on a square lattice, characterized bya Kekul´e-like pattern of hopping parameters t , t (cid:48) , t , t (cid:48) and an on-site potential (cid:15) x , (cid:15) y that is constant on each3sublattice, see Figure 4. In this way we manage to ad-dress several interesting models in a unified fashion. Thesemimetal graphene for example is realized on the line t = t (cid:48) = t (cid:48) , t = 0 = (cid:15) x . (A few other lines obtaingraphene as well. Notice that the honeycomb lattice isappears represented as a brick wall lattice in Figure 4).As a consequence, our model realizes a condensed mat-ter analog of the (2+1)-dimensional parity anomaly ,and our duality transformation provides a superconduct-ing dual representation of this phenomenon. As was thecase in one dimension, the Gaussian duality mapping theinsulating model to a topological p-wave superconductorhas the key property of preserving the locality of the edgemode excitations.The two-dimensional underlying lattice Λ considered inthe following is bipartite with lattice points x = ( x , x )and y = ( y , y ), such that x + x ∈ even and y + y ∈ odd integers. The total number of lattice pointsalong the horizontal direction is L x , and L y along thevertical direction, such that L x × L y defines the size ofthe lattice and where, for simplicity, L x and L y representeven integers. Figure 4 (Top) is an example of a lattice Λ.Consider in particular the lattice shown in Fig. 4 (Top)where, for a given point x = ( x , x ), the correspondinghopping amplitudes of Hamiltonian (28) are given by: t x , y = t , for y = ( x + 1 , x ) t , for y = ( x − , x ) t (cid:48) , for y = ( x , x − t (cid:48) , for y = ( x , x + 1) (90)and with on-site energies (cid:15) x = − (cid:15) y (see Eq. (51)).This model, endowed with periodic (toroidal) bound-ary conditions, has a single-particle energy spectrum(bulk bands) given by E , k σ = − (cid:113) (cid:15) x + A k , + + B k , − , E , k σ = − E , k σ ,E , k σ = − (cid:113) (cid:15) x + A k , − + B k , + , E , k σ = − E , k σ , (91)where the wavevectors k = ( k x , k y ) are defined in theBrillouin zone ( k x = πL x n x , k y = πL y n y ) with n x =0 , , · · · , L x − n y = 0 , , · · · , L y −
1, and A k , ± = ( t + t ) cos (cid:18) k x (cid:19) ± ( t (cid:48) + t (cid:48) ) cos (cid:18) k y (cid:19) B k , ± = ( t − t ) sin (cid:18) k x (cid:19) ± ( t (cid:48) − t (cid:48) ) sin (cid:18) k y (cid:19) . (92)There is a chiral symmetry at work, since the energylevels are symmetrically distributed around zero energyand time reversal is not broken.The Gaussian duality of Eq. (38) with δ = δ = (0 , p -wave supercon-ductor with an antiferromagnetic distribution of π -fluxesper square plaquette (or more precisely, two spin copies ofthis system). The dual (chiral) superconductor is shownin Fig. 4 (Bottom) and corresponds to (see Eqs. (41) and (52)) H D = − (cid:88) (cid:104) r , r (cid:48) (cid:105) ,σ (cid:104) t av r , r (cid:48) c † r ,σ c r (cid:48) ,σ + ( t av r , r (cid:48) ) ∗ c † r (cid:48) ,σ c r ,σ +∆ r , r (cid:48) c † r (cid:48) ,σ c † r ,σ + (∆ r , r (cid:48) ) ∗ c r ,σ c r (cid:48) ,σ ) (cid:105) , (93)where (cid:104) r , r (cid:48) (cid:105) represents nearest-neighbor links of a rect-angular lattice with lattice points r = ( r , r ) and t av r , r (cid:48) = (cid:26) t + t , for r (cid:48) = ( r + 1 , r ) t (cid:48) + t (cid:48) − i ( − r r (cid:15) r , for r (cid:48) = ( r , r + 1) , ∆ r , r (cid:48) = (cid:26) − t − t , for r (cid:48) = ( r + 1 , r ) − t (cid:48) − t (cid:48) − i (cid:15) r , for r (cid:48) = ( r , r + 1) . (94)
1. Mapping of topological boundary modes
Our Kekul´e-type insulator may displays zero energymodes if the on-site potential vanishes. It is instruc-tive to consider explicitly the case of graphene to high-light the differences between zig-zag and armchair edgeterminations from the point of view of the dual super-conductor. So let us take open boundary conditions alongthe r -direction and periodic along the r -direction, i.e.,and open cylinder, and the parameter set t = t (cid:48) = t (cid:48) , t = 0 = (cid:15) x = (cid:15) y . Then, the Gaussian duality mapused above for the toroidal boundary conditions (bulk)also works for the cylinder, since δ = δ = (0 ,
