PRL 114, 170505 (2015)
PHYSICAL
REVIEW
week ending 1 MAY 2015
LETTERS
Many-Body Localization Implies that Eigenvectors are Matrix-Product States M. Friesdorf,1 A. H. Werner,1 W. Brown,2 V. B. Scholz,3 and J. Eisert1 lDahlem Center for Complex Quantum Systems, Freie Universitat Berlin, 14195 Berlin, Germany 2Computer Science Department, University College London, London WC1E 6BT, United Kingdom 3Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland (Received 23 October 2014; published 1 May 2015) The phenomenon of many-body localization has received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at nonzero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalization following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties—a vanishing group velocity and the absence of transport—with entanglement properties of individual eigenvectors. For systems with a generic spectrum, we prove that strong dynamical localization implies that all of its many-body eigenvectors have clustering correlations. The same is true for parts of the spectrum, thus allowing for the existence of a mobility edge above which transport is possible. In one dimension these results directly imply an entanglement area law; hence, the eigenvectors can be efficiently approximated by matrix-product states. DOT: 10.1103/PhysRevLett. 114.170505
PACS numbers: 03.67.-a, 03.65.Ud, 05.30.-d, 72.20.Ee
The concept of disorder-induced localization was intro duced in the seminal work by Anderson [1], who captured the mechanism responsible for the absence of diffusion of waves in disordered media. This mechanism is specifically well understood in the single-particle case, where one can show that, in the presence of a suitable random potential, all eigenfunctions are exponentially localized [1], In addition to this spectral characterization of localization, there is a notion of dynamical localization, which requires that the transition amplitudes between lattice sites decay exponentially [2,3], Naturally, there is great interest in extending these results to the many-body setting [4,5], In the case of integrable systems that can be mapped to free fermions, such as the X Y chain, results on single-particle localization can be applied directly [6,7], A far more intricate situation arises in interacting systems. Such many-body localization [8,9] has received an enormous amount of attention recently. In terms of condensed-matter physics, this phenomenon allows for systems to remain an insulator even at nonzero temperature [10], in principle, even at infinite temperature [11]. In the foundations of statistical mechanics, such many-body localized systems provide examples of systems that fail to thermalize. When pushed out of equilibrium, signatures of the initial condition will locally be measurable even after long times, in contradiction to what one might expect from quantum statistical mechanics [4,12,13], Despite great efforts to approach the phenomenon of many-body localization, many aspects are not fully under stood and a comprehensive definition is still lacking. Similar to the case of single-particle Anderson localization, there are two complementary approaches to capture the phenomenon. On the one hand, probes involving real-time
00 3 1 -9 0 0 7 /1 5 /1 14( 17)/170505(6)
dynamics [14,15] have been discussed, showing excitations “getting stuck,” or seeing suitable signatures in densityautocorrelation functions [16], leading to a dynamic read ing of the phenomena. On the other hand, statistics of energy levels has been considered as an indicator [11] as well as a lack of entanglement in the eigenbasis [5] and a resulting violation of the eigenstate thermalization hypoth esis (ETH) [17-19]. In such a static description of manybody localization, it has, for example, been suggested to define many-body localized states as those states that can be prepared from product states or Slater determinants of single-particle localized states with a finite-depth local unitary transformation. Accordingly, a many-body local ized phase has been suggested to be one where most eigenvectors have this property [5], It seems fair to say that the connection between these static and dynamic readings is still unclear. In fact, many-body localization is presently in its description still marred by conflicting pictures and seemingly based on unrelated definitions. In this work, we make a substantial first step in relating these approaches: If the evolution of an interacting manybody system [20,21] is dynamically localizing (and its spectrum is generic), then all eigenvectors have exponen tially clustering correlations in general and, in one dimen sion, satisfy an area law and can be efficiently written as a matrix-product state (MPS) [22,23] with a low bond dimension. Our proof uses advanced mathematical tools developed for the analysis of interacting many-body systems, such as energy filtering, and does not rely on the precise details of the considered model. We show that short-ranged correlations in individual energy eigenstates necessarily follow from the absence of transport in local izing models. Thus, our result establishes a novel link
170505-1
© 2 0 1 5 American Physical Society
PHYSICAL
PRL 114, 170505 (2015)
REVIEW
between different aspects of many-body localization and provide a deeper understanding of the phenomenon in all of its facets. Setting.—For simplicity, we will state all our results in the language of spin systems of local dimension d, but they can equivalently be derived for fermionic systems. We consider local Hamiltonians H = ^2jhj, where each inter action term hj is supported on finitely many sites of a lattice with N vertices, and the physical system associated with each site is finite dimensional. For most of the argument, we make no assumptions concerning the dimensionality or the type of lattice. In what follows, we will consider observables A and B that are supported on finitely many sites on the lattice. There is a natural distance d(A, B ) in the lattice related to the length of the shortest path of inter actions connecting the supports of A and B. For conven ience of notation, we will choose ||A|| = 1 = ||fl|| from here on, which does not restrict generality [24-28], For any local Hamiltonian of this sort, there is an upper bound v > 0 to the largest group velocity, referred to as Lieb-Robinson velocity. There are several possible formulations of such a bound (see Sec. A of Supplemental Material SM [29]). One convenient and common way is to compare the time evolution of a local observable A(t) = e"HAe~nH under the full Hamiltonian H with its time evolution under a truncated Hamiltonian H'a that only includes interactions hj contained in a region of distance no more than / from the support of A. The Lieb-Robinson theorem then states that there exists a constant c > 0 and a velocity v > 0 such that ||A(f) —e"H^Ae~ltHA|| < is true for all times t > 0. Such a bound limits the speed of information propagation: In any such lattice with shortranged interactions, all interactions are causal in this sense. A related, but in the case of v — 0 weaker, estimate is given by the following commutator bound (see also Sec. A of SM [29]) II[A. B(t)} || < C e - ^ A’B)-V,\.
week ending
LETTERS
1 MAY 2015
them as operators on their respective truncated Hilbert spaces H A and XtB. The smallest gap will be called y. Moreover, the gaps of these Hamiltonians need to be locally independent with respect to each other, in the sense that
Ea —Ea’ = Eb — Ey =>■ a — a' ,
b = b' ,
where a. a' and b, b' label the eigenvalues of HA and EIB, respectively. The smallest difference of these gaps will be called rj. (3) Nonsymmetric gaps (AIII): For all eigenvalues Ek of the full Hamiltonian, the spectrum is no-where symmetric in the sense that min||£,. - Ek\ - \ES - Ek || > £ > 0 r±s
V k.
