A New Approach to Transport Coefficients in the Quantum Spin Hall Effect

We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} does not commute with the spin operator in view of Rashba interactions, as in the typical models for the quantum spin Hall effect. A gapped periodic one-particle Hamiltonian H0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_0$$\end{document} is perturbed by adding a constant electric field of intensity ε≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ll 1$$\end{document} in the j-th direction, and the linear response in terms of a S-current in the i-th direction is computed, where S is a generalized spin operator. We derive a general formula for the spin conductivity that covers both the choice of the conventional and of the proper spin current operator. We investigate the independence of the spin conductivity from the choice of the fundamental cell (unit cell consistency), and we isolate a subclass of discrete periodic models where the conventional and the proper S-conductivity agree, thus showing that the controversy about the choice of the spin current operator is immaterial as far as models in this class are concerned. As a consequence of the general theory, we obtain that whenever the spin is (almost) conserved, the spin conductivity is (approximately) equal to the spin-Chern number. The method relies on the characterization of a non-equilibrium almost-stationary state (NEASS), which well approximates the physical state of the system (in the sense of space-adiabatic perturbation theory) and allows moreover to compute the response of the adiabatic S-current as the trace per unit volume of the S-current operator times the NEASS. This technique can be applied in a general framework, which includes both discrete and continuum models.