1) de-scribes a translation along the periodic direction. Onecan see from Fig. 4 that this situation corresponds toa zig-zag edge, while the parameter set t = t = t (cid:48) , t (cid:48) = 0 = (cid:15) x = (cid:15) y would correspond to an armchair ter-mination. The corresponding dual superconductors rep-resent (two copies of) a stack of horizontal or vertical Ki-taev chains respectively, in the topologically non-trivialregime. When the chains are horizontal, they obtainthe topological superconducting Majorana edge modesdual to zig-zag terminated graphene. When the chainsare vertical, the superconductor does not display edgemodes, and its dual corresponds to the armchair termi-nated graphene.The general duality transformation of Section II Cbreaks down for open boundary conditions in both r and r directions. Nonetheless, all of our conclusions hold justas well in this case were zig-zag and armchair termina-tions coexist. In order to illustrate this point explicitly itbecomes necessary to introduce a different Gaussian du-ality, showcasing once more the fact that exact dualitiesare very sensitive to boundary conditions.Let us focus for simplicity on a particular, spinless, caseof our Kekul´e-type insulator with open boundary condi-tions in both directions. The Hamiltonian of interest isgiven by H = H + H + H v , (95)4where H = − M y (cid:88) r =1 M x (cid:88) r =1 (cid:104) t c † r − , r − c r , r − + t c † r , r − c r +1 , r − + H . c . (cid:105) , (96) H = − M y (cid:88) r =1 M x (cid:88) r =1 (cid:104) t ( c † r − , r c r , r + t c † r , r c r +1 , r + H . c . (cid:105) , (97)[ H , H ] = 0, and H v = − t L y − (cid:88) r =1 L x (cid:88) r =1 (cid:104) c † r ,r c r ,r +1 + H . c . (cid:105) , (98)with L x = 2 M x + 1 and L y = 2 M y , has an appealinginterpretation as a stack of dimerized Peierls chains. Be-cause the Peirls chain are alternating, the bulk transla-tion symmetry in either direction is generated by trans-lations by two sites. As a function of t , the stack in-terpolates between a trivial metal, t = t , and spinlessgraphene, t = 0, with zig-zag vertical and armchair hor-izontal boundaries.The description of H as a stack of Peierls chains sug-gests a natural way to map the system to a superconduc-tor. Let us apply, to each horizontal chain, the duality ofSection III A 2, Eq. (76). Unfortunately, this simplest al-ternative would obtain a non-local dual representation of H v . There is, however, a way to fix this problem. Recallthat the Gaussian dualities of this paper are motivatedby the observation that some systems may be split intoindependent subsystems, and then it is possible to rear-range one subsystem relative to the other. So, given thesplitting of the open Peierls chain into two subsystems,we could rearrange one subsystem for, say, the chainsat odd height 2 r −
1, and the other subsystem for thechains at even height 2 r . This idea is implemented bythe Gaussian duality a r − , r − → a L x − r − , r − ,b r , r − → b L x +1 − r , r − ,a r , r → a L x +1 − r , r ,b r − , r → b L x − r − , r . (99)The Majorana operators that are not explicitly listed re-main unchanged.Now, the alternating structure of the duality mappingin the vertical direction obtains the trivial transformation H Dv = H v . (100)The effect of the first half of the transformation on H follows immediately from the work in Section III A 2. In H , the hoppings t and t are exchanged relative to H .However, the second half of the transformation is also modified relative to the first half in such a way as toprecisely compensate for this exchange. Explicitly, H D = H D + H D + H v , (101)with H D + H D = − L y (cid:88) r =1 L x − (cid:88) r =1 (cid:104) t c † r ,r c r +1 ,r +∆ c † r ,r c † r +1 ,r + H . c . (cid:105) . (102)As before, the dual system may be described as a stackof Kitaev wires, but now with open boundary condi-tions in both directions. It realizes on its vertical bound-aries the Kitaev edge, a one-dimensional p-wave super-conductor robust against statistical translation invariantdisorder and/or interactions. We see that the zig-zagboundary of spinless graphene is dual to the Kitaev edge.
