Information propagation for interacting particle systems
Norbert Schuch, Sarah K. Harrison, Tobias J. Osborne, Jens Eisert
IInformation propagation for interacting particle systems
Norbert Schuch, Sarah K. Harrison, Tobias J. Osborne,
3, 4 and Jens Eisert
4, 5 Institute for Quantum Information, California Institute of Technology, MC 305-16, Pasadena CA 91125, USA Department of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, UK Institut f¨ur Theoretische Physik, Leibniz-Universit¨at Hannover, Appelstr. 2, 30167 Hannover, Germany Institute for Advanced Study Berlin, 14193 Berlin, Germany Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
We show that excitations of interacting quantum particles in lattice models always propagate with a finitespeed of sound. Our argument is simple yet general and shows that by focusing on the physically relevantobservables one can generally expect a bounded speed of information propagation. The argument applies equallyto quantum spins, bosons such as in the Bose-Hubbard model, fermions, anyons, and general mixtures thereof,on arbitrary lattices of any dimension. It also pertains to dissipative dynamics on the lattice, and generalizes tothe continuum for quantum fields. Our result can be seen as a meaningful analogue of the Lieb-Robinson boundfor strongly correlated models.
How fast can information propagate through a system ofinteracting particles? The obvious answer seems: No fasterthan the speed of light. While certainly correct, this is notthe answer one is usually looking for. For instance, in a clas-sical solid, liquid, or gas, perturbations rather propagate atthe speed of sound, which is determined by the way the par-ticles in the system locally interact with each other, withoutany reference to relativistic effects. We would like to under-stand whether a similar “speed of sound” exists for interact-ing quantum systems, limiting the propagation speed of local-ized excitations, i.e., (quasi-)particles. For interacting quan-tum spin systems, such a maximal velocity, known as theLieb-Robinson bound [1–4], has indeed been shown. Whileit seems appealing that there should always be such a bound,systems of interacting bosons can show counterintuitive ef-fects, in particular since the interpretation of excitations interms of particles is no longer fully justified; in fact, an exam-ple of a non-relativistic system where bosons condense into adynamical state which steadily accelerates has recently beenconstructed [5]. This example suggests the disturbing possi-bility that our intuition is wrong, and only relativistic quantumtheory can provide a proper speed limit.There are many important reasons, both theoretical and ex-perimental, to investigate information propagation bounds ininteracting particle systems. It turns out that such bounds leaddirectly to important, general results concerning the cluster-ing of correlations in equilibrium states [2]. Lieb-Robinsonbounds facilitate the simulatability of strongly interactingquantum systems—the mere existence of a Lieb-Robinsonbound for a quantum system can be used to develop general,efficient, numerical procedures to simulate the dynamics oflattice models [6]. From a more practical perspective, new ex-periments allow one to explore the non-equilibrium dynamicsof ultracold strongly correlated quantum particles—bosonic,fermionic, or mixtures thereof—in optical lattices with un-precedented control [7, 8]. In such experiments, it is impor-tant to understand how the particles move: For example, whenstudying instances of anomalous expansion, it is far from clear a priori whether it is possible to identify a meaningful speedof sound at all. t FIG. 1. Schematic