Open XXZ spin chain: Nonequilibrium steady state and a strict bound on ballistic transport

Open XXZ spin chain: Nonequilibrium steady state and a strict bound on ballistic transport

Tomaž Prosen Department of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
July 12, 2019
Abstract

An explicit matrix product ansatz is presented, in the first two orders in the (weak) coupling parameter, for the nonequilibrium steady state of the homogeneous, nearest neighbor Heisenberg XXZ spin 1/2 chain driven by Lindblad operators which act only at the edges of the chain. The first order of the density operator becomes in the thermodynamic limit an exact pseudolocal conservation law and yields – via the Mazur inequality – a rigorous lower bound on the high-temperature spin Drude weight. Such a Mazur bound is found a nonvanishing fractal function of the anisotropy parameter for .

pacs:
02.30.Ik, 03.65.Yz, 05.60.Gg, 75.10.Pq

Introduction. Exactly solvable models which exhibit certain generic physical properties are of paramount importance in theoretical physics, in particular in condensed matter and statistical physics where one of the key open issues is the transport in low dimensional strongly interacting quantum systems. An example par excellence of such models is an anisotropic Heisenberg XXZ spin 1/2 chain with a constant nearest neighbor spin interaction which, in spite of it being Bethe ansatz solvable giamarchi (), still offers many puzzles. For example, at high temperature and vanishing external magnetic field, it is not clear even if the model exhibits ballistic or diffusive spin transport affleck (). The question is of long-lasting experimental interest experiments (). Recently, theoretical study of interacting many-body systems has got a new impetus by invoking the methods of open quantum systems and master equations breuer () in the study of quantum transport far from equilibrium saito (); wichterich ().

We consider the markovian master equation in the Lindblad form lindblad ()

 dρ(t)dt=−i[H,ρ(t)]+∑k2Lkρ(t)L†k−{L†kLk,ρ(t)} (1)

for an open XXZ spin 1/2 chain with the Hamiltonian

 H=n−1∑j=1(2σ+jσ−j+1+2σ−jσ+j+1+Δσzjσzj+1) (2)

where are Pauli operators on a tensor product space , with symmetric Lindblad driving acting on the edges of the chain only

 L1,2=√(1±μ)ε/2σ±1,L3,4=√(1∓μ)ε/2σ±n. (3)

In this Letter we construct an exact nonequilibrium steady state (NESS) solution of this model in the regime of weak coupling (small ) with the method which seems to be un-related to the Bethe ansatz and expresses NESS in the form of a homogeneous matrix product operator. Such an ansatz has been employed earlier for exact solutions of classical many-body stochastic processes blythe (), however with an important distinction that here the constructing auxiliary matrices satisfy cubic instead of quadratic algebraic relations. Our solution gives birth to a spin current related conservation law of an infinite chain, which is almost local (pseudolocal) in the metallic regime , and hence is used in a Mazur inequality mazur () to bound the spin Drude weight and prove ballistic transport for certain values of anisotropy , while for other we use efficient numerical computation. We observe that the graph of the Mazur bound versus exhibits a fractal structure.

Boundary driven XXZ chain. NESS is a fixed point of the flow (1)

where and

 ^D := 12(1+μ)^D++12(1−μ)^D−,with (5) ^D±ρ := 2σ±1ρσ∓1−{σ∓1σ±1,ρ}+2σ∓nρσ±n−{σ±nσ∓n,ρ}.

Note the distinct roles of two bath parameters: coupling strength controls the strength of coupling to the spin baths at the edges of the chain, while driving strength controls the nonequilibrium forcing due to unequal average spin polarizations (or chemical potentials, in the spinless fermion formulation giuliano ()) of the two baths. For example, in the common derivation of the master eq. (1) breuer () weak coupling (small ) is a standard assumption, whereas the linear response physics would be mimicked by taking small at nonsmall (see e.g. pz (); marko ()).

Here we address far-from-equilibrium physics within the regime of weak coupling, so we formally expand NESS in terms of the coupling parameter

 ρ∞=∞∑p=0(iε)pρ(p). (6)

Plugging the ansatz (6) into the fixed point condition (4) results in an operator valued recurrence relation

for the sequence , with the initial condition which is an infinite temperature equilibrium state.

Theorem. Solutions of (7) in the first two orders read:

 2nρ(1) = μ(Z−Z†), (8) 2nρ(2) = μ22(Z−Z†)2−μ2[Z,Z†]. (9)

is a non-Hermitian matrix product operator

 Z=∑(s1,…,sn)∈{+,−,0}n⟨L|As1As2⋯Asn|R⟩n∏j=1σsjj, (10)

where and are triple of near-diagonal matrix operators acting on an auxiliary Hilbert space spanned by the orthonormal basis :

 A0 = |L⟩⟨L|+|R⟩⟨R|+∞∑r=1cos(rλ)|r⟩⟨r|, A+ = |L⟩⟨1|+c∞∑r=1sin(2⌊r+12⌋λ)|r⟩⟨r+1|, (11) A− = |1⟩⟨R|−c−1∞∑r=1sin((2⌊r2⌋+1)λ)|r+1⟩⟨r|,

where and is the largest integer not larger than . Constant is arbitrary, but we adopt a choice for () and for () making the matrices (11) always real.