IV. SUMMARY AND OUTLOOK
In this paper we have developed the general theory ofGaussian duality transformations for fermions, definedas maps that preserve a quadratic Hermitian form. Thedual partner of a system of free fermions is also a systemof free fermions if the duality transformation is Gaussian,and both systems are equally local in space. The the-ory and practice of fermionic Gaussian dualities benefitsfrom “Majorana fermions,” the complex Clifford algebracanonically represented in Fock space. As a consequence,the theory of Gaussian dualities for canonical bosons ismarkedly different and left for a future publication.As transformations of statistical mechanics, duali-ties are valued for mapping strongly-coupled systems toweakly coupled ones. Gaussian dualities seem uncon-ventional from this point of view, since systems of freefermions are, by definition, all weakly coupled. To someextent, the conceptual mismatch is just a matter of choiceof language, that is, of physical representation of the sys-tems under consideration. Take for example the sim-plest model of magnetic ordering, the transverse-fieldIsing chain. The self-duality of this model, a non-localtransformation of spins, is the prototype of a strong cou-pling/weak coupling duality transformation. However,the Jordan-Wigner mapping transforms the Ising modelinto the Majorana chain, and the self-duality of the Isingmodel into a Gaussian self-duality of the Majorana chain.The key property shared by all duality transforma-tions is symmetry transmutation in face of the localityconstraint. This phenomenon occurs when a symmetryand its dual partner, necessarily a symmetry of the dualHamiltonian, have different physical interpretations. ForGaussian dualities in particular we demonstrated trans-mutation of particle number, fermionic parity, transla-tion, spin rotation, and time-reversal symmetry in var-ious space dimensions. Transmutation of particle num-ber is most conspicuous, since it allows for insulators (or5semimetals) and superconductors (with zeroes of the gapfunction) to appear as dual partners.Because of symmetry transmutation, Gaussian duali-ties can establish equivalences of topological insulatorsand superconductors, relating systems classified as in-equivalent from a single-particle viewpoint. Here we in-vestigated in detail two paradigmatic examples of suchpairs of dual partners: the dimerized Peierls/Majoranachain (at vanishing chemical potential), and graphenewhich happens to be dual to a weak topological supercon-ductor. (Transmutation of lattice symmetry is manifestin both examples. In particular, the superconductor dualto graphene resides on a square lattice.)While our list of equivalences is far from exhaustive,our approach constitutes a general framework. Hence,future research should be focused on searching systemat-ically for other classes of Gaussian dualities besides theones presented, aiming to obtain all possible equivalencesacross entries in established classification tables of elec-tronic matter. We can offer two comments as to what“possible” should entail. First, we do not necessarillyexpect equivalences of systems of different space dimen-sionality, since dimensional reduction by duality is pos-sible but uncommon. Second, to obtain equivalencesbetween topologically trivial and non-trivial systems, itwill become necessary to obtain the Gaussian analog of aholographic symmetry. The Gaussian dualities of thispaper do not realize holographic symmetries: boundarysymmetries (zero-energy modes) of a topologically non-trivial system are mapped to equally localized boundarysymmetries of its dual partner, and topologically triv-ial systems are mapped to equally trivial dual partners.In short, our dualities preserve the bulk-boundary corre-spondence (for a useful discussion of this correspondence,see for example Ref. [34] and references therein).Let us now discuss