On the equivalence of the Bott index and the Chern number on a torus, and the quantization of the Hall conductivity with a real space Kubo formula
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec On the equivalence of the Bott index and the Chern numberon a torus, and the quantization of the Hall conductivitywith a real space Kubo formula
Daniele Toniolo
Department of Physics, Norwegian University of Science and Technology, N-7491Trondheim, NorwayE-mail: [email protected]
Abstract.
The equivalence of the Bott index and the Chern number is established inthe thermodynamic limit for a gapped, short ranged and bounded Hamiltonian on a twodimensional torus of linear size L . A Kubo formula as an exact operatorial identity isprovided in real space and used to show the quantization of the transverse conductance withincorrections of order L − . In doing so the physical foundations of the theory that introduces theBott index in the realm of condensed matter as proposed by Hastings and Loring in J. Math.Phys. (51), 015214, (2010) and Annals of Physics 326 (2011) 1699-1759 are recalled.PACS numbers: n the equivalence of the Bott index and the Chern number on a torus
1. Introduction
The integer quantization of the transverse (Hall) conductance (IQHE) for a two dimensionalelectronic gas under an external perpendicular magnetic field has been experimentallydiscovered in 1980 [1], the fractional quantization (FQHE) a couple of years later [2]. Thetheoretical analysis of these phenomena has never stopped since. Initial landmarks have beenestablished by Laughlin [3], Halperin [4] and Thouless [5]. Different schools of thoughtoriginated to explain these phenomena: who focuses on the two-dimensional bulk aspects ofthe sample [6]; who stresses the relevance of the one dimensionality of the edge [7], whoon the interplay between bulk and edge-physics [8]. The attention to a realistic geometricalsetting is particularly relevant in the approach of Buttiker [9], while a rigorous treatment of thestrong disorder needed for the quantization conductance is central in the work of Bellissard[10]. The initial sections of [10] can be used as an introduction to the subject of the IQHE.Another line of research particularly careful on the mathematical physical aspects of IQHEand FQHE is due to Avron, Seiler and Simon [11]. Haldane in 1988 formulated a latticemodel with localized magnetic flux over the corners of a honeycomb lattice but total zeroflux per plaquette that manifests a quantized transverse conductance [12], nowadays calledChern insulator. This model has been relevant to the theoretical formulation [13] [14] andexperimental discovery [15] of the topological insulators.The quantization of the Hall conductance on a torus geometry, that means that periodicboundary conditions are imposed on the two dimensional sample, is determined by atopological invariant called Chern number. This has been showed for the first time in theref. [5], the relation has then been made explicit in the ref. [16] by Kohmoto. Also previoussuggestions were provided in the references [17] and [18]. Countless mathematical booksdescribe the tools of differential geometry needed to understand how the concept of Cherninvariant is used in the literature cited above, two possible references are [19] and [20].Hastings and Loring in a set of articles [21] [22] [23] used several theoretical toolsincluding non commutative topology, C* algebras and K-theory to rigorously search forthe topological invariants of the ten Altland and Zirnbauer symmetry classes [24] in a waythat would be also relevant for numerical computations. The program of classification oftopological invariants of Fermi systems according to their symmetries and dimensionalitystarted with the works of Qi et al. [25], Kitaev [26] and Ryu et al. [27].One of the motivations for this work is to collect in the section 2 the physical foundationsof the theory that leads to the use of the so called Bott index in condensed matter as developedby Hastings and Loring [21] [23]. This index has been used also by other authors for aclassification of topological classes using scattering methods [28]. Moreover it has recentlybeen employed to characterized the topological properties of out of equilibrium systems [29][30] [31]. The section 3 defines and discuss the Bott index according to [21, 23, 32]. Thenovelties are in last two sections: a direct proof of the equivalence of the Bott index andthe Chern number in the thermodynamic limit in section 5, and a Kubo formula presentedin real space as an exact operatorial equality in section 4 that proves the quantization of theHall conductance in finite systems within a correction of order O ( L − ) . For a proof of theBott index - Chern number equivalence based on a “momentum space” approach the reader isdirected to the recent reference [33].