representation of the “light cone” of particles ini-tially placed into a region R of a lattice (yellow circles) and thenpropagating in time t in a way governed by an interacting quantummodel, outside of which the influence of these particles is exponen-tially suppressed. The original Lieb-Robinson bound already applies in a verygeneral setting, namely, to any low-dimensional quantum spinsystem, and to any fermionic system confined to a lattice. Itis therefore tempting to extend the original argument to othersettings, in particular, to systems of interacting bosons; unfor-tunately, all attempts to do so have run into insuperable diffi-culties for systems with nonlinear interactions, including theBose-Hubbard model. The reason for the failure of the orig-inal Lieb-Robinson argument is fundamentally connected tothe unboundedness of the creation operator for bosons: TheLieb-Robinson velocity depends on the norm of the interac-tion, which is unbounded for, e.g., bosons hopping on a lattice,and examples without a speed limit can be constructed [5].In this Letter, we show how these difficulties can be over-come by considering the right question concerning the prop-agation of information. Our approach allows us to determineLieb-Robinson type bounds for the maximal speed at whichinformation can propagate through systems of interacting par-ticles in a very general scenario: In particular, it applies tosystems of interacting bosons, as well as to fermions, spins,anyons, or mixtures thereof, both on lattices and in the con-tinuum. Moreover, it can also be applied beyond Hamiltonianevolution, such as to systems evolving under some local dis-sipative dynamics. a r X i v : . [ qu a n t - ph ] J a n The type of system we have in mind is exemplified by the
Bose-Hubbard model , a model of bosons hopping on an arbi-trary lattice G of any finite dimension and interacting via anon-site repulsion, ˆ H BH = − τ (cid:88) (cid:104) j,k (cid:105) (ˆ b † j ˆ b k + h . c . ) + U (cid:88) j ˆ n j (ˆ n j − − µ (cid:88) j ˆ n j , (1)where the first summation is over neighboring sites on thelattice, ˆ b j is the boson annihilation operator for site j , and ˆ n j = ˆ b † j ˆ b j is the number operator. The natural distance inthe lattice will be denoted by d ( · , · ) , e.g., d ( j, k ) = | j − k | for a one-dimensional chain. While we will, for clarity, focusour discussion on the Bose-Hubbard model, our arguments di-rectly generalize to models of the form ˆ H = − τ S (cid:88) s =1 (cid:88) (cid:104) j,k (cid:105) (ˆ b † s,j ˆ b s,k + h . c . ) + f ( { ˆ n ,j , . . . , ˆ n S,j } j ∈ G ) , (2)where the ˆ b s,j are annihilation operators for bosons, fermions,or even anyons of species s = 1 , . . . , S at site j , and ˆ n s,j =ˆ b † s,j ˆ b s,j ; the species could for instance refer to an internal spindegree of freedom. The interaction between the particles ischaracterized by f which can be an arbitrary function of thelocal densities, and may involve higher moments of the par-ticle number, or even non-local interactions. Moreover, ourargument also applies to time-dependent Hamiltonians of thisform, as long as the tunneling amplitude τ ( · ) is bounded.The scenario we consider is described by the Bose-Hubbardmodel on a lattice G , where in the initial state all sites areempty (i.e., (cid:104) ˆ n j (cid:105) = 0 ) except for the sites in a region R whichcan be in an arbitrary initial state with finite average particlenumber. Note that the region R may very well encompass themajor part of the lattice. What we are interested in is howfast these bosons will travel into the empty part G \ R of thelattice, as a function of the distance d ( · , · ) on the underlyinggraph. In