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Ballistic transport in integrable quantum chains with degenerate spectra

We study the ballistic transport in integrable lattice models, i.e., the spin XXZ and Hubbard chains, close to the noninteracting limit. The stiffnesses of spin and charge currents reveal, at high temperatures, a discontinuous reduction (jump) when the interaction is introduced. We show that the jumps are related to the large degeneracy of the parent noninteracting models. These degeneracies are properly captured by the degenerate perturbation calculations which may be performed for large systems. We find that the discontinuities and the quasilocality of the conserved current in this limit can be traced back to the nonlocal character of an effective interaction. From the latter observation we identify a class of observables which show discontinuities in both models. We also argue that the known local conserved quantities are insufficient to explain the stiffnesses in the Hubbard chain in the regime of weak interaction.


I. INTRODUCTION
It is by now well established that the transport in integrable quantum lattice models is anomalous (for a recent review see [1]) and can remain dissipationless/ballistic even at infinite temperature T . The main interest focused on the conductivities and related ballistic components -the stiffnesses  in the one-dimensional (1D) integrable models, mainly the spin XXZ and the Hubbard chains. Non-vanishing stiffnesses have been demonstrated via the sensitivity to the flux [2] and via the Mazur inequality [4,23] that directly accounts for the conservation laws specific for integrable systems, i.e., the local conserved charges [24]. The latter approach demonstrates that the ballistic transport can be recognized as a manifestation of non-thermal steady states [16,[25][26][27][28][29][30][31][32][33]. While the numerical evidence within the XXZ model at half-filling clearly reveals finite spin stiffness D s (T > 0) [3,6,12], the discovery of quasilocal charges [13][14][15][16] have established lower bound for it. This finding fueled the efforts employing the thermodynamic Bethe Ansatz (TBA) approaches [5,22] and the framework of generalized Hydrodynamics (GHD) [18,19,21]. Both analytical approaches for XXZ model indicate on a discontinuous variation of D s with the anisotropy ∆ [13,14], most pronounced near the noninteracting limit ∆ → 0. Numerical calculations reveal a smaller discontinuity and this disagreement has not been yet resolved [1,17]. Due to the larger Hilbert space, much less explored are the charge and spin stifnesses, D c,s , within the integrable 1D Hubbard model where in general Mazur inequality [4] and GHD [19,34] suggest finite D c > 0 at U > 0 away from half-filling [20].
In this work we present the method for calculation of stiffnesses in the integrable XXZ chain with Hamiltonian, H = H 0 + ∆V , and the Hubbard chain with H = H 0 + U V for small but nonzero interactions, 0 < ∆ 1 and 0 < U 1, respectively. Here, we assume that both H 0 and H are integrable. We employ the observation that the reference systems of noninteracting fermions (NIF), as described by H 0 , exhibit large degeneracies which exponentially increase with system size due to the specific dispersion relation of NIF, k ∝ cos(k). The degeneracy is then lifted by the interaction V , which in the integrable case leads to a sudden decrease of the stiffnesses (to which we will refer as the jump further on). In particular it causes a decrease of the spin stiffness, D s , within the XXZ model, and both charge and spin stiffnesses, D c,s , in the 1D Hubbard model. In the Appendixes A and B, we demonstrate that similar reasoning can be applied also to the case when H 0 describes interacting integrable systems with degenerate spectrum, e.g., the XXZ model at ∆ = 1/2.
The approach holds for any extensive observable A in a system of size L. Namely, the stiffness can be defined as D A = T D A where D A = ĀĀ /L and ... is the infinite-temperature averaging over the Hilbert space. Here, we introduce also averaging over infinite time window,Ā = lim τ →∞ τ 0 dt e iH0t Ae −iH0t /τ . The latter averaging eliminates the matrix elements of the observable A which are outside of the degenerate subspaces of the Hamiltonian where A mn = m|A|n and H 0 |m = E m |m . For clarity, we explicitly distinguish between the time-averagingĀ, performed for the unperturbed Hamiltonian H 0 and time-averagingÃ performed in the same way but for H. Consequently, the stiffnesses for the systems described by H 0 and H equal ĀĀ /L and ÃÃ /L, respectively.
