Dynamics of entanglement and transport in one- dimensional systems with quenched randomness
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Citation Nahum, Adam, et al. “Dynamics of Entanglement and Transport in One-Dimensional Systems with Quenched Randomness.” Physical Review B, vol. 98, no. 3, July 2018.
As Published http://dx.doi.org/10.1103/PhysRevB.98.035118
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/117113
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW B 98, 035118 (2018)
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Dynamics of entanglement and transport in one-dimensional systems with quenched randomness
Adam Nahum,1,2 Jonathan Ruhman,2 and David A. Huse3 1Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 3Department of Physics, Princeton University, New Jersey 08544, USA
(Received 24 June 2017; published 16 July 2018)
Quenched randomness can have a dramatic effect on the dynamics of isolated 1D quantum many-body systems, even for systems that thermalize. This is because transport, entanglement, and operator spreading can be hindered by “Griffiths” rare regions, which locally resemble the many-body-localized phase and thus act as weak links. We propose coarse-grained models for entanglement growth and for the spreading of quantum operators in the presence of such weak links. We also examine entanglement growth across a single weak link numerically. We show that these weak links have a stronger effect on entanglement growth than previously assumed: entanglement growth is subballistic whenever such weak links have a power-law probability distribution at low couplings, i.e., throughout the entire thermal Griffiths phase. We argue that the probability distribution of the entanglement entropy across a cut can be understood from a simple picture in terms of a classical surface growth model. We also discuss spreading of operators and conserved quantities. Surprisingly, the four length scales associated with (i) production of entanglement, (ii) spreading of conserved quantities, (iii) spreading of operators, and (iv) the width of the “front” of a spreading operator, are characterized by dynamical exponents that in general are all distinct. Our numerical analysis of entanglement growth between weakly coupled systems may be of independent interest.
DOI: 10.1103/PhysRevB.98.035118
I. INTRODUCTION These “Griffiths” regions act as bottlenecks or weak links, hindering the propagation both of conserved quantities and A basic question about a many-body quantum system, of quantum information. The effects on transport are fairly closely related to its ability to thermalize, is how effec- well understood: strong enough disorder leads to subdiffusive tively quantum information spreads through it. The dynamical transport [19–21], as observed numerically [21–25]. (Certain generation of quantum entanglement, following a quantum rare-region effects have also been addressed experimentally quench from a weakly entangled state, provides one window [26].) Subballistic entanglement and signal propagation has on information spreading [1–6]. Unitary dynamics typically been observed numerically in the Griffiths phase [23,27]. generates correlations between increasingly distant degrees of Here we provide long-wavelength pictures which expose freedom as time goes on. The resulting irreversible growth in the universal physics underlying entanglement growth and the entanglement entropy of a subsystem reveals differences operator spreading in the Griffiths phase, and yield the uni- between integrable, nonintegrable, disordered, and many-body versal scaling exponents and scaling forms governing these localized (MBL) [3,4,7]systems. processes. In contrast to both clean systems and the MBL Complementary insight into information spreading comes phase, we find that the length scales governing entanglement from considering light-cone-like effects limiting the propa- growth and operator growth (as measured by the so-called gation of signals and disturbances through the system [8]. out-of-time-order correlator) scale with different powers of the This leads to the question of how an initially local quantum time. In a certain sense, entanglement growth is parametrically operator spreads and becomes nonlocal under Heisenberg time slower than operator spreading in the Griffiths phase. We find evolution. Again there are important differences between clean that in a portion of the thermal Griffiths “phase” where the and disordered systems. transport is diffusive and the operator spreading is ballistic, In translationally invariant one-dimensional (1D) systems, so in these two respects the rare regions do not change entanglement growth and operator spreading are both typically the disorder-free behavior, the entanglement growth remains associated with nonzero speeds [1,2,5,6,9]. By contrast, in subballistic due to the rare region effects. The transition the MBL phase, both entanglement growth [2–4,10,11] and from diffusive to subdiffusive transport is clearly exhibited operator spreading [12–18] are associated with length scales in Ref. [25], although to show that required probing quite that grow only logarithmically in time. long length scales using matrix product methods. Our results This paper studies a third situation: 1D systems that are say that the entanglement growth remains subballistic at this disordered, but are in the thermalizing phase. In 1D, quenched transition. Numerically probing this long-time entanglement randomness can strongly affect transport and information behavior is a challenge that might be well beyond presently spreading even in the thermal phase. This is because there known methods and capabilities. The physics of this regime is can exist rare regions where disorder happens to be stronger weak links. Most models that have been numerically studied than average, and which locally resemble the MBL phase. have uniform couplings and randomness only on the local fields
2469-9950/2018/98(3)/035118(16) 035118-1 ©2018 American Physical Society ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) or potentials. To make this weak-link behavior more accessible system does not matter at this stage, but an archetypal example in numerics, it may be worthwhile exploring models where the is the Heisenberg spin chain with random couplings and fields, couplings are random, perhaps even with a power-law bare = · + z distribution of weak links. H Ji Si Si+1 hi Si . (1) We provide a long-distance picture for entanglement en- i i tropy growth in terms of an effective classical surface growth (We could also consider a Floquet spin chain with effective problem. The height of the growing “surface,” S(x), is the discrete time dynamics.) Such chains will thermalize if the amount of entanglement across a cut in the 1D system at randomness is not too strong. However, thermalization may position x; the growth of the surface is deterministic, but be slow as a result of weak links. These may be simply weak it is affected by quenched randomness in the local growth couplings where a single bond Ji is very weak. More robustly, rates. This picture is motivated by an analogy to a simpler the weak links may be due to extended Griffiths regions where quantum dynamics based on a random quantum circuit [6]. the disorder happens to be more severe, so that MBL physics The basic assumptions are also substantiated independently, arises locally [21]. including with a semimicroscopic analysis of entangling across Moving to a coarse-grained description, let us label weak a Griffiths region. links by an index i, with a given weak link located at a The surface growth picture leads to the unexpected conclu- position xi . A key assumption is that each weak link has a sion that entanglement growth is subballistic throughout the well-defined local entanglement rate i , which is the rate at entire thermal Griffiths phase, i.e., whenever Griffiths regions which entanglement is generated across the weak link in the locally resembling the MBL phase are possible. (This appears absence of other weak links. Wewill give arguments supporting to require the absence of exact non-Abelian symmetries, as this assumption in Secs. VI and VII. The local entanglement these appear to prohibit MBL [28–30].) We give expressions rates are assumed to be independent random variables with a for the dynamical exponent governing the growth of the von power-law distribution at small : Neumann entropy and for the probability distribution of this ∼ a − ∞ quantity at late times. P ( ) A , 1
II. ENTANGLEMENT GROWTH In this section, we study entanglement growth starting from a weakly entangled state: this could be the ground state of the FIG. 1. Schematic: entanglement across a cut through bond at pre-quench Hamiltonian in a quantum quench. The specific 1D position x.