particular, we would like to find a “speed of sound”for the bosons, that is, a velocity v such that for any region S in G \ R with d ( S, R ) ≥ l [i.e.: d ( s, r ) ≥ l ∀ s ∈ S, r ∈ R ],and for all times t for which vt < l , the expectation valueof any observable ˆ O S on S is equal to the expectation valueof the vacuum, up to a correction which decays exponentiallyaway from the light cone, e γ ( vt − l ) .To start, we consider the Bose-Hubbard model ˆ H BH andfocus on measurements of the local particle number operators ˆ n j . This corresponds to looking for bosons at the initiallyempty sites, and thus captures the most natural notion of par-ticles propagating into a region. Let us denote the initial stateby ρ (0) , which evolves according to ˙ ρ ( t ) = − i [ ˆ H BH , ρ ( t )] for t ≥ . As we are interested in the speed at which particlesin the Bose-Hubbard model propagate, let us try to understandhow the local particle densities α j ( t ) = tr(ˆ n j ρ ( t )) , j ∈ G , evolve under ˆ H BH . To this end, we derive a bound on the rateat which α j ( · ) changes, which in turn leads to a bound on thevelocity at which particles can propagate through the system.It holds that ˙ α j ( t ) = − i tr (cid:0) ˆ n j [ ˆ H BH , ρ ( t )] (cid:1) = − i tr (cid:0) [ˆ n j , ˆ H BH ] ρ ( t ) (cid:1) = 2 τ (cid:88) (cid:104) j,k (cid:105) Im (cid:104) tr (cid:0) ˆ b † k ˆ b j ρ ( t ) (cid:1)(cid:105) , (3)where the summation runs over all sites k neighboring j , d ( j, k ) = 1 . Since we are only interested in an upper boundon this rate of change, we now consider | ˙ α j ( t ) | and apply thetriangle inequality to obtain | ˙ α j ( t ) | ≤ τ (cid:88) (cid:104) j,k (cid:105) (cid:12)(cid:12) tr(ˆ b † k ˆ b j ρ ( t )) (cid:12)(cid:12) . (4)To bound this term we use the operator Cauchy-Schwarz in-equality, viewing tr(ˆ b † k ˆ b j ρ ( t )) = (cid:104) ˆ b k ρ / ( t ) , ˆ b j ρ / ( t ) (cid:105) as a Hilbert-Schmidt scalar product of ˆ b j ρ / ( t ) and ˆ b k ρ / ( t ) , where ρ / ( t ) is the matrix square root of ρ ( t ) .This gives rise to (cid:12)(cid:12) tr(ˆ b † k ˆ b j ρ ( t )) (cid:12)(cid:12) ≤ (cid:16) tr(ˆ b † k ˆ b k ρ ( t ))tr(ˆ b † j ˆ b j ρ ( t )) (cid:17) / . Combining this with Eq. (4), we obtain a set of coupled dif-ferential inequalities | ˙ α j ( t ) | ≤ τ (cid:88) (cid:104) j,k (cid:105) ( α j ( t ) α k ( t )) / , (5)which, using √ xy ≤ ( x + y ) / , yields the linearized system | ˙ α j ( t ) | ≤ τ (cid:18) D α j ( t ) + (cid:88) (cid:104) j,k (cid:105) α k ( t ) (cid:19) , where D is the maximal vertex degree of the interaction graph.We are interested in the worst-case growth of α j ( t ) as t pro-gresses. This will occur when we have equality in the aboveexpression (i.e., the derivative is as large as possible), and thusa bound γ k ( t ) ≥ α k ( t ) is given by the solution of the linearsystem of differential equations ˙ γ j ( t ) = τ (cid:18) D γ j ( t ) + (cid:88) (cid:104) j,k (cid:105) γ k ( t ) (cid:19) which fulfills γ j (0) = α j (0) . This solution has the form (cid:126)γ ( t ) = e D τt e τMt (cid:126)γ (0) , where M is the adjacency matrix of the lattice, i.e., M j,k = 1 if d ( j, k ) = 1 and otherwise, and (cid:126)γ := ( γ k ) k ∈ L . This yieldsan upper bound (cid:126)α ( t ) ≤ e D τt e τMt (cid:126)α (0) for the expected particle number at time t for any site, for (cid:126)α := ( α k ) k ∈ L .In order to understand how quickly particles propagatefrom the initially occupied region R into a region S with d ( R, S ) ≥ l , we need to consider the off-diagonal block of e D τt e τMt corresponding to those two regions. Thus, in orderto obtain a light cone with an exponential decay exp( vt − l ) outside it, we need to understand how