Integrals of motion for one-dimensional Anderson localized systems

Anderson localization is known to be inevitable in one dimension for generic disordered models. Since localization leads to Poissonian energy level statistics, we ask if localized systems possess"additional"integrals of motion as well, so as to enhance the analogy with quantum integrable systems. We answer this in the affirmative in the present work. We construct a set of nontrivial integrals of motion for Anderson localized models, in terms of the original creation and annihilation operators. These are found as a power series in the hopping parameter. The recently found Type-1 Hamiltonians, which are known to be quantum integrable in a precise sense, motivate our construction. We note that these models can be viewed as disordered electron models with infinite-range hopping, where a similar series truncates at the linear order. We show that despite the infinite range hopping, all states but one are localized. We also study the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended, depending on the strength of a coupling constant. We formulate a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure in the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in localized phase, but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction.


I. INTRODUCTION
The simplest theoretical model to study single particle localization in the presence of disorder was proposed by Anderson 1 . A localized state has a wavefunction that decays exponentially about some point in space over a characteristic localization length. In three dimensions, localized states exist below a certain energy (the mobility edge) for a given strength of disorder. A disordered electronic system is thus localized if its Fermi energy lies below the mobility edge. In one and two dimensions, an infinitesimal amount of disorder is sufficient to localize all single particle states and thus a disordered noninteracting electronic system is always localized 2,3 .
Recent developments in the area of eigenstate thermalization 4-6 relate closely to the above well established notions of Anderson localization. In this context, it is believed that an isolated localized eigenstate does not thermalize, in the sense that no subsystem of it can be brought into thermal equilibrium by exchanging heat with the rest of the system. An analogous statement can be made about information, as defined through an appropriate partial trace of the density matrix. A related feature of such a system is the lack of level repulsion in its energy level spectrum. This can be thought of as arising from the presence of almost degenerate states localized so far apart that they are unable to hybridize to lift the degeneracy.
The effect of interactions on such systems is very interesting. Interactions among the elementary degrees of freedom generically tend to drive the system towards thermalization and delocalization 7,8 . This tendency competes with the the one that causes localization in the presence of disorder. Understanding the resultant phenomenon of many body localization, that is observed for sufficiently strong disorder, is currently a very active area of research [9][10][11][12][13][14] .
Interestingly, a different class of systems that fail to thermalize are integrable systems. These possess a set of dynamical (i.e. depending on interaction or other system parameter, which we denote here as y) integrals of motion in the thermodynamic limit. Standard examples of such systems are the one-dimensional Hubbard and XXZ models 15-17 . In these examples, the integrals of motion I k are polynomial in y with the order of the polynomial increasing with k. An arbitrary linear superposition of all integrals Q = k a k I k -also an integral in its own right -is an infinite power series in y, which is expected to be the generic situation 18 . Gaudin magnets 19,20 on the other hand provide examples of integrable systems where all conserved charges are linear in y.
While the exact enumeration of the conserved charges 21 is a matter of subtlety 22 , their presence greatly constrains the dynamics of the system. As a result, when started off from an arbitrary initial state in isolation, these systems do not evolve in a way that causes thermalization in the sense of the above paragraph 6,23 . Additionally the usual space time symmetries result in degeneracies in the energy level spectrum, and hence a lack of level repulsion 24 . The addition of perturbations destroys such conservation laws and restores level repulsion, although the strength of the perturbations has a non-trivial finite-size dependence [25][26][27] .
In this context, it is natural to ask in what ways are localized systems similar to integrable ones. In particular we may ask if (parameter dependent) conservation laws, similar to those in integrable systems exist for localized systems. It has been argued in the context of manybody localization that they do, and results related to the growth of entanglement in these systems are predicated on their existence 28 . However, obtaining the structure of the conserved charges in terms of microscopic parameters remains a challenge 11,29 . The situation is less complicated in the absence of interactions since the Hamiltonian is that of a single particle. Nevertheless, obtaining the charges systematically and analytically in terms of the microscopic parameters of the Hamiltonian is nontrivial. In this paper we outline the procedure to do so. We also elucidate the connection between localization and conserved charges.