Similar reasonable conditions have already been con sidered in the context of equilibration of closed systems and are assumed to hold when a small amount of random noise is added to a local Hamiltonian [35], Especially in the context of localizing systems that are typically based on large random terms in the Hamiltonian, these assumptions are very natural indeed. Our assumptions are, however, considerably weaker than assuming fully nondegenerate energy gaps [35], Another quantity that will be important later on is the number of states up to energy E given by 0 ( £ ) := E /,£, 0, where A is an arbitrary local observable, cloc > 0 a constant independent of the system size, and HA denotes a Hamiltonian which includes all interactions contained in a region of distance no more than / from the support of A. This corresponds to setting the Lieb-Robinson velocity v to zero in Eq. (1). Note that this definition captures the transport properties of one specific Hamiltonian. In the context of disordered systems, this means that we consider a fixed realization as it would, for example, arise in an experiment. For this fixed realization, this definition also implies that information cannot be transported by encoding it locally in the chosen Hamiltonian. For noninteracting systems, such as the XY chain, this strong suppression of
170505-2
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
transport does occur with high probability [6,36], As such, however, strong dynamical localization does not allow for a growth of entanglement entropies logarithmically in time, a phenomenon that has been numerically observed in spin chains with random on-site noise [15]. To this end, we now relax our dynamical assumption to a weaker variant of zero velocity Lieb-Robinson bounds and restrict it to a suitable subspace. It is natural to assume that the transport properties of an electronic system are closely connected to the energy available in the system. In fact, often the notion of a mobility edge is introduced [36,37]. The intuition is that for all energies below this mobility edge, transport is fully blocked, while it might be possible for higher energies. The following definition makes this precise and also relies on a rather weak version of zero velocity Lieb-Robinson bounds (see Sec. A of SM [29]). Definition 2.—(Mobility edge): A Hamiltonian H is said to have a mobility edge at energy Emob if and only if its time evolution satisfies for all times t, V p e span{|/)(/t|: E), Ek < Emob}, |tr(p[A(f).B])| < min( t,\) c mohe ~ ^ A-B\
(2)
with constants cmob,;/ > 0 independent of t. That is, all transport is suppressed for states supported only on the low-energy sector below energy £ mob. We expect this condition to be fulfilled for much more general systems than Definition 1. In particular, Heisenberg-type models with suitable chemical potential have, in the Jordan-Wigner picture, low-energy states that contain only few fermions. Perturbation arguments [8,9] then suggest the existence of a mobility edge in the sense of Definition 2. While our results will be applicable to all eigenvectors below the mobility edge, they can naturally also be used to capture the ground state, yielding an extension of the previous results [36]. In experiments, a natural setting for dynamical locali zation is one where a system is excited locally. For example, in the case of a ground state vector |0), this corresponds to the action of a unitary operator U with local support |y/) = f/|0) = e“'G|0), where G denotes its gen erator. In this setting, suppression of transport means that the excitation cannot be measured at far distances, even for long times, in the sense that |(0|A(f) - eiGA{t)e~iG|0)| < min(t, l ) ||G ||C e ^ ( ‘4'f/), for arbitrary observables A. Here C,/a > 0 are constants independent of the time t and the system size. If excitations are stuck in the above sense in an entire certain energy subspace, this implies our zero velocity Lieb-Robinson estimate in Eq. (2), thus relating Definition 2 to a quantity that can, in principle, be experimentally measured (see Sec. A of SM [29]).