It is clear that the last term in Eq. (1) may contribute to D A only for models with degenerate spectra, as it is the case for NIF, and vanishes when the degeneracy is lifted. In order to account for lifting of the degeneracy, one can diagonalize the perturbation V within the degenerate spectrum of H 0 . Therefore, all relevant matrix elements of V are contained in the time-averaged operator,V . We solve the eigenproblem, V |α = v α |α , and express the time-averaged quantities,Ā, in the {|α } basis. Then, one can splitĀ =Ā +Ā ⊥ into two orthogonal operators When the degeneracy of H 0 is (at least partially) lifted bȳ V , only the former operator has nonzero matrix elements for states with equal energies. Consequently onlyĀ contributes to the stiffness in the perturbed model. For all models considered in this work we have found that, up to high accuracy, the stiffness for weakly perturbed Hamiltonian equals whereas the jump in the stiffness equals to Ā ⊥Ā⊥ /L. The relevant question is to what extent the perturbative approach applies also for nonzero ∆, when the energy levels start to cross. We demonstrate that due to integrability, it remains valid also beyond the level crossing. Therefore the integrability is essential for our approach, but the analysis is applicable to any other quantum-lattice model near points with macroscopic degeneracies. We stress that the macroscopic degeneracies discussed in the present work should not be confused with the pairs of eigenstates which in a real-valued Hamiltonian (i.e. without an external flux) are doubly degenerate and related to each other by time reversal [35].
Calculating the left-hand-side of Eq. (4) is a complex problem which involves diagonalization of the many-body Hamiltonian H. However,V is a block-diagonal matrix, thus the complexity of calculating the right-hand-side of Eq. (4) is determined by the dimension of the largest degenerate subspace of H 0 . Note that in the case when H 0 represents NIF, the degenerate subspaces as well as the explicit form ofV can be (in principle) obtained via analytic calculations. Thus, Eq. (4) is an essential advantage for the numerical calculations. Namely, the diagonalization of degenerate subspaces is feasible up to a large system sizes, otherwise reachable only by approximate methods. It also allows to identify observables which commute (do not commute) withV and, consequently do not show (do show) jump when the degeneracy is lifted.

II. XXZ MODEL
We first consider the spin XXZ chain, H = H 0 + ∆V , with Here we use the fermionic representation, periodic boundary condition (PBC) are assumed, J = 1, and we expressV using the momentum representation where k = cos k. The above equation yields (apart from a trivial choice q = 0) two options: (1) q = p − k when ∆V represents the Hartree-Fock term, and (2) p = π − k =k, i.e., p + k = 0, for all q. The latter one is specific for the dispersion k = cos k and leads to large and nontrivial pairing degeneracy for NIF. This finding is crucial for further analysis and eventually for the jump in the investigated spin stiffnesses at ∆ > 0. One may resolve the conservation of energy imposed by the δ-function and find the explicit form ofV ,V We stress that the crucial term Q † 1 Q 1 can be expressed as a product of local charges which were introduced in Refs. [36,37]. They do not conserve the particle number and are not translationally invariant At this stage we distinguish between two classes of local conserved charges valid for the NIF system, One may show that [Q 1 ,Ē 2m+1 ] = 0 and [Q 1 ,F 2m ] = 0. Consequently,Ē 2m+1 =Ē 2m+1 as well asF 2m =F 2m , and the corresponding stiffnesses do not exhibit a jump at ∆ → 0. We note that these quantities correspond to the known (strictly local) charges in the XXZ model in the limit of ∆ → 0, e.g., F 2 represents the energy current. On the other hand, E 2m and F 2m+1 do not commute with Q 1 and the corresponding stiffnesses are expected to have a jump. In particular, it holds for the spin current J s =J s = F 1 . Interestingly, the latter class corresponds to the symmetry sectors where stiffnesses are determined by the quasilocal charges [38,39].
