On the relation of the entanglement spectrum to the bulk polarization
OOn the relation of the entanglement spectrum to the bulk polarization
Carlos Ortega-Taberner
1, 2 and Maria Hermanns
1, 2 Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden Nordita, KTH Royal Institute of Technology and Stockholm University, SE-106 91 Stockholm, Sweden (Dated: February 8, 2021)The bulk polarization is a Z topological invariant characterizing non-interacting systems in onedimension with chiral or particle-hole symmetries. We show that the bulk polarization can alwaysbe determined from the single-particle entanglement spectrum, even in the absence of symmetriesthat quantize it. In the symmetric case, the known relation between the bulk polarization and thenumber of virtual topological edge modes is recovered. We use the bulk polarization to computeChern numbers in 1D and 2D, which illuminates their known relation to the entanglement spectrum.Furthermore we discuss an alternative bulk polarization that can carry more information about thesurface spectrum than the conventional one and can simplify the calculation of Chern numbers. I. INTRODUCTION
Topological phases of matter have attracted a lot ofattention during the last decades, not the least becausea large variety of relevant systems have been realized ex-perimentally. The early focus was mainly on topologi-cally ordered systems [1], where interaction effects arecrucial for stabilizing the phases. The most notable ex-amples are the fractional quantum Hall liquids [2] andquantum spin liquids [3]. However, since 2005 [4, 5]the focus has shifted to symmetry-protected topologicalphases (SPT) where symmetries are necessary to pro-tect the topological phases and determine which distincttopological phases can be realized for a given dimension-ality. These can be implemented as free-theories and canbe characterized in terms of topological invariants [6].Entanglement has played an important role in the un-derstanding and characterization of topological systems[7, 8]. One can distinguish two types of states depend-ing on their entanglement. Short range entangled statescan be continuously transformed into a direct productstate, while long range entangled states cannot. Thelatter correspond to topologically ordered states. Cer-tain short range entangled states cannot be continuouslytransformed between themselves unless certain symme-tries are broken. These are the SPT phases that we focuson in this paper.There are different tools based on entanglement thathave been used to characterize topological phases. Avery efficient one is the entanglement entropy [9], whichallows one to determine the total quantum dimension ofthe underlying topological quantum field theory [10, 11].However, it can only be used for topologically orderedphases and it cannot uniquely characterize the topolog-ical phase at hand. Another, closely related, tool is theentanglement spectrum (ES), originally introduced forfractional quantum Hall systems [12]. It provides infor-mation about the edge spectrum and has proven use-ful for other topologically ordered phases such as frac-tional Chern insulators [13] and certain quantum spinliquids [14].For non-interacting topological insulators and super- conductors the ES for (gapped) periodic systems canbe computed very efficiently, using methods developedby Peschel and others [15]. The ES in these systems isequivalent to the flat-band energy spectrum of the cor-responding system with open boundaries [16]. The samecorrespondence was also found for closely related gaplesssystems [17]. However, even for non-interacting systems,it is unclear which information (beyond the protected‘edge’ spectrum) is encoded in the ES. The aim of thispaper is to show that other physical properties are alsoencoded in the ES, in particular the bulk polarization.The bulk, or macroscopic, polarization is a fundamen-tal concept in physics, primarily used to describe the re-sponse of matter to electric fields. The modern theoryof polarization [18–21] related the bulk polarization to ageometric phase, which is nothing but the Zak phase fortranslationally invariant systems. Due to this, it has alsofound its way into topological physics [22]. In certainsymmetry classes the bulk polarization is quantized andserves as a Z topological invariant, where it is known tobe related to certain feature of the ES [23]. Because ofits relation to a geometric phase, the bulk polarizationis also related to other topological invariants in higherdimensions such as the Chern number.We show that there is much more information encodedin the single-particle ES than was previously known. Inparticular, we show how the bulk polarization can be de-coded from the single-particle ES, even when it is notquantized by the symmetries. This is a general propertyof non-interacting 1D gapped systems. We also applythis method to compute Chern numbers directly fromthe single-particle ES. The differences in the bulk polar-ization, which can be any real number, are the relevantquantities to study. They can, however, be difficult tocompute from the bulk polarization itself, as it is onlydefined modulo 1. We show how, using our method, onecan define an alternative bulk polarization, computed byusing open boundary conditions, which is continuous in R for gapped paths in parameter space, simplifying thecalculation of polarization differences. The bulk polar-ization defined this way can give more information aboutthe edge spectrum than the conventional one, and can a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b be used to simplify the computation of Chern numbers.We also discuss the relation between this alternative bulkpolarization and another topological invariant known asthe trace index [24].The relation between the ES and the bulk polariza-tion has already been studied. In Ref. [25], the au-thors show that the Zak phase can be computed fromthe Schmidt decomposition of a translationally invariant,infinite chain. The latter is related to the ES. This sug-gests that, for non-interacting systems, there should bea direct relation between the Zak phase and the single-particle ES, which is substantially easier to compute thanthe ES. A particular limit of our result was derived fora fully dimerized SSH chain [23]. The similarity betweenthe behavior of the Zak phase and the ES of certain Cherninsulators, which was observed in Ref. [26, 27], is also ex-plained by our results. Outline of the paper : In section II we introduce the1D model used throughout the paper and we examineits phase diagram. In section III we discuss the bulkpolarization; the different ways one can define it and itsrelation to the geometric phases. The ES is introduced insection IV, where we also set our notation. In section Vwe present our method and show how we can reproducethe bulk polarization for systems with and without trans-lational invariance. In section VI we introduce an alter-native bulk polarization constructed from the ES and useit to compute Chern numbers in 1D and 2D. Finally insection VII we conclude and discuss possible extensionsof our work.