Introduction and main results
The aim of this paper is to shed some light on the theory of spin transport in gapped (non-interacting) fermionic systems, a problem which is highly relevant to the research on topological insulators (see the end of Section 1.2).The theory of spin transport, as compared to charge transport, is still in a preliminary stage.First, despite two decades of scientific debate, no general consensus has been reached yet about the correct form of the operator representing the spin current density.Denoting by H 0 the unperturbed Hamiltonian operator, by X = (X 1 , . . ., X d ) the position operator and by S z the operator representing the z-component of the spin, one may consider (1)   (i) the "conventional" spin current operator which has been used e. g. in [64,60]; (ii) the "proper" spin current operator proposed in [62,72].
Whenever [H 0 , S z ] = 0, the two above definitions agree and the theory of spin transport reduces to the theory of charge transport.However, in general [H 0 , S z ] = 0 in topological insulators, as it happens e. g. in the model proposed by Kane and Mele in view of the so-called Rashba term [33,26].As we will explain now (see also [43]), the lack of commutativity poses technical and conceptual problems for the theory of spin transport, and the main objective of our paper is to clarify some of these issues.Related to the different possible choices for the spin current operators, first note that whenever H 0 is periodic, J Sz conv is periodic while J Sz prop is not, which "leads to technical difficulties, but also questions its physical relevance" [58].Moreover, in most of the tight-binding models, thanks to the underlying ultraviolet cutoff, J Sz conv provides a bounded operator, while J Sz prop is generically unbounded.On the other hand, as emphasized in [62,72], J Sz prop yields a spin current density associated with a mesoscopic sourceless continuity equation and to Onsager relations, in contrast to J Sz conv .As a second issue, whenever [H 0 , S z ] = 0 the spin conductivity is given, in analogy with charge transport, by a double commutator formula, namely (1) All over the paper we use Hartree units, so that the reduced Planck constant , the mass of the electron m e and the charge of the positron e are equal to 1.With this choice, both the unit of charge conductivity e 2 h and of spin conductivity e 2π reduce to 1 2π .
where τ is the trace per unit volume and Π 0 the Fermi projector of the gapped system.Formula (1.3), equivalently rewritten in terms of Bloch orbitals, has been considered as the starting point for further analysis of the robustness of the spin conductivity [64,60], or for a mathematical comparison of spin conductivity and spin conductance [43].
In this paper we address two foundational questions in spin transport theory: (Q1) is it possible to derive from the first principles of Quantum Theory, in the general case [H 0 , S z ] = 0, a double commutator formula for the spin conductivity similar to (1.3)? (Q2) to which extent is such a formula affected by a different choice of the spin current operator, namely J Sz conv versus J Sz prop ?Moreover, any formula for spin transport coefficients should satisfy the so-called Unit Cell Consistency (UCC), namely the requirement that any prediction on macroscopic transport must be independent of the choice of the fundamental cell [69].
In order to answer these questions, we reconsider the whole approach to quantum transport theory.
1.1.Two paradigms for quantum transport.The usual paradigm is based on the adiabatic switching-on of the perturbing electric field.More specifically, one considers the time-dependent Hamiltonian operator where f : R → [0, 1] is a smooth function such that f (s) = 0 for all s ≤ −1 and f (s) = 1 for all s ≥ 0, i. e. the Hamiltonian describes the process where the perturbation is switched on during the finite time interval [−1/η, 0], for η > 0. As η → 0 + , the process becomes adiabatic.One assumes that the system is prepared, at some time t ≤ −1/η, in the equilibrium state Π 0 and that the switching occurs adiabatically.The state ρ ε,η (s) at macroscopic time s = ηt is given by the solution to the time-dependent Schrödinger equation The linear response coefficient σ A of an extensive observable A is defined by comparing the expectation value of A at time t * ≥ 0 (when the perturbation is completely switched-on) and in the far past (when the system is in the unperturbed equilibrium state).By considering the adiabatic limit, one defines σ A by setting The real part appears in the formula since one does not know a priori whether the conditional cyclicity of the trace per unit volume can be invoked. (2)The standard approach for obtaining a tractible fomula for σ A is to first approximate ρ ε,η (0) by first-order time-dependent perturbation theory, and then to formally exchange the small field limit and the adiabatic limit, see e. g. [1,28].Choosing f (ηt) = e ηt χ (−∞,0] (t) + χ (0,+∞) (t), this results in Kubo's formula [40] for the linear response coefficients (3)   σ Kubo dt e ηt τ e −iH 0 t [X j , Π 0 ] e iH 0 t A . (1.6) In the case of charge transport, one considers the response of a charge current in the i-th direction, i ∈ {1, . . ., d}, whose corresponding quantum mechanical operator is J c i := i[H 0 , X i ] , and from (1.6) one obtains the formula σ Kubo The importance of the double commutator formula (1.7) (sometimes dubbed Kubo-Chern formula) cannot be overstated, as it implies e. g. quantization of Hall conductivity in 2-dimensional systems [5,10,38,22].When considering spin transport, we had to face the fact that even the algebra which leads formally to (1.7) becomes cumbersome for spin currents, whenever [H 0 , S z ] = 0.Moreover, the fact that the formula is intrinsic (i.e. does not depend on the choice of the switching function appearing in (1.4) and on the choice of t * ≥ 0) is not obvious as far as spin currents are concerned.Thus we propose an alternative way of computing linear response coefficients based on the non-equilibrium almost-stationary states (NEASS), a concept related to the almost-invariant subspaces in space-adiabatic perturbation theory [52,53,67].In a nutshell, the NEASS Π ε is the unique almost-invariant state for the perturbed stationary Hamiltonian H ε = H 0 − εX j that is ε-close to the equilibrium state Π 0 of the unperturbed Hamiltonian H 0 .More precisely, the asymptotic expansion of Π ε in powers of ε is uniquely determined by the conditions where details on the precise norms will be given later.Assuming, for the moment, that the state of the system at times when the perturbation has been turned on is (2) A similar phenomenon appears in Quaternionic Quantum Mechanics, where the Hilbert space trace fails to be cyclic [49]. (3)Note that the specific choice f (ηt) = e ηt for t ≤ 0 has the computational advantage that the integral in (1.6) becomes the inverse Liouvillian, i. e. that the right hand side of (1.6) for finite η > 0 equals, at least formally, τ ((L H0 − iη) −1 ([X j , Π 0 ])A).However, in the adiabatic limit any other integrable and smooth choice for the switching function f leads to the same value for σ Kubo approximately given by the NEASS Π ε , we find a simple prescription for computing linear (and also higher order) response coefficients: let Π .
(1.8)In this paper we will show how to compute formulas for the spin-conductivities based on formula (1.8) for linear response coefficients, instead of (1.6).The advantage of this method is that the operator Π 1 is rather explicit, namely Π 1 = I [X j , Π 0 ] , where the overline denotes the operator closure and I is the inverse of the Liouvillian operator B → [H 0 , B], with integral representation (4.2).
Of course one expects, and formally it is also easy to see, that the two expressions (1.6) and (1.8) agree.However, in the present setting -where expectations are obtained via a trace per unit volume which is only conditionally cyclic -this is not straightforward to prove.Moreover, both formulas are somewhat heuristic: for (1.6) we assumed applicability of time-dependent perturbation theory also for long adiabatic time-scales, while for (1.8) we just postulated that the perturbed system is in the state Π ε .
In order to reconcile and justify both approaches, one needs to prove that in the adiabatic regime the dynamical switching drives the initial equilibrium state Π 0 approximately into the NEASS Π ε , i. e. that the state ρ ε,η (t) is close to Π ε .Indeed, it is shown in [42] that for times t ≥ 0 and any n, m uniformly on bounded intervals in (macroscopic) time.Here I m,ε = [ε m , ε 1/m ] is an intervall of admissible time-scales for the switching.Too slow switching (η ≪ ε m for all m ∈ N * ) must be excluded, because due to tunneling the NEASS decays on such long times-scales, while too fast switching (1 ≫ η ≫ ε 1/m for all m ∈ N * ) would merely yield an error o(1) on the right hand side of (1.9).In other words, the initial equilibrium state Π 0 dynamically evolves into the NE-ASS independently of the shape of the switching function up to lower order errors.A proof of a similar statement in the context of interacting models on lattices is provided in [68] and the issue of justifying linear response and Kubo's formula is briefly reviewed in [28].
1.2.Main results on spin transport and conductivity.By using the NEASS paradigm, we will answer the questions (Q 1 ) and (Q 2 ) stated before, at least in the periodic setting.Let us shortly summarize the main results in the paper.
We consider a crystalline system of non-interacting fermions, whose one-body Hamiltonian H 0 is periodic.This operator acts on the Hilbert space H = L 2 (X ) ⊗ C N , where either X = R d (continuum case) or X ⊂ R d is a discrete set (discrete case), and N is the number of internal degrees of freedom of the particle, which may include spin; periodicity of H 0 is understood with respect to (magnetic) translations along vectors in a Bravais lattice Γ ≃ Z d .We assume that the Hamiltonian H 0 has a spectral gap, and that the initial state of the system is given by the spectral projection Π 0 on the bands below this gap (Fermi projector).The system is driven out of equilibrium by applying a constant electric field of intensity ε ≪ 1 pointing in the j-th direction, j ∈ {1, . . ., d}.Hence, the stationary Hamiltonian of the perturbed system is H ε = H 0 − εX j , where X j is the j-th component of the position operator.
