Entanglement area laws for long-range interacting systems
Zhe-Xuan Gong, Michael Foss-Feig, Fernando G. S. L. Brandão, Alexey V. Gorshkov
EEntanglement area laws for long-range interacting systems
Zhe-Xuan Gong,
1, 2, ∗ Michael Foss-Feig, Fernando G. S. L. Brand˜ao, and Alexey V. Gorshkov
1, 2 Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA Joint Center for Quantum Information and Computer Science,NIST/University of Maryland, College Park, Maryland 20742, USA United States Army Research Laboratory, Adelphi, MD 20783, USA IQIM, California Institute of Technology, Pasadena CA 91125, USA
We prove that the entanglement entropy of any state evolved under an arbitrary /r α long-range-interacting D -dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area forany α > D + 1 . We also prove that for any α > D + 2 , the ground state of such a Hamiltonian satisfiesthe entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known tosatisfy the area law. These results significantly generalize their existing counterparts for short-range interactingsystems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presenceof long-range interactions. PACS numbers: 03.65.Ud, 03.67.Bg, 75.10.Dg
Quantum many-body systems often have approximately lo-cal interactions, and this locality has profound effects on theentanglement properties of both ground states and the statescreated dynamically after a quantum quench. For example,the entanglement entropy, defined as the entropy of the re-duced state of a subregion, often scales as the boundary areaof the subregion for ground states of short-range interactingHamiltonians [1]. This “area law” of entanglement entropy isin sharp contrast to the behavior of thermodynamic entropy,which typically scales as the volume of the system. While thestudy of area laws originates from black hole physics [2, 3],area laws have received considerable attention recently in thefields of quantum information and condensed matter physics.In particular, area laws are known to be closely related to thevelocity of information propagation in quantum lattices [4],quantum critical phenomena and conformal field theory [5],the efficiency of classical simulation of quantum systems [6],topological order [7], and many-body localization [8].However, the description of many-body systems in termsof local interactions is often only an approximation, and notalways a good one; in numerous systems of current inter-est, ranging from frustrated magnets and spin glasses [9, 10]to atomic, molecular, and optical systems [11–16], long-range interactions are ubiquitous and lead to qualitatively newphysics, e.g. giving rise to novel quantum phases and dynam-ical behaviors [17–24], and enabling speedups in quantuminformation processing [25–29]. Particles in these systemsgenerally experience interactions that decay algebraically ( ∼ /r α ) in the distance ( r ) between them. As might be expected, α controls the extent to which the system respects notions oflocality developed for short-range interacting systems: For α sufficiently small, it is well established [18] that locality maybe completely lost, and for α sufficiently large there is am-ple numerical and analytical evidence [30–33] that area lawsmay persist. However, there is currently no general and rigor-ous understanding of when area laws do or do not survive thepresence of long-range interactions.The modern understanding of area laws draws heavily from several rigorous proofs, all of which require some restric-tions on the general setting discussed above. As the mostnotable example, Hastings [34] proved that ground states ofone-dimensional (1D) gapped Hamiltonians with finite-rangeinteractions satisfy the area law. A subsequent developmentwas made later in Refs. [35, 36], which proved that statesin 1D with exponentially decaying correlations between anytwo regions (a set that includes the ground states of gappedshort-range interacting Hamiltonians) must satisfy the arealaw. Generalizing these proofs to include long-range inter-acting Hamiltonians is, however, rather difficult. For ex-ample, it is a well-known challenge to generalize Hastings’proof of the area law [34] to higher dimensions [37], andlong-range interacting systems are in some sense similar tohigher-dimensional short-range interacting systems. In addi-tion, since ground states of gapped long-range interacting sys-tems can have power-law decaying correlations [38–40], onewould need to relax the condition of exponentially decayingcorrelations in the proof of Refs. [35, 36] to algebraically de-caying correlations. However, this relaxation invalidates theproof, as there exist 1D states with sub-exponentially decay-ing correlations that violate the area law [41].To circumvent these challenges in proving area laws forlong-range interacting systems, here we employ a “dynam-ical” approach. Specifically, we prove that a state satisfiesthe area law if it can be dynamically created in a finite timeby evolving a state that initially satisfies the area law undera long-range interacting Hamiltonians [42]. We then use thepowerful formalism of quasi-adiabatic continuation [43] to