2. Physical setting
The physical system under investigation is a set of free fermions, with in general N internaldegrees of freedom, described by a short ranged, bounded and gapped Hamiltonian on a lattice n the equivalence of the Bott index and the Chern number on a torus H = ∑ i , j Ψ † i H i , j Ψ j (1)A particle at a given site i can hop at most within a distance equal to R , this is the range ofthe Hamiltonian. This can alternatively be stated writing H = ∑ Z H Z with H Z supported on aregion Z of linear size R . The Hamiltonian is bounded, this means that k H k ≤ J .The Bott index, as described in the following section 3, is an index of matrices. A coupleof unitary, or quasiunitary, matrices is arbitrarily closed to a couple of commuting unitary(quasi unitary) matrices if and only if their Bott index is vanishing. The index is also welldefined for Hermitian and other classes of matrices [35], [32], [36]. See in particular the latterreference for a discussion of all the symmetry classes. The definitions 26 and 30 make itclear that the Bott index could be non vanishing only when the dimension of P is bigger thanone, but this is surely not a problem in a many body setting. It is important to note that thequantization of the index 30 is exact so there is no obstacle to its application to a few bodyproblem as well.To construct the torus we glue together the opposite sides of a rectangle of linear sizes L x and L y that are supposed of the same order L . For every lattice site i on the rectangle of position ( x i , y i ) we build the diagonal matrix X with elements X i , j = x i δ i , j and the corresponding matrix Y , Y i , j = y i δ i , j . The matrices X and Y have L x L y diagonal elements. Note that points that areclose on the lattice may have corresponding values of x (or y ) far in the matrix X but at most(of the order of) L x elements distant. We define the diagonal unitary matrices U and V withelements: U i , i = exp (cid:18) i π L x X i , i (cid:19) , V i , i = exp (cid:18) i π L y Y i , i (cid:19) (2)The real diagonal matrices X , X , Y and Y are defined as: L x U = X + iX and L y V = Y + iY . They satisfy X + X = L x and Y + Y = L y , moreover X i , i ≤ | X i , i | + | X i , i | and Y i , i ≤ | Y i , i | + | Y i , i | . With θ ∈ [ , π ] this is like to say: θ π ≤ | cos θ | + | sin θ | . Taking T equalto any of the matrices X , X , Y , Y and representing the Hamiltonian matrix H i , j accordingto the construction of X i , j and Y i , j , the previous assumptions imply: k [ T , H ] k ≤ O ( RJ ) (3)and k [ T , P ] k ≤ O (cid:18) RJ ∆ E (cid:19) (4)Let us sketch the proof of 3. The value of the elements of T ranges from 0 to L . T isdiagonal then only the off diagonal part of H contributes to the commutator. Since the matrix H connects only sites that are at most at a distance equal to R , the commutator of eq. 3 is theoff diagonal part of H with elements multiplied at most for a factor, in modulus, equal to R ,then eq. 3 follows. n the equivalence of the Bott index and the Chern number on a torus P on the occupied energylevels can be written as P = π i I Γ E dz ( z − H ) − (5)(6)then: [ T , P ] = π i I Γ E dz (cid:2) T , ( z − H ) − (cid:3) (7) (cid:2) T , ( z − H ) − (cid:3) = ( z − H ) − ( z − H ) T ( z − H ) − − ( z − H ) − T ( z − H )( z − H ) − = ( z − H ) − [ T , H ] ( z − H ) − = R ( z ) [ T , H ] R ( z ) (8)The resolvent R ( z ) ≡ ( z − H ) − has been introduced. Then: [ T , P ] = π i I Γ E dzR ( z ) [ T , H ] R ( z ) (9) k [ T , P ] k ≤ π | Γ E |k [ T , H ] k sup z ∈ Γ E k R ( z ) k (10) | Γ E | denotes the length of the contour Γ E enclosing the