rapidly the off-diagonalelements of the banded matrix M grow under exponentiation e τMt . This can be done by applying Theorem 6 from Ref. [9],which yields for the ( i, j ) -th element of exp( τ M t ) the bound [exp( τ M t )] i,j ≤ Ce v t − d ( i,j ) with velocity v = χ ∆ τ , where χ ≈ . is the solution of χ ln χ = χ + 1 , ∆ = (cid:107) M (cid:107) ∞ / depends on the lattice dimen-sion, and C = 2 χ / ( χ − ≈ . Together with the prefactor exp( D τ t ) , this gives a Lieb-Robinson velocity v = v + D τ [10]. For the scenario of an empty lattice with particles ini-tially placed in a region R , this implies that for any j with d ( j, R ) ≥ l , α j ( t ) ≤ Ce vt − l (cid:88) k ∈ R α k (0) = CN e vt − l , (6)i.e., up to an exponentially small tail, the particles propagatewith a speed no faster than v , independent of their initial state.Here, N = (cid:80) k ∈ R α k (0) = (cid:104) ˆ N (cid:105) is the total number of parti-cles in the system (i.e., the expectation value of the total par-ticle number operator ˆ N = (cid:80) j ˆ n j ). Note that while this (un-surprisingly) means that the strength of the signal observedmay depend on the number of bosons initially put into thesystem, the maximum propagation speed v does not dependon N . In fact, for a purely harmonic one-dimensional modelfor U = 0 , the exact speed of sound is indeed linear in τ , sothe above bound is tight up to a small constant prefactor.Having understood how to obtain a bound on the propa-gation speed of particles, we now turn to more general ob-servables. First, let us show how we can bound the highermoments of the particle number operator. For p ≥ , α ( p ) j ( t ) = tr (cid:0) ˆ n pj ρ ( t ) (cid:1) = (cid:88) N tr (cid:0) ˆ n j ˆ n p − j P N ρ ( t ) P N (cid:1) ≤ (cid:88) N tr (cid:0) ˆ n j N p − P N ρ ( t ) P N (cid:1) (7) (6) ≤ (cid:88) N N p − (cid:0) CN e vt − l (cid:1) tr( ρ ( t ))= C (cid:104) ˆ N p (cid:105) e vt − l , where P N projects onto the subspace with a total of N par-ticles, and we have used that Eq. (6) applies to each sub-space with fixed particle number independently as the Hamil-tonian commutes with P N . Here, (cid:104) ˆ N p (cid:105) denotes the (time-independent) expectation value of the p -th moment of the total particle number operator. This proves a Lieb-Robinson boundfor the higher moments of the particle number operator.Let us now turn our attention towards arbitrary local ob-servables ˆ A j . Any such observable can be written as ˆ A j = (cid:80) p,q c p,q (ˆ b † j ) p ˆ b qj , and we have thus that (cid:12)(cid:12) tr( ˆ A j ρ ( t )) (cid:12)(cid:12) ≤ (cid:88) p,q | c p,q | (cid:12)(cid:12) tr[(ˆ b † j ) p ˆ b qj ρ ( t )] (cid:12)(cid:12) (8) ≤ (cid:88) p,q | c p,q | (cid:16) tr (cid:2) (ˆ b † j ) p ˆ b pj ρ ( t ) (cid:3) tr (cid:2) (ˆ b † j ) q ˆ b qj ρ ( t ) (cid:3)(cid:17) / . In turn, for p > , tr (cid:2) (ˆ b † j ) p ˆ b pj ρ ( t ) (cid:3) = tr (cid:2) ˆ n j (ˆ n j − · · · (ˆ n j − p + 1) ρ ( t ) (cid:3) = p (cid:88) r =1 d r,p α ( r ) j ( t ) ≤ ˜ C p e vt − l (9)by virtue of Eq. (6), for some constant ˜ C p . If p = 0 , wetrivially have tr[ ρ ( t )] = 1 . Together, this yields a bound (cid:12)(cid:12) tr( ˆ A j ρ ( t )) (cid:12)(cid:12) ≤ C (cid:48) e vt − l if c ,q = c p, = 0 for all p and q , and (cid:12)(cid:12) tr( ˆ A j ρ ( t )) (cid:12)(cid:12) ≤ C (cid:48) e ( vt − l ) / otherwise, where we have assumed that (cid:80) | c p,q | is finite, andused that w.l.o.g. c , = 0 . In both cases, this means thatoutside the light cone given by vt = l , tr( ˆ A j ρ ( t )) decays ex-ponentially; however, the decay is on double the length scalein the latter case.Finally, observables acting on more than one site can bebounded analogously to the