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The surface growth picture takes into account two crucial physical constraints. First, the growth rate at a weak link i, which is at position xi , is constrained by the local rate i . Second, the spatial slope of the entropy profile S(x,t)is constrained by the density of active degrees of freedom in the spin chain:
∂S(xi ,t)/∂t i , |∂S(x,t)/∂x| seq. (4)
Here, seq is the entropy density of the thermal state to which the system is locally equilibrating. The second inequality follows from subadditivity of the von Neumann entropy, together with the assumption that local reduced density matrices (for O(1) adjacent spins) thermalize at late times (no local high-entropy FIG. 2. Surface growth picture for entanglement S(x) across a “hot spots”). In a lattice spin model at infinite temperature, seq cut at x in a finite chain, following Eq. (6). The lines show successive is the logarithm of the local Hilbert space dimension per unit equally spaced times. Weak links with larger rates are successively length; this version of the inequality follows rigorously from “dominated” by weaker links with smaller . [This sample was subadditivity. In the following, we will rescale x so that the generated with a = 0, see Eq. (2).] inequality becomes |∂S(x,t)/∂x| 1. (5) −1/(a+1) of order min ∼ D ,sothetwotermsin i t +|x − xi | If the system has any conserved densities, such as energy, will scale as D−1/(a+1)t and D, respectively. Minimizing with that have a spatial distribution that is away from thermal equi- respect to D gives librium, then these equations for entanglement production are coupled to the transport equations for the conserved densities, a + 2 S(x,t) ∼ t1/zS ,z= . (7) and i and seq in general depend on the local densities. For S a + 1 simplicity, we will assume that any such conserved densities are close to equilibrium and it is only the entanglement The dynamical exponent zS sets the typical lengthscale for that is far from equilibrium. (This is natural for a quench entanglement at time t: for example, the typical time for the from an initial state with a spatial distribution of conserved entanglement profile to saturate in a finite system of size L quantities that is statistically homogeneous: at times much will be of order LzS . It also governs the typical distance to the larger than microscopic timescales, but much smaller than L, locally dominant weak link, the conserved density will locally be close to equilibrium, but the entanglement across a given cut can still be far from its equi- D ∼ t1/zS . (8) librium value.) More generally, since there are regimes where the transport is slower than the entanglement dynamics, in This is the lengthscale for the dynamical coarsening of the some circumstances the late time entanglement dynamics can entanglement pattern visible in Fig. 2. be slowed due to being constrained by the more slowly relaxing A key feature of Eq. (7) is that the entropy grows “subballis- conserved density (if this is sufficiently spatially nonuniform). tically” even for arbitrarily weak disorder, i.e., zS exceeds one As a result of these inequalities, each weak link i imposes for any finite a. This is despite the fact that operator growth is the constraint S(x,t) S(xi ,0) + i t +|x − xi |. We propose ballistic for large enough a, as we will explain in Sec. III. that at late times the entropy is essentially as large as it can be Equation (7) differs from earlier results which effectively given these constraints: assumed the timescale for entanglement saturation of a large chain was equal to a timescale associated with the weakest link S(x,t) = min {S(xi ,0) + i t +|x − xi |}. (6) i in the chain [19,20]. These previous works took a viewpoint The entanglement across a bond at x is therefore determined by of the “spreading” of entanglement, while we now argue that a single locally “dominant” weak link. The spatial boundaries this process is more accurately viewed as the constrained are taken into account by treating them as weak links with = local production of entanglement. The situation with numerical 0. For a pure state, the final profile at asymptotically late times, studies of entanglement production (and its relation to operator once the system has fully thermalized, is the pyramid S(x,t) = spreading) in this Griffiths regime is still unclear: Ref. [27] min{x,L − x}. These formulas, of course, neglect subleading reports exponents quite different from what we find here, corrections (see Sec. II A); for example, we know that the but then their exponents also appear to be inconsistent with entanglement near the center of the chain will depart from Ref. [34], which studies the entanglement of the unitary time the maximal value by an O(1) correction even as t →∞[33]. evolution operator rather than of initial product states. It seems Figure 2 shows growth according to the rule in Eq. (6). In likely that the finite size effects are still very strong for the sizes the next section, we will see how it emerges at large lengths that have been studied numerically. and timescales from a semimicroscopic toy model. Since S in Eq. (7) is the minimum of a set of uncorrelated A simple optimization argument tells us how S(x,t) scales random variables, it is straightforward to find its full probability with time if we start from a pure product state with S(x,0) = 0. distribution in the limit of an infinite chain initiated in a Let D be the typical distance to the dominant weak link at time pure product state. If the density of weak links is ρ and the t. The weakest link within this distance scale will have a rate distribution of local rates is (2)atsmall , the cumulative
035118-3 ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) rate U S(x) S(x) Γ(x)/2 x