In this work we study a general one dimensional model with on-site disorder that can interpolate between models with long-range hopping and the more standard Anderson-type one. The starting point is a type 1 Hamiltonian reviewed in 22,30,31 . This was introduced as the most simple model of quantum integrability in finite dimensional spaces. This model has infinite ranged hopping, and as such has no inbuilt metric or length scale. We first show by calculating its Participation ratio (PR) 32,33 the perhaps surprising result that all states except one are localized. This is done as follows: an eigenstate |ψ of the Hamiltonian is expanded in a basis of position eigenstates on the lattice as |ψ = k c k |k , where k labels the position eigenstates and c k are the coefficients in the expansion. The PR for this state is then defined as It is usually understood that PR ψ ∼ O(1) indicates localization while PR ψ ∼ O(N ) -delocalization, where N is the number of sites. While this definition is valid for a fixed wave function, we may also define the PR at a given energy, as later in the paper, where an averaging over disorder realizations is carried out, at a fixed energy. The type 1 model has a known set of conservation laws, which inspire the construction of a generic Anderson-type model, which has only nearest-neighbor hopping. In 1d it is well known that for this model, all single particle eigenstates are localized for any strength of the disorder. The conserved charges of this model are then constructed by analogy with the type 1 Hamiltonian. These charges are expressed as a power series in the hopping, whose coefficients we determine by means of an algorithm. We also show that the series, upon disorder averaging over a "non-resonant" ensemble-defined below, is convergent. This provides numerical evidence that the ensemble chosen and the procedure of averaging the coefficients in the conserved charges over the ensemble is meaningful.
We then turn our focus to a model which contains both localized and delocalized phases (i.e. phases in which all single particles states are either localized or delocalized). This is the Aubry-Andre model 34 , in which the random potential is replaced by a quasi-periodic one. This allows us to test our criterion for the convergence of the power series and clearly elucidate the connection of the conserved charges to localization. Thus, the convergence (divergence) of the power series representation of conserved charges can indeed be identified with the presence (or absence) of localization and the localization-delocalization transition can be located using the charges. Finally, we investigate the effect of interactions and argue that a power series in the interaction becomes intractable and thus obtain the the conserved charges only to first order in it.

II. LATTICE MODELS
We consider a general Hamiltonian of non-interacting particles hopping on a one dimensional lattice with an on-site potential where c † i and c i are fermionic creation and annihilation operators, n i = c † i c i is the number operator, ǫ i is the onsite disordered potential, and t ij is the hopping between sites i and j. The parameter y is a real number introduced for convenience, which; it allows us to perform an expansion of the conserved charges in its powers.
Our general strategy to construct construct conserved charges for this models will be to first consider the 'unperturbed' Hamiltonian which only has the on-site potential. The conserved charges for this Hamiltonian are simply the operators n i , which are independent and commute with each other and the Hamiltonian. It can also be readily seen that the eigenstates of this Hamiltonian are completely localized on the individual sites. Thus the zeroth order Hamiltonian trivially describes a localized system with conserved charges. We now show that upon introducing the hopping, new conserved charges Q i appear, which can still be labeled by the site indices i while the system remains localized. To do this, we consider different types of hopping parameters t ij .

III. TYPE 1 HAMILTONIANS
We now show that for a particular type of Hamiltonian, the conserved charges Q i are linear in the hopping. We explicitly construct Q 0 and the construction of the Q i 's for the other sites i follows in a similar manner. We demand that Q 0 have the general form where α 0 k and β 0 k are parameters that have to determined. The commutator of Q 0 and H is then given by where The requirement that [Q 0 , H] = 0 is satisfied by the following form of t ij . A. PR for type 1 Hamiltonians All single particle states of type 1 Hamiltonians (6), except possibly the ground state for y > 0 or the highest energy state for y < 0 are localized, see e.g. Fig. 1.
This can be understood in more detail from the exact solution for the spectrum of these models 31 . Exact single particle eigenstates of the Hamiltonian (6) are and the corresponding eigenvalues E (energies) are solutions of the equation Suppose ǫ i are ordered in the ascending order. By plotting the left hand side of Eq. (8) as a function of E, one can verify that it has N − 1 real roots The remaining root E 0 is also real and is below ǫ 0 (ground state) for y > 0 and above ǫ N −1 for y < 0 (highest excited state). Eqs. (7) and (8) also provide an exact solution for one fermion (Cooper) pair and one spin flip sectors of the BCS and Gaudin models, respectively, where c iσ are spin-full fermions and s i are quantum spins of arbitrary magnitudes s i , see 31 for details. For the Our results for the PR of type 1 Hamiltonians therefore also apply to these sectors of these models.