We consider a generalized spin operator in the form S = 1 L 2 (X ) ⊗s and -denoting by the corresponding conventional and proper S-current operator -we define the conventional and proper S-conductivity, respectively, as (4)   Re τ In view of the controversy on the choice of the spin current operator discussed at the beginning of the Introduction, we find convenient the decomposition where we have defined the operators (1.12) In this decomposition, the S-orbital term O contains the contribution associated with the conventional S-current operator, while the S-rotation terms X i R contains corrections related to the replacement of the latter with the proper S-current operator.More precisely, we prove in Theorem 5.7 that splitting (1.11) leads to where with A D (resp.A OD ) referring to the diagonal (resp.off-diagonal) part of the operator A with respect to the orthogonal decomposition induced by Π 0 , and the rotation S- Notice that the first line of (1.14) is in the form of a current-current correlation at the equilibrium, involving the conventional S-current and the charge current, while (4) Notice that the j-dependence of σ S conv/prop,ij is hidden on the right-hand side of the following definition in Π 1 , see its definition in Proposition 4.1(ii).
the second line involves Π 1 .Moreover, the trace per unit volume in (1.14) and (1.15) can be replaced with the ordinary trace of the operator restricted to the fundamental cell, up to a volume factor, as in the statement of Theorem 5.7, even if the operator appearing in (1.15) is not periodic.
In order to analyze the S-rotation contribution σ S rot,ij , we preliminary prove in Proposition 5.5(i) that for any bounded periodic observable B, satisfying suitable regularity properties, the expectation of the B-torque operator i[H 0 , B] on Π 1 is given by a double commutator formula, namely where the operator T B may be dubbed B-torque response in agreement with [43].If, in addition, [B, X j ] = 0 then τ i[H 0 , B] Π 1 = 0, as stated in Proposition 5.5(ii).Physically, this result means that even if i[H 0 , B] = 0, the fact that B commutes with the perturbation −εX j implies the mesoscopic conservation of the observable B, at least within first order approximation in the NEASS.In particular, when B = S z (or for any generalized spin operator S, see Corollary 5.6), we have that the expectation of the spin-torque on Π 1 equals the expectation of the spin-torque response T Sz and that the latter vanishes, in agreement with [43,Theorem 2.8].Notice that the vanishing of the expectation of the spin-torque response is a condition singled-out in [43] to obtain the equality of spin conductivity and spin conductance in 2-dimensional systems.On the other hand, since X j (resp.X j S) is unbounded, Proposition 5.5 does not provide any information on charge (resp.spin) conductivity, whose analysis requires an additional technical effort.
As a further step, we consider the Unit Cell Consistency (UCC) of both the contributions to the proper S-conductivity appearing in (1.13).We prove in Proposition 5.9 that σ S conv always satisfies UCC, while for the additional contribution σ S rot we can prove UCC only if the model enjoys a discrete rotational symmetry, in agreement with the claim in [62] that the use of J Sz prop is "possible for systems where the spin generation in the bulk is absent due to symmetry reasons."In Proposition 5.10 we isolate a subclass of discrete models, enjoying a discrete rotational symmetry and a further property, such that σ S conv = σ S prop .Remarkably, the paradigmatic model proposed by Kane and Mele is in this class.A crucial consequence is that, for this class of models, the choice of the spin current operator (either J S conv or J S prop ) is immaterial as far as the S-conductivity is concerned.
While the paper is focused on transport theory, one of our long-term goals is to clarify the relation between the spin transport coefficients and the topological invariants associated to Quantum Spin Hall (QSH) insulators.These materials, theoretically predicted in [33,34] and soon experimentally realized [39,65], display dissipationless edge spin currents, which are robust against continuous deformations of the model and disorder [60].A crucial issue, both for fundamental understanding and for potential applications, is whether there exists a bulk topological invariant "protecting" the QSH effect.Two candidates have been extensively investigated in the literature.First, the Z 2 -valued index proposed by Fu, Kane and Mele [34,21], whose definition and geometric properties rely on the fermionic time-reversal symmetry of the system [23,20,15].Second, the (half-)integer-valued spin-Chern number, introduced in [61] via spin dependent boundary conditions, and later intrinsically redefined by Prodan as a bulk invariant [56], which relies instead on the almostconservation of spin, and is associated to robust spin edge currents [58,55,59,37].Our analysis establishes a direct relation between the bulk spin conductivity and the spin-Chern number, in agreement with the (recent) discovery that QSH plateaux may persist under broken time-reversal symmetry [18].Indeed, whenever spin is conserved, our results yield that the (bulk) spin conductivity equals the spin-Chern number (Remark 5.13).Moreover, the result is robust: if spin is approximately conserved, with errors of order O(λ), then the mentioned equality holds true up to a correction of order O(λ) (Proposition 5.14), in analogy with the persistence of edge spin currents proved in [58].
In summary, our paper contributes to put spin transport theory on a firm mathematical ground: We derive a new formula for the spin conductivity which covers both the choice of the conventional and the proper spin current operator; we isolate conditions under which UCC is satisfied and additional conditions which guarantee that σ S conv = σ S prop ; we make connection with the spin-Chern number.We hope that our mathematical investigations will contribute to clarify some of the controversies in the emerging and promising field of spintronics, and will stimulate a fruitful exchange of ideas between mathematicians and solid state physicists.While, for technical reasons, this paper focuses on the case of periodic non-interacting systems, we are confident that our approach can be suitably generalized to random and interacting systems.
1.3.Further reference to the literature.We conclude this introduction with some comments on the existing literature.The mathematical literature on the quantum Hall effect and on the justification of linear response theory in the context of quantum transport of charge is by now prodigious, and we will not embark in the task of giving a full account of it here.Indeed, several different mathematical problems have been labeled "proving Kubo's formula" and a short review highlighting the differences will appear elsewhere [28].Here we only mention a few works without going into any detail: A similar approach to the one we use was employed in [66], where Kubo's formula for the Hall conductivity of simple isolated bands is derived using semi-classical methods.The rigorous derivation of Kubo's formula for interacting fermionic systems on the lattice has recently been done in [7,46,68], where [7,46] consider only situations where the perturbation does not close the spectral gap.A similar result for non-interacting fermions in the continuum is in preparation [42], generalizing a previous result [19] which also assumes a non-closing-gap condition.In many other works Kubo's formula for the Hall conductivity is taken as a starting point and the objective is to prove quantization of the Hall plateaux also in presence of disorder, assuming the Fermi energy lies in a mobility gap [10,1,12], or including interaction effects [27,24,6], with the aim of proving universality of the Hall conductivity.Moreover, the linear response to a quenched perturbation has been recently analyzed in [14].Finally, in [13,17] (and references therein) mathematical frameworks are developed, within which the applicability of linear response theory in very general random resp.interacting systems can be established.However, a rigorous justification of Kubo's formula for the quantum Hall conductivity in situations with mobility gap is still a completely open problem, even in the case of non-interacting systems on the lattice.
Linear response theory can also be considered in the case of heat or charge fluxes induced by thermodynamical (i.e. non-mechanical) driving forces, such as deviations of temperature or chemical potential from their equilibrium values.In this context, the validity of the Green-Kubo formula has been extensively investigated in algebraic quantum statistical mechanics, by relating it to the structure of non-equilibrium steady states [29,30,31,32].
The field of spintronics and of quantum transport of spin is relatively new, but has already attracted a lot of attention both in the physics and mathematics communities.Results concerning the quantization and robustness of spin Hall currents in the presence of disorder [56,58] and of interactions [3,44] also rely, to some extent, on a Kubo-like formula.We foresee the possibility of adapting the techniques developed in [46,68] to derive such formulas from first principles also in the context of interacting fermions on a lattice.
A further development of the present line of research consists in pushing the validity of Kubo-like formulas for adiabatic S-currents to arbitrarily high orders in the adiabatic parameter ε.This has been achieved in [38] for quantum Hall systems.