FIG. 3. Random circuit model: each bond x receives Haar- x x random unitaries at its own rate (x)/2. FIG. 4. (Left) Maximum possible slope for the entanglement probability distribution at long time is profile in the quantum circuit model, ∂S/∂x = 1. (Right) A staircase with a defect. After coarse-graining, a finite density ρ of defects results a+ S 2 = − P (entropy >S) = exp −c (9) in a slope ∂S/∂x 1 2ρ. ta+1 with (restoring the equilibrium entropy density sth) constant. If the surface is flat, ∂S/∂x = 0,thegrowthrateis 2Aρ /4[35]. The important regime for us will instead be where c = . (10) the slope |∂S/∂x| is close to the maximal value of unity. s (a + 1)(a + 2) th Microscopically, this means that the surface is close to the The surface growth picture is restricted to the entanglement perfect staircase configuration shown in Fig. 4 (left). Note that across a single cut; to generalize to regions with multiple the growth rate vanishes in this configuration. However, when endpoints, or to periodic boundary conditions, we may use |∂S/∂x| is slightly smaller than unity, there is a small density the “directed polymer” description in Sec. II B. ρ of “defects” in the staircase, involving a local minimum of the height, see Fig. 4 (right). These defects allow growth. Each A. Random circuit model time a unitary hits a defect, the local height increases by two units. The coarse-grained growth rate is thus ∂S/∂t ρ .The Quantum circuit dynamics, with randomly chosen two-site coarse-grained slope is given by |∂S/∂x|=1 − 2ρ,so unitaries, capture many universal features of entanglement growth in translationally invariant systems [6]. This setup is ∂S ∂S easily adapted to give a toy model for entanglement growth 1 − . (11) in the presence of weak links. This leads to a concrete ∂t 2 ∂x surface growth problem defined on the lattice, from which the coarse-grained rule (6) can be seen to emerge in the scaling Note that, microscopically, each time a unitary hits a defect, limit. This is not a derivation of the surface growth picture the defect moves up the staircase by one step, so defects run up for isolated 1D systems with time-independent Hamiltonian the staircase at an average speed /2. The growth rate ∂S/∂t or Floquet dynamics, whose microscopic dynamics are very can be also thought of as (twice) the “current” of these defects. different from the toy model; however, it motivates the surface Next, consider entanglement growth in a system with a growth picture, some of whose predictions can subsequently single weak link at the origin, with rate 1 . At a typical be checked by other means. time t, the local configuration resembles Fig. 5 (left). Micro- Consider a chain of spins, each with a large local Hilbert scopically, the weak bond is almost always a local minimum space dimension, q 1. The chain is initially in an unentan- of the height profile. Unitaries are applied there at a rate 1/2, gled product state. Random two-site unitaries are then applied causing growth at rate 1. Each such event launches one defect 1 up each staircase on the two sides of the weak link. The growth to adjacent spins in a Poissonian fashion, at rates 2 (x) that depend on the bond x (Fig. 3). These rates are distributed at rate of the adjacent regions is therefore set by the growth rate = small as in Eq. (2). at the weak link, ∂S/∂t 1. The coarse-grained slope of the | |= − At large q, the dynamics of the entanglement S(x,t) across adjacent regions is fixed using (11): ∂S/∂x 1 2 1/ . bond x maps exactly to a classical surface growth model For small 1, the deviation of the slope from unity is small. [6]. We absorb a factor of log q into the definition of S Neglecting this deviation, (by defining the von Neumann entropy using a logarithm base q). The entanglement S(x,t) then obeys a very simple S(x,t) min{ 1t +|x|, t/4}, (12) dynamical growth rule: each time a unitary is applied to bond x, S(x,t) increases to the maximal value allowed by the general constraint that S(x,t) can exceed S(x ± 1,t)forthe neighboring bonds by at most 1. With this growth rule, the differences between adjacent heights are always ±1atlate times [6]. The resulting dynamics is microscopically stochastic. How- FIG. 5. Growth at a weak link location (random circuit model). ever, we may neglect the noise-induced fluctuations, since they Unitaries are applied at the weak link at a small rate 1. Each such are negligible in the long-wavelength limit. event increases the height at the origin by two units. Two “defects” First, take to be constant (no weak links), and consider then travel up the sides of the double staircase at an average speed the growth of a region whose coarse-grained slope ∂S/∂x is /2.
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S(x) t/4
t/4
1t x
FIG. 6. (Left) Entanglement growth around a weak link charac- FIG. 7. Minimal cut configurations determining S(x,t)inan terized by a small rate 1. (Right) Entanglement growth around a pair infinite system with a single weak link of strength 1. The grey of weak links, showing how the weaker link dominates at late times. patch represents the unitary circuit and the thick line represents the − The figure shows eight equally spaced times. (coarse-grained) minimal cut: (left) for x< t (1 4 1/ ) and (right) 4 (1−2 1/ ) − for x> t (1 4 1/ ) . 4 (1−2 1/ ) when the initial state is a pure product state. This profile is shown in Fig. 6 (left). The weak link influences a region of In the context of entanglement, the directed polymer may size t/4 on either side. be viewed as a coarse-grained “minimal cut” through a unitary It is straightforward to generalize to multiple weak links. circuit representing the dynamical evolution [6]. We briefly Consider two with rates and that are separated by a 1 2 summarize the main features of this coarse grained picture as distance l [Fig. 6 (right)]. We take < . The regions 1 2 it applies to the random circuit model. In the present case, with of influence of the two weak links meet at a time 2l/ , weak links, we obtain a directed polymer subject to pinning by giving a profile with a central peak. The region of influence of vertical defect lines [37]. the weaker link then gradually expands at the expense of the The entanglement S(x,t) is given by the “energy” of a stronger link. At time t l/( − ), the central peak hits the 2 1 minimal energy cut which splits the space-time slice into two link with the rate and disappears. Subsequently, the link with 2 disconnected pieces—see Fig. 7. (In this section we treat the the larger rate has no effect on the coarse-grained configuration, time t as a spatial dimension.) One endpoint of this cut must be being “dominated” by the weaker link. (The link with rate at position x on the top boundary, and the cut must disconnect does not affect the slope, or equivalently the density of 2 the parts of the top boundary to the left and right of x.Inan defects, in this regime, except precisely at its location: the infinite system, this means that the other endpoint of the cut flow of defects is limited only by the slower rate .) Again, 1 must be at the bottom boundary. In the absence of weak links, we may write the minimal energy such cut is vertical, and the energy per unit