II. THE MODEL
To illustrate our results we consider a model in the BDIclass [28]. Although simple, it supports a rich phase di-agram containing topological phases with winding num-bers ν = 0 , H = (cid:88) iα,jβ c † iα H ij,αβ c jβ , (1)where H ij =( mσ x + κσ z ) δ ij + 12 i κ (cid:48) σ z ( δ i − j, − δ i − j, − )+ 12 t [( σ x + iσ y ) δ i,j +1 + ( σ x − iσ y ) δ i,j − ]+ 12 t (cid:48) [( σ x + iσ y ) δ i,j +2 + ( σ x − iσ y ) δ i,j − ] , (2)and the corresponding Bloch Hamiltonian is H ( k ) = (cid:18) κ + κ (cid:48) sin( k ) t (cid:48) e i k + te ik + mt (cid:48) e − i k + te − ik + m − κ − κ (cid:48) sin( k ) (cid:19) =( t (cid:48) cos(2 k ) + t cos( k ) + m ) σ x (3)+ ( − t (cid:48) sin(2 k ) − t sin( k )) σ y + ( κ + κ (cid:48) sin( k )) σ z . FIG. 1: Phase diagram for the system in the
BDI classshowing the winding number of each region — plottedfor parameters t = 1 , κ = κ (cid:48) = 0. Three cuts of thephase diagram, which will be used in later figures, areshown with continuous, dashed and dotted lines.For κ = κ (cid:48) = 0, it is in the BDI class, i.e. it has time-reversal ( T ), particle-hole ( C ), and chiral symmetry ( S ): T = K ; T H ( − k ) T − = H ( k ) C = σ z K ; CH ( − k ) C − = − H ( k ) S = σ z ; SH ( k ) S − = − H ( k ) . (4)The corresponding phase diagram [28] is shown in Fig. 1for t = 1.The BDI class in one spatial dimension is character-ized by a Z invariant, the winding number. When openboundary conditions are imposed on the system, thenumber of symmetry-protected zero-energy edge modesis equal to the winding number. The behavior of theedge modes is shown in Fig. 2, using the dotted path( t = 1 , t (cid:48) = −
2) marked in the phase diagram in fig-ure 1. Figure 2(a) shows the edge spectrum in the BDIclass, with 4 (resp. 2) symmetry-protected zero modesfor winding number 2 (1).For κ (cid:48) (cid:54) = 0, only particle-hole symmetry C is preservedand the system belongs to symmetry class D . In one spa-tial dimension, the latter is characterized by a Z invari-ant. Consequently, adding such a term causes the zeromodes in the ν = 2 phase to split pairwise and the phasebecomes topologically trivial, indistinguishable from thephase with ν = 0. This is shown in Fig. 2(b), where oneclearly sees that the zero modes below m = 1 are split.For κ (cid:54) = 0 only time-reversal T is preserved and thesystem is in the AI class. In 1D this class is trivial and,thus, all zero-energy modes split from zero energy. Withboth κ (cid:54) = 0 and κ (cid:48) (cid:54) = 0 the system has no local symmetry,thus belonging to the A class. This symmetry class is alsotopologically trivial in one dimension. Fig. 2(c) and (d)show the edge spectrum for class AI and A, respectively.In both cases, the edge modes are split from zero for allvalues of m . - - ( a ) - - ( b ) - - ( c ) - - ( d ) FIG. 2: Energy spectrum with open boundaryconditions along the dotted path (i.e. t = 1 , t (cid:48) = −
2) ofFig. 1 plotted for different values of thesymmetry-breaking terms (a) κ = κ (cid:48) = 0 , (b) κ = 0 , κ (cid:48) = 0 .
3, (c) κ = 0 . , κ (cid:48) = 0 and (d) κ = 0 . , κ (cid:48) = 0 . Model
T C S
Class inv. κ = 0 , κ (cid:48) = 0 1 1 1 BDI Z κ = 0 , κ (cid:48) (cid:54) = 0 0 1 0 D Z κ (cid:54) = 0 , κ (cid:48) = 0 1 0 0 AI 0 κ (cid:54) = 0 , κ (cid:48) (cid:54) = 0 0 0 0 A 0 TABLE I: Different Cartan classes for the model inEq. (2), with the correspondent symmetries andtopological invariants in 1D.
III. GEOMETRIC PHASES AND BULKPOLARIZATION
The bulk polarization is a property which characterizestopological insulators in 1D. The bulk polarization is pro-portional to the surface charges of the system with openboundary conditions. In a topological insulator these arequantized because of the appearance of zero-energy edgestates. Because of this, the bulk polarization is itselfquantized, and therefore serves as a topological invari-ant. In this section we review some relevant aspects of thebulk polarization and the modern theory of polarization,which gives a geometrical description of the polarization[18–21].Consider a generic, quadratic Hamiltonian in one di-mension H = (cid:88) ij,αβ c † iα H ij,αβ c jβ . (5)We obtain its single-particle eigenstates by (cid:88) jβ H ij,αβ ψ jβpµ = E pµ ψ iαpµ , (6)where [ U ] jβ,pµ = ψ jβpµ is the unitary matrix that diagonal-izes H . In translational invariant systems ψ jβkµ = e ikj u βkµ ,where u βkµ are the components of the eigenstates, | u kµ (cid:105) ,of the Bloch Hamiltonian and k is the momentum. Forsimplicity we consider a unit cell with a single site. Onecan now compute the Zak phase [29]. For a two-bandmodel, it is defined as the geometric phase acquired bythe occupied state | u k (cid:105) as it winds around the Brillouinzone, γ = (cid:90) π d k iA k , (7)where A k = (cid:104) u k | ∂ k | u k (cid:105) is the Berry connection [30]. Thegeneralization to multi-band models is straightforward.The Zak phase is only gauge invariant modulo 2 π .However, for two different states defined by a parameter λ — assuming A k ( λ ) is smooth in the path connectingthem — the change in the Zak phase can be computedas ∆ γ λ i λ f = (cid:90) λ f λ i dλ (cid:90) π dk Ω λk , (8)where Ω λk = ∂ λ A k ( λ, k ) − ∂ k A λ ( λ, k ) (9)is the Berry curvature. The Berry curvature is fully gaugeinvariant and, therefore, the change in the Zak phase inEq. (8) is defined in R . The Zak phase itself, as de-fined in Eq. (7), can be shown to be proportional to thewinding number for a certain gauge (see Appendix C),which means that it carries physical information beyondmodulo 2 π . Note that other ways of computing the Zakphase rely on computing e iγ instead, such that the resultis always defined only modulo 2 π .The derivative of the polarization with respect to λ was obtained in Ref. [19] as ∂ λ P Bloch = (cid:90) π dk π Ω λk , (10)which allows us to make the identification P Bloch = γ/ π .The polarization itself is however not an observable. Therelevant quantities are the derivatives or changes in thepolarization with respect to external parameters. Theseresult in currents and charge transport, which are theonly measurable quantities. Similar to how only differ-ence in the Zak phases between two states can be mea-sured [31].Considering a system with periodic boundary condi-tions, one can also compute the geometric phase obtainedby threading a U (1) flux through the ring. This allows usto define a polarization in the absence of translational in-variance. Depending on how the flux is introduced we ob-tain different polarizations with different physical mean-ings. These different polarizations, their relations andphysical interpretations were studied in a recent articleby Watanabe and Oshikawa [32]. We focus on two po-larizations, P and ˜ P , obtained by introducing