LETTERS
week ending I M AY 2015
Main result.—Vie are now in a position to state our main result. It provides a clear link between the dynamical properties of the system and the static correlation behavior of the corresponding eigenvectors and can be applied to quantum systems of arbitrary dimension. In one dimension it directly implies an area law and the existence of an approximating MPS (see Corollary 2). Theorem 1.—(Clustering of correlations of eigenvectors): The dynamical properties of a local Hamiltonian imply clustering correlations of its eigenvectors in the follow ing way. (a) If the Hamiltonian shows strong dynamical locali zation and its spectrum fulfils assumptions AI and All, then all its eigenvectors |k) have exponentially clustering correlations, i.e., |(*|AB|*> - (lc\A\k)(k\B\k)\ < 4cloce-"d(A-BV2. (b) If the Hamiltonian has a mobility edge at energy £ mob and its spectrum fulfils assumptions AI and AIII, then all eigenvectors |£) up to that energy £ mob cluster exponen tially, in the sense that for all k > 0, \(k\AB\k)-(k\A \k)(k\B \k)\ e - fid ( A .B )
< ( l2 *©(£* + K)cmob + lng^ (A’J ) g4+2^
2n
where ©(E) is the number of eigenstates up to energy E and k can be chosen arbitrarily to optimize the bound. Part (b) of the theorem only requires Lieb-Robinson bounds on a subspace and is thus perfectly compatible with a growth of entanglement entropies following quenches. In fact, for the proof it is sufficient to assume Eq. (2) on the level of the individual eigenstates. It further allows us to freely choose an energy cutoff k that serves as an artificial energy gap and can be used to optimize the bound. For typical local models, we expect the density of states to behave like a Gaussian, an intuition that can be numerically tested for small systems (see Fig. 1) and rigorously proved in a weak sense [38], In this case the number of states will behave like a low-order polynomial in the system size at energies close to the ground state energy Ek + k. Thus, fixing a k > 0 independent of the system size will lead to a prefactor that scales like a loworder polynomial in the system size where the order of the polynomial will increase as one moves to higher energies. In the bulk of the spectrum, the number of states will grow exponentially with the system size, thus rendering our bounds useless as one moves to high energies. Interestingly, this feature of stronger correla tions, associated with a larger entanglement in the state, at higher energies seems to be shared by the Heisenberg chain with random on-site magnetic field (see Fig. 1). Naturally, our theorem can also be applied to the ground state where it extends previous results [36] and, for
170505-3
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
week ending
LETTERS
1 MAY 2015
FIG. 1 (color online). We look at a one-dimensional Heisenberg model with random on-site magnetic field of the form H = J2i(x ix i+1 + LL-t-i + Z,ZM + HiZi), where /«,• is drawn uniformly from the interval [~h,h\. (a) Shown is the number of eigenstates for 14 sites as a function of the discretized energy for h = 1 averaged over 100 realizations, and the variance of this averaging is indicated by vertical error bars. The resulting function is close to a Gaussian, (b) Here, the integrated density of states of this model with the same parameters is shown. The corresponding error bars are too small to appear. This quantity is related to the number of states 0 by an exponential factor 2N. This plot indicates that indeed the number of states will have small tails and thus few states at low energies, (c) Depicted is the (von Neumann) entanglement entropy of the eigenvectors as a function of the energy. The results were obtained for the disorder strength h = 4 on a system of 12 sites, and for each individual state an average over the different cuts through the ID chain was taken. This plot corroborates the intuition that the entanglement entropy and thus the associated bond dimension of the corresponding MPS increase with the energy. exam ple, for the case o f an alm ost degenerate ground state will be a substantial im provem ent. Naturally, our results can also be applied to the highly excited states at the other end o f the spectrum . The proofs for both parts o f the theorem rely on energy filtering techniques [37,39]. This is a versatile tool espe cially in the study o f perturbation bounds for Hamiltonian system s and allows us to partially diagonalize a local observable in the energy eigenbasis while still keeping some locality structure. Energy filtering o f an observable is defined as
proof of part (b) of the theorem is more involved and relies on a high-pass filter (see Sec. B o f SM [29]). Implications on area laws and matrix-product states.— In one dimension, the conclusions o f our main theorem about the correlation behavior o f eigenvectors can be turned into a statem ent about their entanglem ent structure. It has been noted before that m any-body localization should be connected to eigenvectors fulfilling an area law (see Conjecture 1 in Ref. [5]) and eigenvectors being well approxim ated by m atrix-product state vectors o f the form D
|MPS) =
/ ? ( A ) = / “ dtf{t)A{t).
^
ti(M hM i2...M iN)\iu ...,i N),
it....,iN=l
Here, / : [R -> R + is a filter function that can be used to interpolate between locality o f the resulting observable and the strength o f the off-diagonal elem ents in the Ham iltonian basis. In order to show part (a) o f the theorem , we will make use o f energy filterings of the local observables A and B with respect to the full Ham iltonian H as well as with respect to local restrictions HA and H B. Those restrictions contain the support o f the corresponding observable and are chosen as large as possible, while still satisfying [Ha ,H b\ — 0. The m ain idea is to choose a suitable (Gaussian) filter function /(/ and use the gap assumptions AI and A ll to show that the joint filter o f the observables AB decouples into energy filters for A and B separately. Then, strong dynam ical localization can be used to make these filters local. Finally, choosing the width o f the energy filter a small enough, and in particular exponentially small in the system size, allows us to conclude the proof. The
where D is the bond dim ension and M j € CHx/) for all j. Our main theorem allows us to rigorously prove this connection. Corollary 1.—(Area laws and m atrix-product states): An eigenvector | k) o f a localising Ham iltonian can be approxi mated by an MPS with fidelity |(fc)MPS| > 1 —e, where the bond dim ension for a sufficiently large system is given as follows. (a) If the Ham iltonian shows strong dynamical locali zation and its spectrum fulfills assum ptions AI and A ll, then the statem ent holds for all eigenvectors | k) and, for some constant C > 0, the approxim ation has a bond dimension
D = C (iV /e)16^ log2L (b) If the Hamiltonian has a mobility edge at energy Emob, and its spectrum fulfils assum ptions AI and AID, then
170505-4
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