S(x,t) min{ 1t +|x − x1|, 2t +|x − x2|, t/4}. (13) height is /4 (in the notation of Ref. [6] this is the entanglement rate vE). In a finite system, the cut can terminate on the left- The same logic extends to arbitrary numbers of weak links. or right-hand spatial boundary. The t/4 term can be dropped since at large times every point To begin with it is sufficient to consider only horizontal x is within the domain of influence of some weak link. This and vertical cuts. The energy of a horizontal cut is equal to gives Eq. (6). its extent in the x direction. If we consider S(x,t) in a clean We may also quantify the subleading corrections to Eq. (6): semi-infinite system with a boundary at position 0, the minimal the average gradient |∂S/∂x| of the straight sections1 is cut is vertical and of energy t/4 for early times, while at late reduced from the maximal value of unity by an amount of times the horizontal cut with energy x is favorable; this gives − + − order D 1/(a 1) when a>0 and of order D 1 when a<0. S(x,t) = min{ t/4,x}. Since D ∼ t1/zS 1, these corrections are indeed small. A single weak link corresponds to a vertical defect line where the energy density per unit height is reduced and equal to B. “Minimal cut” interpretation i .Forlarget, it is worthwhile for the polymer to travel a large horizontal distance to take advantage of this favorable energy There is a general relationship between surface growth in + density. It must of course “pay” in energy for the nonvertical 1 1D and the statistical mechanics of a directed polymer in section required to reach the defect. For a crude picture, we a two-dimensional environment [36]. The results discussed can consider only horizontal and vertical segments; it is easy above may also be understood in this language, and this allows to see that in an infinite system we then recover Eq. (12). them to be generalized to more complex geometries. In more detail, the energy of a segment of horizontal extent x remains equal to x even if the segment is at a finite angle to the horizontal, so long as this angle is small enough2 (this is true for slopes 2/ ). This means that the true minimal 1For large t, the profile consists of staircases of typical length D ∼ t 1/zS with almost-maximal coarse-grained |∂S/∂x|. The minima between such staircases are weak links with typical strength min ∼ − + D 1/(a 1) [see above Eq. (7)]. The gradient at position i is less than 2This can be seen from the microscopic picture of the polymer as a the maximum by O( min/ i ). Summing this, the total height of the “minimal cut” through the large-q unitary circuit. The energy of the staircase is reduced from that of a perfect staircase by S of order polymer is equal to the number of bonds it cuts. A horizontal cut of D −1 + min i=1 i (note i min x/t), which gives the scaling in the length x cuts x bonds. This horizontal cut can be deformed to one text. with a finite coarse-grained slope, with the same energy, so long as
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1/zO value. The shape of the operator turns out to be qualitatively t 1/z t W different for weak and strong disorder. For weak disorder C zW >zO ; in this regime, the front is much smaller than the “plateau” region in which C has saturated to its maximum value. Conversely, for strong disorder, zW = zO , so that C does x not have a well-defined plateau when lengths are scaled by the spreading length. FIG. 8. Definition of dynamical exponents zO and zW governing Our starting point is a picture for operator spreading in the size of a spreading operator and the size of the front [as measured 1D systems developed on the basis of calculations in random using the OTO correlator C(x,t), Sec. III]. The front is only well- circuits in Ref. [31] (see also parallel work [32]).Itisshown = defined in the weak-disorder regime (a>1), where zO 1and there that for x>0 and for long times, we may write (we zW < 1. switch to a continuum notation) ∞ cut configuration for small x is as shown in Fig. 7.This C(x,t) = dx ρ(x ,t), (16) corresponds to taking subleading corrections in the slope of x S(x) into account, and gives S(x,t) = t + (1 − 2 / )|x| 1 1 where ρ(x,t) is a conserved density in agreement with the previous section. ∞ This picture may be extended to multiple weak links. The = ∼ ρ(x,t) 1, (17) typical transverse excursion for a given cut is of order D 0 t1/zS , so much smaller than t [Eq. (8)]. It may also be extended and where—in a random circuit without weak links—ρ(x,t) to a region with multiple endpoints. Consider the entanglement behaves essentially like the probability density of a random of a finite region of length x in an infinite system. At early times, walker with a bias in favor of rightward steps. The average the minimal cut configuration involves two disconnected cuts, speed v of the walker sets the butterfly velocity. For x
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2 2 ρ At times 1/ (but small compared to vB / D, when diffusive spreading becomes comparable), the packet ρ(x)is of width ∼vB / and of height /vB [see Fig. 9 (top)]. x C B. Operator spreading across many weak links To understand spreading across multiple weak links, it is useful to think of each one as performing a linear transforma- x tion on the “wave packet” ρ(x,t). We go into a frame moving at speed vB . The foregoing tells us that if the initial wave packet FIG. 9. Operator spreading caused by passage through a weak is link (schematic). (Top) The spreading of the “wave packet” ρ(x,t) = − [Eq. (16)]. (Bottom) The corresponding spreading of the front of the ρ0(x) δ(x x0), (25) commutator C(x,t) [Eq. (14)]. then the new wave packet is (we define γ = /vB ) 3 right of x do contribute. Specifically, we expect this fraction − ρ (x) = γeγ (x x0) (x x + = − (t−t∗) ρ(0 ,t) e . (23) − −1 vB x vB t γk , (29) = (Walkers hop across at rate , and are whisked away at speed k 1 vB .) Since we neglect spreading of ρ within the homogeneous where we have made the approximation that all parts of the region, the density to the right of the weak link is related to wave packet have passed through the same number of weak the above by ρ(x,t) = ρ(x − vB t): for 0 3A given string can either have X,Y,Z or 1 at site x (two of which 4For example, for one choice of initial condition, iterating with = (−γx)k γx commute with X), and we expect all options to be equally likely [31] the same value of γ gives ρk γ k! e (x<0), which becomes deep in the interior of the operator. Gaussian for large k. 035118-7 ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) whereas for a<0 it is dominated by the smallest γ , which is entanglement is in general smaller than that for the spreading of order x−1/(a+1). This gives of operators, even in clean systems [6,56]. The Griffiths phase is an extreme example, where the two length scales grow with a>0: x ∼ t, zO = 1, (30) different powers of time. To summarize, operators have a well-defined front and a ∼ a+1 = + a<0: x t ,zO 1/(a 1), (31) nonzero butterfly velocity only when a>0. In all regimes the where we have introduced the dynamical exponent zO