Periodic operators and trace per unit volume
In condensed matter physics it is customary to describe crystalline solids by means of periodic Hamiltonian operators.The appropriate trace-like functional used to compute thermodynamic expectations of periodic observables is given by the trace per unit volume.This Section is devoted to recall some generalities about this framework.
Let X denote the configuration space of a d-dimensional crystal.We will treat both continuum models, in which X = R d equipped with the Lebesgue measure, and discrete models, in which X ⊂ R d is a discrete set of points arranged in a crystalline structure, equipped with the counting measure (in d = 2 think of the square lattice Z 2 or of the honeycomb structure, for example).In general "crystalline structure" means that we assume the existence of a Bravais lattice that acts on X by translations, i. e. T γ x := x + γ for γ ∈ Γ defines a group action T : Γ × X → X .We consider the one-particle Hilbert space for a particle moving on X and having N internal degrees of freedom (e. g. spin).
In the following we will write elements of H as C N -valued functions on X .We assume that there is a unitary representation T of Γ on H by (magnetic) translation operators where M : Γ × X → U(C N ) are unitaries satisfying the cocycle condition (5)   M(γ Position operators for j ∈ {1, . . ., d} are defined via The following simple observation is very useful.
Lemma 2.1.For any periodic operator A, the operator [A, X j ] is also periodic.
Notice that, in general, the operator [A, X j ] might be non-densely defined or even defined on the trivial subspace {0}, as pointed out in [35,.This pathology will not appear for the specific operators we will consider in the following Sections.
The analysis of periodic operators is best performed in the so-called (magnetic) Bloch-Floquet-Zak representation (see e. g. [41,45,50] and references therein).The (magnetic) Bloch-Floquet-Zak transform is initially defined on compactly supported 2), hence it defines an element in the Hilbert space where the norm refers to a fundamental cell C 1 for Γ (see (2.7)).As functions of k, elements in the range of U BF are not periodic with respect to the reciprocal lattice Γ * , but rather ̺-equivariant, namely where (6)  ̺ : defines a unitary representation of Γ * on H f .The map defined by (2.4) extends to a unitary operator where 6) We denote by U(H f ) the group of the unitary operators on H f .
At least formally, a periodic operator A on H becomes a covariant fibered operator on H ̺ .More precisely, taking into account the following inclusion and natural isomorphism one has where each A(k) acts on H f and satisfies the covariance property A(k Most relevant extensive observables in crystalline systems are periodic self-adjoint operators.However, in an infinite system neither these periodic extensive observables nor translation invariant states are trace class.The appropriate functional is instead given by the trace per unit volume τ , which is well suited to take into account invariance or covariance by discrete lattice translations in the setting of periodic or more generally ergodic operators [9,54,12,2].The trace per unit volume is defined as follows (compare [12,Prop. 3.20]).Denote by χ Ω the orthogonal projection on H which multiplies by the characteristic function of Ω ⊂ X .For any L ∈ 2N + 1, we set and χ L := χ C L .The set C 1 is called a fundamental or primitive (unit) cell.It is not unique since the choice of the spanning vectors {a j } 1≤j≤d for Γ (see (2.1)) is not unique.Notice that, restricting to odd integers L ∈ 2N + 1, one has the convenient decomposition (7) We call an operator A acting in H trace class on compact sets if χ K Aχ K is trace class for all compact sets K ⊂ X (8) .