The PR defined through Eq. (14) reads For concreteness we take y > 0. Then, the ground state is E 0 < ǫ 0 . We assume that most γ i are of the same order of magnitude and consequently the vector with components γ i is delocalized. Further, we take ǫ i to lie in a fixed interval that does not scale with N , e.g. from −w to w. For excited states E k is between ǫ k−1 and ǫ k . The summations in the numerator and denominator of Eq. (10) both come from ǫ i in a small vicinity of ǫ k for large N and converge as n n −2 and n n −4 , respectively, where n = |i − k|. The numerator and the denominator scale as [γ 2 k /δ 2 ] 2 and γ 4 i /δ 4 , where δ ∝ 1/N is the mean level spacing between ǫ i in the vicinity of ǫ k . Therefore, PR E k is of order 1 (much smaller than N ) meaning excited states are always localized. Fig. 1 shows PR for N = 10 3 uncorrelated random ǫ i uniformly drawn from an interval (−1, 1) and the same distribution of γ i . Consistent with our numerical results, we estimate the largest PR for excited states to scale as ln N , i.e.
for large N , where α depends on N much weaker than ln N . Such values of PR come from clustering in ǫ i . Indeed, suppose spacings δ i = ǫ i+1 − ǫ i between m of ǫ i for i from k to k + m are all much smaller than δ k−1 and, moreover, ǫ k+m − ǫ k ≤ δ k−1 . It follows from Eq. (10) that PR E k ≈ PR E k+m+1 ≈ m because the above ǫ i contribute most to these PRs. Normalized spacings s i = δ i /δ are distributed according to the Poisson distribution P (s)ds = e −s ds. The probability of having m spacings between 0 and s 0 ≪ 1 is then roughly s m 0 . We need ms 0 ≤ 1 and also N s m 0 = 1 so that at least one such clustering occurs 36 . This implies m ≈ ln N/ ln(ln N ) and Eq. (11) follows. Numerically we find that typical values of α ≈ 1 − 3 and averaged over disorderᾱ ≈ 1.7, at least for N = 2 4 − 2 12 . Note that according to this argument such large values of PR typically come in pairs spaced by m + 1, roughly equal to the value of the PR itself. We also stress that, in contrast to the largest PR, a typical (and average) PR is something between one and three for any N (does not scale) as can be seen from Fig. 1.
It is interesting to compare this ln N behavior to the flat band localization studied earlier 37,38 . The latter leads to a (weakly) divergent PR in the localized regime, a phenomenon that is viewed as corresponding to critical (power law type) localization. The type-1 Hamiltonian kinetic energy may also be viewed as a "flat band" model, with a flat dispersion for all except one state. Indeed, for t ij = γ i γ j all but one eigenvalues of the second term in Eq. (2) are zero. The non-zero eigenvalue (ground state for y > 0) corresponds to the eigenstate γ i c † i |0 . Let us consider limits y → 0 and y → ∞ separately. When y → 0 all states are localized as expected. Indeed, Eq. (8) implies E k → ǫ k , summations in Eq. (10) are dominated by the i = k term and we obtain PR E k = 1 for all k. When y → ∞ excited states are localized as before because E k for k ≥ 1 remains trapped in the interval (ǫ k−1 , ǫ k ). The ground state energy on the other hand diverges -Eq. (8) implies E 0 → −y i γ 2 i . Then, ǫ i are negligible as compared to E 0 in Eq. (10) and which is of order N according to our choice of γ i . The ground state is therefore delocalized for y → ∞. It undergoes a localization-delocalization crossover at a certain y c , which we estimate below in this section. It is possible to evaluate the PR analytically to leading order in 1/N for distributions of ǫ i and γ i with negligible short range fluctuations (such that the spacing δ i = ǫ i+1 −ǫ i changes slowly with i -|δ i+1 −δ i |/δ i is of order 1/N for all i -and similarly for γ i ). For simplicity, let us take constant γ i , which we can set to one with no loss of generality, and equally spaced ǫ i , i.e. δ i = δ = 2w/N .