the statement holds for all eigenvectors below this energy E mob, and the bond dimension is given by D = poly [Q(Ek + k ),N], for any fixed k which enters in the precise form of the polynomial. The proof is a direct consequence of our main theorem together with the fact that exponential clustering in one dimension implies strong bounds on entanglement entro pies between any bipartite cut of the chain [40]. Using techniques from Ref. [41], this leads to an efficient MPS approximation with the above bounds on the bond dimension (see Sec. F of SM [29]). Summary and outlook.—Despite considerable progress in understanding the effects of random potentials on quantum many-body systems, a precise definition of the phenomenon of many-body localization continues to be elusive. Attempts to capture the phenomenology can largely be classified into two complementary approaches. One of them puts characteristic properties of the eigen functions into the focus of attention and asks for a lack of entanglement and a violation of the ETH. The other one takes the suppression of transport as the basis, which seems closer to being experimentally testable. In this work, we have established a clear link between these two approaches by showing that dynamical locali zation implies that eigenvectors cluster exponentially. This result, together with the existence of an approximating MPS description in one dimension, reminds of the defi nition of Ref. [5], that defines many-body localization in terms of matrix-product state approximations of eigen states. In contrast, in our work, this feature is shown to directly follow from an absence of transport. Note that the converse statement is by no means obvious. It is well possible to imagine a setting where eigenstates have little entanglement, but there is no signature of dynamical localization. In this context a further clarification of the precise connection between dynamical and static properties and their implication for numerical simulations based on tensor network states seems like a worthwhile endeavor. A different approach towards approximating the indi vidual eigenstates with a matrix-product state could poten tially be provided by constructing meaningful local constants of motion that give a set of local quantum numbers [42], It seems likely that tools using energy filtering will again prove useful in this context, a prospect that we briefly discuss in the Supplemental Material [29]. In contrast to our work, such a construction will unlikely be energy resolved and it seems entirely unclear how the intuition of a mobility edge could be incorporated in this. On the practical side, further numerical effort will be needed to understand the behavior of individual models and to fully understand the transport properties for differ ent energy scales. Our work could well provide a first
LETTERS
week ending 1 MAY 2015
stepping stone for further endeavors in this direction. Showing that eigenstates are well approximated by matrix-product states implies that not only ground states, but in fact also excited states, can efficiently be described in terms of tensor networks. Our result, as such, does not yet provide an efficient algorithm to find the respective matrix-product states: this reminds one of the situation of the existence of lattice models for which the ground states are exact matrix-product states, but it amounts to a computationally difficult problem to find them [43]. Still, this appears to be a major step in the direction of formulating such numerical prescriptions of describing the low-energy sector of many-body localizing systems. Eventually, the leading vision in any of these endeavors is a rigorous proof of many-body localization in the spirit of the original results by Anderson. For this, creating a unifying framework and linking the possible definitions is certainly a key step. We thank H. Wilming, C. Gogolin, T. J. Osborne, F. Verstraete, and F. Pollmann for insightful discussions and M. Goihl for collaboration on the numerical code. We would like to thank the EU (SIQS, AQuS, RAQUEL, COST), the ERC (TAQ), the BMBF, and the Studienstiftung des deutschen Volkes for support. V. B. S. is supported by the Swiss National Science Foundation through the National Centre of Competence in Research “Quantum Science and Technology” and by an ETH postdoctoral fellowship. W. B. is supported by EPSRC.
[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [2] A. Klein, arXiv:0708.2292. [3] G. Stolz, in Entropy and the Quantum II, edited by R. Sims and D. Ueltschi, Contemporary Mathematics Vol. 552 (American Mathematical Society, Arizona, 2011). [4] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015). [5] B. Bauer and C. Nayak, J. Stat. Mech. (2013) P09005. [6] C. K. Burrell and T. J. Osborne, Phys. Rev. Lett. 99, 167201 (2007). [7] E. Hamza, R. Sims, and G. Stolz, Commun. Math. Phys. 315, 215 (2012). [8] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006). [9] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, arXiv:condmat/0602510. [10] I. L. Aleiner, B. L. Altshuler, and G. V. Shlyapnikov, Ann. Phys. (Amsterdam) 321. 1126 (2006). [11] V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007). [12] A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). [13] C. Gogolin, M. P. Muller, and J. Eisert, Phys. Rev. Lett. 106, 040401 (2011). [14] A. Nanduri, H. Kim, and D. A. Huse, Phys. Rev. B 90, 064201 (2014). [15] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012).
170505-5
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
[16] M. Znidaric, T. Prosen, and P. Prelovsek, Phys. Rev. B 77, 064426 (2008). [17] M. Srednicki, Phys. Rev. E 50, 888 (1994). [18] J.M . Deutsch, Phys. Rev. A 43, 2046 (1991). [19] M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008). [20] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015). [21] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). [22] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992). [23] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007). [24] M. B. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006). [25] B. Nachtergaele and R. Sims, Commun. Math. Phys. 265, 119 (2006). [26] J. Eisert and T. J. Osborne, Phys. Rev. Lett. 9 7 , 150404 (2006). [27] S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006). [28] B. Nachtergaele, Y. Ogata, and R. Sims, J. Stat. Phys. 124, 1 (2006). [29] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/Phy sRevLett. 114.170505 for a in-depth description of the analytical details, which includes Refs. [30-34].
week ending 1 MAY 2015
LETTERS
[30] K. Them, Phys. Rev. A 89, 022126 (2014). [31] T. Barthel and M. Kliesch, Phys. Rev. Lett. 108, 230504 ( 2012).
[32] M. Kliesch, C. Gogolin, and J. Eisert, arXiv: 1306.0716. [33] B. Nachtergaele and R. Sims, in Entropy and the Quantum, Contemporary Mathematics Vol. 529, edited by R. Sims and D. Ueltschi (American Mathematical Society, Arizona,
2010). [34] U.
Schollwoeck,
Ann.
Phys.
(Amsterdam)
326, 96
(2011). [35] N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009). [36] E. Hamza, R. Sims, and G. Stolz, Commun. Math. Phys. 315, 215 (2012). [37] M .B . Hastings, arXiv: 1001.5280. [38] J.P. Keating, N. Linden, and H. J. Wells, arXiv:1403.1121. [39] S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims, Commun. Math. Phys. 309, 835 (2012). [40] F. G. S.L . Brandao and M. Horodecki, Nat. Phys. 9, 721 (2013). [41] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006). [42] A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Phys. Rev. B 91, 085425 (2015). [43] N. Schuch, I. Cirac, and F. Verstraete, Phys. Rev. Lett. 100, 250501 (2008).