Definition 2.2 (Trace per unit volume).
Let A be an operator acting in H such that A is trace class on compact sets.The trace per unit volume of A is defined as whenever the limit exists. (7)The symbol denotes the disjoint union up to zero-measure sets. (8)This condition is automatically satisfied in the discrete case for any operator A, since the range of χ K is finite-dimensional.

Let us denote
We will refer to operators in B τ 1 as the operators of trace-per-unit-volume class or τclass for simplicity.Moreover, in view of [12,Proposition 3.17] we have (2.10) The following Lemma recalls some useful properties of τ -class operators. and (2.12) In particular, we have that A is trace class on compact sets.
Proof.Inequality (2.11) is proved in [12, Lemma 3.10] and its proof easily generalizes to obtain also (2.12), as follows.Let A = U |A| be the polar decomposition of A.
Notice that for any γ, ν ∈ Γ The next result allows to compute the trace per unit volume of operators which are periodic and trace class on compact sets.Proposition 2.4.(i) Let A be periodic and trace class on compact sets (9) .Then τ (A) is well-defined and

.13)
(ii) Let A be a periodic and bounded operator acting on H. Denoting by dk A(k) (9) The condition that A is trace class on compact sets is satisfied whenever A is in B τ 1 , as proved in Lemma 2.3.
its Bloch-Floquet-Zak decomposition, assume that A(k) is trace class and that Proof.(i) In view of the decomposition (2.8) and the hypotheses on A, one has (ii) This is proved e. g. in [51, Lemma 3].
In the following result, we introduce a class of operators which are not necessarily in B τ 1 , but have finite trace per unit volume.Proposition 2.5.Let A be periodic and trace class on compact sets (9) .Then (i) the operator X j A for j ∈ {1, . . ., d} has finite trace per unit volume and Tr (χ 1 X j Aχ 1 ) . (2.15) (ii) If, in addition τ (A) = 0, then τ (X j A) does not depend on the exhaustion (10)  C L ր X set in Definition 2.2 and on the choice of the origin, in the sense that As already pointed out, in general it might happen that X j A is not densely defined, or even that it is trivially defined only on the zero vector of the Hilbert space.
Proof.(i) Since χ L X j χ L is bounded for every L ∈ 2N + 1 and A is trace class on compact sets by hypothesis, we have that Therefore, in view of the decomposition (2.8), one has that Using that A is periodic, that [T γ , X j ] = −γ j T γ , and the result from Proposition 2.4(i), we obtain that . (10) Notice that this particular choice of the exhaustion C L ր X is such that C L ∩Γ is symmetric with respect to the involution x → −x.
Consequently, we get that Since both γ and −γ are in Γ ∩ C L for all L ∈ 2N + 1, the sum in brackets on the right-hand side of the above vanishes, and the thesis follows immediately.
A property which will be fundamental for all the following analysis is the conditional cyclicity of the trace per unit volume.We state it in the following Lemma, whose proof can be found in [12,Lemma 3.22].

Lemma 2.6 (Conditional cyclicity of the trace per unit volume). If
The trace per unit volume is defined in (2.9) through a specific choice of the cell C L , which in turn depends via (2.7) on the choice of a particular linear basis {a 1 , . . ., a d } for Γ.The term Unit Cell Consistency refers to the requirement that physically relevant quantities are independent of the latter choice.Precisely, one considers a different linear basis {ã 1 , . . ., ãd } for Γ and the corresponding cell, defined by and sets χ L := χ C L .Denoting by τ ( • ) and τ ( • ), respectively, the trace per unit volume induced by the choice of the primitive cells C 1 and C 1 , we prove in Proposition A.2(i) that for any periodic operator A, which is trace class on compact sets, one has that When a contribution to the transport coefficient is in the form τ (X i A) for a periodic operator A, as in formula (1.15), a more careful analysis is needed, as discussed at the end of Section 5.1 and in Appendix A.

The unperturbed model
Our goal is to study the linear response of a crystalline system to the application of an external electric field of small intensity.Before considering the perturbed system, we state our assumptions on the unperturbed one.Assumption 3.1.We assume the following: (H 1 ) the Hamiltonian H 0 of the unperturbed system is a self-adjoint periodic operator on H, bounded from below, such that in Bloch-Floquet-Zak representation its fibration is a smooth equivariant map taking values in the self-adjoint operators with dense domain Here L(D f , H f ) denotes the space of bounded linear operators from D f , equipped with the graph norm of H 0 (0) denoted by • D f (11) , to ) be in a spectral gap (12) of H 0 .We denote by Π 0 = χ (−∞,µ) (H 0 ) the corresponding spectral projector (Fermi projector).We assume that its fibration k → Π 0 (k) takes values in the finite-rank projections on H f (13) .