some possible applications of ourresults. It is interesting to think of dual partner systems,with at least one open boundary, as scattering regionsand ask what happens at the level of dualities if we at-tach leads to the system. The remarkable answer is thatit is often possible to extend the Gaussian duality to ap-ply to the whole system (scattering region plus leads), sothat its dual partner also has an unambigous interpreta-tion as a scattering region with leads attached. Depend-ing on the nature of the leads and Gaussian duality, thedual leads may be rearranged in space and/or becomesuperconducting. In any case it is now possible to ob-tain quantitative dual mappings of transport propertiesto further extend the equivalence of insulators and super-conductors. Notice that it is no problem to add disorder,but symmetry transmutation of the disorder ensembleshould be expected. That is, if the disorder ensemble ofone system displays some statistical symmetry, the dualsystem will be distributed according to a dual ensemblewith a dual statistical symmetry of possibly very differentnature. These brief comments set the ground for inves-tigating equivalences of statistical topological insulatorsand superconductors. Topologically non-trivial insulators and superconduc-tors display zero-energy boundary modes associated tothe degeneracy of their many-body ground-state energylevel. In general, it is possible to trace the superconduc-tor ground degeneracy back to some discrete symmetryleft over from the breaking of particle conservation. Thispicture does not apply to insulators and so the originof their topological degeneracy is harder to unveil. Butour equivalence can help here. In terms of its super-conducting equivalent, the ground degeneracy of a topo-logical insulator is explained by spontaneous symmetrybreaking. Moreover, it can be detected and character-ized numerically by investigating the system in terms ofJosephson-like physics.Finally, we would like to point out a very importantapplication of our duality approach to topological mat-ter. Since our duality transformations are not restrictedto free fermion/boson systems (as shown in a couple ofexamples of this paper), it is apparent that they maybecome a powerful tool for a potential classification ofinteracting fermion/boson systems.
ACKNOWLEDGMENTS
We gratefully acknowledge discussions with J. I. Cirac,M. Diez, G. van Miert, C. Morais Smith, C. Ortix, B.Seradjeh, and M. Tanhayi. This work is part of theDITP consortium, a program of the Netherlands Organ-isation for Scientific Research (NWO) that is funded bythe Dutch Ministry of Education, Culture and Science(OCW). GO thanks the Institute for Nuclear Theoryat the University of Washington for its hospitality andthe Department of Energy for partial support during thecompletion of this work.
Appendix A: The m -merized Peierls chain Let us describe briefly the superconducting model asso-ciated to the general m -merized Peierls case with m > m = 3 (trimerized) Peierls chain with hop-pings t , t , and t and periodic boundary conditions( L = 2 M is divisible by m = 3). Then, the dual super-conducting Hamiltonian is given by H D = − L (cid:88) j =1 (cid:16) w j c † j c j +1 + ∆ j c † j c † j +1 + H . c . (cid:17) , (A1)with coupling constants w = t + t , w = t + t , w = t + t , = ± t − t , ∆ , = ± t − t , ∆ , = ± t − t , and the rest of the couplings periodically repeat. Repeat-ing the same line of reasoning, the case m = 4 leads to adual Hamiltonian such as Eq. (A1), where the coupling6constants are given by w = t + t , w = t + t , w = t + t , w = t + t , ∆ = t − t , ∆ = t − t , ∆ = t − t , ∆ = t − t . This clearly shows the difference be-tween the cases where m is odd from those where m iseven. In the latter case, the periodicity is always m lat-tice constants, except for the particular dimerized m = 2case where the periodicity is one lattice constant, whilethe periodicity becomes 2 m when m is odd. Appendix B: Insulating dual partner of the BCSsuperconductor