local case: Any two-site operatoracting on sites j , k can be written as the sum of terms ˆ A j ˆ A k ,and (cid:12)(cid:12) tr( ˆ A j ˆ A k ρ ( t )) (cid:12)(cid:12) ≤ (cid:16) tr( ˆ A † j ˆ A j ρ ( t ))tr( ˆ A k ˆ A † k ρ ( t )) (cid:17) / . The terms on the r.h.s. are local observables which can bebounded as before by exp( vt − l ) , yielding the same expo-nential bound for two-site—and recursively for many-site—observables. (Note that there exist cases where terms whichare bounded by exp[( vt − l ) / only appear, and in additionone of the ˆ A ’s above could be the identity. Thus, bounds ofthe form exp(( vt − l ) /κ ) can occur, where κ can grow expo-nentially in the block size. This, however, still implies that thesignal is exponentially small outside the light cone.)While we have illustrated our arguments for the Bose-Hubbard model, they generalize straightforwardly to the moregeneral class of models described by Eq. (2). First, it is clearthat we can replace the on-site replusion and chemical po-tential in the Bose-Hubbard model by any type of interac-tion (even a non-local one) which only depends on the par-ticle numbers, since any such term vanishes in the commu-tator [ˆ n j , ˆ H ] in Eq. (3). Second, for systems that containseveral types of bosons the same arguments apply: Such sys-tems can be modelled using multiple copies of the originalgraph, each of which supports the hopping of one individ-ual boson species, and one obtains independent differentialinequalities for the particle densities α j,s ( t ) = tr[ˆ n j,s ρ ( t )] for each species.Beyond general bosonic models, our arguments also applyto fermions and mixtures of bosons and fermions [11], and infact even to anyonic systems. Again, in a first step one candecouple the individual species of particles (which mutuallycommute) to hop on independent graphs. Then, it is easy tocheck that our arguments work independently of the statisticsof the particles, since [ˆ n j , ˆ H ] in Eq. (3) evaluates to the sameexpression in terms of the fermionic (anyonic) creation andannihilation operators. Even better, fermionic and anyonicsystems yield stronger bounds for the higher moments, andthus for the scenario of general local observables: In Eq. (7), ˆ n p − j can be bounded by instead of ˆ N p − , which yields abound α ( p ) j ( t ) ≤ CN e vt − l on the higher moments. Corre-sponding results also follow for spin systems, as these can bedescribed as hardcore bosons.Our arguments work not only for unitary theories, but alsofor certain types of dissipative (Markovian) models, extend-ing [12] to bosonic systems. For instance, in the practicallyrelevant case of a bosonic system with particle losses, we havethat ˙ ρ ( t ) = − i [ ˆ H BH , ρ ] − λ (cid:88) j (cid:16) { ˆ b † j ˆ b j , ρ ( t ) } − b j ρ ( t )ˆ b † j (cid:17) . Therefore, ˙ α j ( t ) = − i tr([ˆ n j , ˆ H BH ] ρ ( t )) − λ tr (cid:0) ˆ n j ρ ( t ) (cid:1) , which shows that the contribution from the dissipative term to ˙ α j is negative; thus, tighter differential inequalities and thusa lower speed of sound than in the Hamiltonian case can beobtained.To conclude, we have proven that there is a speed limit forthe propagation of information in a system of interacting par-ticles. This result is particularly relevant for the case of bosonson a lattice, as bosonic systems cannot be assessed using theestablished techniques of Lieb-Robinson bounds due to theunboundedness of the bosonic hopping operator. Our argu-ment applies equally to bosonic, fermionic, anyonic, and spinsystems, as well as mixtures thereof, with arbitrary interactionterms between the particles, and can be generalized to also ad-dress systems with dissipation.The key