170505-6
Copyright of Physical Review Letters is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
PHYSICAL
REVIEW
week ending 1 MAY 2015
LETTERS
Many-Body Localization Implies that Eigenvectors are Matrix-Product States M. Friesdorf,1 A. H. Werner,1 W. Brown,2 V. B. Scholz,3 and J. Eisert1 lDahlem Center for Complex Quantum Systems, Freie Universitat Berlin, 14195 Berlin, Germany 2Computer Science Department, University College London, London WC1E 6BT, United Kingdom 3Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland (Received 23 October 2014; published 1 May 2015) The phenomenon of many-body localization has received a lot of attention recently, both for its implications in condensed-matter physics of allowing systems to be an insulator even at nonzero temperature as well as in the context of the foundations of quantum statistical mechanics, providing examples of systems showing the absence of thermalization following out-of-equilibrium dynamics. In this work, we establish a novel link between dynamical properties—a vanishing group velocity and the absence of transport—with entanglement properties of individual eigenvectors. For systems with a generic spectrum, we prove that strong dynamical localization implies that all of its many-body eigenvectors have clustering correlations. The same is true for parts of the spectrum, thus allowing for the existence of a mobility edge above which transport is possible. In one dimension these results directly imply an entanglement area law; hence, the eigenvectors can be efficiently approximated by matrix-product states. DOT: 10.1103/PhysRevLett. 114.170505
PACS numbers: 03.67.-a, 03.65.Ud, 05.30.-d, 72.20.Ee
The concept of disorder-induced localization was intro duced in the seminal work by Anderson [1], who captured the mechanism responsible for the absence of diffusion of waves in disordered media. This mechanism is specifically well understood in the single-particle case, where one can show that, in the presence of a suitable random potential, all eigenfunctions are exponentially localized [1], In addition to this spectral characterization of localization, there is a notion of dynamical localization, which requires that the transition amplitudes between lattice sites decay exponentially [2,3], Naturally, there is great interest in extending these results to the many-body setting [4,5], In the case of integrable systems that can be mapped to free fermions, such as the X Y chain, results on single-particle localization can be applied directly [6,7], A far more intricate situation arises in interacting systems. Such many-body localization [8,9] has received an enormous amount of attention recently. In terms of condensed-matter physics, this phenomenon allows for systems to remain an insulator even at nonzero temperature [10], in principle, even at infinite temperature [11]. In the foundations of statistical mechanics, such many-body localized systems provide examples of systems that fail to thermalize. When pushed out of equilibrium, signatures of the initial condition will locally be measurable even after long times, in contradiction to what one might expect from quantum statistical mechanics [4,12,13], Despite great efforts to approach the phenomenon of many-body localization, many aspects are not fully under stood and a comprehensive definition is still lacking. Similar to the case of single-particle Anderson localization, there are two complementary approaches to capture the phenomenon. On the one hand, probes involving real-time
00 3 1 -9 0 0 7 /1 5 /1 14( 17)/170505(6)
dynamics [14,15] have been discussed, showing excitations “getting stuck,” or seeing suitable signatures in densityautocorrelation functions [16], leading to a dynamic read ing of the phenomena. On the other hand, statistics of energy levels has been considered as an indicator [11] as well as a lack of entanglement in the eigenbasis [5] and a resulting violation of the eigenstate thermalization hypoth esis (ETH) [17-19]. In such a static description of manybody localization, it has, for example, been suggested to define many-body localized states as those states that can be prepared from product states or Slater determinants of single-particle localized states with a finite-depth local unitary transformation. Accordingly, a many-body local ized phase has been suggested to be one where most eigenvectors have this property [5], It seems fair to say that the connection between these static and dynamic readings is still unclear. In fact, many-body localization is presently in its description still marred by conflicting pictures and seemingly based on unrelated definitions. In this work, we make a substantial first step in relating these approaches: If the evolution of an interacting manybody system [20,21] is dynamically localizing (and its spectrum is generic), then all eigenvectors have exponen tially clustering correlations in general and, in one dimen sion, satisfy an area law and can be efficiently written as a matrix-product state (MPS) [22,23] with a low bond dimension. Our proof uses advanced mathematical tools developed for the analysis of interacting many-body systems, such as energy filtering, and does not rely on the precise details of the considered model. We show that short-ranged correlations in individual energy eigenstates necessarily follow from the absence of transport in local izing models. Thus, our result establishes a novel link
170505-1
© 2 0 1 5 American Physical Society
PHYSICAL
PRL 114, 170505 (2015)
REVIEW
between different aspects of many-body localization and provide a deeper understanding of the phenomenon in all of its facets. Setting.—For simplicity, we will state all our results in the language of spin systems of local dimension d, but they can equivalently be derived for fermionic systems. We consider local Hamiltonians H = ^2jhj, where each inter action term hj is supported on finitely many sites of a lattice with N vertices, and the physical system associated with each site is finite dimensional. For most of the argument, we make no assumptions concerning the dimensionality or the type of lattice. In what follows, we will consider observables A and B that are supported on finitely many sites on the lattice. There is a natural distance d(A, B ) in the lattice related to the length of the shortest path of inter actions connecting the supports of A and B. For conven ience of notation, we will choose ||A|| = 1 = ||fl|| from here on, which does not restrict generality [24-28], For any local Hamiltonian of this sort, there is an upper bound v > 0 to the largest group velocity, referred to as Lieb-Robinson velocity. There are several possible formulations of such a bound (see Sec. A of Supplemental Material SM [29]). One convenient and common way is to compare the time evolution of a local observable A(t) = e"HAe~nH under the full Hamiltonian H with its time evolution under a truncated Hamiltonian H'a that only includes interactions hj contained in a region of distance no more than / from the support of A. The Lieb-Robinson theorem then states that there exists a constant c > 0 and a velocity v > 0 such that ||A(f) —e"H^Ae~ltHA|| < is true for all times t > 0. Such a bound limits the speed of information propagation: In any such lattice with shortranged interactions, all interactions are causal in this sense. A related, but in the case of v — 0 weaker, estimate is given by the following commutator bound (see also Sec. A of SM [29]) II[A. B(t)} || < C e - ^ A’B)-V,\.
week ending
LETTERS
1 MAY 2015
them as operators on their respective truncated Hilbert spaces H A and XtB. The smallest gap will be called y. Moreover, the gaps of these Hamiltonians need to be locally independent with respect to each other, in the sense that
Ea —Ea’ = Eb — Ey =>■ a — a' ,
b = b' ,
where a. a' and b, b' label the eigenvalues of HA and EIB, respectively. The smallest difference of these gaps will be called rj. (3) Nonsymmetric gaps (AIII): For all eigenvalues Ek of the full Hamiltonian, the spectrum is no-where symmetric in the sense that min||£,. - Ek\ - \ES - Ek || > £ > 0 r±s
V k.