There is an elementary Gaussian duality of an s -waveBCS superconductor to a trivial insulator. Let us de-note by ˆ c k ,σ the annihilation fermionic field in momen-tum space k , and assume (cid:15) k = (cid:15) − k . Then we may definethe Gaussian dualityˆ c k , ↑ (cid:55)→ ˆ c k , ↑ , ˆ c k , ↓ (cid:55)→ ˆ c †− k , ↓ , (B1)acting non-trivially only on the spin-down fermions. Themean-field BCS Hamiltonian is H = (cid:88) k ,σ ( (cid:15) k − µ ) ˆ c † k ,σ ˆ c k ,σ − (cid:88) k ∆ (ˆ c † k , ↑ ˆ c †− k , ↓ + H . c . ) , (B2)and gets mapped to H D = (cid:88) k (cid:2) (cid:15) k ˆ c † k ,α σ zα,β ˆ c k ,β − h ν ˆ c † k ,α σ να,β ˆ c k ,β (cid:3) + C, (B3)with C = (cid:80) k ( (cid:15) k − µ ), ν = x, y, z , and h x = ∆ , h y = 0 , h z = µ. (B4)Particle conservation is restored in H D at the expense ofbroken (by symmetry transmutation) time-reversal andspin-rotation symmetry. Appendix C: A Bosonic insulator and its dualsuperfluid
In recent years, lattice models of bosons have acquirednew relevance thanks to the spectacular experimental de-velopment of ultracold atom physics. In this section wewill focus on a case of practical importance in which onlythree states {| ¯ n − (cid:105) , | ¯ n (cid:105) , | ¯ n + 1 (cid:105)} per lattice site j arephysically active. Then the particle operators g j , g † j , n j act on these states as g j | ¯ n − (cid:105) = 0 , g † j | ¯ n − (cid:105) = √ ¯ n | ¯ n (cid:105) ,g j | ¯ n (cid:105) = √ ¯ n | ¯ n − (cid:105) , g † j | ¯ n (cid:105) = √ ¯ n | ¯ n + 1 (cid:105) ,g j | ¯ n + 1 (cid:105) = √ ¯ n | ¯ n (cid:105) , g † j | ¯ n + 1 (cid:105) = 0 , (C1)and n j | ¯ n − (cid:105) = (¯ n − | ¯ n − (cid:105) ,n j | ¯ n (cid:105) = ¯ n | ¯ n (cid:105) ,n j | ¯ n + 1 (cid:105) = (¯ n + 1) | ¯ n + 1 (cid:105) . (C2) These are bosonic operators restricted to a finite(three)-dimensional Hilbert space per site. Their algebra andapplication to optical lattices has been extensively in-vestigated in Refs. 37 and 23. Here it will suffice to noticethat there is a connection to spin S = 1 operators that,in the present paper and because of the duality we willapply later on, we take to be (cid:114) n g † j = S zj + i S xj = S + j , n j = S yj + ¯ n. (C3)Here we will focus on a system of bosonic atoms dis-tributed with (integer) average particle density ¯ n ≥ H = (cid:88) j (cid:104) − t g † j +1 g j + H . c . ) − µ n j + V n j n j +1 (cid:105) . (C4)The chemical potential µ fixes ¯ n , the interaction V > N = (cid:80) j n j is conserved. This bosonic t - V model displays a Mottinsulating phase.As a result of Eq. (C3), H has an interesting interpre-tation as an XXZ spin S = 1 model in a magnetic field, H = (cid:88) j (cid:104) − ¯ n t S xj S xj +1 + S zj S zj +1 ) − ( µ − nV ) S yj + V S yj S yj +1 (cid:105) (C5)(up to an additive constant). In the following the chem-ical potential will be kept fixed at µ = 2¯ nV in order toeliminate the magnetic field term.The quantum phase diagram of the S = 1 XXZ is wellknown. For t = 0, the ground state is antiferromag-netically (Ising) ordered in the spin language and in aMott insulating state in the boson language. There isa mass gap in the quasiparticle spectrum. According tothe Haldane gap conjecture, the mass gap remains as theHeisenberg antiferromagnetic line t = − V / ¯ n is reached.For the bosonic t - V model this means that the atomicsystem remains in a Mott insulating state. Next we willshow that the bosonic t - V model is dual to a particlenon-conserving Hamiltonian with an unconventional su-perfluid ground state.To establish the equivalence of the bosonic t - V modelto a superfluid, we will exploit a duality transformationfirst investigated in the context of spin S = 1 models, seeRef. 39 and references therein. 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