Proof. First order. From (5) it follows that , and , hence the first order (8) satisfies (7) for if

 [H,Z]=−σz1+σzn. (12)

It is important to observe that the ansatz (10) does not contain any operator, while can only contain terms with a single . Let us write with . Eq. (12) implies that all the terms of where appear in the bulk should vanish. Looking locally at each triple of sites around this means

 ∑s1,s2,s3∈{±,0}tr(σr1j−1σzjσr3j+1[hj−1,j+hj,j+1,σs1j−1σs2jσs3j+1]) ×As1As2As3=0,forallr1,r3∈{±,0}

resulting in eight independent 3-point algebraic relations

 [A0,A+A−]=0,{A0,A2+}=2ΔA+A0A+, (13) [A0,A−A+]=0,{A0,A2−}=2ΔA−A0A−, 2Δ[A20,A+]=[A−,A2+],2Δ[A20,A−]=[A+,A2−], 2Δ{A20,A+}−4A0A+A0={A−,A2+}−2A+A−A+, 2Δ{A20,A−}−4A0A−A0={A+,A2−}−2A−A+A−.

Sufficient additional conditions that the boundary terms of containing result in the exact RHS of (12), when is of the form (10), can be expressed in terms of auxiliary boundary vectors

 ⟨L|A− = ⟨L|A+A−A+=⟨L|A+A2−=0, A+|R⟩ = A−A+A−|R⟩=A2+A−|R⟩=0, (14) ⟨L|A0 = ⟨L|,A0|R⟩=|R⟩.⟨L|A+A−|R⟩=1.

Verifying (13) and (14), which imply (12), for the representation (11) results in trivial trigonometric identities.

Second order. To prove that (9) satisfies (7) for it is sufficient to show (a) and (b). These relations are implied by (a’) and (b’), using the identity , together with (12), which we have just proven, Hermitian (anti)symmetrization, and the property . As and (anti)commutator with only act on sites and , and since on these sites can only contain terms with and , respectively (see (14)), it is sufficient to check (a’) and (b’) for all , which reduce to trivial algebra. QED.

The theorem constitutes the rigorous part of this Letter. Its essential ingredient, namely the conservation law property (12), shall be applied later to the long standing problem of the spin Drude weight. However, for establishing the existence of a unique NESS several mathematical issues still have to be addressed: (i) It is s not clear a-priori whether recurrence (7) should have a unique solution. Suppose uniqueness can be proven up to some order . Then, the solution of a linear equation (7) is certainly non-unique up to the addition of an arbitrary linear combination of conserved quantities , namely . However, as has been checked explicitly by means of computer algebra for small and any , a unique set of coefficients exists such that is in the image of , and so the solution of (7) for the next order exists. For general this statement remains a conjecture. (ii) For (9) to give the complete second order of NESS, one thus needs to check in addition that . This has been verified explicitly again only for , and is conjectured to be true for any . (iii) Convergence properties of perturbation series (6) are unknown, although it should be useful even only as an asymptotic series.

Computation of observables. We note that for any finite , due to their near-diagonality the matrices (11) can be truncated to the dimensional auxiliary space spanned by vectors only, reproducing exactly the same matrix product operator (10). Let us now describe how to compute physical observables in NESS, , to second order in . For example, the spin current from site to , results directly from (14), More interesting are expectations of local magnetization and spin-spin correlations which are, since does not contain any , nonvanishing only in the second order in , hence we define and . The multiple sums appearing in such as can be evaluated using the transfer matrices (TMs), , , where , as e.g. , and similarly for and higher correlations. However, evaluation of such TM products can be drastically simplified by observing that, since the matrices (11) have only a single element in each row, the subspace of diagonal vectors spanned by , is invariant under the action of TMs , . Using identification and defining reduced TMs , , reading explicitly

 T =|L⟩⟨L|+|R⟩⟨R|+12(|L⟩⟨1|+|1⟩⟨R|) (16) + ∞∑r=1{cos2(rλ)|r⟩⟨r|+c22sin2(2⌊r+12⌋λ)|r⟩⟨r+1| +c−22sin2((2⌊r2⌋+1)λ)|r+1⟩⟨r|}, V =|L⟩⟨1|2−|1⟩⟨R|2+∞∑r=1{c22sin2(2⌊r+12⌋λ)|r⟩⟨r+1| −c−22sin2((2⌊r2⌋+1)λ)|r+1⟩⟨r|},

we obtain efficient TM product expressions

 Mj = μ⟨L|Tj−1VTn−j|R⟩, (17) Cj,k = μ2⟨L|Tj−1VTk−j−1VTn−k|R⟩,j

etc, where, for any finite , the operators (16,16) can again be truncated to -dimensional matrices over . Note that with our aforementioned choice of constant the matrix elements of are always positive.