point that allowed us to make statements about thepropagation of information in bosonic systems beyond Lieb-Robinson bounds was first to focus on a subset of observablesrelevant to detecting the propagation of particles, namely thenumber of particles present at each site, and second to de-vise a closed system of inequalities bounding the evolution oftheir expectation values. This allowed us to reduce the prob-lem of characterizing the full dynamics of the system, whichtakes place in a superexponentially large Fock space, to sim-ply keeping track of the dynamics of a relatively small num-ber of parameters. This considerably reduced the complexity of the problem and gave rise to an exactly solvable worst-casebound.The idea of studying information propagation by restrictingto a specific set of observables and investigating the result-ing worst-case differential equation can also be applied to thestudy of continous systems. This can be done either by tak-ing an appropriate continuum limit of a lattice model, or bydirectly considering a corresponding differential equation forthe particle density which is continuous in space. Acknowledgements.—
This work was supported by the EU(COMPAS, MINOS, QESSENCE), the EURYI, the Gordonand Betty Moore Foundation through Caltech’s Center forthe Physics of Information, the National Science Foundationunder Grant No. PHY-0803371, and the ARO under GrantNo. W911NF-09-1-0442. Part of this work was done at theMittag-Leffler-Institute. [1] E. H. Lieb and D. W. Robinson, Commun. Math. Phys. , 251(1972).[2] M. B. Hastings, Phys. Rev. B , 104431 (2004); B. Nachter-gaele and R. Sims, Commun. Math. Phys. , 119 (2006).[3] B. Nachtergaele and R. Sims, arXiv:1004.2086.[4] J. Eisert and T. J. Osborne, Phys. Rev. Lett. , 150404 (2006);S. Bravyi, M. B. Hastings, and F. Verstraete, ibid. , 050401(2006); C. K. Burrell and T. J. Osborne, ibid. , 167201(2007); A. Hamma, F. Markopoulou, I. Premont-Schwarz, andS. Severini, ibid. , 017204 (2009); J. Eisert, M. Cramer, andM. B. Plenio, Rev. Mod. Phys. , 277 (2010).[5] J. Eisert and D. Gross, Phys. Rev. Lett. , 240501 (2009).[6] T. J. Osborne, Phys. Rev. Lett. , 157202 (2006); M. B. Hast-ings, Phys. Rev. B , 144302 (2008).[7] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. , 885(2008); S. Trotzky et al, arXiv:1101.2659.[8] L. Hackerm¨uller et al. , Science , 1621 (2010).[9] M. Cramer and J. Eisert, New J. Phys. , 71 (2006).[10] There are two ways to obtain better bounds on the velocity.First, we can bound √ xy ≤ ( λx + y/λ ) , which gives a ve-locity bound λv + D τ /λ for any λ > . Second, one cansolve the non-linear differential inequality (5) by substituting α j ( t ) =: β j ( t ) , which gives linear inequalities ˙ β j ( t ) ≤ τ (cid:88) (cid:104) j,k (cid:105) β k ( t ) with initial conditions β (0) = (cid:112) α (0) . The worst-case solutionof this system is (cid:126)β ( t ) = e τMt (cid:126)β (0) , which, using the previ-ous estimate of e τMt , yields a velocity v . In order to obtainbounds on α j ( t ) , we need to square this bound. On the onehand, this implies that the correlations outside the light conedecay as e vt − l ) ; however, it also yields an unfavorable depen-dence of the prefactor on the initial conditions, ( (cid:80) j (cid:112) α j (0)) ,which can diverge for a fixed number of bosons as the region R grows.[11] A. Albus, F. Illuminati, and J. Eisert, Phys. Rev. A , 023606(2003); H. P. B¨uchler and G. Blatter, ibid. , 063603 (2004).M. Cramer, J. Eisert, and F. Illuminati, Phys. Rev. Lett. ,190405 (2004); M. Lewenstein et al., ibid. , 050401 (2004).[12] D. Poulin, Phys. Rev. Lett. , 190401 (2010); C. K. Burrell,J. Eisert, and T. J. Osborne, Phys. Rev. A80