⋄
We shortly discuss sufficient conditions implying that Assumption (H 1 ) holds true.As far as discrete models are concerned, H f is finite dimensional and H 0 (k) are selfadjoint matrices.The smoothness of the map k → H 0 (k) follows from the fact that the hopping amplitudes in the model {t γ } γ∈Γ decay sufficiently fast as |γ| → ∞.In all the most popular discrete models of topological insulators [25,33] the hopping amplitudes have finite range, namely t γ = 0 if |γ| > R for some R, hence assumption (H 1 ) is automatically satisfied.
As for the continuum case X = R d , we first consider a Bloch-Landau operator in the form acting in L 2 (R d ), where A and V Γ are the magnetic and electrostatic potentials, respectively (the charge Q of the particle is reabsorbed in A and in V ).For the sake of simplicity, we consider only d ≤ 3 and we ignore the "spin space" C N , but similar results holds true if for example V Γ is matrix-valued and acts non-trivially on these degrees of freedom.With the help of Kato's theory [35], and arguing as in [45] on the basis [11], it is not difficult to prove that, if A = A Γ is Γ-periodic, and C 1 denotes the fundamental cell of the lattice, then for the validity of (H 1 ) it is sufficient to assume either of the following two sets of hypotheses: (11) From now on D f is understood to be equipped with the norm • D f . (12)In the following, when we refer to "the" spectral gap of H 0 , we will refer to this specific gap. (13)From the smoothness assumption (H 1 ), it follows that Rank(Π 0 (k)) = m ∈ N ∪ {+∞} is independent of k.Therefore, in view of the fact that Π 0 is an orthogonal projection, m < +∞ is equivalent to the assumption Π 0 ∈ B τ 1 . ( If instead A = A b is a linear potential inducing a constant uniform magnetic field, it is enough to assume that V Γ is infinitesimally form bounded with respect to −∆ on H f , and that the magnetic flux per unit cell is a rational multiple of the magnetic flux quantum.As a further example, we consider the Hamiltonian In the above, p ≡ (p 1 , p 2 ) = −i∇ denotes the momentum operator, E ≡ (E 1 , E 2 , E 3 ) is a constant vector (which we interpret as a constant electric field), and s ≡ (s x , s y , s z ) denotes the vector of spin matrices (half of the Pauli matrices).Thus, the first term in H 0 represents the kinetic energy, the second one a spin-orbit coupling, and the third one is a Rashba term: this Hamiltonian represents a continuous analogue of the Kane-Mele Hamiltonian proposed in [33].One can argue that the above operator can be fibered via Bloch-Floquet-Zak transform leading to a family of fiber Hamiltonians H 0 (k) as in Assumption (H 1 ).Moreover, since , and since the momentum operator is relatively bounded with respect to the Laplacian, one can see that [H 0 , S] is relatively bounded with respect to H 0 , and hence the assumptions on S (compare Definition 5.1 and Remark 5.2) are satisfied as well.
The following spaces of operators and functions turn out to be useful for our analysis.
Since the Fréchet derivative follows the usual rules of the differential calculus, we have that P(H 1 , H 2 ) is a linear space, P(H f ), P(D f ) and P(H f , D f ) are normed algebras, and e. g. for A ∈ P(H f , D f ) and B ∈ P(H f ) we have Since we are interested in computing [A, X j ] where A is in one of the above spaces of operators, the following Proposition will be relevant.It states their invariance under the derivation [ • , X j ], where the overline denotes the operator closure.
Proof.Notice that ), the commutator appearing on the right-hand side of the first equality is densely defined on C ∞ ̺ (R d , H 1 ) and so by unitary conjugation the commutator on the left-hand side is densely defined as well.Thus, Lemma 2.1 implies that [A, X j ] acting on . Therefore, as H 2 is a Banach space, the extension principle implies the thesis.Lemma 3.5.Under Assumption 3.1 we have that (H 0 − z1) −1 ∈ P(H f , D f ) for every z ∈ ρ(H 0 ), and that Π 0 ∈ P(H f , D f ).
Proof.The first claim is evident because of (H 0 − z1) −1 (k) = (H 0 (k) − z1) −1 .Since D f is a Banach space, the second one follows from Riesz's formula where C is a positively-oriented complex contour intersecting the real axis at the Fermi energy (so, in the gap) and below the bottom of the spectrum of H 0 .
Corollary 3.6.Under Assumption 3.1 we have that Proof.(i) By Lemma 3.5, one has that Π 0 ∈ P(H f , D f ).Proposition 3.4 implies the statement.
(ii) In view of Lemma 3.5, one has that Π 0 ∈ P(H f , D f ).Using an argument similar to the one presented in the proof of Proposition 3.4 one deduces the thesis.
(iii) Since by hypothesis H 0 ∈ P(D f , H f ), Proposition 3.4 concludes the proof.
For the sake of readability, we introduce the concise notation

Non-equilibrium almost-stationary states
Now that we have established the model for the unperturbed system, we consider the perturbed Hamiltonian where ε ∈ [0, 1] is the strength of the external electric field pointing in the j-direction.
As discussed in the introduction, we are interested in the linear response of the system to such a perturbation when it starts initially in the zero-temperature equilibrium state Π 0 .While it is clear that the perturbation given by the linear electric potential has the effect of driving the system out of equilibrium, the perturbation is slowly varying and thus acts locally merely as a shift in energy.Hence it is expected that the initial equilibrium state Π 0 changes continously into a nearby nonequilibrium almost stationary state (NEASS).A detailed discussion and justification of the concepts of NEASS can be found in [68,42].
For the following construction of the NEASS in the present setting we only need to know that the operator Π ε , representing the NEASS, is determined uniquely (up to terms of order O(ε M +1 )) by the following two properties: (SA 1 ) Π ε = e −iεS ε Π 0 e iεS ε for some bounded, periodic and self-adjoint operator Here O(ε M +1 ) is understood in the sense of the operator norm.(i) Let S := i I([[X j , Π 0 ], Π 0 ]), where with C a positively-oriented contour in the complex energy plane enclosing the part of the spectrum of H 0 below the gap.Then S is in Then and Π ε r are in P(H f , D f ), and the map We postpone the proof of the above Proposition to Section 6.3.It is already clear from the statement that the map I( • ) plays a crucial role: its properties are summarized in Section 6.2, where we recall in particular the well-known fact from perturbation theory that I(A) is the unique solution to the equation [H 0 , I(A)] = A whenever A is off-diagonal in the orthogonal decomposition induced by Π 0 , namely

Results on the S-conductivity
As stated in the previous sections, we want to investigate quantum S-currents induced by the perturbation given by an external electric field, and compute their S-conductivities as linear response coefficients.To fix the ideas, the reader can think of the case S = 1 H (which corresponds physically to the charge current, in appropriate units, e. g. in quantum Hall systems) or to S = 1 L 2 (X ) ⊗ s z , where s z = σ z /2 is half of the third Pauli matrix (which corresponds to the spin current e. g. in Quantum Spin Hall systems).Definition 5.1 (S-current and S-conductivity).Let S = 1 L 2 (X ) ⊗ s be a selfadjoint operator on H = L 2 (X ) ⊗ C N .Furthermore, assume that S is periodic (14)  and its fibration The conventional and the proper S-current operator are defined respectively as where H 0 satisfies Assumption 3.1.The conventional and proper S-conductivity are defined respectively as

⋄
Remark 5.2.Some comments are in order here.
(i) By (3.6) and S ∈ P(D f ), it is evident that prop,i is densely defined as the operator on the right-hand side is so.Similarly, we have that 2) (14) Notice that this assumption is not automatically satisfied since the (magnetic) translation operators (see (2.2)) may act non-trivially on the factor C N .Obviously, for either the standard magnetic translations [71] or translation operators with a trivial action on the factor C N the periodicity of S is ensured.
In order to prove that Re τ (J S prop,i Π 1 ) = σ S prop,ij according to (5.1), it suffices to show that all the traces per unit volume above are well-defined and finite, and that the term carrying a prefactor ε 2 is uniformly bounded in ε.While the control of the remainder term will be done in Section 6.4, we focus now on the linear response coefficient, namely Re τ (J S prop,i Π 1 ).