Examples. For the isotropic case , the TMs have an effective rank , i.e. they close on , so magnetization profile is - interestingly - found explicitly to be linear, with constant long-range correlator , . Similarly, for any of the form , , i.e. for rational , we find that is invariant under , since the off-diagonal chains of (16,16) are cut, i.e. matrix elements at position either or vanish, so can be replaced by dimensional matrices of , , independently of the system size . This allows us to find explicit results for some small , say for , . For , however, the TMs have always infinite rank, with exponentially growing matrix elements, but (17) allow for stable and efficient numerical computations, say of in only steps. In Fig. 1 we plot magnetization profiles, ranging from flatlike (for ) indicative of ballistic transport to kink-shaped (for ) with superexponential (in ) growth of near the baths indicating the superexponential shrinking of the perturbative border for , and consequently also of the spin current , since . Note that such kink-shaped profiles have been observed numerically for strong driving in conjunction with the negative differential conductance giuliano (). One has to stress that our result, being perturbative in , cannot be used to address the question of conductivity and spin diffusion, namely the conductivity does not exist in the weak coupling limit, i.e. the limits and do not commute in the thermodynamic limit (TL) .

Mazur bound on Drude weight. Nevertheless, our result offers very interesting physical application. Namely, as it is clear from eq. (12), the operator commutes with the XXZ Hamiltonian, apart from the boundary terms. For , the matrix product operator (10) makes sense even for , where becomes a translationally invariant pseudolocal conservation law in the spirit of Ref. prosen (). Since almost all matrix elements (11) are smaller than for , it is easy to prove that the coefficients of expansion of in Pauli spin operator clusters decay exponentially with the length of the cluster.

Ref. zotos () used the Mazur inequality mazur () to rigorously estimate the lower bound on the Drude weight at high temperature note () in terms of some conservation laws

 Dn=limt→∞β2nt∫t0dt′⟨J(t′)J⟩≥β2n∑k(J,Qk)2(Qk,Qk), (18)

where , , is the inverse temperature, and is Hilbert-Schmidt inner product in the operator space, which defines the norm , and conserved quantities are chosen to be mutually orthogonal . However, for XXZ chain all the known local, or ‘normalizable’, conservation laws mathieu () are orthogonal to the spin current , so the RHS of (18) vanishes. Nevertheless, many numerical computations (e.g.fabian ()) suggested that for , should be positive, although doubts have been raised recently affleck (), making the issue quite controversial. Here we propose to use the first-order-term in NESS (8), as Hermitian operator, which is conserved in TL, since clearly , and is relevant to the current, since . Putting and performing TL on both sides note2 () of (18), we obtain where . Computing by iterating TM we get efficiently computable expression for the Mazur bound

 DZ=14limn→∞n⟨L|Tn|R⟩. (19)

For , we derive from the fact that the matrix elements in the tridiagonal band of (16) are all positive and exponentially growing, which allows for numerically observed spin diffusion at infinite temperature pz (); gemmer (). Consider now . For a dense set of values of the anisotropy , TM can be, as discussed earlier, replaced by a finite dimensional matrix , and (19) can be evaluated exactly by means of Jordan decomposition of . For example, for , we find Iterating , for large the off-diagonal element of the Jordan block starts dominating, hence , so . Similarly, we calculated for all (note ):

 λDZ(cosλ)λDZ(cosλ)π/21π/55(5−√5)/64π/39/162π/55(5+√5)/64π/41/3π/63/20 (20)

For , we have , so . For all other values of we estimated (19) numerically and plotted the graph of in Fig. 2 which clearly exhibits a fractal structure. This seems to be the first appearance of fractal behavior in an infinite strongly correlated system and calls for further analysis.

Our numbers (20) have also been compared to the Bethe ansatz result zotos2 () , for , which overshoots the values of the table (20) for and for and , respectively. However, since the assumptions used in Ref. zotos2 () have been debated, it seems interesting to speculate whether our bound may be saturating.

Conclusion. We developed an original method to construct explicit matrix product ansatz for the weak coupling limit of NESS in the XXZ spin chain which is driven out of equilibrium by Lindblad baths attached at the ends of the chain. As a byproduct of this result we discovered a pseudolocal conserved quantity which – unlike previously known local conservation laws – is not orthogonal to the spin current in the zero magnetization sector. Employing this conserved quantity enabled us to rigorously estimate the spin Drude weight in the metallic regime and prove the ballistic high-temperature transport there. Since the XXZ model can be mapped to a - model of interacting fermions, our result should be equally relevant for understanding the charge transport.

We acknowledge fruitful comments by I. Affleck, T. Barthel, J. Eisert, E. Ilievski, M. Horvat, P. Prelovšek, K. Saito, M. Žnidarič and grants P1-0044, J1-2208 of ARRS (Slovenia), as well as the Bessel prize of Alexander von Humboldt foundation.

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