the flux intwo different ways.If the flux is introduced homogeneously through a vec-tor potential A x = Φ /L we can define the polarization P = (cid:90) π d Φ2 π i (cid:10) Ψ Φ (cid:12)(cid:12) ∂ Φ (cid:12)(cid:12) Ψ Φ (cid:11) + 12 π Im ln (cid:10) Ψ (cid:12)(cid:12) e πi ˆ P (cid:12)(cid:12) Ψ π (cid:11) , (11)where (cid:12)(cid:12) Ψ Φ (cid:11) is the ground state in the presence of flux andˆ P = L (cid:80) jα j ˆ n jα is the polarization operator, with ˆ n jα being the number operator. The second term is neededto make the expression gauge invariant. The derivativeof this polarization gives the average current along thechain.This is equivalent to the polarization obtained by Resta[21] as P = 12 π Im ln (cid:10) Ψ Φ=0 (cid:12)(cid:12) e πi ˆ P (cid:12)(cid:12) Ψ Φ=0 (cid:11) . (12)Expression (12) is particularly useful because it can beeasily expressed in terms of single-particle eigenstates as P = 12 π Im ln det (cid:48) S, (13)where the matrix S is given by S pµ,qν = (cid:88) jα ψ jα ∗ pµ e i πL j ψ jαqν , (14)and det (cid:48) indicates that the determinant is restricted tothe space of occupied single-particle states. Below wewill compare the bulk polarization obtained using ourmethod to the one obtained using equation (13).If the flux is introduced via twisted boundary condi-tions at the seam, i.e. in the bond between sites j = 1and j = L , this is equivalent to performing the gaugetransformation (cid:12)(cid:12)(cid:12) ˜Ψ Φ (cid:69) = e i Φ ˆ P (cid:12)(cid:12) Ψ Φ (cid:11) , (15)which makes (cid:12)(cid:12)(cid:12) ˜Ψ Φ (cid:69) fully periodic in Φ. We can now definethe bulk polarization˜ P = (cid:90) π d Φ2 π i (cid:68) ˜Ψ Φ (cid:12)(cid:12)(cid:12) ∂ Φ (cid:12)(cid:12)(cid:12) ˜Ψ Φ (cid:69) , (16) FIG. 3: Schematic of the chain with periodic boundaryconditions considered, where the different sites arelabeled by their position. The bipartition into regions A and B is shown. Other subregions of A considered inthe text are also shown.whose derivative gives the current flowing through theseam [32]. This bulk polarization is related to the previ-ous one by P = ˜ P + ¯ P , (17)where ¯ P = (cid:90) π d Φ2 π (cid:68) ˜Ψ Φ (cid:12)(cid:12)(cid:12) ˆ P (cid:12)(cid:12)(cid:12) ˜Ψ Φ (cid:69) . (18)Similar to the discussion about the Bloch polarizationbelow Eq. (10), both P and ˜ P are defined modulo 1,while their changes are defined in R . For translationallyinvariant systems one finds that P = P Bloch + L − ν mod 1 (19)where here ν denotes the number of occupied bands [32]. IV. ENTANGLEMENT SPECTRUM
Let us now proceed to discuss the ES. An importantquantity to consider is the correlation matrix, which inposition space is defined by the ground state expectationvalue C αβij = (cid:68) c † iα c jβ (cid:69) . (20)For a system where all single-particle eigenstates withenergies E < C αβij = 12 (cid:104) I − H/ ( H ) / (cid:105) ij,αβ . (21)In this paper, we are mainly interested in the spectrumof the correlation matrix when restricted to a spatial sub-region A (its complement will be denoted by B in thefollowing), see figure 3. When restricted to sub-regionA, the boundaries of A act as virtual boundaries to thesystem. Following Ref.s [26, 27], we refer to this spec-trum as the entanglement occupancy spectrum (EOS).We denote the eigenvalues of the EOS by ξ j . Topologicalphases are characterized by (symmetry-protected) zero-energy modes in the edge spectrum and ξ = virtualedge modes in the EOS.Following Ref. [15, 33] , one can write the reduced den-sity matrix as ρ A = K exp( −H ) , (22)where K is a normalization constant and H is a quadraticHamiltonian, referred to as entanglement Hamiltonian.The spectrum of the reduced density matrix is the ES.The single-particle eigenvalues (cid:15) j of H , referred to as en-tanglement energies, form the single-particle ES. Theyare related to those of the EOS by ξ j = ( e (cid:15) j + 1) − , (23)the corresponding eigenstates are the same. Conse-quently, the EOS is in one-to-one correspondence to thesingle-particle ES [16]. Eq. (23) implies that, for non-interacting systems, the full information of the ES iscontained in the spectrum of the subsystem correlationmatrix. The latter is much simpler to interpret. Con-sequently, we will focus on the EOS in the remainder ofthe paper and only mention the ES when necessary.One property of the subsystem correlation matrix wewill make use of later is that, for sufficiently large sys-tems, its eigenstates are found to be exponentially local-ized on either virtual edge if ξ is away from 0 and 1, orthey are found to be bulk modes if the correspondenteigenvalues are exponentially close to ξ = 0 , A as half the system in the remainder of themanuscript. The results (in the thermodynamic limit)do not depend on this choice. V. BULK POLARIZATION IN THE EOS
Let us first review previous results on the relation be-tween the EOS and the bulk polarization (or Zak phase).For systems in 1D protected by chiral or particle-holesymmetries the Zak phase is a topological invariant. Itis zero whenever there is an even number of eigenvaluesper virtual edge at ξ = 1 /
2, and π when the number isodd [34]. In the absence of symmetry-protection, muchless is known. Ryu and Hatsugai considered the fullydimerized SSH chain with broken chiral symmetry [23].This model is rather special in that there is only a sin-gle pair of eigenvalues in the EOS that is not strictlyidentical to 0 or 1: these are ξ and 1 − ξ , related due to translational symmetry. The authors could show thatthe value of one of these ‘midgap states’ is identical tothe Zak phase divided by 2 π — or alternatively ˜ P Bloch —although they do not specify to which of the two eigen-values it corresponds. Their result is a particular limitof equation (24), which we discuss in the next section.In Ref.s [26, 27] the authors considered a two-dimensional Chern insulator and noted that there wasa similarity between the Zak phase and the virtual topo-logical edge states that connect the EOS values at 0 tothose at 1. Interpreting this as a one-dimensional sys-tem with a parameter, we can explain this behavior byEq. (26), noting that there is a single pair of eigenvaluesthat dominates the sum. Note that the observation ofRef.s [26, 27] is particular to systems with Chern number0 or ±
1, i.e. where there are only few midgap states inthe EOS. It fails for systems with higher Chern numbers,for which there are several terms in Eq. (26) with com-parably large contributions and, consequently, the Zakphase deviates considerably from the virtual topologicaledge states.
A. Systems with equispaced entanglement energies