govern- width of the front increases with time, and there is a change ing the size of a spreading operator. in the exponent governing this width at a = 1. The various Now let us estimate the spreading of the wave packet ρ(x,t) dynamical exponents are summarized below in Table I. within a given sample. The variance formula (28)gives x D. Operator entanglement 2 ∼ −2 width γk . (32) We may also consider the entanglement of a spreading k=1 operator, viewing the operator as a state in a “doubled” system This formula will give the correct scaling in the regime where [57]. The growth of this operator entanglement within the all parts of the wave packet have passed through O(x) links. region in between the two “fronts” of the spreading operator This is the case for a>0, where (32) gives a width much is governed by essentially the same physics as that governing smaller than x. Interestingly, there are two distinct behaviors the growth of the entanglement of states. Thus we expect the within the a>0regime:fora>1 the sum is of order x, while operator entanglement to grow with dynamic exponent zS . − ∞ for 0 035118-8 DYNAMICS OF ENTANGLEMENT AND TRANSPORT IN … PHYSICAL REVIEW B 98, 035118 (2018) numerical work using long chains argues for a regime in the TABLE I. Dynamical exponents governing lengthscales for en- disordered XXZ model where energy transport is diffusive but tanglement growth (zS ); spreading of conserved quantities (zC ); charge transport is subdiffusive [58]. As we discuss at the end spreading of quantum operators under Heisenberg time evolution of this section, it is also possible to artificially design a chain (zO ); width of the “front” of a spreading operator (zW ). Here, we assume a single exponent a governs the distribution of timescales with a U(1) charge for which acharge is clearly smaller than [see Eq. (2)] for the various processes at a weak link, see text. both a and aenergy, although this example involves a power law distribution of microscopic weak links so may not be relevant − to typical models in the Griffiths phase. In the following, we 1 035118-9 ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) ACb weak link [61–63]. This bound states that in a pure state dS Ab1 || || c Hb1b2 ln d. (46) ACb1 b2 dt Here, c is a numerical constant given in Ref. [61]; d is the ACb1 b2 smaller of the Hilbert space dimensions of b1 and b2, here = || || given by d 2. Most importantly, Hb1b2 is the magnitude of the largest eigenvalue of Hb1b2 , here equal to Jweak. For small Jweak, the physical rate of entanglement growth FIG. 10. Two types of microscopic weak link. Left: an infinite (i.e., in a typical state) may be much smaller than the rigorous spin chain with a weak bond connecting spins b1 and b2. (Top right) upper bound (46). Indeed, numerically we find that in a Floquet A chain in which the Z component of spin b is almost conserved. Ising spin chain the von Neumann entropy growth rate is of This can be mapped to a chain with a weak bond by duplicating b order (bottom). dS ∼ J 2 ln 1/J (47) dt weak weak VI. BOUNDING ENTANGLEMENT GROWTH rather than of order Jweak, see Sec. VII. But for our purposes ACROSS A GRIFFITHS REGION in this section the upper bound (46) will suffice. Microscopically, a Griffiths region does not consist of In previous sections we assumed that a Griffiths region weakly coupled degrees of freedom, so we cannot immediately could be characterized by a local entanglement growth rate, apply (46). A better cartoon is that the Griffiths region consists , which vanishes as the region becomes large. Here, we show of “slow” degrees of freedom. To see this, recall that deep in analytically that the rate for entanglement growth across a the MBL phase the Hamiltonian may be formulated in terms Griffiths region is exponentially slow when the length of of l-bits—dressed spin variables whose Z components are the Griffiths region is large. strictly conserved [11,60]. This picture is also a useful starting We will start by considering a trivial kind of weak link—a point for considering strongly disordered regions which locally weak bond in a spin chain. For this simple case, a standard resemble the MBL phase. Since these regions are finite, the rigorous result provides a bound on the entanglement growth dressed spin variables are not strictly conserved, but are instead rate. A Griffiths region is more complicated: although it acts “slow.” as a weak link in a coarse-grained sense, it is not equivalent We can learn how to treat such slow degrees of freedom to a simple weak bond, and the degrees of freedom within the in the context of a toy model, where the Griffiths region is Griffiths region are strongly coupled. Nevertheless, we show replaced by a single central spin whose Z component is almost below that the bound can be extended—nonrigorously—to conserved [see Fig. 10 (right)]. We label the left and right this case by making use of the “l-bit” picture for many-body regions by A and C, respectively, and the central slow spin by localized systems [11,60]. b. We denote the Pauli operators for the slow spin by X, Y , Z, First consider a spin chain in which one bond has a very and take a Hamiltonian whose O(1) terms all commute with Z: small coupling. Label the degrees of freedom as in Fig. 10 (left). b1 and b2 denote the spins to the left and right of the H = HAb(Z) + HbC(Z) + hweakX. (48) weak bond respectively, and A and C contain the other spins Here the notation H (Z) means that this term acts on the on the left and the right, respectively. The Hamiltonian may be Ab central spin only via its Z operator; it can act arbitrarily on written as the spins in A. For concreteness, we take the weak term which breaks conservation of Z to be a transverse field. H = H + H + H , (42) Ab1 b1b2 b2C It is easy to find an example showing that the instantaneous rate of entanglement growth between A and the rest, dS /dt, reflecting the fact that the two sides are coupled only via A can be O(1).7 However, the time-averaged rate is small when spins b and b . For example, for an infinite chain with Ising 1 2 h is small. To see this we relate the physical system to a interactions and longitudinal and transverse fields, we would weak reference system Ab b C in which the spin b is replaced with have 1 2 two spins. There is a mapping from the Hilbert space of AbC into that of Ab1b2C given by H = J Z Z , (43) b1b2 weak 0 1 |↑ →|↑ |↑ , |↓ →|↓ |↓ (49) b b1 b2 b b1 b2 = + + HAb1 (JZi−1Zi hXi gZi ), (44) in the Z basis. This mapping commutes with the time evolution i<0 if we choose the following Hamiltonian for the reference system: Hb B = (JZi− Zi + hXi + gZi ), (45) 2 1 = + + i>1 Hreference HAb1 (Z1) Hb2C (Z2) hweakX1X2, (50) with Jweak J . This is a weak link of a simple kind: the 7 systems Ab1 and b2B are coupled only by a small term in the Take C to be empty and A to consist of a single spin, take HbC Hamiltonian. In this simple situation there is a rigorous bound to be an Ising coupling, and take the initial state to have both spins on the rate at which entanglement can be generated across the polarized in the X direction. 