value of the energy E in the complexplane, | Γ E | ≤ O ( ∆ E ) . Being k R ( z ) k = dist ( z , σ ( H )) − , then sup z ∈ Γ E k R ( z ) k ≤ O ( ∆ E ) − , sothe eq. 4 is proven. From now on L x and L y are identified with L . From above, in particularusing X i , i ≤ | X i , i | + | X i , i | , it follows that: k [ X , H ] k ≤ O ( RJ ) (11) k [ X , P ] k ≤ O (cid:18) RJ ∆ E (cid:19) (12) k [ e ( i π L X ) , P ] k ≤ O (cid:18) RJL ∆ E (cid:19) (13)Assuming that the energy gap ∆ E does not vanish increasing the size of the system, the rhs ofeq. 13 in the thermodynamic limit is of order O ( L − ) . Let us consider the Baker-Campbell-Hausdorff formula to write: [ e ( i π L X ) , H ] e ( − i π L X ) = e ( i π L X ) H e ( − i π L X ) − H (14) = π L [ iX , H ] + (cid:18) π L (cid:19) [ iX , [ iX , H ]] + h . o . (15)Applying the argument employed to prove eq. 3 we can show that: (cid:18) π L (cid:19) k [ iX , [ iX , H ]] k ≤ O (cid:18) L (cid:19) (16)then the only term of order O ( L − ) contributing to k [ e ( i π L X ) , H ] k is the first one of eq. 15. n the equivalence of the Bott index and the Chern number on a torus
3. Bott index
The Bott index has been introduced in the context of condensed matter physics by Hastingsand Loring in a set of papers [21], [22], [23]. They used this index to study in the ref. [21] adisordered Bernevig-Hughes-Zhang model [34], the prototype of topological insulators. Fora discussion of the Bott index in a mathematical settings and for references to the originalworks see [32].Let us start recalling the definition of the Bott index given for unitary matrices andthen extend it to the case relevant in our condensed matter system of the projected operators P e ( i π XL ) P and P e ( i π YL ) P .The Bott index is a winding number of unitary matrices. Given two unitary matrices U and V , it is: k UVU † V † − k = k [ U , V ] k , this is shown using VUU † V † = . With k [ U , V ] k = δ the spectrum of UVU † V † is such that: σ (cid:0) UVU † V † − (cid:1) ⊆ { z ∈ C : | z − | ≤ δ } .This is nothing but the definition of the operatorial norm as the eigenvalue of maximummodulus. With δ < UVU † V † is out of the real negative axis in fact thiscorresponds to exclude the value z = −
1, then we can employ the complex log with the branchcut on the real negative axis to define:Bott ( U , V ) ≡ π i Tr log (cid:0)
UVU † V † (cid:1) (17)Being (cid:0) UVU † V † (cid:1) unitary, within the given hypothesis the spectrum of log (cid:0) UVU † V † (cid:1) ispurely imaginary then the index is real. It is moreover an integer, in fact given A non singular(if we want to use the principal branch of the log we also require that σ ( A ) ∩ R − = /0) thenTr log A = log det A + π mi with m ∈ Z. In our case det (cid:0)
UVU † V † (cid:1) =
1, then Bott ( U , V ) is an integer. It is immediate to see also that when U and V are commuting their index isvanishing. On the other hand it is possible to show that if the index of U and V is vanishingthere exists a couple of commuting unitary matrices U and V that are arbitrary closed to U and V : k U − U k + k V − V k ≤ ε . See Theorem 2.6 of [32] for the precise statement and aproof.When Bott ( U , V ) = U and V are at a finite distance from any commuting coupleof unitary matrices, see Proposition 2.5 of [32].Consider an homotopy t → ( U t , V t ) , t ∈ [ , ] of unitary matrices that deforms ( U , V ) to ( U , V ) , with k [ U , V ] k < k [ U , V ] k < ( U , V ) = Bott ( U , V ) thereexists ˜ t ∈ ( , ) such that k [ U ˜ t , V ˜ t ] k =