For excited states, we write E k = ǫ k − α k δ, where 0 < α k < 1, and solve Eq. (8) for α k to the leading order in 1/N as described in Appendix B of 39 . This yields We note that λ = y/δ is the proper dimensionless coupling constant in the sense that it must stay finite in the N → ∞ limit. This is because the second summation in Eq.
(2) scales as N 2 for t ij = γ i γ j and our choice of γ i . Therefore, we need y ∝ δ ∝ 1/N so that both terms in Eq.
(2) are extensive in the thermodynamic limit. For the BCS Hamiltonian in Eq. (9), so defined λ is the dimensionless superconducting coupling 40 . Eq. (10) becomes to leading order in 1/N which evaluates to This answer is in good agreement with numerics already for N = 20, see also Fig. 1. Note that 1 ≤ PR E k ≤ 3. We saw above that the ground state energy E 0 → −∞ as y → ∞, while E 0 → ǫ 0 for y → 0. Let y be large enough that E 0 is well separated from ǫ 0 . Then, we can replace summation in Eq. (8) with integration and obtain Performing the same replacement in Eq. (10) and using Eq. (16), we derive Note that in the limit y → ∞, PR E0 = N in agreement with Eq. (12). This expression also allows us to estimate the value y c beyond which the ground state becomes extended. We obtain λ c = y c /δ ≈ 1/ ln N . This also corresponds to the coupling for which the gap in the spectrum ∆ = E 1 − E 0 ≈ −w − E 0 becomes comparable to the spacing δ. For a superconductor described by the BCS model (9) this localized-extended crossover translates into a normal-superconducting one 41,42 . As N → ∞ this crossover becomes a quantum phase transition at λ = 0, i.e. any infinitesimal coupling is sufficient to make the ground state extended (superconducting).

IV. A MODEL WITH FINITE-RANGED HOPPING
We now consider the following Anderson-type model in one dimension with nearest neighbor hopping.
This corresponds to the case with t ij = t for |i − j| = 1 and 0 otherwise for the general Hamiltonian in Eq. (2). H 0 is the zeroth order Hamiltonian with only the onsite potential and H 1 contains the hopping. It is known that all single particle eigenstates of this Hamiltonian are localized 1,3 .

A. Construction of the conserved charges
Proceeding as for the case of type 1 Hamiltonians, we focus on the conserved charge Q 0 corresponding to the site i = 0, which to lowest order is equal to n 0 . However, in this case Q 0 is not simply linear in y. In fact, it can be argued that the an expansion of Q 0 in the hopping does not truncate at any finite order in the thermodynamic limit. Indeed, as explained in the Introduction, conserved charges are generally infinite power series in y. We thus assume Q 0 of the form where P 0 = n 0 and P 1 , P 2 . . . are operators to be determined in terms of the microscopic parameters subject to the condition [Q 0 , H] = 0 ∀i. For concreteness, we first take our one dimensional system to be a finite-sized ring of N + 1 sites going from 0 to N . Since the Hamiltonian H and and all the zero order charges n i are quadratic in the creation and annihilation operators, we take all the operators P 1 , P 2 . . . to be similarly quadratic, i.e.
where the coefficients η m ij are to be determined. We have The requirement that the commutator vanishes to all orders in y yields a recursion relation among η ij .
[P m−1 , Equating the coefficient at (c † r c s − c † s c r ) on both sides, we obtain for r < s, η m rs = η m sr , η 0 00 = 1, and η 0 ij = 0 ∀ i, j = 0. It can be verified that terms present in P m are of the form Fig. 2 for the first few P i .

is odd) This is shown schematically in
The other Q i are obtained by translating all site indices in the above relations by an appropriate number. By construction, they all commute with H. Since H is generally non-degenerate, this implies Q i also commute among themselves, [Q i , Q j ] = 0 ∀ i, j. To see this, first recall that for Hermitian matrices [A, B] = [A, C] = 0 implies [B, C] = 0 as long as eigenvalues of A are nondegenerate. All operators involved in the above construction of Q i are of the formÂ = ij A ij c † i c j , where A ij is a Hermitian N × N matrix, which represents operatorÂ in the sector with total particle number n = 1. Moreover, the commutativity of any two such operators is equivalent to that of the underlying matrices. Eigenvalues of the Hamiltonian in the n = 1 sector at y = 0 are ǫ i , which are assumed to be distinct, i.e. the corresponding matrix is non-degenerate at y = 0. By continuity of the eigenvalues in y, it remains non-degenerate in some finite interval (until the first level crossing) of the real axis containing y = 0. Thus, [Q i , Q j ] = 0 ∀ i, j in this interval of y. But, as can be seen e.g. from the above construction of Q 0 , commutativity of Q i on any finite interval of values of y implies that they commute for all y.