035118-10 DYNAMICS OF ENTANGLEMENT AND TRANSPORT IN … PHYSICAL REVIEW B 98, 035118 (2018) where HAb1 (Z1)issimplyHAb(Z) with Z replaced by Z1. apply the duplication trick to b to give a system Further, if two states |ψ and |ψ are related by the phys ref Aab b cC. (58) mapping, they yield the same density matrix on A. Therefore 1 2 we are guaranteed that We now consider generation of entanglement across the cut Aab |b cC.ThetermsinH and H act only within A or C, Sphys(t) = Sref(t) (51) 1 2 A C A A respectively, and can be neglected. Next, we have the terms at all times. coming from HB . These act on all the Zsina,b,c.Wehave Since the weak field has now become a weak interaction some freedom in how we represent these terms, since any Z in b between Ab1 and b2B, we can use Eq. (46) to bound the change can be represented either by a Z in b1 or by the corresponding SAb1 in the entanglement between Ab1 and b2B: Z in b2. Using this freedom, any term in HB that does not ref act on both a and c may be represented with a term which SAb (t) (c ln 2) hweakt. (52) 1 does not cross the cut Aab1|b2cC. The largest remaining terms ref ∼ Of course, S is not meaningful in the physical system. from HB , which act on both a and c, are of magnitude Ab1 But subadditivity of the von Neumann entropy, together with exp (− /3ζ ). These terms typically act on O( ) spins in region (51), guarantees that it is close to the quantity of interest: b but only O(1) spins in a and c, so can be represented by terms |Sref (t) − Sphys(t)| ln 2. This gives the desired result involving only O(1) spins on one side of the cut and O( ) Ab1 A on the other. Finally, we have the terms from HAB and HBC physical + SA (t) (chweak t 2) ln 2. (53) which couple across the boundary of the Griffiths region. After the unitary transformation, HAB couples the boundary spin α We see that at long times the time-averaged dSA/dt is at most in region A to all of the l-bits in the Griffiths regions, with of order hweak, and that the coefficient remains of O(1) even exponentially decaying couplings. Terms in which α is coupled if the size of A or C diverges. [See Sec. VII for a numerical to the leftmost spins in region b are of size ∼ exp (− /3ζ ). analysis of this problem in a Floquet spin chain, showing that − /3ζ A term like e Zα(...)Xi , where i is a site in region b, the growth rate is even smaller than the maximum allowed by becomes a coupling involving the corresponding sites in both this bound.] b1 and b2. Again, these terms involve O( ) spins on one side Finally, we turn to a spin chain with subsystems ABC, of the cut and O(1) on the other. where B is a Griffiths region consisting of a large number Altogether, the strongest Hamiltonian terms coupling across of consecutive spins. We will consider a strongly disordered − /3ζ 8 the cut Aa1b1|b2cC have size ∼e and involve O( ) spins. Griffiths region which locally resembles the fully many body For a nonrigorous application of the bound (46), we must localized phase. Write the Hamiltonian as estimate the norm of the entangling Hamiltonian. This may − /3ζ H = HA + HB + HC + HAB + HBC, (54) be larger than e by an exponentially large combinatorial factor, since the number of terms is large. We take the region to where HAB and HBC each act on a single bond. Now we make be sufficiently strongly disordered (i.e., ζ<ζ0, where ζ0 is an use of the l-bit picture for the MBL phase [11,60]. We expect order one constant) that the exponential decay of the couplings that a unitary transformation on B can reduce its Hamiltonian wins, giving to a form which only depends on the Zi operators for spins i ∈ B, dSAab1 −α e (59) † dt UHB U = H ,H= hi Zi + Jij Zi Zj + ..., B B for some α that does not depend on . The difference between i ij S and the physically meaningful entropy S (or S )is (55) Aab1 A Aa O( ), and is unimportant at long times. where the couplings decay exponentially with distance, and This shows that the entanglement growth rate across a where the unitary transformation preserves the locality of oper- strongly disordered Griffiths region is exponentially small in ators up to exponential tails. For the purposes of entanglement the size of the region, regardless of the size of the complete † growth we can work with H = UHU instead of H, since U chain. (Assumptions made in previous work [19,20], where a only introduces O(1) entanglement. H is of the form local timescale associated with the weakest link in a finite chain † was taken as the thermalization time of the chain, are equivalent H = HA + HC + H + U(HAB + HBC)U . (56) B to taking the entanglement growth rate to grow with the size of If HAB = ZαZβ , where α, β are the boundary spins in A and the adjacent regions. We see this is not the case.) This is enough B, respectively, to show that there is a power law distribution of local rates. In turn this suffices to prove that S grows subballistically, by the = † = X + Y + Z HAB UHAB U Zα ui Xi ui Yi ui Zi ∈ i B ⎞ 8 These terms can be split into two groups, each of which acts on ZZ ⎠ only O(1) spins from one side. Since (46) can be applied to each + k Zi Zj + ... . (57) ij group separately and the results added, the appropriate log d factor i,j∈B (46)isO(1) rather than O( ). In any case, this polynomial correction Now we split the Griffiths region B into the left-hand, central, is negligible in comparison to the error in estimating the coefficient and right-hand regions, a, b, and c, of length /3 each, and in the exponential. 