We first discuss systems for which the entanglementenergies of either virtual edge are equispaced. That is,they are given in the thermodynamic limit by ε nα = ε α + nδ α , where α = L, R labels the two virtual edges. This isa feature of integrable systems [35], e.g. nearest neighborhopping models such as the SSH chain, and it applies tothe Hamiltonian in Eq. (2) when t (cid:48) = 0. In this case,one can obtain the Zak phase in a very simple fashion:We reorder the eigenvalues of the EOS by magnitude, ξ < ξ < ... < ξ L A M , and compute χ = L A M (cid:88) j =1 ξ j − mod 1 , (24)where L A is the length of subsystem A and M is thenumber of orbitals per site. In the thermodynamic limit, χ becomes identical to either ˜ P Bloch or 1 − ˜ P Bloch ,lim L →∞ (cid:12)(cid:12)(cid:12) ˜ P Bloch (cid:12)(cid:12)(cid:12) = lim L →∞ χ mod 1 . (25)In the special case where there is only one eigenvalueper edge that is (cid:54) = 0 , π .In Fig. 4 we show the EOS along the dashed cut inthe phase diagram of Fig. 1, for t (cid:48) = 0. Along this line,our model is equivalent to an SSH chain. In the presenceof chiral and translation symmetry, all eigenvalues of theEOS are doubly degenerate: one eigenvalue from eachvirtual edge, such that χ from Eq.(24) will pick up oneeigenvalue from each pair. Because of chiral symmetry,the sum of the two eigenvalues related by the symmetrygives always 1 so they do not contribute to χ . The onlycontribution to χ is the one from the eigenstates with ξ = 1 / ξ = 1 / < ξ < m , there aretwo dominant midgap eigenvalues, and the bulk polariza-tion follows the lower one very closely. When this midgapeigenvalue approaches 0, the contribution from the othermodes becomes more relevant and the bulk polarizationstarts to deviate from the dominant eigenvalue, see fig-ure 4(c). In Fig. 4(d) we show how the difference between χ and the bulk polarization decreases with system size,showing that they are equal in the thermodynamic limit.Note that their discrepancy is very small already for smallsystem sizes. B. General case
For general non-interacting gapped systems we find(see Appendix A) that it is only the eigenstates that lo-calize on the left virtual edge ( A L in Fig.3) that shouldcontribute to χ . The inclusion of bulk modes is irrelevantsince they have ξ = 0 ,
1. We can then compute χ = (cid:88) µ ∈ L ξ µ mod 1 . (26)the sum is performed over the subspace L formed by theeigenstates with (cid:104) ˆ x (cid:105) < L/
4, where L/ A . It does not matter what threshold weuse as long as we include all left-edge states and excludeall right-edge states. In practice only a few eigenvaluesgive a significant contribution to χ .In the thermodynamic limit we obtainlim L →∞ ˜ P = lim L →∞ χ mod 1 . (27)Note that in order to compute χ correctly the eigen-states with ξ (cid:54) = 0 ,
0. 0.5 1. 1.5 2.0.0.20.40.60.81. ξ m ( a )
0. 0.5 1. 1.5 2.0.0.20.40.60.81. ξ m ( b )
1. 1.2 1.40.0.050.10.15 ξ m ( c )
2. 3. 4. 5.4.6.8.10. - l og χ - P ˜ log [ L ] ( d ) FIG. 4: In (a) and (b) we show the EOS (black), χ (green) and ˜ P Bloch (red) for a cut in the phase diagramthrough the ν = 1 → ν = 0 transition (dashed line inFig. 1, for t = 1 , t (cid:48) = 0), for parameters (a) κ = κ (cid:48) = 0and (b) κ = 0 . , κ (cid:48) = 0. Computed for L = 40. In (c)we zoom in on a region of the plot in (b) to show thesplitting between ˜ P Bloch and the eigenvalue closest to ξ = 1 /
2. In (d) we show a scaling plot of the differencebetween χ and ˜ P Bloch for parameters t = 1 , t (cid:48) = 0 , m = 1, κ = 0 . , κ (cid:48) = 0 and increasingsystem size to show that the difference vanishes in thethermodynamic limit.degeneracy is four-fold or higher we must obtain the lo-calized eigenstates. Localizing the bulk eigenstates thatare exponentially close to ξ = 0 , (cid:15) on the right vir-tual edge, there is a corresponding value − ε on the leftone. Since the spectra are equispaced, this implies thatthe two virtual edge spectra are merely shifted with re-spect to each other. If the shift is zero, e.g. in presenceof a symmetry, summing every other eigenvalue is equiv-alent, modulo 1, to summing all the left eigenvalues. Fora finite shift, summing all odd eigenvalues corresponds toeither summing all left eigenvalues or all right eigenval-ues. In general, this implies that χ , as defined in Eq. (24),only gives the Zak phase up to an overall sign. This is-sue can be resolved by determining the localization of asingle eigenstate.In Fig. 5 we show two examples that highlight the im-portance of the localization structure of the EOS. In fig-ures 5(a) and (b), we compute the EOS for varying m at t = 1, t (cid:48) = 2, indicated by the solid line in the phase-diagram in Fig. 1, in the presence of two different sym-metry breaking terms: (a) κ = 0 . κ (cid:48) = 0 and (b) κ = 0, κ (cid:48) = 0 .
3. For κ = κ (cid:48) = 0, there is a phase transition ν = 2 → m = 2, whereas both choices of symmetry
0. 1. 2. 3. 4.0.0.20.40.60.81. ξ m ( a )
0. 1. 2. 3. 4.0.0.20.40.60.81. ξ m ( b )
0. 1. 2. 3. 4.0.0.20.40.60.81. ξ m ( c )
0. 1. 2. 3. 4.0.0.20.40.60.81. ξ m ( d )
0. 1. 2. 3. 4.0.0.20.40.60.81. ξ m ( e ) FIG. 5: We show the EOS (black), χ (green) and ˜ P (red) computed for L = 40 sites. (a) and (b) for thecontinuous line in the phase diagram, t = 1 , t (cid:48) = 2 withthe symmetry-breaking parameters (a) κ = 0 . , κ (cid:48) = 0and (b) κ = 0 , κ (cid:48) = 0 .
3. (c) and (d) for the dotted linein the phase diagram, t = 1 , t (cid:48) = − κ = 0 . , κ (cid:48) = 0 and(d) κ = 0 , κ (cid:48) = 0 .
3. (e) is the same as (c) but with aposition dependent κ i = ( i + L/ L ) /L . We alsoplot P + 1 / χ is indeed equalto ˜ P and not P .breaking render the system trivial for all m . However, thetwo symmetry breaking terms split the four-fold degener-ate ξ = 1 / κ (cid:48) splits thefour-fold degenerate states into two two-fold degeneratepairs at opposite edges whose contribution cancels (mod-ulo 1), κ split them into two lef t − lef t and right − right pairs, with a finite contribution to the Zak phase. In fig-ures 5(c) and (d), we consider the same symmetry break-ing terms, but now for t = 1 and t (cid:48) = −
2, where theBDI systems shows a phase transition ν = 2 → → m . Note that in Fig.s 5(a) and (b) theobservation made in Ref. [26], i.e. that the there is asimilarity between the Zak phase and the EOS, does notapply anymore.Since χ is defined in position space our results are alsovalid for systems without translationally invariance. Weshow this in Fig. 5(e) where we show the EOS, χ , P and˜ P along the dotted cut in the phase diagram with anadded position dependent symmetry breaking term κ i = ( i + L/ L ) /L . We show that Eq. (27) holds withouttranslational invariance, where P now differs from χ and˜ P .This relation between the Zak phase and the EOS is aconsequence of an identity between the Zak phase and themany-body ES previously found for infinite chains [25].In Appendix A we modified this derivation to account forthe periodic boundary conditions we use and show howtheir result simplifies greatly when expressing it in termsof the EOS, resulting in equation (27). VI. ALTERNATIVE BULK POLARIZATIONAND CHERN NUMBERS