If the jump J ⊥ s J ⊥ s /L remains nonzero for L → ∞, then the operator J ⊥ s is (at least) quasilocal, in view of the definition in Ref [40]. It order to show the latter property, we note that J s J ⊥ thus J ⊥ s has nonzero projection on local and normalized operator J s / √ L and is quasilocal. Here, we recall that J s is an extensive operator hence its norm is ||J s || 2 = J s J s ∝ L.
Next, we construct all many-body states for N f fermions. The studied basis is diagonal in the NIF occupations, n k . It is crucial to single-out the pairs of states p andp = π − p which are created by Q † 1 . We distinguish between pair-states when d p = n p + np ∈ {0, 2}, and singly occupied states with We consider systems with even L where both p,p exist. Obviously, N f = 2N d + N s , where N d , N 0 , N s are number of occupied pair-states, empty pairstates and singly occupied states, respectively. At fixed N f , For even L we can consider only |p| < π/2, so that |p| > π/2 (note that |p| = π/2 cannot form a pair), and the relevant perturbation can be written in terms of sums running only over The final step is then exact diagonalization (ED) of Eq. (12) for each subspace with a fixed number N d , N 0 of the pairstates. This restricts our numerical studies to L ≤ 28, where the degeneracy can reach N deg ∼ 3000, but the dimension of the whole Hilbert space is N st ∼ 10 8 . Here, we concentrate on (canonical) results obtained at fixed filling n av = N f /L = 1/2 and on the systems with L = 4K. The details of numerical calculations are discussed in the Appendix A.
The applicability of the perturbative approach for nonzero ∆ is checked in two ways. First, we compare the stiffnesses which occur on both hand-sides of Eq. (4), whereby ÃÃ /L is ED result for the XXZ chain with ∆ = 0.01. Fig. 1a shows that the perturbative approach perfectly reproduces ED results, but it allows also to study much larger systems, when the L-dependence of D s becomes negligible. Calculated D s exceeds the Mazur lower-bound and GHD result in Refs. [14,18] and is well consistent with numerical result obtained via typicality approach [1,17]. In Fig. 1b we confirm our analytical predictions concerning the quantities for which corresponding stiffnesses D A show a jump at ∆ → 0. In particular, D A for E 2 and F 3 , see Eq. (9), show a clear jump which is even more evident than for J s = F 1 .
In order to identify artifacts originating from improper order of limits, L → ∞ and t → ∞, we have calculated the dynamical spin conductivity, σ s (ω). We employ the microcanonical Lanczos method (MCLM) [41] for L = 28, where we reach the energy resolution δω < 10 −3 . We present also the integrated conductivity I s (ω) = 2 π ω 0 dω σ s (ω ) which allows to extract both the value of D s as well as the emergent incoherent part. The incoherent part is particularly important, as it contain the spectral weight which is eliminated from the stiffness, i.e., from the δ(ω)-peak, by lifting the degeneracy. When ∆ increases, the incoherent peak shifts towards larger frequencies, as shown in the inset in Fig. 1c. In the main panel we show the renormalized conductivity I(ω/∆). It confirms that the incoherent contribution scales linearly with ∆, at least for weak ∆ ≤ 0.2. Deviations become apparent only at substantial ∆ ∼ 0.4. Also, I s (ω → 0) > 0.09 at ∆ ≤ 0.05. Both results confirm the relevance of the perturbative approach for finite ∆ > 0.

III. HUBBARD CHAIN
In analogy to the XXZ chain, we analyze the 1D Hubbard model, where we assumed the PBC, set t h = 1, and consider 0 < U 1. Again, one may find the time-averaged perturbation The final step consists in full ED of degeneracies. Here, we can study systems with L ≤ 14 sites with up to N deg < 4000 and total N st ∼ 10 8 basis states, see the Appendix A.