035118-11 ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) A B 10 vN 5 S +1 FIG. 11. Setup for the numerical simulation. A chain of L = 0 + 1 0 5 10 15 20 2l 1 spin- 2 degrees of freedom is subjected to Floquet dynamics with Ising couplings and transverse and longitudinal fields. The Z x component of the central spin is almost conserved due to the weak = transverse field. FIG. 12. SvN as a function of position in systems with L 21 and ε = 0.3. The different curves show times from t = 0 to 500 in logic in Sec. II. We focused on strongly disordered regions increments of t· = 10 (the subsystems are separately entangled prior to t = 0). with ζ<ζ0, while in practice the prevalent weak links with agiven may be less strongly disordered. However, this will where x1 is the location of the weak link. If we start from a state only decrease the value of the exponent a. in which the two subsystems are separately fully entangled, S(x,t) = min{ t +|x − x |,x,(L − x)}. (62) VII. ENTANGLING ACROSS A WEAK LINK: NUMERICS 1 We now present a numerical study of entanglement growth In particular, either of these forms implies that for large L and across a microscopic weak link. The setup for the numerical x, the entanglement S(x1,t) at the location of the weak link has 1 = + the simple piecewise linear scaling form: simulation is a spin- 2 chain of length L 2l 1, shown schematically in Fig. 11. We use Floquet dynamics considered S(x1,t) = min{ 1t, x1}. (63) previously in Ref. [64]: an Ising spin chain with longitudinal and transverse fields. This system has been shown to thermalize Although simple, these scaling forms are not aprioriobvious: rapidly in the absence of a weak link. A simplifying feature they are nontrivial predictions about the universal physics of the Floquet case is that energy is not conserved [64]. stemming from the results of this paper. We perform exact time evolution using the online package We now give details of the numerical simulation. We use a ITENSOR [65]. Floquet time evolution operator The weak link we consider is formed by the spin at the U( ) = U U ( ), (64) central site, whose Z-component is almost conserved. The Z X commutator of this operator with the one-step time evolution which includes half a period of evolution with the Ising operator is of order , where will be taken to be small. As interactions and longitudinal fields, discussed in Sec. VI, this is a toy model for a Griffiths region: ⎛ ⎡ ⎤⎞ the central spin is an “almost” l-bit, whereas all other spins L−1 L = ⎝− ⎣ + ⎦⎠ are strongly nonconserved under the evolution. The slow spin UZ exp iτ J Zj Zj+1 h Zj , (65) bottlenecks the growth of entanglement between the rapidly j=1 j=1 thermalizing regions surrounding it. We focus on the dynamics preceded by half a period of evolution with the transverse fields, of the Renyi entropies (including the von Neumman entropy ⎛ ⎡ ⎤⎞ SvN ≡ S1) ⎝ ⎣ ⎦⎠ UX( ) = exp −iτg Xl+1 + Xj . (66) 1 n S (t) = log Trρ (60) j=l+1 n 1 − n 2 A Note that the transverse field on the central spin is weakened across the bond connecting the spin at site l to the nearly by a (variable) factor . The other couplings are fixed at the conserved spin (ρ = Tr ρ and the regions A and B are A B values J = 1, h = 0.809, g = 0.9045, and τ = 0.8, which defined in Fig. 11). were shown in Ref. [64] to yield rapid thermalization. Here we will give results only for the model with a slow spin, We start from the product state |↑ ...↑ in the Z basis. But but we have also simulated a spin chain with a weak bond. As before beginning the entangling dynamics, we evolve the left might be expected from the mapping between a slow spin and and right subsystems separately, i.e., with = 0, for a time a weak link in Sec. VI, the results are extremely similar. T to ensure that they are separately strongly entangled. In the Our main goal will be to validate two key elements of the 0 simulations, we have used T = 5, however, we note that our picture for entanglement growth discussed above. Firstly, that 0 results depend very weakly on T . The state at time t 0is the entanglement growth rate in the vicinity of a weak link, x , 0 i therefore defined to be can be characterized by a local rate i which is independent of the size of the surrounding regions. Secondly, that in the scaling |ψ(t) =U( )t U(0)T0 |↑ ...↑ . (67) limit the growth of entanglement entropy is captured by the = simple scaling forms discussed in Sec. II. For the present case, Note that the entanglement between A and B is zero when t the relevant scaling form, if we start from a product state, is 0. In Fig. 12, we plot a typical spatial dependence of SvN(x,t) for a system of size L = 21 and for successive times t starting S(x,t) = min{ t +|x − x1|,x,(L − x),vEt}, (61) from t = 0. Figure 13 shows the same for S2(x,t). 035118-12 DYNAMICS OF ENTANGLEMENT AND TRANSPORT IN … PHYSICAL REVIEW B 98, 035118 (2018) 10 2 5 S 0 0 5 10 15 20 x FIG. 13. As Fig. 12, but for the second Renyi entropy S2. Let us now examine the validity of the scaling forms. In Fig. 14, we plot the von Neumann and second Renyi entropies, FIG. 15. The initial growth rates n as a function of (calculated rescaled by their asymptotic late time values, against the using L = 21). The stars (blue) and circles (red) correspond to von rescaled time variable nt/Smax. Here, n is the growth rate Neumann and second Renyi entropy, respectively. of Sn(t) obtained by fitting the linear growth near t = 0for L = 21. Smax is the value of the entropy as t →∞, which is determined numerically and which depends on n and L. It is clear from Fig. 14 that the early time growth rate, n,is independent of the length of the system. This is consistent with We note that in the case of the von Neumann entropy Smax converges very quickly with increasing L to the prediction of our coarse-grained picture. By numerically fitting the data, we Ref. [33] for a random state. find that for small <0.1 the growth rates are given by B An = A 2 log , = 2, (68) vN vN 2 n n − 1 where AvN,B, and A are -independent constants. The rates for larger values of = 0.1,0.2,...,1 are shown in Fig. 15. This confirms that indeed the growth rate is small in and that a nearly conserved quantity causes a bottleneck for entanglement growth. Note that for small , the growth rate is parametrically smaller than the rigorous upper bound in Sec. VI. Interestingly, Fig. 15 suggests that for = 1 (the homogeneous case) the growth rates for SvN and S2 may be equal. We also find that the transition from linear growth to saturation becomes sharper with increasing L, consistent with the expected scaling form Eq. (63). This is much clearer for the second and higher Renyi entropies (Fig. 14). For the von Neumann entropy, a more careful finite size analysis is required, which we give below. For translationally invariant Floquet systems with no con- served quantities, the late time saturation of the entropy has been found to be exponential [64]. We also obtain a good fit at late times with SvN = Smax − (t),(t) = exp(− |t − t∗|), (69) where the parameters and t∗ of the fit are extracted separately for each L and . We find that at small the saturation rate behaves as = A 2. (70) The timescale t∗ for saturation is of order L. Assume that there is a simple crossover between linear and exponential behavior at a time tl. Matching S and ∂S/∂t at this FIG. 14. The von Neumann (top) and second Renyi (bottom) time gives the following gluing conditions: entropies scaled by their maximal value at large time Smax vs the Smax 