Note that commutation relations (21) and consequently recursion relations (22) do not constrain the diagonal part of the coefficients η m , i.e. η m rr , for m ≥ 1.
These are in fact arbitrary and represent a certain "gauge freedom" in the construction of Q i . The choice of η m rr however affects the off-diagonal part of η k for k > m. In our construction of Q i we set η m rr = 0 for all m ≥ 1. Conserved charges Q i resulting from any other choicẽ η m rr = a m r of the diagonal part uniquely relate to our Q i , and similarly for other Q i . Indeed,η m rr = a m r follows from η m rr = δ m,0 δ r,0 and analogous diagonal elements of other Q i , while the off-diagonal part ofη m is uniquely fixed by Q 0 = n 0 at y = 0 and recursion relations (22) (whichη m rs must also satisfy). The advantage of our choice of a gauge is in a simple relationship between the Hamiltonian (18) and the conserved charges, namely, To see this, consider the difference where R i are y-independent operators. Note that the zeroth order term cancels in the difference. Since H commutes with all Q i , the right hand side (RHS) of Eq. (25) must also commute. This implies in particular [R 1 , n i ] = 0 for all i (from the coefficient at the lowest power of y in the commutator of the RHS with Q i ), which in turn means that R 1 = i r 1 i n i . Now note that the left hand side (LHS) has zero diagonal matrix elements, i.e. no terms of the form c † r c r . This is because the zeroth order term is absent, while higher order terms have no diagonal matrix elements since η m rr = 0 for all m ≥ 1 in our gauge (and similarly the diagonal is absent in other Q i ). Then, the diagonal matrix elements must vanish on the RHS as well, to all orders in y. In particular, r 1 i = 0, i.e. R 1 = 0 and Applying the same argument to the RHS of this equation we similarly obtain R 2 = 0 etc., until we finally arrive at Eq. (24). While we have constructed the conserved charges here explicitly for the case with only nearest neighbor hopping, it is straightforward to see that it can be generalized to a case with any type of finite-range hopping: the charges will always be power series in y with the operators P 1 , P 2 , . . . always quadratic in the creation and annihilation operators. However, the coefficients η m ij will depend on the specific form of the hopping. Thus, in contrast to the case of type 1 Hamiltonians, here the conserved charges are power series in the hopping.

B. Convergence of the power series
Having constructed the power series, we ask whether and in what sense it is convergent. A reasonable condition for convergence is a sufficiently rapid decay of the coefficients η m ij with increasing m. However, this is complicated by the fact that there are energy difference denominators in the coefficients η m ij that can cause them to blow up when the on-site energies at two different sites are equal. To avoid this, we restrict ourselves to a particular type of disorder that may be termed "non-resonant". By this we mean any ensemble of ǫ i , which shows "level repulsion", i.e. the probability of finding ǫ i very close to each other is very small. From the random matrix theory, we know that the eigenvalues of a generic matrix display level repulsion in their eigenvalues of various degree, the Gaussian Orthogonal Ensemble (GOE) 43 of real symmetric matrices has the least level repulsion. This condition ensures that perturbative resonances from small denominators, that would otherwise cause individual terms in the expansions of the conserved charges to diverge, are prohibited. A similar strategy has been recently adopted in the context of many-body localization 44 . The on-site energies ǫ i are drawn from the eigenvalues of a real symmetric matrices whose elements are taken from a Gaussian random distribution with fixed variance. The eigenvalues of these matrices are assigned randomly to different sites. Different random assignments then constitute different realizations of disorder, which can then be averaged over to check for convergence. The result of this procedure is shown in Fig. 3, where ǫ i are drawn from the eigenvalues of real symmetric matrices whose elements are taken from a Gaussian distribution of variance σ=0.1, 0.25 and 0.4. It can be seen that the η m decrease rapidly with increasing order of power series m indicating convergence. We have also checked the convergence of the power series for ǫ i drawn from the eigenvalues of non-integrable t − t ′ − V model, which also follow a GOE distribution 25,45 .