2. See Lemma 2.4 of [32].This fact is important for applications of the Bott index to the study of out of equilibriumsystem where it has been recognize that the index can change following the unitary timeevolution [29]. This is investigated also in [31].It might be interesting to express the Bott index as a winding number, also to possiblyrecognize its relation with other indexes in the physics literature. In section IV of [22], seealso [32] and references therein, the Bott index is defined as the winding number of the loop: γ ( t ) = det ( tUV + ( − t ) VU ) , with t ∈ [ , ] . According to the definition of winding numberof a loop we have Bott ( U , V ) ≡ π i I γ dzz (18) = π i Z dt γ − ( t ) ∂ t γ ( t ) (19) = π i Z dt ∂ t log det ( tUV + ( − t ) VU ) (20) n the equivalence of the Bott index and the Chern number on a torus A , it holds: ∂ φ log det A = Tr ∂ φ log A . Before using this formulawe want to put in evidence UVU † V † as the argument of the log. So, with:det ( t ( UV ) + ( − t ) VU ) = det ( t [ U , V ] + VU ) (21) = det [( t [ U , V ] U † V † + ) VU ] (22) = det ( t [ U , V ] U † V † + ) det ( VU ) (23)we have: Bott ( U , V ) = π i Tr log ( t [ U , V ] U † V † + ) (cid:12)(cid:12)(cid:12) (24)We see that the two definitions 17 and 18 are equivalent.The Bott index is still well defined relaxing the condition of unitarity of matrices, thisis where the physical applications come in. Let us consider the projector P on the occupiedstates of the Hamiltonian 1, we recall that the existence of an energy gap is supposed. Then P = W † (cid:18) n (cid:19) W (25)with n = dim P and W the unitary matrix with columns equal to the eigenvectors of H .Let us consider the matrices P e ( i π XL ) P and P e ( i π YL ) P , using 25 : P e ( i π XL ) P = W † (cid:18) U (cid:19) W (26) P e ( i π YL ) P = W † (cid:18) U (cid:19) W (27)The matrices U and U almost commute and are quasi unitarity: k [ U , U ] k ≤ O ( L − ) (28) k U a U † a − n k ≤ O ( L − ) a ∈ { , } (29)The proof is done following the methods of Lemma 5.1 of [22] and using eq. 13.The Bott index of U and U is defined as:Bott ( U , U ) ≡ π ImTr log (cid:16) U U U †1 U †2 (cid:17) (30)Let us show three facts, as stated in section 5.3 of [23]: k U U U †1 U †2 − k ≤ O ( L − ) (31) k log ( U U U †1 U †2 ) − ( U U U †1 U †2 − ) k ≤ O ( L − ) (32)Bott ( U , U ) − π ImTr ( U U U †1 U †2 ) ≤ O ( L − ) (33)Eq. 31 follows by 28 and U U U †2 U †1 = + O ( L − ) . Eq. 32 follows by the Mercator series:given k A − k < A = ∞ ∑ n = ( − ) n + ( A − ) n n (34)Then, assuming also A normal: k log A − ( A − ) k = k ∞ ∑ n = ( − ) n + ( A − ) n n k ≤ k A − k ∞ ∑ n = k A − k n − n With A = U U U †1 U †2 eq. 32 follows. Eq. 33 follows by noticing that the trace of a matrixis less equal than the norm of the matrix itself times the dimension of the matrix that in ourcase is n = dim P . When the number of internal degrees of freedom of a particle is N then thematrices appearing in eq. 26 have dimension NL that is O ( L ) , then:Tr h log ( U U U †1 U †2 ) − ( U U U †1 U †2 − ) i ≤ O ( L − ) O ( L ) (35) n the equivalence of the Bott index and the Chern number on a torus
4. Kubo formula
The transverse conductivity is computed as the long time transverse response of the systemto an electric field that is adiabatically switch on at early times [38], [37]. The responsecould be equivalently evaluated, in a two dimensional setting, as the response to a “pierced”magnetic field [39]. The approach that I pursue, that will be the tool to prove the quantizationof the Hall conductance on a torus using the Bott index, is to show that the formula forthe transverse conductivity obtained in the context of perturbation theory arises as an exactoperatorial