Similar reasonable conditions have already been con sidered in the context of equilibration of closed systems and are assumed to hold when a small amount of random noise is added to a local Hamiltonian [35], Especially in the context of localizing systems that are typically based on large random terms in the Hamiltonian, these assumptions are very natural indeed. Our assumptions are, however, considerably weaker than assuming fully nondegenerate energy gaps [35], Another quantity that will be important later on is the number of states up to energy E given by 0 ( £ ) := E /,£, 0, where A is an arbitrary local observable, cloc > 0 a constant independent of the system size, and HA denotes a Hamiltonian which includes all interactions contained in a region of distance no more than / from the support of A. This corresponds to setting the Lieb-Robinson velocity v to zero in Eq. (1). Note that this definition captures the transport properties of one specific Hamiltonian. In the context of disordered systems, this means that we consider a fixed realization as it would, for example, arise in an experiment. For this fixed realization, this definition also implies that information cannot be transported by encoding it locally in the chosen Hamiltonian. For noninteracting systems, such as the XY chain, this strong suppression of
170505-2
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
transport does occur with high probability [6,36], As such, however, strong dynamical localization does not allow for a growth of entanglement entropies logarithmically in time, a phenomenon that has been numerically observed in spin chains with random on-site noise [15]. To this end, we now relax our dynamical assumption to a weaker variant of zero velocity Lieb-Robinson bounds and restrict it to a suitable subspace. It is natural to assume that the transport properties of an electronic system are closely connected to the energy available in the system. In fact, often the notion of a mobility edge is introduced [36,37]. The intuition is that for all energies below this mobility edge, transport is fully blocked, while it might be possible for higher energies. The following definition makes this precise and also relies on a rather weak version of zero velocity Lieb-Robinson bounds (see Sec. A of SM [29]). Definition 2.—(Mobility edge): A Hamiltonian H is said to have a mobility edge at energy Emob if and only if its time evolution satisfies for all times t, V p e span{|/)(/t|: E), Ek < Emob}, |tr(p[A(f).B])| < min( t,\) c mohe ~ ^ A-B\
(2)
with constants cmob,;/ > 0 independent of t. That is, all transport is suppressed for states supported only on the low-energy sector below energy £ mob. We expect this condition to be fulfilled for much more general systems than Definition 1. In particular, Heisenberg-type models with suitable chemical potential have, in the Jordan-Wigner picture, low-energy states that contain only few fermions. Perturbation arguments [8,9] then suggest the existence of a mobility edge in the sense of Definition 2. While our results will be applicable to all eigenvectors below the mobility edge, they can naturally also be used to capture the ground state, yielding an extension of the previous results [36]. In experiments, a natural setting for dynamical locali zation is one where a system is excited locally. For example, in the case of a ground state vector |0), this corresponds to the action of a unitary operator U with local support |y/) = f/|0) = e“'G|0), where G denotes its gen erator. In this setting, suppression of transport means that the excitation cannot be measured at far distances, even for long times, in the sense that |(0|A(f) - eiGA{t)e~iG|0)| < min(t, l ) ||G ||C e ^ ( ‘4'f/), for arbitrary observables A. Here C,/a > 0 are constants independent of the time t and the system size. If excitations are stuck in the above sense in an entire certain energy subspace, this implies our zero velocity Lieb-Robinson estimate in Eq. (2), thus relating Definition 2 to a quantity that can, in principle, be experimentally measured (see Sec. A of SM [29]).
LETTERS
week ending I M AY 2015
Main result.—Vie are now in a position to state our main result. It provides a clear link between the dynamical properties of the system and the static correlation behavior of the corresponding eigenvectors and can be applied to quantum systems of arbitrary dimension. In one dimension it directly implies an area law and the existence of an approximating MPS (see Corollary 2). Theorem 1.—(Clustering of correlations of eigenvectors): The dynamical properties of a local Hamiltonian imply clustering correlations of its eigenvectors in the follow ing way. (a) If the Hamiltonian shows strong dynamical locali zation and its spectrum fulfils assumptions AI and All, then all its eigenvectors |k) have exponentially clustering correlations, i.e., |(*|AB|*> - (lc\A\k)(k\B\k)\ < 4cloce-"d(A-BV2. (b) If the Hamiltonian has a mobility edge at energy £ mob and its spectrum fulfils assumptions AI and AIII, then all eigenvectors |£) up to that energy £ mob cluster exponen tially, in the sense that for all k > 0, \(k\AB\k)-(k\A \k)(k\B \k)\ e - fid ( A .B )
< ( l2 *©(£* + K)cmob + lng^ (A’J ) g4+2^
2n
where ©(E) is the number of eigenstates up to energy E and k can be chosen arbitrarily to optimize the bound. Part (b) of the theorem only requires Lieb-Robinson bounds on a subspace and is thus perfectly compatible with a growth of entanglement entropies following quenches. In fact, for the proof it is sufficient to assume Eq. (2) on the level of the individual eigenstates. It further allows us to freely choose an energy cutoff k that serves as an artificial energy gap and can be used to optimize the bound. For typical local models, we expect the density of states to behave like a Gaussian, an intuition that can be numerically tested for small systems (see Fig. 1) and rigorously proved in a weak sense [38], In this case the number of states will behave like a low-order polynomial in the system size at energies close to the ground state energy Ek + k. Thus, fixing a k > 0 independent of the system size will lead to a prefactor that scales like a loworder polynomial in the system size where the order of the polynomial will increase as one moves to higher energies. In the bulk of the spectrum, the number of states will grow exponentially with the system size, thus rendering our bounds useless as one moves to high energies. Interestingly, this feature of stronger correla tions, associated with a larger entanglement in the state, at higher energies seems to be shared by the Heisenberg chain with random on-site magnetic field (see Fig. 1). Naturally, our theorem can also be applied to the ground state where it extends previous results [36] and, for