As mentioned above, the bulk polarization is only de-fined modulo 1. This is obvious when it is defined interms of a geometric phase, which is only defined mod-ulo 2 π . In our formulation in Eq. (27), this is reflectedin an ambiguity in the number of bulk modes with ξ = 1that are included in χ . Being defined only modulo 1, thebulk polarization cannot differentiate the phases ν = 0and ν = 2, even though the EOS of both phases is dis-tinct. Another issue due to it being defined in R mod 1appears when we try to follow its evolution over a path C λ . Between any two points λ and λ (cid:48) in this path it is notpossible to tell if the bulk polarization has increased ordecreased by only looking at ˜ P ( λ ) at these specific points.This latter drawback becomes relevant when computingChern numbers.In an attempt to solve these issues we introduce analternative bulk polarization given by˜ P o = (cid:90) π d Φ2 π i (cid:68) ˜Ψ Φo (cid:12)(cid:12)(cid:12) ∂ Φ (cid:12)(cid:12)(cid:12) ˜Ψ Φo (cid:69) . (28)where (cid:12)(cid:12)(cid:12) ˜Ψ Φ=0o (cid:69) is the ground state when the (previouslyperiodic) chain is opened between sites j = L/ j = L/ (cid:12)(cid:12)(cid:12) ˜Ψ Φo (cid:69) = e − i Φ ˆ N A (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) . (29)When introducing the flux in the system with periodicboundary conditions (See Eq. (A5)) there is an ambiguitydue to the bulk modes. This ambiguity disappears whenwe introduce the flux as above in the open chain — wesimply include all the bulk modes. Note that Φ is nota magnetic flux but should be treated as an additionalparameter, and therefore ˜ P o is not the polarization of theopen chain. ˜ P o can be obtained in an even simpler waythan ˜ P as ˜ P o = (cid:90) π d Φ2 π (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) ˆ N A (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) = (cid:88) α,j ∈ A (cid:68) ˜Ψ (cid:12)(cid:12)(cid:12) c † jα c jα (cid:12)(cid:12)(cid:12) ˜Ψ (cid:69) = (cid:88) α,j ∈ A C o , j α, j α =Tr[ C A, o ] , (30)where we used Eq. (29) and ( C A, o ) C o denotes the (sub-system) correlation matrix of the open chain.The idea is that in the process of opening the chain,which can be done adiabatically, all eigenvalues that arerelated to the right virtual edge ( A R in Fig. 3) are pushedto ξ = 0 ,
1. As a result, the sum of all eigenvalues of C A,o (modulo 1) is equal to χ (26). [36] Therefore, in thethermodynamic limit, we havelim L →∞ ˜ P = lim L →∞ ˜ P o mod 1 . (31)The advantage of using ˜ P o is that the contributionfrom the ’bulk’ modes is constant as long as no statescross the Fermi energy, i.e. the system remains gapped.The quantity χ (26), on the other hand, is only inde-pendent of the bulk mode contribution when the mod-ulo 1 is included. The consequence is that for paths C λ in parameter space, for which the energy spectrum isgapped, ˜ P o ( λ ) is defined in R , such that the total changein ˜ P o along this path can be obtained knowing ˜ P o only inthe initial and final states, ∆ ˜ P o ( C λ ) = ˜ P o ( λ f ) − ˜ P o ( λ i ).Note, however, that ˜ P o is gauge-invariant only whentaken modulo 1. This is similar to the case of the Zakphase, which is known to be equal to the winding numberfor certain gauges (see Appendix C). It encodes more in-formation than what is accessible when considered mod-ulo 2 π .This method resolves most of the issues found in sec-tion V, but unfortunately it cannot be applied whenthe system is in a non-trivial topological phase. Whenwe open the chain for a system in a topological non-trivial state, zero-energy modes will appear. The groundstate of the closed chain has several equivalent degener-ate ground states for the open chain, so that C A, o is notwell-defined at these points. One might think that thisproblem can be circumvented by introducing symmetry-breaking term that render the system trivial and extrap-olating ˜ P o towards the topological phase. However, thisextrapolation is again not necessarily unique, as theremight be integer jumps in ˜ P o when the edge-modes crosszero-energy. As we will see below these jumps in ˜ P o arebranch cuts that provide the correct description of thesystem. A. SSH chain
Consider first the simple case of the SSH chain. Whenwe include the chiral-symmetry breaking parameter κ theHamiltonian has the symmetry SH ( κ ) S † = − H ( − κ ) , SC A ( κ ) S † = I − C A ( − κ ) , (32)so ˜ P o ( − κ ) = L − ˜ P o ( κ ). Unless ˜ P o (0) = L/
2, which isthe result expected for a trivial insulator, the result willbe discontinuous at κ = 0. We can see this in Fig. 6(a)where we show ˜ P o in the parameter space ( m, κ ). Thepresence of the zero-mode for m < P o as abranch cut. The appearance of a branch cut in the ( m, κ )parameter space is natural but its position depends onthe choice of gauge (see Appendix C).The relation between the zero-modes and discontinu-ities in Tr[ C A,o ] was already observed in the context ofChern insulators [24]. The appearance of the branch cutis the result of the degenerate point at ( m = 1 , κ = 0)having an associated Chern number C = 1 [34].For simplicity, assume that the branch cuts appearalong κ = 0, such as depicted in Fig.s 6 (a) or (b). Thegeneralization to more complicated scenarios is straight-forward. Consider a counterclockwise loop around thedegenerate point at ( m = t, κ = 0) starting from thebranch cut parametrized by θ , m ( θ ) = 1 + 12 sin (cid:16) θ − π (cid:17) κ ( θ ) = − sin( θ ) . The Chern number is then defined as C = 12 π (cid:73) dθ (cid:73) d Φ Ω θ Φ , (33)with the Berry curvature Ω θ Φ defined in equation (9).From Eq. (28), it follows that the Berry connection forthe flux is simply A Φ ( θ, Φ) = ˜ P o ( θ ), so we now need tocompute A θ ( θ, Φ).We can expand (cid:12)(cid:12)(cid:12) ˜Ψ Φ=0 o (cid:69) in the eigenbasis of the numberoperator ˆ N A , denoted by {| j (cid:105)} , as (cid:12)(cid:12)(cid:12) ˜Ψ Φ=0 o ( θ ) (cid:69) = (cid:88) j c j ( θ ) | j (cid:105) , (34)where c j ( θ ) = (cid:68) j (cid:12)(cid:12)(cid:12) ˜Ψ Φ=0 o ( θ ) (cid:69) . The flux can then be intro-duced as (cid:12)(cid:12)(cid:12) ˜Ψ Φ o ( θ ) (cid:69) = (cid:88) j c j ( θ ) e − i Φ N jA | j (cid:105) . (35)Computing the derivative of the ground state with re-spect to the parameter θ gives ∂ θ (cid:12)(cid:12)(cid:12) ˜Ψ Φo ( θ ) (cid:69) = (cid:88) j [ ∂ θ c j ( θ )] e − i Φ N jA | j (cid:105) . (36)The correspondent Berry connection then gives (cid:68) ˜Ψ Φo ( θ ) (cid:12)(cid:12)(cid:12) ∂ θ (cid:12)(cid:12)(cid:12) ˜Ψ Φo ( θ ) (cid:69) = (cid:88) lj c ∗ l ( θ )[ ∂ θ c j ( θ )] (cid:104) l | e i Φ( N lA − N jA ) | j (cid:105) = (cid:88) j c ∗ j ( θ ) ∂ θ c j ( θ )= (cid:68) ˜Ψ ( θ ) (cid:12)(cid:12)(cid:12) ∂ θ (cid:12)(cid:12)(cid:12) ˜Ψ ( θ ) (cid:69) , (37)i.e. it is independent of Φ.The integral in Eq.(33) is performed on the surfaceof the torus defined by ( θ, Φ). Because of the branchcut, the Berry connection A Φ ( θ ) cannot be made smoothfor the whole torus and therefore we cannot directlyapply Stokes theorem. Instead, we split the torus intwo cylinders, S with θ ∈ [ θ , π − θ ] and S with θ ∈ [2 π − θ , π + θ ] and we change the gauge in A ( θ )such that it is smooth in each cylinder. We can now ap-ply Stokes theorem to the two cylinders independently[37]. For S we have C = 12 π (cid:90) S d S · ( ∇ × A )= 12 π (cid:90) ∂S d l · A ( θ, Φ)= 12 π (cid:90) π − θ θ dθA θ ( θ,