5.1.
The linear response coefficient.In order to compute the linear response coefficient, we employ directly Definition 2.2 for τ (J S prop,i Π 1 ), and start by localizing this operator on the cell C L , defined in (2.7), through the projection χ L which multiplies by the characteristic function of C L .It is convenient to notice at this point the following Thus, in view of Remark 5.2(i), we have that and this allows for the following simple manipulations.Using the Leibniz rule, we obtain the following chain of equalities on Ran(χ L ): where the operators O and R have been defined in (1.12).We call O the S-orbital part and R the S-rotation part of the operator related to linear response of the proper S-current J S prop,i .The latter terminology is due to the fact that, when S is the spin operator and [H 0 , S] = 0, the spin transport is the result of two contributions, that is, the S-orbital part coming from the center-of-mass drift and the S-rotation part due to the spin non-conservation.Notice that both O and R are periodic, instead obviously X i R is not periodic and thus a more careful analysis of its trace per unit volume is required.We begin by handling the S-orbital part O.In view of the defining relation Π 1 = I([X j , Π 0 ]) (compare Propositions 4.1 and 6.3), we obtain on Ran(χ L ): where we have defined the operators and We call C the Chern-like term and E ℓ the ℓ-th extra or beyond-Chern-like term for ℓ ∈ {1, . . ., 4}.This terminology is motivated by the fact that, whenever the spin is conserved, for d = 2, i = 1 and j = 2 in quantum (spin) Hall systems the Chern-like term C corresponds to the (spin) Chern number (see Remark 5.13).In general, whenever [H 0 , S] = 0, all extra terms have trace per unit volume zero (see Subsection 5.2) and obviously the S-rotation part vanishes.
In the following Proposition, we analyze the trace per unit volume of the operators resulting from the previous algebraic manipulations.
Proposition 5.4.Under Assumption 3.1 and hypotheses on S in Definition 5.1, we have that the Chern-like term C, the extra terms E ℓ for any ℓ ∈ {1, . . ., 4} and X i R, defined in (1.12) and (5.6) have finite traces per unit volume.Moreover, one has ) and τ (E 2 ) = 0 = τ (E 4 ), (5.10)where the diagonal and off-diagonal parts of the above operators refer to the orthogonal decomposition induced by the Fermi projection Π 0 .
The proof of the above Proposition is postponed to Section 6.5.
We are going to prove that trace per unit volume of the operator X i R is welldefined and finite.In view of Proposition 2.5(ii), it suffices to show that τ (R) is zero.The latter result is an immediate consequence of the following Proposition 5.5.If H 0 satisfies Assumption 3.1, and B is in P(H f ) ∩ P(D f ) (in particular, B is a bounded periodic operator) the following holds: The above Proposition, whose proof is deferred to Section 6.5, immediately implies the following Corollary 5.6.Under Assumption 3.1 and hypotheses on S in Definition 5.1, we have that We are now in position to state one of our main results.
Theorem 5.7 (General formula for the S-conductivity).Let H ε = H 0 − εX j be acting in L 2 (X )⊗C N , with H 0 and Π 0 as in Assumption 3.1.Let Π ε be the NEASS defined in Section 4. Consider the conventional (resp.proper) S-conductivity σ conv,ij (resp.σ prop,ij ) as in Definition 5.1.Then where and the rotation contribution to the proper S-conductivity is defined as (5.13) Moreover, the trace per unit volume appearing in (5.13) does not depend on the particular exhaustion (10) C L ր X chosen in Definition 2.2 and on the choice of the origin, in the sense that τ (X i R) = τ ((X i + α)R) for every α ∈ R.
The proof of the above Theorem is postponed to Section 6.
Remark 5.8.Some comments about the above result.
(i) Notice that one can rewrite the above formula for the proper S-conductivity σ S prop,ij , summing the two contributions σ S conv,ij and σ S rot,ij as follows: (5.15) While (5.11) emphasizes the splitting between the drift contribution coming from the center-of-mass momentum and the one resulting from the spin rotation, the latter decomposition isolates the contribution coming from a periodic operator, in (5.14), and the one deriving from a non-periodic operator, in (5.15).(ii) The real part is needed in both (5.12) and (5.13), even if on Ran(χ 1 ) one has that prop,i Π ε ) is generally not zero, since the argument of the trace per unit volume is not periodic, due to [H 0 , S] = 0, and thus we are not allowed to use cyclicity.On the other hand, if [H 0 , S] = 0 then τ (Π ε⊥ J S prop,i Π ε ) = 0 and therefore τ (J S prop,i Π ε ) is automatically real.Moreover, for systems with a fermionic time-reversal symmetry Θ such that ΘSΘ −1 = −S, the number τ (J S prop,i Π 1 ) is real, so the real part is redundant.It is worth to investigate how the contributions to the proper S-conductivity, appearing in (5.11), behave under a change of primitive cell.The next Proposition shows that in some discrete models with discrete rotational symmetry, one has that σ S rot,ij = 0, and hence the choice between J S prop,i and J S conv,i becomes immaterial.Remarkably, the Kane-Mele model is in this class.

Proposition 5.10 (Equality of conventional and proper S-conductivity).
Let H 0 be a discrete Hamiltonian with finite range hopping amplitudes and S be as in Definition 5.1.Assume that the model satisfies the hypotheses of Proposition A.3 and Rankχ Pγ = 1, where {P γ } γ∈I ⊂ X is the family of subsets defined in (A.1).Then σ S rot,ij = 0 or equivalently σ S prop,ij = σ S conv,ij .

5.2.
When S is (approximately) conserved.The computation of the linear response coefficient σ S ij simplifies considerably if we assume that S is a conserved quantity of the system, namely that [H 0 , S] = 0. (5.16) Under this assumption, then [Π 0 , S] = 0 as well, since Π 0 is a spectral projection associated to H 0 , and thus S is diagonal in the decomposition induced by Π 0 .If (5.16) holds, then ) by Corollary 3.6(iii) and the hypothesis S ∈ P(D f ).Hence, since Π ε ∈ P(H f , D f ) by Proposition 4.1(ii), we have that J S i Π ε ∈ P(H f ) and furthermore applying Proposition 6.6(i), we deduce that J S i Π ε is τ -class.Thus, the trace per unit volume of J S i Π ε is welldefined and only the Chern-like term contributes to it.Indeed, by Proposition 5.4 the extra term E ℓ does not contribute for ℓ ∈ {2, 4} and the next Lemma shows that τ (E 1 ) = 0. Obviously, E 3 = 0 = R whenever (5.16) holds.The proof of Lemma 5.11 is also deferred to Section 6.5, but it is easily seen to imply the following Theorem 5.12 (S-conductivity in the S conserved case).Let H ε = H 0 − εX j be acting in L 2 (X ) ⊗ C N , with H 0 and Π 0 as in Assumption 3.1.Let Π ε be the NEASS defined in Section 4 and J S i be as in Definition 5.1.Assume moreover that [H 0 , S] = 0. Then the S-conductivity is Proof.In view of Lemma 5.11, the extra terms E 1 and E 3 , and the S-rotation part σ S rot,ij do not contribute to the trace per unit volume of J S i Π 1 .Therefore, using Proposition 2.4(ii) we are going to compute the k-space representation of the trace of To this end, it suffices to notice that the fiber operator associated to [X j , Π 0 ] in the Bloch-Floquet-Zak representation is given by i∂ k j Π 0 (k) (Proposition 3.4) and that This Theorem applies in particular to the transverse charge current in quantum Hall systems (S = 1 H ), and to the transverse spin current in quantum spin Hall systems (S = 1 L 2 (X ) ⊗ s z ) whenever the z-component of the spin is conserved.In particular, in the latter case we recover the formula for the spin conductivity proposed in [16,63], which was derived assuming that the unperturbed Hamiltonian H 0 has an identically degenerate Bloch band, where the degeneracy comes from the spin degrees of freedom.So in that model effectively and (5.16) is in particular satisfied after projection to the relevant spectral subspace.Our argument used only (5.16) and no spectral assumption (other than the gap condition) on the Hamiltonian.Remark 5.13 (Spin conductivity and spin-Chern number).Let S = 1 L 2 (X ) ⊗ s z with s z a spin operator for non-integer spin r, i. e. with spectrum {−r, −r + 1, . . ., r − 1, r}, acting on C N with N = 2r + 1 (e.g. half the third Pauli matrix σ z for r = 1 2 and N = 2).Denote by s z = 2r ℓ=0 (ℓ − r)p ℓ its spectral decomposition.Then the commutation relation [Π 0 , S] = 0 implies that Π 0 admits a splitting in the decomposition induced by S: The formula for the S-conductivity σ S ij in Theorem 5.12 simplifies then to (5.17) This spin-Chern number S-Chern(Π 0 ) ij , proposed in [61] and intrinsically defined in [56], is in general a half integer. (15)It becomes an integer if the system enjoys time-reversal symmetry.Even in time-reversal invariant systems it can be different from zero while the Chern number necessarily vanishes.Our approach to spin transport shows then that for d = 2 the bulk spin Hall conductivity (measured in units of e 2π ≡ 1 2π ) equals the spin-Chern number, as long as [H 0 , S] = 0. On the other hand, when S = 1 L 2 (X ) ⊗ 1 C N and d = 2 the integral in Theorem 5.12 computes, up to a factor 1/2π, the Chern number Chern(Π 0 ) of the family of projections {Π 0 (k)} k∈R 2 , implying the quantization of the Hall conductivity measured in units of e 2 h ≡ 1 2π (see [22] and references therein).⋄ Abstracting from the previous Remark, we consider now any operator in the form is sufficiently small, then one can still define a spin-Chern number related to Π 0 [56,58] and the S-conductivity is still approximately given by the spin-Chern number.To see this, let s = k ℓ=1 s ℓ p ℓ be the spectral representation of s (we need no assumptions on the spectrum of s here) and with a constant C s that depends only on S.
The spin-conserving Hamiltonian H 0 is H 0 -bounded with relative bound λC s .For λ < 1  Cs , H 0 is thus self-adjoint on the domain of H 0 , and for λ small enough, by standard perturbation theory, the Fermi energy µ lies also in a gap of H 0 .Thus we (15) The normalization we use here agrees with [56] and with the most recent physics literature, but differs by a factor 2 from the original formula in [61].
can define the gapped Fermi projection Π 0 := χ (−∞,µ] ( H 0 ) of H 0 and, in analogy with Remark 5.13, its associated spin-Chern number.More precisely, let and thus It is straightforward to see that (16) are well defined and integer.The S-Chern number of Π 0 is finally defined as We now show that the S-conductivity is given at leading order in λ by S-Chern(Π 0 ), a result which coherently complements the robustness of edge spin currents proved by Schulz-Baldes [58].