170505-3
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
week ending
LETTERS
1 MAY 2015
FIG. 1 (color online). We look at a one-dimensional Heisenberg model with random on-site magnetic field of the form H = J2i(x ix i+1 + LL-t-i + Z,ZM + HiZi), where /«,• is drawn uniformly from the interval [~h,h\. (a) Shown is the number of eigenstates for 14 sites as a function of the discretized energy for h = 1 averaged over 100 realizations, and the variance of this averaging is indicated by vertical error bars. The resulting function is close to a Gaussian, (b) Here, the integrated density of states of this model with the same parameters is shown. The corresponding error bars are too small to appear. This quantity is related to the number of states 0 by an exponential factor 2N. This plot indicates that indeed the number of states will have small tails and thus few states at low energies, (c) Depicted is the (von Neumann) entanglement entropy of the eigenvectors as a function of the energy. The results were obtained for the disorder strength h = 4 on a system of 12 sites, and for each individual state an average over the different cuts through the ID chain was taken. This plot corroborates the intuition that the entanglement entropy and thus the associated bond dimension of the corresponding MPS increase with the energy. exam ple, for the case o f an alm ost degenerate ground state will be a substantial im provem ent. Naturally, our results can also be applied to the highly excited states at the other end o f the spectrum . The proofs for both parts o f the theorem rely on energy filtering techniques [37,39]. This is a versatile tool espe cially in the study o f perturbation bounds for Hamiltonian system s and allows us to partially diagonalize a local observable in the energy eigenbasis while still keeping some locality structure. Energy filtering o f an observable is defined as
proof of part (b) of the theorem is more involved and relies on a high-pass filter (see Sec. B o f SM [29]). Implications on area laws and matrix-product states.— In one dimension, the conclusions o f our main theorem about the correlation behavior o f eigenvectors can be turned into a statem ent about their entanglem ent structure. It has been noted before that m any-body localization should be connected to eigenvectors fulfilling an area law (see Conjecture 1 in Ref. [5]) and eigenvectors being well approxim ated by m atrix-product state vectors o f the form D
|MPS) =
/ ? ( A ) = / “ dtf{t)A{t).
^
ti(M hM i2...M iN)\iu ...,i N),
it....,iN=l
Here, / : [R -> R + is a filter function that can be used to interpolate between locality o f the resulting observable and the strength o f the off-diagonal elem ents in the Ham iltonian basis. In order to show part (a) o f the theorem , we will make use o f energy filterings of the local observables A and B with respect to the full Ham iltonian H as well as with respect to local restrictions HA and H B. Those restrictions contain the support o f the corresponding observable and are chosen as large as possible, while still satisfying [Ha ,H b\ — 0. The m ain idea is to choose a suitable (Gaussian) filter function /(/ and use the gap assumptions AI and A ll to show that the joint filter o f the observables AB decouples into energy filters for A and B separately. Then, strong dynam ical localization can be used to make these filters local. Finally, choosing the width o f the energy filter a small enough, and in particular exponentially small in the system size, allows us to conclude the proof. The
where D is the bond dim ension and M j € CHx/) for all j. Our main theorem allows us to rigorously prove this connection. Corollary 1.—(Area laws and m atrix-product states): An eigenvector | k) o f a localising Ham iltonian can be approxi mated by an MPS with fidelity |(fc)MPS| > 1 —e, where the bond dim ension for a sufficiently large system is given as follows. (a) If the Ham iltonian shows strong dynamical locali zation and its spectrum fulfills assum ptions AI and A ll, then the statem ent holds for all eigenvectors | k) and, for some constant C > 0, the approxim ation has a bond dimension
D = C (iV /e)16^ log2L (b) If the Hamiltonian has a mobility edge at energy Emob, and its spectrum fulfils assum ptions AI and AID, then
170505-4
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
the statement holds for all eigenvectors below this energy E mob, and the bond dimension is given by D = poly [Q(Ek + k ),N], for any fixed k which enters in the precise form of the polynomial. The proof is a direct consequence of our main theorem together with the fact that exponential clustering in one dimension implies strong bounds on entanglement entro pies between any bipartite cut of the chain [40]. Using techniques from Ref. [41], this leads to an efficient MPS approximation with the above bounds on the bond dimension (see Sec. F of SM [29]). Summary and outlook.