0) + 12 π (cid:90) π d Φ2 π A Φ (2 π − θ , Φ) − π (cid:90) π − θ θ dθA θ ( θ, π ) − π (cid:90) π d Φ2 π A Φ ( θ , Φ) . (38)In our case, A θ is independent of Φ so the terms involvingit cancel and we have C = (cid:90) π d Φ2 π [ A Φ (2 π − θ , Φ) − A Φ ( θ , Φ)] . (39)Similarly computing the integral for the other cylinderwe obtain C = (cid:90) π d Φ2 π [ A (cid:48) Φ (2 π + θ , Φ) − A (cid:48) Φ (2 π − θ , Φ)] . (40)The Chern number is then C = C + C . Noting againthat A (cid:48) Φ ( θ ) is continuous in S taking the limit of θ → + makes C vanish and the Chern number is given by C = (cid:90) π d Φ2 π [ A Φ (2 π − , Φ) − A Φ (0 + , Φ)] , (41)or C ( m ) = ˜ P o ( m, κ = 0 + ) − ˜ P o ( m, κ = 0 − ) , (42)where C ( m ) is the Chern number of any loop around thedegenerate point crossing κ = 0 at m and another pointin the trivial region ( m > P ˜ o - L / - - ˜ o - L / - - ˜ mod 1 - - FIG. 6: (a) and (b) show the bulk polarization˜ P o − L/ t = 1 , κ (cid:48) = 0 and L = 40 sites,with (a) t (cid:48) = 0 and (b) t (cid:48) = −
2. ˜ P o exhibits branch cutsthat indicate the presence of zero-energy modes. ˜ P o isable to differentiate between the phases ν = 2 and ν = 0. In (c) we show ˜ P mod 1 for the same parametersof (b) to illustrate the advantage of using ˜ P o .polarization described above (see the discussion belowEq. (32)) gives C ( m ) = 2 ˜ P o ( m, κ = 0 + ) − L, (43)which results in +1 when m < m >
1, asit is known for the SSH chain. The only contribution tothe Chern number comes from the discontinuity of ˜ P o atthe branch cut. B. ν = 2 and disordered case. The analysis we did above of ˜ P o for the SSH chainseems rather trivial as ˜ P o does not seem to have moreinformation than ˜ P modulo 1. This is no longer truewhen t (cid:48) (cid:54) = 0 allowing for a ν = 2 phase. In Fig. 6(b) wesee ˜ P o for the dotted cut in the phase diagram with anadded symmetry breaking κ term. We can see that ˜ P o is continuous for any path avoiding the branch cut eventhough the range of values for ˜ P o is larger than 1. Wecan see that a loop around the two degenerate points willhave a Chern number of C = 2.Jumps in ˜ P o originate from zero modes changing theiroccupation. Thus, a jump of n in ˜ P o implies the existenceof at least n zero-energy modes at that point. For thismodel this equivalence is exact and the jump of ˜ P o at κ = 0 gives the number of topological zero-modes, i.e.the winding number. Note, however, that there are manyways of breaking the chiral symmetry and for some ofthem, ˜ P o may not be related to the winding number anylonger. For instance, replacing κ in Eq. (2) by κ i =0 κ ( − i , the zero-energy modes split in such a way that˜ P o is continuous at κ = 0. It cannot differentiate betweenthe phases ν = 0 and ν = 2, and a loop like the onediscussed above results in a Chern number C = 0.To illustrate the advantage of ˜ P o we also show ˜ P mod1 in the ( m, κ ) parameter space in figure 6(c). As men-tioned above the only contribution to the Chern numbercomes from discontinuities in the bulk polarization. How-ever, when it is computed using ˜ P we no longer know inadvance where this discontinuities are located, due to itbeing defined modulo 1. One needs to look at the fullpath to know how the bulk polarization evolves alongthe loop.Since ˜ P o is defined in position space we can also usethis method for a disordered system. Assume now thatwe add disorder in the t and m parameters t i = t + W ω i m i = m + W ω (cid:48) i (44)where ω i and ω (cid:48) i are selected from a uniform distributionwith range [ − / , / P o will present many branchcuts as we approach the limit lim κ → , which makes it im-practical. In order to avoid this, we break chiral symme-try locally in the edge of the open chain so it only affectsthe edge modes. If any of the other states cross E = 0they do it in chiral-symmetric pairs such that ˜ P o is notaffected by it. We show ˜ P o in Fig.7(a) using the localsymmetry breaking term for a strong disorder, W = 3 t ,where the gap is indeed filled with states. We see that itworks just as in the case without disorder, where now dueto the disorder a region of ν = 1 opens up [28]. The maindifference is that since we break chiral symmetry only forthe edge modes, ˜ P o is quantized to multiples of 1 /
2. Notethat, since the states that fill the gap are localized in thebulk, they will not appear in the EOS and one can stilleasily identify the appearance of virtual topological edgestates (see figure 7(b)). In contrast, the topological edgestates in the energy spectrum are completely masked bythe (localized) bulk modes that fill the gap.
C. Chern Number in 2D
We now discuss the computation of Chern numbers in2D systems with ˜ P o . Consider the Chern insulator withinversion symmetry given by the Hamiltonian H = (cid:88) iα,jβ (cid:88) k y c † iα ( k y ) H iα,jβ ( k y ) c jβ ( k y ) , (45) H iα,jβ ( k y ) = 12 ( iσ x − σ z ) δ i,j +1 + 12 ( − iσ x − σ z ) δ i,j − +(sin( k y ) σ y + [2 − m − cos( k y )] σ z ) δ ij . (46)We can treat the Hamiltonian with elements H iα,jβ ( k y ) as a 1D Hamiltonian with an extra param-eter k y and compute ˜ P o , or equivalently A Φ (Φ , k y ), as in P ˜ o - L / - - ξ m ( b ) FIG. 7: (a) Bulk polarization ˜ P o computed for t = 1 , t (cid:48) = 2 , κ (cid:48) = 0, with an added disorder to t and m with strength W = 3, and κ is applied only at theedges. We compute it for L = 400, as the disorderincreases the finite size effect. The disorder opens aregion with ν = 1 as it has been observed in [28]. (b)EOS for the same parameters as (a), with κ = 0, wherewe have highlighted the 4 eigenvalues closest to ξ = 1 / / m = 1 and 3, respectively. For 0 < m < ξ = 1 / k y = 0.In this region the occupied bands have a Chern number C = +1. For 2 < m < ξ = 1 / k y = π and the system has C = −