Since η m ij contain more than one term for each m, we checked the convergence of a typical term, which is of the form t m (ǫa 1 −ǫ b 1 )(ǫa 2 −ǫ b 2 )....(ǫa m −ǫ bm ) . Here ǫ i are chosen randomly from [ǫ N −(m−1) , ǫ m ] (ǫ 0 is at the center) such that ǫ ai = ǫ bi , ∀i and max It is legitimate to ask whether this slightly nonstandard choice of disorder distribution produces localization. We have verified this through numerical exact diagonalization by calculating the PR. We find that the PR for different eigenstates is indeed close to zero for systems of size N = 337 as shown in Fig. 4, consistent with localization. We thus conclude that our model with onsite energies taken from a GOE distribution does indeed produce a localized phase.
A similar exercise to construct the conservation laws for the above model has been carried out in Ref. 46 . In that work too, the conserved charges have been constructed as infinite operator series but whose coefficients correspond to the amplitudes of a particle to be on the sites of a square lattice whose sides are the physical one dimensional lattice. The recursion relation obtained is between conserved charges on different sties and the convergence of the series is assumed to follow from the exponential decay of the eigenfunctions of the Hamiltonian. In our calculations, we construct the conserved charges directly in terms of the microscopic parameters of the Hamiltonian and our convergence criterion is not based on any assumption about the nature of the eigenstates of the Hamiltonian. In fact, as we show in the next section, the convergence of the series for the conserved charges can be used to identify the delocalized and localized phases instead of the eigenfunctions.

V. AUBRY-ANDRE MODEL
Having constructed the conserved charges for a model with finite-range hopping and defined a condition for convergence of the power series for them, we can further investigate the meaning of our convergence criterion. In particular, since our goal is to identify the validity of our construction of the conservation laws with the presence of localization, the power series should fail to converge according to our criterion in a delocalized phase.
We thus require a non-interacting model with disorder in one dimension which has a delocalized phase. While any model with finite-range hopping and an on-site random potential in one dimension always produces localization 1,3 , a quasi-periodic potential can produce localized and delocalized phases. Such a model is the Aubry-Andre model 34 given by the Hamiltonian where β is an irrational number. The parameter h can be tuned to effect a transition from a localized phase (for h > 1) to a delocalized phase (for h < 1) 34 . We note that this model is usually studied with an additional term that introduces a p-wave pairing gap 47 , but we set it equal to zero for our analysis. The localized phase here is one in which all single particle states are localized and similarly all single particle states are delocalized in the delocalized phase. The transition between these phases happens at h = 1. Since the Hamiltonian in Eq. (27) is also of the form (18), we can use the expressions obtained for the η m ij in the previous section to construct the conserved charges. These will now depend on the parameter h (i.e. y → (2h) −1 in the previous section) and if the criterion for convergence postulated by us is a valid one to detect localization, we should observe the power series to converge in the localized phase (h > 1) and diverge in the delocalized phase (h < 1). This is indeed the case as we see e.g. from Fig. 5, which shows that a typical matrix element of η m goes to zero quite rapidly with increasing m for h > 1 but diverges for h < 1. Thus, we have established that our convergence criterion is valid for identifying the localization-delocalization transition.

VI. INTERACTIONS
We now turn to systems with interactions. The simplest way to introduce interactions to models we studied here is through a nearest neighbor density-density term. Let us, for example, add such a term to Eq. (18), where we redefined H 0 as compared to Eq. (18). We assume that the particles here are spineless fermions. It is tempting to try a construction of the conserved charges starting from a zeroth order Hamiltonian that combines the on-site and interaction terms since they commute with each other and their eigenstates are localized at every site. However, the interaction term is quartic in creation and annihilation operators and so the conserved charges can no longer be assumed to be power series in the hopping with each term quadratic in the creation and annihilation operators. Such an assumption leads to no solution for the coefficients since the commutators keep producing terms with increasingly longer trails of creation and annihilation operators as one goes to higher orders in the hopping. A more profitable exercise is to try to obtain the conserved charges as power series in the hopping but only to the first order in the interaction. While these are not exact, they offer a reasonable approximation in the limit of small interaction strength. Weak interactions typically should not destroy the localization present in the non-interacting limit and thus conserved charges should continue to exist.