It is clear thatV in the Hubbard chain, Eq. (14), closely resembles Eq. (7) for the XXZ model. However here, Q 0 annihilates spin singlets on a single lattice site. Again, Q 0 is a conserved charge of H 0 which does not conserve the particle number and is not translationally invariant, while the Q † 0 Q 0 term is crucial for explaining the discontinuities of stiffnesses emerging for U → 0. Performing additional summation over spin degrees of freedom in Eqs. (9), one obtains spinsymmetric conserved charges of H 0 analogous to E 2m+1 and F 2m which do commute withV in the Hubbard chain. Such quantities coincide with local charges known for the Hubbard chain at U → 0 [4,24]. In this limit known local charges commute with Q 0 and their D A do not reveal a jump at U → 0.
A particular interest concerns the charge and spin currents Previous studies of the Hubbard chain focused on the halffilling, n av = 1 and m av = 0 [1,[18][19][20], where both stiffnesses were found to vanish. Our results confirm this finding and show that lifting degeneracy explains vanishing of D c (and more evident D s = 0). However, for intermediate n av , m av results are also nontrivial and have been not presented so far. In Fig. 2b we present two characteristic cases for U → 0: D c (n av ) at fixed m av 0 (including also results for m av = 1/L), and D s (m av ) for n av 1 (allowing also for n av = 1 + 1/L). One observes, that D c,s remain in general large for U → 0 while the finite-size effects are rather insignificant. It should be also noted that J c,s have no overlap with known local charges, I l , in the Hubbard model for U → 0, especially with the lowest local charge I 3 [4]. However, it holds true also for other I l>3 [24], for which U → 0 are spin-symmetric generalizations of E 2m+1 and F 2m . On the other hand, quasilocal charges plausibly responsible for D c,s > 0 have not been explicitly identified yet, although such charges should be accounted for within the GHD [19,34]. Due to similarity between Eqs. (7) and (14), we expect that J c,s are quasilocal, as it is the case in the XXZ chain.

IV. CONCLUSIONS
It has previously been established that the spin stiffness, D s , in the XXZ chain reveals a jump, when the many-body interaction, ∆, is turned on. We have shown that this occurs as a manifestation of a general property of integrable systems with degenerate spectra and, most importantly, that the jumps may be explained accurately by perturbative lifting of degeneracy.
We have demonstrated that these jumps occur in the XXZ chain at ∆ → 0 as well as in the 1D Hubbard model for U → 0 for a broad class of operators. Such operators do not commute with additional conserved charges which are specific for NIF with the nearest-neighbor hopping [36,37]. Interestingly, the jumps in XXZ chain occur in the symmetry sectors, where stiffnesses are known to originate from the presence of quasilocal conserved quantities.
The same reasoning can be applied to other integrable lattice models near points of macroscopic degeneracies. Results shown in the Appendix B suggest analogous behavior for the XXZ chain with selected values ∆ = cos(π/n) with n ∈ Z, although then the jumps are much smaller hence the quantitative interpretation of numerical results is less clear. Perturbative lifting of the degeneracy allowed us also to calculate D s for large XXZ chains at ∆ → 0, where we have found that D s is visibly larger than the known analytical TBA/GHD lower bounds. It is plausible that the non-commuting charges [42] might be relevant for explaining the stiffnesses in the XXZ chains with ∆ → cos(π/n), as discussed further on in the Appendix C. Finally, we have found that the known local charges in the Hubbard chain are insufficient to explain the stiffnesses at U → 0, supporting the presence of quasilocal charges in this model.