Proofs
In this Section we present the proofs for a number of core technical results used in the body of the paper.We will exploit the operator spaces defined in Section 3 and in particular their properties with respect to the trace per unit volume.
6.1.Diagonal and off-diagonal operators.The Fermi projection Π 0 of the unperturbed Hamiltonian H 0 clearly induces a decomposition of L 2 (X ) ⊗ C N into Ran Π 0 ⊕ (Ran Π 0 ) ⊥ .Correspondingly, operators acting in L 2 (X ) ⊗ C N will admit a block decomposition.We review in this Section some properties of this decomposition, heading towards the proof of a well-known formula from asymptotic perturbation theory which allows to invert the Liouvillian [H 0 , • ] acting on operators which only have off-diagonal blocks.Definition 6.1 (Diagonal and off-diagonal parts).Given an operator A and an orthogonal projection Π, i. e. Π = Π * = Π 2 , such that A Π is densely defined (17) , (17) The operator A may be unbounded and thus a careful analysis is required.In particular, we want to avoid pathological examples and have A ⊇ A D + A OD on a dense domain.Later, we will for example apply the block decomposition with respect to the Fermi projection Π 0 to the operator X i for i ∈ {1, . . ., d} (see Subsection 5.1), and X i Π 0 is densely defined under Assumption 3.1 on the unperturbed Hamiltonian (compare Corollary 3.6).
one defines its diagonal and off-diagonal parts as We collect in the following Lemma two simple properties of diagonal and offdiagonal operators in a general Hilbert space H. Lemma 6.2.Let A be an operator acting in H and Π an orthogonal projection on H such that AΠ is densely defined.6.2.Inverse Liouvillian.We study here the Liouvillian (super-)operator B → [H 0 , B] associated to the unperturbed Hamiltonian, and in particular the possibility to invert it away from its kernel.We look in other words for the solution B to the equation [H 0 , B] = A, where A ∈ P(H f ) is off-diagonal with respect to the decomposition H = Π 0 H ⊕ (1 − Π 0 )H.We state in the following Proposition the solution to this problem, which traces back at least to [47,48].
Proposition 6.3.Under Assumption 3.1, let A ∈ P(H f ) be such that A = A OD with respect to Π 0 .Then the operator I(A), defined in (4.2), is the unique off-diagonal solution in P(H f , D f ) to the equation Proof.From the very definition (4.2) and our hypotheses on A, we have that I(A) is off-diagonal and is in P(H f , D f ) by Lemma 3.5.Thus we need only to prove (6.1).Since H 0 (k) ∈ L(D f , H f ) and Ran(H 0 (k) − z1) −1 ⊂ D f for any z ∈ ρ(H 0 ), applying [70, §V.5 Corollary 2] we have that on D f Hence, we obtain that on the domain using the Riesz formula (compare (3.5)) and Lemma 6.2(ii).Finally, notice that I(A) is the unique off-diagonal solution in P(H f , D f ) to equation (6.1) for any off-diagonal operator A ∈ P(H f ).Indeed, if B ∈ P(H f , D f ) is another solution to (6.1), then I(A) − B commutes with H 0 , and hence with Π 0 .By Lemma 6.2(i), I(A) − B is diagonal, and hence B = B OD = I(A) OD = I(A).
6.3.NEASS.This Section is devoted to the proof of Proposition 4.1 and thus to the explicit construction of the NEASS Π ε satisfying (SA 1 ) and (SA 2 ).In order to give this proof, we first need the following preparatory Lemma.Proof.Clearly, (e εA − 1) is periodic as ] and in k ∈ K for any compact set K ⊂ R d due to the assumption that A ∈ P(H f , D f ).Therefore we are allowed to interchange the derivation in k and the series in n.Iterating this argument implies that (e εA − 1) is in P(H f , D f ) and that its norm in this space is uniformly bounded with respect to ε ∈ [0, 1].Thus by Proposition 3.4 we deduce that [e εA , X j ] = [e εA − 1, X j ] ∈ P(H f , D f ) again with uniform bounds on its norm for ε ∈ [0, 1].
We are now ready to tackle the Proof of Proposition 4.1.In this proof, we will abbreviate the expression "the map [0, 1] ∋ ε → A ε ∈ P is uniformly bounded" for some space of operators P by just saying that "A ε is in P uniformly in ε ∈ [0, 1]".
(i) By Corollary 3.6(i) one has that [[X j , Π 0 ], Π 0 ] is in P(H f , D f ) ⊂ P(H f ) and is off-diagonal with respect to Π 0 , hence Proposition 6.3 implies that S ∈ P(H f , D f ).Self-adjointness of S is evident.
The first of the terms above satisfies, in view of (6.3), extends to an operator in P(H f ) applying again Proposition 3.4.It remains only to show that [X j , Π ε r ] , where the right-hand side is bounded uniformly in ε ∈ [0, 1] by Lemma 6.4 and Proposition 3.4.This concludes the proof.
6.4.Well-posedness of the proper S-conductivity.Here we prove that Re τ (J S prop,i Π 1 ) = σ S prop,ij , by using (5.2) and combining the following results.Lemma 6.5.Under Assumption 3.1 we have that Π 0 , Π 1 , Π ε r ∈ B τ 1 .Proof.By Assumption 3.1 Π 0 (k) is a finite-rank projection on H f with rank m (independent of k).Thus, in view of Proposition 2.4, we get hence Π 0 ∈ B τ 1 .In view of Propositions 4.1 and 6.3, we have that Π 1 = Π OD 1 is off-diagonal with respect to the orthogonal decomposition induced by Π 0 , and hence Lemma 6.2(i) implies that Π ∞ and Π 0 ∈ B τ 1 by (6.5), we deduce that the right-hand side of the above is in , S] e iεS for some ε ∈ (0, ε).As we just proved Π 1 ∈ B τ 1 , and the other operators which appear on the right-hand side of the above are in P(H f ) ⊂ B τ ∞ , we conclude that Π ε r ∈ B τ 1 .Proposition 6.6.Under Assumption 3.1 and the hypotheses on S in Definition 5.1, for Π From a similar argument it follows that [H 0 , S]Π ε r ∈ B τ 1 owing to (6.7).(ii) The conclusion follows by applying part (i) of the statement (which we just proved), Lemma 2.3, Proposition 2.4(i), and Proposition 2.5(i).
In the following Lemma, we prove that some expectation values of J S conv,i which are relevant to transport theory, reduce to the real part of the expectation value of J S naive,i := i[H 0 , X i ]S.Lemma 6.7.Under Assumption 3.1 and the hypotheses on S in Definition 5.1, let
Proposition 6.8 (Bounds on the remainder terms).Under Assumption 3.1, there exist Proof.We begin by showing the first inequality.In view of Lemma 6.7 and the triangle inequality, we obtain that (6.12) By using the inequalities in (2.10) and Lemma 6.4, the first summand on the righthand side can be bounded uniformly in ε as where C is a constant independent of ε.Applying again the inequalities in (2.10), we get for the second and third summand in (6.12), respectively in view of Lemma 6.5.Now, to obtain the second inequality of the thesis, notice that The first summand on the right-hand side is bounded uniformly in ε, as it is shown before, while for the second one we proceed as follows.Since [H 0 , S]Π ε r is τ -class by Proposition 6.6(i), Proposition 2.5(i) implies that Applying the inequality |Tr(AB)| ≤ A Tr(|B|) for a bounded operator A and a trace class operator B, estimate (2.11) and Proposition 2.4(i), we have that 1 Finally, since [H 0 , S] ∈ P(D f , H f ) as shown in (6.7), one can reason as in (6.12) to conclude that [H 0 , S]Π ε r 1,τ ≤ D, for a constant D independent of ε, which yields the second inequality in the thesis.6.5.Chern-like and extra contributions to the S-conductivity.In this subsection we prove Proposition 5.4, Proposition 5.5, Theorem 5.7 and Lemma 5.11.
Proof of Proposition 5.4.We are going to show that C and E ℓ are in B τ 1 for any ℓ ∈ {1, . . ., 4} and then equality (5.7) follows from Proposition 2.4(i).We begin by looking at the Chern-like term C. We can write on Ran(χ L ) where in the last equality we have used Remark 5.3 and (3.6).In view of Corollary 3.6(i) and Lemma 6.5, we have that In view of the cyclicity of the trace per unit volume and the off-diagonality of [X i , Π 0 ], one can rewrite τ (C) as We now analyse the first extra term E 1 .Similarly to the previous computation, we have on Ran(χ L ) In view of Corollary 3.6(iii), Lemma 3.5, Lemma 6.5 and Proposition 6.6(i), we get that Then, noticing that Π 1 = Π OD 1 by construction (see Propositions 4.1(ii) and 6.3) and applying Lemma 2.6, we obtain the final expression for τ (E 1 ) in (5.9).
We now move to E 2 .Analogously, we get on Ran(χ L ) By Corollary 3.6(i), Lemma 6.5 and Proposition 4.1(ii), we deduce that ).Now we analyse E 3 .Similarly, we obtain on Ran(χ L ) Analogously, for E 4 we have that on Ran(χ L ) Finally, by applying Proposition 6.6(ii) we infer that X i R has finite trace per unit volume for any i ∈ {1, . . ., d} and equality (5.7) is implied by Proposition 2.5(i).Finally, Lemma 2.6 implies that τ (E 2 ) = 0 = τ (E 4 ).Indeed, for the first identity observe that τ (X OD i SΠ 1 H 0 ) = τ (Π 1 H 0 X OD i S) = τ (H 0 X OD i SΠ 1 ), and for the second one just notices that E 4 is the commutator of [X i , Π 0 ] ∈ B τ ∞ and Π 0 S[Π 0 , X j ] ∈ B τ where in the first equality we have used the cyclicity of the trace per unit volume , based on the fact that [H 0 , B](H 0 − z1) −1 ∈ B τ ∞ by Lemma 3.5, and that Now we are going to compute n−1 k=0 R −k ϑ,(ij) X j R k ϑ,(ij) .The rotation of angle ϑ acts nontrivially only in the plane (x i , x j ), which we parametrize with the complex coordinate z := x i + ix j .In this parametrization, the rotation of angle ϑ is implemented as Rϑ z := e iϑ z.Introducing the complex position operator Z := X i + iX j , one has then As n−1 k=0 e i2πk/n = 0, we deduce that the term in (A.7) vanishes.This concludes the proof.
Remark A.4.In general, even exploiting the peculiar discrete rotational symmetries in the hypotheses of Proposition A.3, it is not obvious that the persistent proper S-current vanishes, i. e. τ (J S prop,i Π 0 ) = 0, since the argument we used relies on the periodicity of the operators involved and J S prop,i is not periodic.Nevertheless, in the Kane-Mele model this property holds true thanks to the specific structure of the model.A.3.Vanishing of persistent S-current when S is conserved.When S is a conserved quantity, namely when [H 0 , S] = 0, the vanishing of the persistent Scurrent J S i = J S prop,i = J S conv,i holds true without any symmetry assumption on H 0 (compare [10,12], where similar results are deduced in the case S = 1, and [8], which offers a proof in the context of many-body quantum spin systems).

1 .
Introduction and main results 1.1.Two paradigms for quantum transport 1.2.Main results on spin transport and conductivity 1.3.Further reference to the literature 2. Periodic operators and trace per unit volume 3. The unperturbed model 4. Non-equilibrium almost-stationary states Date: April 2, 2020.Version submitted to arxiv.org.

Proposition 5 . 9 (
Unit Cell Consistency of the S-conductivity).Under the hypotheses of Theorem 5.7, we have that (i) σ S conv,ij satisfies UCC.(ii) If, in addition, the model enjoys a discrete rotational symmetry satisfying the hypotheses of Proposition A.3, then σ S rot,ij satisfies UCC.Proof.(i) Since all operators involved in the trace per unit volume computing σ S conv,ij are periodic, Proposition A.2(i) implies the thesis.(ii) By applying Proposition A.3(i) along with Proposition A.2(ii), the conclusion follows.

Lemma 6 . 4 .
If A ∈ P(H f , D f ), then both (e εA − 1) and [e εA , X j ] are in P(H f , D f ) and their norms in this space are bounded uniformly in ε ∈ [0, 1].