identity. This approach was presented by Avron and Seiler in the ref. [40]. Theywere considering a setting where the Hamiltonian has a differential dependence on a “flux”parameter. The transverse conductance averaged over the “flux” torus T reads σ H = i π Z T Tr ˆ R E dHP E dH ˆ R E (36)The identification of σ H with the Chern number occurs proving the identity Q E dP E P E dP E Q E = ˆ R E dHP E dH ˆ R E (37) P E and Q E are the projector and its orthogonal on the ground state energy E , ˆ R E ≡ Q E R ( E ) Q E is the reduced resolvent. Note that the resolvent R ( z ) is singular on the spectrum of theHamiltonian unless when evaluated on E ∈ σ ( H ) it acts on Q E , the reason is Q E sends to zerothose vectors that would make R ( E ) to diverge.I prove eq. 37 in a modified form suitable for our system resembling the structure of theChern number as defined below 44. Dropping the subscript E of P and Q , I want to provethat: Q [ − iX , P ] P [ − iY , P ] Q = ˆ R E [ − iX , H ] P [ − iY , H ] ˆ R E (38)To show the equality we start writing P in the commutators on the lhs of 38 as a contourintegral, see eq. 9 Q [ − iX , P ] P [ − iY , P ] Q = (39) Q π i I Γ E dzR ( z ) [ − iX , H ] R ( z ) P π i I Γ ′ E dz ′ R ( z ′ ) [ − iY , H ] R ( z ′ ) Q (40) = π i I Γ E dz ∑ n = P n z − E n [ − iX , H ] Pz − E π i I Γ ′ E dz ′ Pz ′ − E [ − iY , H ] ∑ n ′ = P n ′ z ′ − E ′ n (41) = ∑ n = P n E − E n [ − iX , H ] P [ − iY , H ] ∑ n ′ = P n ′ E − E ′ n (42) = ˆ R E [ − iX , H ] P [ − iY , H ] ˆ R E (43)Note that: Γ E and Γ ′ E are contours encircling E in the complex plane, so, given E ≡ E , allthe eigenvalues E n with n = Γ E and Γ ′ E . In the first passage Q and P act ontheir neighboring resolvents, then the contour integrals are performed.
5. Equivalence of the Bott index and the Chern number in the thermodynamic limit
We start this section introducing an expression of the Chern number that is suitable for theproof of the equivalence with the Bott index. This appeared as eq. (19) in the remarkable n the equivalence of the Bott index and the Chern number on a torus ( P E ) ≡ − π ImTr u . a . Q E [ − iX , P E ] P E [ − iY , P E ] (44)The trace Tr u . a . stays for the unit area trace that is: Tr u . a . ≡ lim A → ∞ Tr A A . Tr A is the traceover the Hilbert space of functions with periodic boundary conditions over the area A andnormalization equal to A . The definition 44 as given in the references above does not includethe factor Q E . Its presence is due to the equality [ A , P ] = P [ A , P ] Q + Q [ A , P ] P with A any“physical” operator. This will turn out useful in the proof of the equivalence of the Chernnumber and the Bott index.The equivalence of the usual formulation of the Chern number involving momentum(fibered) projectors P ( k ) and that given in eq. 44 in absence of disorder is particularlyemphasized in the ref. [42].An alternative perspective to the quantization of the Hall conductance that leads to thesame results of [10] is that of Avron et al. [11] that considered the Fredholm index of a coupleof projectors.Let us see how the Bott index as in eq. 33 reduces to the expression of the Chern numberof eq. 44 within a correction of order O (cid:0) L − (cid:1) . It is worth to stress that the correspondence,as proven here, holds within the given hypothesis on the system’s Hamiltonian, that is shortranged, bounded and gapped leading to the set of bounds of eqs. 12 and 13. The Bott indexas given by eq. 33 is:Bott = π ImTr (cid:16) P e i θ x P e i θ y P e − i θ x P e − i θ y P (cid:17) + O (cid:0) L − (cid:1) (45)We start inserting two identities = e − i θ x e i θ x and = e − i θ y e i θ y , then it readsBott = Im2 π Tr (cid:16) P e i θ x P e − i θ x e i θ x e i θ y P e − i θ x e − i θ y e i θ y P e − i θ y P (cid:17) + O (cid:0) L − (cid:1) (46)Then considering that:e i θ x P e − i θ x = P + [ i θ x , P ] + O (cid:0) L − (cid:1) (47)e i ( θ x + θ y ) P e − i ( θ x + θ y ) = P + [ i ( θ x + θ y ) , P ] + O (cid:0) L − (cid:1) (48)e i θ y P e − i θ y = P + [ i θ y , P ] + O (cid:0) L − (cid:1) (49)we plug 47, 48, 49 into 46 discarding the resulting Hermitian operators, in fact their traceis real, and the terms of order O (cid:0) L − (cid:1) or higher inside the trace that gives a contribution oforder O (cid:0) L − (cid:1) or higher. Only two terms are remaining such that:Bott = Im2 π Tr ( P [ i ( θ x + θ y ) , P ] [ i θ y , P ] + P [ i θ x , P ] [ i ( θ x + θ y ) , P ]) + O ( L − ) Then employing the useful property of the commutator of any (physical) operator A with aprojector P , with Q ≡ − P , such that: [ A , P ] = P [ A , P ] Q + Q [ A , P ] P (this equality reaffirmsthat the trace of a commutator is vanishing) we get: Bott = π ImTr2 P [ i θ x , P ] Q [ i θ y , P ] + O (cid:0) L − (cid:1) (50)that reproduces the eq. 44 at the order O (cid:0) L − (cid:1) , using θ x = π L X , θ y = π L Y , and the definitionof trace per unit area.Note that if we wanted to keep track of the discarded terms in the Baker-Campbell-Hausdorff expansion of eqs. 47, 48 and 49 we might employ, also recursively, the formula:e i θ x P e − i θ x = P + Z du e iu θ x [ i θ x , P ] e − iu θ x (51) n the equivalence of the Bott index and the Chern number on a torus P ( t ) is replacedby U ( t , t ) P ( t ) U ( t , t ) † then the norm of the commutators 12 and 13 grows with time. Thishappens with a general out of equilibrium Hamiltonian H ( t ) .A proof of the Bott index - Chern number equivalence based on a “momentum space”approach has been recently discussed in [33].
6. Discussion and perspectives
It is relevant to stress some consequences and perspectives of the equivalence among the Bottindex and the Chern number. The formulation of the theory that leads to the construction ofthe Bott index admits the presence of disorder in the system. In fact the first application ofit was on a disordered BHZ model in ref. [21]. The role of the disordered in system withtopological features should not be overlook in fact it is the presence of strong disorder thatmakes possible the presence of plateaus in the shape of the Hall conductance as the externalmagnetic field is varied. See the introduction of [10] for a discussion. The Chern numberitself admits a formulation developed in [10], based on the use of tools from non commutativegeometry, that shows its quantization even in the presence of a disorder as strong as to closethe band gap, see for example Fig 1 of ref. [43]. The presence of a mobility gap is certainlystill required, otherwise the system would lose its insulating nature. The starting formula torealize that the Chern number is well defined also with strong disorder is precisely eq. 44.In the formulation of the Bott index a band gap has been assumed. It seems natural toask if the same extension of the Chern index to strong disorder, that for the quantization ofthe index requires just an average over all the possible disorder configurations [10, 44] mightshow a quantization of the Bott index as well.
7. Acknowledgements
It is a pleasure to thank Yang Ge and Marcos Rigol for correspondence, Yosi Avron fordiscussions and Hermann Schulz-Baldes for correspondence and for spotting some typos of aprevious version.
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