1. For m < m > C = 0. When we open the chain, theeigenvalue of the right virtual topological edge mode willbe pushed to ξ = 0 or 1. This is seen in figure 8(a) and(b) for intermediate points in the process of opening thechain. This eigenvalue presents a jump whenever the cor-respondent edge-mode crosses zero energy, going between ξ = 1 and ξ = 0, i.e. the number of eigenvalues at ξ = 1is not constant across k y . In this model, due to inversionsymmetry, this is only possible at k y = 0 , π .We can now take a look at ˜ P o in the parameter space( m, k y ), shown in figure 8(c). We see, again, that ˜ P o re-mains continuous except for the branch cuts that appearat the points where the physical edge-modes cross zeroenergy. The Chern number can be obtained as C ( m ) = (cid:90) π − π dk y π ∂ k y ˜ P o ( m, k y ) . (47)However, since we know the position of the branch cutswe can integrate avoiding them and obtain C ( m ) =[ ˜ P o ( m, k y = 0 − ) − ˜ P o ( m, k y = − π + )]+ [ ˜ P o ( m, k y = π − ) − ˜ P o ( m, k y = 0 + )] . (48)The expression in the right hand side is actually a knowntopological invariant, the trace index [24], defined in1 - - ξ k y / π ( a ) - - ξ k y / π ( b ) P ˜ o - L / - - FIG. 8: (a) and (b) EOS of the Chern insulator in the C = 1 ( m = 1) and C = − m = 3) phases,respectively. In gray and light gray lines we plot twointermediate points of the adiabatic process of openingthe chain between sites 1 and L , showing how themidgap state of the right virtual cut evolves towards thebulk bands, introducing a discontinuity in momentum.(c) ˜ P o − L/ L = 40 sites.terms of the trace of C A,o . In reference [24] it is shown tobe equal to the Chern number by relating it to the Hallcurrent. Here, the connection between the trace indexand the Chern number is seen by framing Tr[ C A,o ] as ageometric quantity
VII. CONCLUSION
It was previously known that the bulk polarization(Zak phase, or geometric phase for U (1) flux insertion)was encoded in the ES. Here we have shown how it can beobtained from the single-particle ES of a non-interactinggapped system, even when it is not quantized, showingthat there is substantially more information in the single-particle ES than previously realized. Our formulation isboth simpler than the one involving the ES, which ismuch more difficult to compute, and it provides addi-tional insight into the single-particle ES and EOS. Inparticular in the topological case, the relation between the quantized bulk polarization (or the Chern number in2D) and the number of virtual topological edge states isdemonstrated in a particularly transparent way. It alsoprovides a new simple method for computing the bulk po-larization for systems without translational invariance.We also define a novel bulk polarization that is con-tinuous in R for gapped paths. This provides additionalinformation about the edge spectrum, similar to how theZak phase is equal to the winding number when com-puted in a particular gauge. Our new bulk polarizationsimplifies the calculation of changes in the bulk polar-ization, and provides a new route to the computation ofChern numbers. Outlook:
In the application of our results to topologicalsystems in 1D we have focused on systems with transla-tional invariance even though our results do not requireit. For general spatially inhomogeneous systems the samebulk can support different boundaries so the relation be-tween bulk and boundary is not as straightforward as inthe translational invariant case. Our results might givemore insight into this relation.As it stands, the relation derived in this paper onlyworks for non-interacting systems. With interactions, thecorrelation matrix does not have the full information ofthe ground state. However, there might be certain sys-tems, perhaps for weakly interacting systems, where thecorrelation matrix encodes the topological information.This would be worth exploring since, even for interactingsystems, the correlation matrix is much easier to computethan the density matrix.In the field of higher-order topological insulators somesystems can be characterized by a physical generaliza-tion of the bulk polarization to two dimensions, thequadrupole moment [38, 39]. This quadrupole moment isdefined via the position operator and it is an open ques-tion whether there is a formulation in terms of twistedboundary conditions. In this case, one might be able toconstruct this quadrupole moment using the ES, as wedid here for the bulk polarization.
ACKNOWLEDGMENTS
Acknowledgments.–
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A ShortCourse on Topological Insulators: Band Structure andEdge States in One and Two Dimensions (Springer In-ternational Publishing, Cham, 2016) pp. 55–68.[35] Ingo Peschel, Matthias Kaulke, and ¨Ors Legeza,“Density-matrix spectra for integrable models,” Annalender Physik , 153–164 (1999).[36] This is strictly speaking only true in the thermodynamiclimit, where we avoid the finite size effects due to openingthe chain.[37] Mahito Kohmoto, “Topological invariant and the quanti-zation of the hall conductance,” Annals of Physics ,343 – 354 (1985).[38] Wladimir A. Benalcazar, B. Andrei Bernevig,and Taylor L. Hughes, “Quantized electric mul-tipole insulators,” Science , 61–66 (2017),https://science.sciencemag.org/content/357/6346/61.full.pdf.[39] Byungmin Kang, Ken Shiozaki, and Gil Young Cho,“Many-body order parameters for multipoles in solids,”Phys. Rev. B , 245134 (2019). Appendix A: Computing the Bulk polarization fromthe Schmidt decomposition
In this appendix we review how to compute the bulkpolarization from the Schmidt decomposition [25] andrelate it to our own results.Consider a closed chain of length L and a bipartition3into regions A , for i ∈ [1 , L/ B , for i ∈ [ L/ , L ].Using a Schmidt decomposition the ground state gives (cid:12)(cid:12) Ψ (cid:11) = (cid:88) p,q s p s (cid:48) q | p, q (cid:105) A | q, p (cid:105) B , (A1)The Schmidt indices p and q label the fluctuations atthe left ( i = 1) and right ( i = L/
2) cuts, respectively.The convention used for the states | p, q (cid:105) R is that the first(second) index labels the state near the left-most (right-most) region of R , where R can be A or B . Assumingthe system is large enough compared with the correlationlength the fluctuations across the two cuts are indepen-dent of each other. The reduced density matrix can becomputed as ρ A = (cid:88) p,q s p s (cid:48) q | p, q (cid:105) A (cid:104) p, q | A . (A2)As mentioned in section IV it is fully determined by thecorrelation matrix C A . In the state | p, q (cid:105) A , the Schmidtindex p ( q ) labels a set of occupation numbers of theeigenstates of C A related to the cut at i = 1 ( i = L/ A L ( A R ). Note that the remaining subspace A bulk is com-posed of eigenstates that have eigenvalues exponentiallyclose to ξ = 0 ,