Appendix A: Numerical calculations for the XXZ and Hubbard chains
While the definition and treatment of degenerate NIF subspaces is described in the main text, the final step for each of r different multiplets is the numerical exact diagonalization (ED) of Eq. (12) which yields (in general) nondegenerate states |α with E α = E r + ∆E rl . We turn now to the analysis of the spin current, expressed in NIF basis as J s = k sin k(n k − n av ). Subtracting n av is useful at halffilling, n av = 1/2, since then only π-pair states contribute, i.e., J s = p sin p (2n p − 1). It is in contrast to, e.g., the energy current J E = k sin(2k)n k where pair-state contribution vanishes and no discontinuity appears at ∆ → 0. Of interest is the spin stiffness D s , defined (at T → ∞) as where N st is total number of many-body states, and we can consider here only the canonical average (at given N f and filling n av = N f /L). We calculate D s as the sum of contributions over all degenerate multiplets where all J αβ are evaluated and a small fraction of level may still remain degenerate E α = E β even after ED. In this study we handle exactly the degenerate subspaces with up to N deg ∼ 3000 states, whereby the full summation over all NIF states N st has to be performed subsequently. This restricts our numerical results to L ≤ 28, where N st ∼ 10 8 . It should be recognized that, in general, the even-size systems L = 2M with odd number of electrons N f can yield quite different (unphysical) result for D s . In particular this appears at half-filling N f = L/2 for L = 4K where one gets anomalous result D s = 0 for ∆ > 0, with the origin in additional degeneracies [11]. We therefore present results for D s and its scaling with 1/L for canonical case and L = 4K. One can partly avoid these problems by performing the full grand canonical (GC) calculation, but results still suffer from quite appreciable finite-size dependence in spite of L ≤ 28. Much more stable results can be achieved by taking into GC sum only sectors with even N f . 1D Hubbard model. Full diagonalization of degeneracies to the lowest order in U follows the procedure analogous to the one described for XXZ model. Within the NIF many-body basis |n = ks (c † ks ) n ks |0 , we determine first the degenerate multiplets having the same total momentum q = ks k n ks . The multiplets emerge due to allowed spin flip and/or due to the π-pair occupied and empty states. The calculation is then performed within the canonical spaces, fixing N ↑ , N ↓ for given L. By allowing up to N deg = 4000 degenerate states within each multiplet we can reach L ≤ 14 for arbitrary density n av and magnetization m av , with up to N st < 10 8 states.
The energy spectra in the XXZ model are highly degenerate also for selected values of ∆ = ∆ c . This degeneracy gives rise, e.g., to the presence of noncommuting quasilocal chargess derived in Ref. [42]. Moreover, the analytical lower bound on the spin stiffness in the XXZ chain reveals jumps in stiffnesses at ∆ → ∆ c [1]. The latter jumps at ∆ c = 0 are much smaller then those discussed in the main text for ∆ → 0, making the numerical studies much more demanding. While we are not able to provide a reliable finite-size scaling towards L → ∞, in this section we demonstrate for finite systems, that these jumps may be accurately described by the perturbative approach based on lifting the degeneracy. In particular, we discuss the case with the largest jump (apart from the case ∆ = 0 ), i.e., we study the XXZ model for ∆ → ∆ c = 0.5.