—Despite considerable progress in understanding the effects of random potentials on quantum many-body systems, a precise definition of the phenomenon of many-body localization continues to be elusive. Attempts to capture the phenomenology can largely be classified into two complementary approaches. One of them puts characteristic properties of the eigen functions into the focus of attention and asks for a lack of entanglement and a violation of the ETH. The other one takes the suppression of transport as the basis, which seems closer to being experimentally testable. In this work, we have established a clear link between these two approaches by showing that dynamical locali zation implies that eigenvectors cluster exponentially. This result, together with the existence of an approximating MPS description in one dimension, reminds of the defi nition of Ref. [5], that defines many-body localization in terms of matrix-product state approximations of eigen states. In contrast, in our work, this feature is shown to directly follow from an absence of transport. Note that the converse statement is by no means obvious. It is well possible to imagine a setting where eigenstates have little entanglement, but there is no signature of dynamical localization. In this context a further clarification of the precise connection between dynamical and static properties and their implication for numerical simulations based on tensor network states seems like a worthwhile endeavor. A different approach towards approximating the indi vidual eigenstates with a matrix-product state could poten tially be provided by constructing meaningful local constants of motion that give a set of local quantum numbers [42], It seems likely that tools using energy filtering will again prove useful in this context, a prospect that we briefly discuss in the Supplemental Material [29]. In contrast to our work, such a construction will unlikely be energy resolved and it seems entirely unclear how the intuition of a mobility edge could be incorporated in this. On the practical side, further numerical effort will be needed to understand the behavior of individual models and to fully understand the transport properties for differ ent energy scales. Our work could well provide a first
LETTERS
week ending 1 MAY 2015
stepping stone for further endeavors in this direction. Showing that eigenstates are well approximated by matrix-product states implies that not only ground states, but in fact also excited states, can efficiently be described in terms of tensor networks. Our result, as such, does not yet provide an efficient algorithm to find the respective matrix-product states: this reminds one of the situation of the existence of lattice models for which the ground states are exact matrix-product states, but it amounts to a computationally difficult problem to find them [43]. Still, this appears to be a major step in the direction of formulating such numerical prescriptions of describing the low-energy sector of many-body localizing systems. Eventually, the leading vision in any of these endeavors is a rigorous proof of many-body localization in the spirit of the original results by Anderson. For this, creating a unifying framework and linking the possible definitions is certainly a key step. We thank H. Wilming, C. Gogolin, T. J. Osborne, F. Verstraete, and F. Pollmann for insightful discussions and M. Goihl for collaboration on the numerical code. We would like to thank the EU (SIQS, AQuS, RAQUEL, COST), the ERC (TAQ), the BMBF, and the Studienstiftung des deutschen Volkes for support. V. B. S. is supported by the Swiss National Science Foundation through the National Centre of Competence in Research “Quantum Science and Technology” and by an ETH postdoctoral fellowship. W. B. is supported by EPSRC.
[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [2] A. Klein, arXiv:0708.2292. [3] G. Stolz, in Entropy and the Quantum II, edited by R. Sims and D. Ueltschi, Contemporary Mathematics Vol. 552 (American Mathematical Society, Arizona, 2011). [4] R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015). [5] B. Bauer and C. Nayak, J. Stat. Mech. (2013) P09005. [6] C. K. Burrell and T. J. Osborne, Phys. Rev. Lett. 99, 167201 (2007). [7] E. Hamza, R. Sims, and G. Stolz, Commun. Math. Phys. 315, 215 (2012). [8] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006). [9] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, arXiv:condmat/0602510. [10] I. L. Aleiner, B. L. Altshuler, and G. V. Shlyapnikov, Ann. Phys. (Amsterdam) 321. 1126 (2006). [11] V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007). [12] A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). [13] C. Gogolin, M. P. Muller, and J. Eisert, Phys. Rev. Lett. 106, 040401 (2011). [14] A. Nanduri, H. Kim, and D. A. Huse, Phys. Rev. B 90, 064201 (2014). [15] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys. Rev. Lett. 109, 017202 (2012).
170505-5
PRL 114, 170505 (2015)
PHYSICAL
REVIEW
[16] M. Znidaric, T. Prosen, and P. Prelovsek, Phys. Rev. B 77, 064426 (2008). [17] M. Srednicki, Phys. Rev. E 50, 888 (1994). [18] J.M . Deutsch, Phys. Rev. A 43, 2046 (1991). [19] M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008). [20] J. Eisert, M. Friesdorf, and C. Gogolin, Nat. Phys. 11, 124 (2015). [21] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). [22] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun. Math. Phys. 144, 443 (1992). [23] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Inf. Comput. 7, 401 (2007). [24] M. B. Hastings and T. Koma, Commun. Math. Phys. 265, 781 (2006). [25] B. Nachtergaele and R. Sims, Commun. Math. Phys. 265, 119 (2006). [26] J. Eisert and T. J. Osborne, Phys. Rev. Lett. 9 7 , 150404 (2006). [27] S. Bravyi, M. B. Hastings, and F. Verstraete, Phys. Rev. Lett. 97, 050401 (2006). [28] B. Nachtergaele, Y. Ogata, and R. Sims, J. Stat. Phys. 124, 1 (2006). [29] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/Phy sRevLett. 114.170505 for a in-depth description of the analytical details, which includes Refs. [30-34].
week ending 1 MAY 2015
LETTERS
[30] K. Them, Phys. Rev. A 89, 022126 (2014). [31] T. Barthel and M. Kliesch, Phys. Rev. Lett. 108, 230504 ( 2012).
[32] M. Kliesch, C. Gogolin, and J. Eisert, arXiv: 1306.0716. [33] B. Nachtergaele and R. Sims, in Entropy and the Quantum, Contemporary Mathematics Vol. 529, edited by R. Sims and D. Ueltschi (American Mathematical Society, Arizona,
2010). [34] U.
Schollwoeck,
Ann.
Phys.
(Amsterdam)
326, 96
(2011). [35] N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009). [36] E. Hamza, R. Sims, and G. Stolz, Commun. Math. Phys. 315, 215 (2012). [37] M .B . Hastings, arXiv: 1001.5280. [38] J.P. Keating, N. Linden, and H. J. Wells, arXiv:1403.1121. [39] S. Bachmann, S. Michalakis, B. Nachtergaele, and R. Sims, Commun. Math. Phys. 309, 835 (2012). [40] F. G. S.L . Brandao and M. Horodecki, Nat. Phys. 9, 721 (2013). [41] F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006). [42] A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Phys. Rev. B 91, 085425 (2015). [43] N. Schuch, I. Cirac, and F. Verstraete, Phys. Rev. Lett. 100, 250501 (2008).
170505-6
Copyright of Physical Review Letters is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.