1 1 vN rescaled time nt/S for different values of L and for = 0.2. The = − = + max tl ,t∗ tl log2 , (71) value of Smax is determined numerically. vN 035118-13 ADAM NAHUM, JONATHAN RUHMAN, AND DAVID A. HUSE PHYSICAL REVIEW B 98, 035118 (2018) (a) L = 21 (b) VIII. SUMMARY We have examined the dynamics of isolated one- dimensional quantum systems that thermalize, but have a distribution of weak links produced by quenched randomness. This occurs in systems that sustain (but which are outside) the 101 102 L = 21 many-body-localized (MBL) phase, due to Griffiths regions that are locally in this phase. We developed coarse-grained pictures for the production of entanglement and the spreading FIG. 16. (a) The solid lines show the normalized deviation of operators and of conserved densities. We found that such systems have multiple dynamic length from maximal entropy /Smax defined in Eq. (69) vs time t. The different colors correspond to different values of ranging scales that diverge with different powers of the time. These between 0.1,0.2,...,0.7. The dashed curves are the fits to the length scales are for the growth of entanglement, the spreading linear form: 1 − t/Smax (red dashed) and to the exponential form: of conserved densities, the spreading of initially local oper- 1 − exp [− (t − t∗)]/Smax (black dashed). The red dots mark the ators, and the width of the “front” of a spreading operator. crossover points between linear behavior, at short times, and ex- These quantities have effective classical dynamics of distinct ponential, at long times. These crossover points are used to extract types. Entanglement growth maps to classical surface growth the timescale tl is extracted. (b) Comparison between the extracted with inhomogeneous rates. Transport of conserved quantities is crossover time tl (red dots) and the expected form Eq. (71) (blue solid governed by random walk dynamics with weak links. Finally, curve). the front of a growing operator can be related to the passage of a biased random walk through weak links. Contrary to systems without quenched disorder, the process which means that of entanglement saturation (either for states or operators) is always parametrically slower than the spreading of operators as S(tl) = Smax − vN/ . (72) measured by the norm of their commutator with a local operator or by the out-of-time-order correlator (OTOC). Entanglement In Fig. 16, we test the assumption underlying the previous. growth is subballistic even in the weak-disorder portion of In the first panel, we present the linear (red dashed lines) and the Griffiths phase: sub-ballistic growth is inevitable whenever exponential (black dashed lines) fits to (t)/Smax. The red dots weak links of arbitrary length can be created, no matter how are crossover times between these two behaviors, determined small the probability is for such weak links. By contrast, by eye. In the second panel, we compare these crossover times when weak links are sufficiently rare, both the OTOC and with the prediction in Eq. (71), finding good agreement. conserved densities are relatively insensitive to them: there The previous equation shows that the departure from linear is a regime where conserved quantities spread diffusively and behavior occurs when the entropy is within the operators spread ballistically, as in a clean nonintegrable chain. vN AvN B We also examined the exchange of quantum information SvN ∼ ∼ log (73) A 2 between two clean systems coupled by a single weak link, giving universal scaling forms for the growth of entanglement bits of its maximum. The S ∼ log 1/ behavior for the 2 and the spreading of operators from one system into the other. von Neumann entropy indicates that the finite-size rounding Various questions remain for the future. Each of the is more severe for weaker weak links; this conforms with quantities we have considered (entanglement, the OTOC, what we find numerically. For the higher Renyi entropies, this and conserved densities) is affected by weak links with a logarithmic factor is absent, as a result of Eq. (68). Since S is power law distribution of timescales (with exponent denoted independent of L, the above confirms the scaling form Eq. (63) a in the text). An important task is to clarify the relations for a system of size L with a weak link whose strength is between the exponents a for distinct quantities, as discussed in fixed as L →∞. (The deviation from linearity for the last Sec. IV. For example, the exponents a pertaining to conserved S ∼ log 1/ bits of entanglement is also negligible if we C 2 quantities may differ between different conserved quantities, take to scale as a negative power of L, which is the natural as suggested by recent numerics [58] (this issue could also be scaling in the Griffiths regime.) probed with the methods of Refs. [54,66]). In a system with Note that if the relative logarithmic factor in the growth rates conserved energy, can the exponent a for entanglement exceed for S and S [Eq. (68)] also applies for coarse-grained vN n>1 a for energy? weak links, then the von Neumman entropy in the Griffiths energy It has been suggested on the basis of simulations [67] that phase will exceed the higher Renyi entropies by a factor of subballistic entanglement growth may occur in the thermal order log t at late times (since the weak links that are relevant − + phase of 1D chains with quasiperiodic detuning rather than at time t have strength ∼ t 1/(2 a)). disorder. The relation of such systems to rare-region physics We have confirmed the two key assumptions: is inde- n remains to be understood. pendent of L, and the scaling form Eq. (6) is indeed obtained The pictures described here are set up to deal with dynamics in the large L limit for a microscopic weak link. If we allow in the thermal phase. It would be interesting to understand ourselves to extrapolate the above results to Griffiths regions the crossover effects at the MBL phase transition [19,20,68– (coarse-grained weak links), these conclusions support the 71] in more detail by relating the present considerations to scaling forms derived from the surface growth picture. 035118-14 DYNAMICS OF ENTANGLEMENT AND TRANSPORT IN … PHYSICAL REVIEW B 98, 035118 (2018) real-space RG approaches. The results above are expected to ACKNOWLEDGMENTS apply at sufficiently large length and timescales no matter We thank S. Gopalakrishnan, J. Haah, C. Jonay, and S. how close the system is to the MBL transition; however, Vijay for collaboration on related work. AN acknowledges the scales required will diverge as the MBL transition is support from the Gordon and Betty Moore Foundation under approached. 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