The unperturbed Hamiltonian with degenerate spectrum reads whereas the degeneracy is lifted in the full Hamiltonian for In the following, we present numerical evidence that where the stiffnesses ÃÃ /L and Ā Ā /L are defined in the same way as in the main text, but with appropriately modified Hamiltonians H 0 and H. For simplicity, here we apply GC averaging over the entire the Hilbert space, whereas in the main text we have used the canonical averaging over the subspace with a fixed number of particles. Figure 3 shows the spin stiffness, D s , in the XXZ model obtained from ED of the Hamiltonian for ∆ = ∆ c and ∆ = ∆ c + 0.01. Since the difference between both results (i.e., the jump) is small while its L-dependence is quite significant, we are unable to judge whether the jump remains nonzero also in the thermodynamic limit. Nevertheless, based on the analytical lower bound on D s given in Ref. [1], we conjecture that it remains nonzero also for L → ∞. In Fig. 3b we show the relative difference between expressions on the both hand sides of Eq. (B3). Namely, we show whereby ÃÃ is calculated with small but nonzero perturbation |∆ − ∆ c | for A = F 1 = J s and A = F 3 , see Eq. (9) in the main text. Results for the XXZ model with ∆ → ∆ c   One may observe for all studied systems that the jump may be accurately described by the perturbative approach even for quite significant perturbations. It supports one of our main claims that lifting degeneracy is a universal mechanism which explains the jumps of stiffnesses in various integrable models. Numerical results shown in the main text in Fig. 1a indicate that the spin stiffness, D s , in the XXZ model for ∆ → 0 exceeds the analytical predictions obtained from the Mazur bound as well as the results from the generalized hydrodynamics (GHD). The latter is unexpected, since the Drude weight obtained from the GHD is widely believed to be exact in the thermodynamic limit. Our results agree with other numerical studies in Ref. [1,17]. We are not aware of any argument suggesting that L-dependence of D s changes above a certain length-scale L * , which is beyond the reach of numerical approaches. Moreover, Fig. 1c does not indicate that there exists a low-frequency scale below ω < ω * ∝ ∆ which could lead to an irregular frequency-dependence of the optical conductivity σ(ω → 0), so that our numerical results might be affected by incorrectly applied limits, L → ∞ and t → ∞.
The stiffness of an extensive operator A is bounded by the Mazur inequality, where the summation runs over all orthogonal conserved charges, C. Typically, one considers only local and quasilocal charges for which [40] 1 L in the thermodynamic limit. In the main text, we have introduced additional local charges, Q n , which do not conserve the total S z , are not translationally invariant and, in general, do not commute with charges which conserve S z . It is known that analogous quasilocal charges exist non only for ∆ = 0 but also for other commensurate values of ∆ [42]. In other words, they exist exactly for these model parameters for which one obtains analytical results from the Mazur bound and the GHD. It is clear that for an observable A which conserves total S z , one obtains AQ n = 0. For this reason, such Q n have not been included in the analytical Mazur bound neither they are accounted for in the GHD. However, it is possible that for properly selected Q n and Q m , one may introduce a Hermitian nonlocal conserved operator such that The 1/L dependence on the right hand side of Eq. (C4) comes from nonlocality of X nm . Such operators may contribute to the stiffness, provided that for local A there exists an extensive number of orthogonal X mn which fulfill Eq. (C4). The general proof for commensurate ∆ = 0 goes far beyond the scope of the present work. However, for ∆ = 0 one may rather easily show that an extensive number of X mn satisfies Eq. (C4) and the details are presented below.
In the main text we have studied the noninteracting limit of the XXZ model and introduced local charges which do not conserve the total S z Beacuse of the 1/ √ L factor, these operators have Lindependent norm, i.e., Q n Q † n ∼ 1. We have also introduced a sequence of orthogonal current-like operators, whereby the first one represents the spin current, F 1 = J s . In order to facilitate comparison with other studies, F n have been chosen as extensive operators with, F n F m = δ mn L/8. We study also X mn defined via Eqs. (C3) and (C5) and consider here only odd difference m − n. We use also a particle-hole transformation defined by Consequently for a single local operator, J s , there exist an extensive (∼ L) number of nonclocal charges, e.g., X m,m+1 for m = 1, 2, 3, ... and ech of them has a projection that scales as 1/L. The above calculations should not be considered as a proof that the difference between the numerical results and the GHD calculations originate from the presence of charges which do not conserve S z [to this end one needs to derive Eq. (C13) for commensurate ∆ = 0 and ensure that charges in the Mazur bound are mutually orthogonal]. Nevertheless, these calculations demonstrate that such origin of the discussed discrepancy cannot be excluded. We note, that noncommuting charges are present also in the integrable Hubbard chain, but only in the noninteracting limit, so the discussed scenario does not apply to the latter model.