We know from our previous calculation that the operator of the form Q 0 = n 0 + ijm η m ij y m c † i c j commutes with H 0 . Let us now define a new operator Q = Q 0 + V δQ to linear order in V and calculate the commutator.
Since, only η 1 i,i+1 and η 1 i,i−1 are non-zero, the non-zero A rstv 1 are given by the following equations: A k,k,k+2,k+1 1 = A k,k,k+1,k+2 The corresponding expression for δQ to order y is Other approaches to construct conservation laws for interacting systems have been proposed including a recent one where the interacting problem is mapped onto a non-Hermitian problem on a lattice in operator space 46 . A convergence criterion for the resultant series based on the operator norm is then used to identify localized and delocalized phases.

VII. CONCLUSIONS AND DISCUSSION
We have demonstrated a scheme to obtain the conserved charges for non-interacting disordered models displaying localization in one dimension. Our motivation was an observation of similarities between localized and integrable systems, such as the absence of level repulsion and thermalization. We constructed that these conserved charges as power series in the hopping. Further, by a choice of a suitable convergence criterion, the divergence (convergence) of conserved charges identifies with the presence of localization (delocalization).
We should emphasize that this does not mean that the non-interacting delocalized systems are not integrable. In fact, when such a system has both localized and delocalized phases, it is natural to expect that the integrals of motion can be analytically continued into the delocalized phase. Equivalently, one can construct "dual" conserved charges in the delocalized phase in the momentum basis as power series in the potential taking the hopping term as the zeroth order Hamiltonian. To illustrate this, we note that the Andre-Aubry model is self dual 48,49 and one can introduce a new set of fermionic operators, ck = 1 √ L n exp(i2πkβ)c n , which are eigenstates of the momentum operator with eigenvalue: k =kF n−1 mod F n , where F n is the n-th Fibonacci number and L = F n . In terms of these new set of fermionic operators the Hamiltonian (27)  The Hamiltonian satisfies the duality relation: H(h)/h = H(1/h). We have shown that for the Aubrey-Andre Hamiltonian written in real space, one can construct set of conserved charges that converge for 0 < h < 1. Because of the duality of the model one can construct similar conserved charges in terms of ck and c † k . The power series of these charges converge when 0 < 1/h < 1 and both sets of charges diverge at h = 1.
This can be better understood by noting that localization is a basis dependent concept. We have been using localization (as is the standard practice) to mean localization in real space. To obtain the conserved charges for such a localized phase, we start from a Hamiltonian whose eigenstates are perfectly localized in real space and then add terms perturbatively in the hopping. Similarly, the delocalized phase of the Aubry-Andre model is localized in momentum space and one can then obtain its conserved charges by starting with a Hamiltonian perfectly localized in momentum space (tight binding model) and then add terms perturbatively in the on-site potential. This is the essence of the duality outlined above. Thus, the conserved charges also carry labels indicating the space (real or momentum) where the system is localized. What is important though is that once the basis in which the system is localized is identified and the conserved charges are constructed accordingly, they are sensitive to the onset of delocalization in that basis and can be used to locate localization-delocalization transitions.
The importance of the basis can be further understood when one compares the behavior hard-core bosons with that of spinless fermions in the Aubry-Andre model 50,51 . The duality between the localized and delocalized phases is destroyed for hard-core bosons. As a result, the relaxation of real space local observables in the localized phase is different from their conjugates in momentum space in the delocalized phase. This feature is absent for spineless fermions where the duality holds and as a consequence, conserved charges of the type derived in this work exist in both phases.
While it is only possible to construct these charges to lowest order in the interaction using our procedure, their fate upon the introduction of interactions can in principle be investigated numerically, which we defer to a future work. 52

VIII. ACKNOWLEDGEMENTS
The work at UCSC was supported by DOE under Grant No. FG02-06ER46319. E. A. Y. was financially supported in part by the David and Lucile Packard Foundation. RM acknowledges support from the UGC-BSR Fellowship. SM thanks the DST, Government of India and the India-Israel joint research programme for funding.