1. The eigenvalues of ρ A , λ pq = s p s (cid:48) q canbe obtained as [24] λ pq = (cid:89) µ ∈ A (1 − ξ µ ) (cid:18) ξ µ − ξ µ (cid:19) n pqµ = (cid:89) µ ∈ A L (1 − ξ µ ) (cid:18) ξ µ − ξ µ (cid:19) n pµ × (cid:32) (cid:89) ν ∈ A R (1 − ξ ν ) (cid:18) ξ ν − ξ ν (cid:19) n qν (cid:33) , (A3)from which we identify s p = (cid:89) µ ∈ A L (1 − ξ µ ) (cid:18) ξ µ − ξ µ (cid:19) n pµ s (cid:48) q = (cid:89) ν ∈ A R (1 − ξ ν ) (cid:18) ξ ν − ξ ν (cid:19) n qν . (A4)We are now in position to introduce the U (1) flux viaa twisted boundary condition as (cid:12)(cid:12) Ψ Φ (cid:11) = (cid:88) pq s p s (cid:48) q e − i Φ N pAL | p, q (cid:105) A | q, p (cid:105) B , (A5)where N pA L = (cid:80) µ ∈ A L n pµ . Note that there is an ambi-guity in how the flux is introduced as one can alwaysinclude bulk modes into A L . As mentioned above, thelabel p describes the set of occupation numbers { n pµ } .Equation (A5) means that an electron crossing the vir-tual cut at i = 1 between region B and A L will acquire a phase − Φ, the convention on the sign is the same as theone used in reference [32].The polarization can be now computed as˜ P = (cid:90) π d Φ2 π i (cid:10) Ψ Φ (cid:12)(cid:12) ∂ Φ (cid:12)(cid:12) Ψ Φ (cid:11) = (cid:90) π d Φ2 π i (cid:88) p,q s p s (cid:48) q e i Φ N pAL ∂ Φ e − i Φ N pAL = (cid:90) π d Φ2 π (cid:88) p,q s p s (cid:48) q N pA L = (cid:88) p s p (cid:88) µ ∈ A L n pµ , (A6)where we used that (cid:80) q s (cid:48) q = 1. Inserting now the ex-pression for s p we obtain˜ P = (cid:88) { n i ∈ AL } =0 , (cid:88) j ∈ A L n j (cid:32) (cid:89) k ∈ A L λ k (cid:33) , (A7)where we defined λ k = (1 − ξ k ) (cid:18) ξ k − ξ k (cid:19) n k . (A8)If we expand the sum for one particular occupationnumber n p we have˜ P = (cid:88) { n i (cid:54) = p ∈ AL } =0 , n p =0 (cid:88) j (cid:54) = p ∈ A L n j (1 − ξ p ) (cid:89) k (cid:54) = p ∈ A L λ k + (cid:88) { n i (cid:54) = p ∈ AL } =0 , n p =1 (cid:88) j (cid:54) = p ∈ A L n j + 1 ξ p (cid:89) k (cid:54) = p ∈ A L λ k = (cid:88) { n i (cid:54) = p ∈ AL } =0 , (cid:88) j (cid:54) = p ∈ A L n j (cid:89) k (cid:54) = p ∈ A L λ k + ξ p (cid:88) { n i (cid:54) = p ∈ AL } =0 , (cid:89) k (cid:54) = p ∈ A L λ k (A9)If we continue expanding the sum in the first term wewill get terms like the second one for all the other k (cid:54) = p eigenvalues. If we further expand the sum in the factoraccompanying ξ p we have(1 − ξ p (cid:48) ) (cid:88) { n i (cid:54) = p,p (cid:48)∈ AL } =0 , n p (cid:48) =0 (cid:89) k (cid:54) = p,p (cid:48) ∈ A L λ k + ξ p (cid:48) (cid:88) { n i (cid:54) = p,p (cid:48)∈ AL } =0 , n p (cid:48) =1 (cid:89) k (cid:54) = p,p (cid:48) ∈ A L λ k = 1 . (A10)4If we continue this procedure for all other occupationnumbers we arrive at˜ P = (cid:88) p ∈ A L ξ p mod 1 . (A11)In terms of the spectrum of C A the bulk polarizationsimplifies greatly. As mentioned above, the subspace A L is not well-defined, as one can always include bulk modes,however since the bulk polarization is defined modulo1, this issue is irrelevant. In practice we extend A L toinclude all eigenstates whose average position lies in theleft half of A (which we denote by L ) and we finally get˜ P = (cid:88) i ∈ L ξ i mod 1 . (A12) Appendix B: Alternative expression for thecorrelation matrix
In this appendix, we show how to express the correla-tion matrix in terms of the Hamiltonian, used in Eq. (21).We consider the generic, quadratic Hamiltonian in onedimension of Eq. (5), which is diagonalized by a unitarymatrix U with H = U DU † with D = diag( E pµ ) . (B1)In terms of the fermionic operators that diagonalize theHamiltonian, γ iα = (cid:88) jβ ψ iα ∗ jβ c jβ , (B2)we find that the correlation matrix can be written as C αβij = (cid:88) pq,µν ψ qνjβ ψ pµ ∗ iα (cid:10) γ † pµ γ qν (cid:11) = (cid:88) pµ ψ pµjβ ψ pµ ∗ iα (cid:10) γ † pµ γ pµ (cid:11) = (cid:88) pµ ψ pµjβ ψ pµ ∗ iα [1 − sign( E pµ )] / . (B3)The first term of the last line is simply a kronecker deltabetween both sets of indices. The second term can berewritten, using Eq. (B1), as U D ( | D | ) − U † = U DU † U ( D ) − / U † (B4)We can rewrite this expression further by noting that[ U ( D ) − / U † ] = U ( D ) − / U † U ( D ) − / U † = U ( D ) − / ( D ) − / U † = U ( D ) − U † =( U D U † ) − . (B5)Therefore, U ( D ) − / U † =( U D U † ) − / (B6) and we conclude that U D ( | D | ) − U † = U DU † ( U D U † ) − / = H ( H ) − / . (B7)Combining Eq.s (B3), and (B7), we arrive at the finalexpression of the correlation matrix in Eq. (21). Appendix C: Winding number in the SSH chain
Consider the Bloch Hamiltonian of the Rice-Melemodel H = (cid:32) κ f ( k ) f † ( k ) − κ (cid:33) , (C1)which can also be written as H = h · σ where h =(Re[ f ( k )] , − Im[ f ( k )] , κ )= (cid:113) | f ( k ) | + κ (sin( θ ) cos( φ ) , sin( θ ) sin( φ ) , cos( θ ))In the gauge where the second component of the eigen-states remains real they are given by | + (cid:105) = 1 N + (cid:32) cot( θ/ e − iφ (cid:33) (C2) |−(cid:105) = 1 N − (cid:32) − tan( θ/ e − iφ (cid:33) , (C3)where N + = (cid:113) ( θ/ N − = (cid:113) ( θ/ . The Berry connections of the occupied state, A λ = (cid:104)−| ∂ λ |−(cid:105) , can be obtained as A λ = i cos( θ ) − ∂ q φ (C4)= i cos( θ ) −
12 cos( φ ) ∂ q tan( φ )= i κ (cid:113) | f ( k ) | + κ − (cid:18) Re[ f ( k )] | f ( k ) | (cid:19) ∂ λ (cid:18) − Im[ f ( k )]Re[ f ( k )] (cid:19) = − κ (cid:113) | f ( k ) | + κ − f ( k ) † ∂ λ f ( k ) − f ( k ) ∂ λ f ( k ) † | f ( k ) | The Berry connection with respect to momentum gives A k = it [ t + m cos( k )][ κ − (cid:112) κ + m + t + 2 mt cos( k )]2[ m + t + 2 mt cos( k )] (cid:112) κ + m + t + 2 mt cos( k ) , (C5)5 γ Z / π FIG. 9: γ/ π obtained in the smooth gauge thatprovides the winding number in the chiral symmetriclimit. and the resulting Zak phase can be seen in Fig. 9, whichpresents a branch cut for m = 0 , κ < κ = 0, theZak phase is γ = (cid:90) π dk f ( k ) † ∂ k f ( k ) − f ( k ) ∂ k f ( k ) † | f ( k ) | = (cid:90) π dk q ( k ) † ∂ k q ( k ) , (C6)where q ( k ) = f ( k ) / ||
12 cos( φ ) ∂ q tan( φ )= i κ (cid:113) | f ( k ) | + κ − (cid:18) Re[ f ( k )] | f ( k ) | (cid:19) ∂ λ (cid:18) − Im[ f ( k )]Re[ f ( k )] (cid:19) = − κ (cid:113) | f ( k ) | + κ − f ( k ) † ∂ λ f ( k ) − f ( k ) ∂ λ f ( k ) † | f ( k ) | The Berry connection with respect to momentum gives A k = it [ t + m cos( k )][ κ − (cid:112) κ + m + t + 2 mt cos( k )]2[ m + t + 2 mt cos( k )] (cid:112) κ + m + t + 2 mt cos( k ) , (C5)5 γ Z / π FIG. 9: γ/ π obtained in the smooth gauge thatprovides the winding number in the chiral symmetriclimit. and the resulting Zak phase can be seen in Fig. 9, whichpresents a branch cut for m = 0 , κ < κ = 0, theZak phase is γ = (cid:90) π dk f ( k ) † ∂ k f ( k ) − f ( k ) ∂ k f ( k ) † | f ( k ) | = (cid:90) π dk q ( k ) † ∂ k q ( k ) , (C6)where q ( k ) = f ( k ) / || f ( k ) ||