re-late such a state to the ground state of a spectrally gappedlong-range interacting Hamiltonian. This strategy is madepossible by the recent proof of Kitaev’s small incremental en-tangling (SIE) conjecture [42, 44], and by significant recentimprovements in Lieb-Robinson bounds [4] for long-range in-teracting systems [45, 46].The manuscript is divided into two proofs of two differentarea laws, the latter of which builds on the former. The firstarea law states that for any initial state, the entanglement en- a r X i v : . [ qu a n t - ph ] F e b tropy of a subsystem cannot change faster than a rate propor-tional to the subsystem’s area. This statement is known to holdfor short-range interacting systems [42, 47], and we have gen-eralized it to systems with interactions decaying faster than /r D +1 . A direct implication of this new area law is thatmatrix-product-state calculations of quench dynamics are ex-pected to remain efficient at relatively short times for most /r α Hamiltonians with α > D + 1 . Moreover, the proof ofour area law also suggests that for α ≤ D , it might be possi-ble for the entanglement entropy to change from an area lawto a volume law in a finite time, thus indicating the onset of adynamical phase transition [48].Our second area law states that if a Hamiltonian has in-teractions decaying faster than /r D +2 , then its ground statesatisfies the area law if it can be connected to an area-law stateby adiabatically deforming the Hamiltonian. Here adiabatic-ity implies a finite energy gap during the adiabatic evolutionand requires interactions to still decay faster than /r D +2 .This area law leads to two new insights: (1) The entangle-ment area law for the ground state of a gapped short-range in-teracting Hamiltonian will remain stable if we add long-rangeinteractions without closing the gap. For certain frustration-free Hamiltonians, including Kitaev’s toric code [49] and theLevin-Wen model [50], the area law is strictly implied for α > D + 2 due a proven stability of the gap for inter-actions decaying faster than /r D +2 [51]. Thus the short-range nature of interactions, believed to be crucial for arealaws, is in fact not necessary. (2) The entanglement area lawmight be violated without destroying the energy gap or mak-ing the energy non-extensive by using /r α interactions with D < α < D + 2 . Thus there may exist exotic quantumphase transitions between gapped phases, challenging the con-ventional wisdom that quantum phase transitions cannot takeplace between gapped phases in an adiabatic evolution [52]. Main results .— In this manuscript, we consider the follow-ing Hamiltonian H on a D-dimensional finite or infinite lattice H = (cid:88) i,j h ij , (cid:107) h ij (cid:107) ≤ /r αij ( i (cid:54) = j ) . (1)Here, h ij is an operator acting on sites i and j that can betime-dependent, (cid:107) h ij (cid:107) denotes the operator norm (largest-magnitude of an eigenvalue) of h ij , and r ij represents thedistance between sites i and j . The maximum Hilbert spacedimension for any site is assumed to be finite and denoted by d . The strength of the on-site interaction h ii can be arbitrary,and is unimportant in the following area laws and proofs.We define the entanglement entropy of a state | ψ (cid:105) with re-spect to a subregion V by S V ( | ψ (cid:105) ) ≡ − tr [ ρ V log ρ V ] , where ρ V = tr ¯ V ( | ψ (cid:105)(cid:104) ψ | ) and ¯ V is the complement of V . We willuse ∂V to denote the set of sites at the boundary of V , and | V | to denote the number of sites in the set V . To clarify thepresentation without sacrificing rigor, we will frequently usethe identification g ( x ) = O ( x ) if there exists finite positiveconstants c and x such that g ( x ) ≤ cx for all x ≥ x . Theconstants c and x may be different each time the O -notationappears, but will not depend on anything other than the lattice geometry and fixed parameters α , D , d , and ∆ (introducedlater). We now state our first area law: Theorem 1. (Area law for dynamics) For any state | ψ (cid:105) underthe time evolution of H defined in Eq. (1) with α > D + 1 , (cid:12)(cid:12)(cid:12)(cid:12) dS V ( | ψ ( t ) (cid:105) ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( | ∂V | ) . (2)To prove Theorem 1, let us introduce the following lemma,which can be directly obtained from the Kitaev’s SIE conjec-ture recently proven in Ref. [42]. Lemma 1. If H = (cid:80) Z h Z with h Z acting on a set of sites Z ,then for any state | ψ (cid:105) (cid:12)(cid:12)(cid:12)(cid:12) dS V ( | ψ ( t ) (cid:105) ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤
18 log( d ) (cid:88) Z, Z ∩ V (cid:54) = ∅ & Z ∩ ¯ V (cid:54) = ∅ (cid:107) h Z (cid:107) | Z | . (3)Roughly speaking, this lemma tells us that the entanglemententropy at most changes at a rate proportional to the totalstrength of interactions that cross the boundary of V .With the help of Lemma 1, the proof of Theorem 1 reducesto the proof of (cid:80) i ∈ V,j / ∈ V (cid:107) h ij (cid:107) ≤ O ( | ∂V | ) . Let us now as-sign a coordinate ( x i , r i ) to each site i ∈ V , with x i measur-ing the directions parallel to the boundary, and r i measuringthe distance of i to the boundary (rounded down to the nextinteger). Upon bounding the sum by a D-dimensional inte-gral, it is straightforward to show that for a given i ∈ V , (cid:80) j / ∈ V (cid:107) h ij (cid:107) ≤ O ( r D − αi ) . Since for a given value of r i ,the possible choices of x i is at most proportional to | ∂V | , itfollows that (cid:80) i ∈ V,j / ∈ V (cid:107) h ij (cid:107) ≤ O ( | ∂V | ) (cid:80) ∞ r =1 r D − α . The-orem 1 is then proven because (cid:80) ∞ r =1 r D − α converges for α > D + 1 . Note that the method used here is an improve-ment over a similar method used in Ref. [42], which if usedwill lead to the condition α > D + 2 instead.To connect from this dynamical area law to a ground-statearea law, we now introduce the formalism of quasi-adiabaticcontinuation. Assume that there is a continuous family ofHamiltonians H ( s ) = (1 − s ) H (0) + sH (1) , (4)parameterized by s ∈ [0 , with each H ( s ) being a time-independent Hamiltonian satisfying Eq. (1) and having aunique ground state | ψ ( s ) (cid:105) and a finite energy gap of at least ∆ . As shown in Ref. [43], the evolution (or continuation)of | ψ ( s ) (cid:105) from s = 0 to s = 1 is governed by an effec-tive Hamiltonian D ( s ) , given by the “Schrodinger equation” d | ψ ( s ) (cid:105) /ds = − i D ( s ) | ψ ( s ) (cid:105) . We emphasize that the evo-lution of | ψ ( s ) (cid:105) is not governed by H ( s ) , because despite theexistence of a finite gap ∆ , to adiabatically evolve under H ( s ) from | ψ (0) (cid:105) to | ψ (1) (cid:105) exactly requires an infinite evolutiontime, in contrast to the unity time needed for the evolution un-der D ( s ) . As a result, the evolution of | ψ ( s ) (cid:105) under D ( s ) isusually called quasi-adiabatic continuation [43].For a given H ( s ) , the choice of D ( s ) is not unique, andhere we choose a convenient form given in Ref. [53], D ( s ) = − i (cid:90) ∞−∞ f (∆ t ) e iH ( s ) t ∂H ( s ) ∂s e − iH ( s ) t dt. (5)Here, f ( x ) belongs to a family of sub-exponentially decay-ing functions, meaning that for any δ < , there exists an x -independent constant c δ such that | f ( x ) | ≤ c δ exp( −| x | δ ) [the explicit form of f ( x ) is not important]. The D ( s ) givenin Eq. (5) has a remarkable feature: if H ( s ) is a short-rangeinteracting Hamiltonian [Eq. (4) in the α → ∞ limit], then D ( s ) contains interactions that decay sub-exponentially withdistance, approximately inheriting the locality of the under-lying interactions [43]. For a finite but suitably large α , it isreasonable to expect that D ( s ) contains interactions that decayas a power-law in distance, as inherited from H ( s ) . If so, thenwe expect to be able to prove a result analogous to Theorem 1,guaranteeing that the entanglement entropy S V ( | ψ ( s ) (cid:105) ) sat-isfies the dynamical area law | dS V ( | ψ ( s ) (cid:105) ) /ds | ≤ O ( | ∂V | ) for α larger than a certain critical value. Upon integratingfrom s = 0 to s = 1 , this would lead immediately to ourTheorem 2 [54]: Theorem 2. (Area law for ground states) For H ( s ) definedin Eq. (4) with α ≥ D + 2 , | ψ (0) (cid:105) satisfying the area lawimplies that | ψ ( s ) (cid:105) satisfies the area law for any s ∈ [0 , . Here the assumption that | ψ (0) (cid:105) satisfies the area law maycome from the scenario where H (0) contains only short-rangeinteractions. The proof of this area law is much more chal-lenging than the proof of Theorem 1. To see the challenge, letus write H ( s ) = (cid:80) ij h ij ( s ) and D ( s ) = (cid:80) ij D ij ( s ) , then D ij ( s ) = − i (cid:90) ∞−∞ f (∆ t )˜ h ( s ) ij ( t ) dt, (6)with ˜ h ( s ) ij ( t ) ≡ e iH ( s ) t ˜ h ij e − iH ( s ) t and ˜ h ij ≡ h ij (1) − h ij (0) .Unlike h ij ( s ) , which acts only on sites i and j , in general D ij ( s ) acts on the entire lattice. Thus we cannot directly ap-ply Lemma 1 to constrain the growth of S V ( | ψ ( s ) (cid:105) ) , as we didfor Theorem 1. To overcome this challenge, we need to derivesome locality structure of the interaction D ij ( s ) despite thefact that it acts on the entire lattice. As mentioned above, ourintuition is that D ij ( s ) should be similar to h ij ( s ) , in that it“mostly” acts on sites close to i and j while its interactionstrength should still decay as /r αij . In order to turn this in-tuition into a precise statement, we need to first look at thelocality structure of A ( t ) = e iHt Ae − iHt for A acting on a setof sites X and H defined in Eq. (1).Formally, we will define A ( t, R ) = (cid:82) dµ ( U R ) U R A ( t ) U † R ,with U R being a unitary operator acting on all sites with dis-tance larger than or equal to R from any site in X and µ ( U R ) being the Haar measure for U R . By this definition, A ( t, R ) only acts on sites within a distance R from ∂X . Let us firstobtain some intuition in the α → ∞ limit, where H is anearest-neighbor Hamiltonian. It is reasonable to expect that A ( t, R ) is a good approximation of A ( t ) if we choose R (cid:29) t ,because it takes a time t ∝ R to “spread” the operator A to sites a distance R from its boundary. More precisely, one canapply the Lieb-Robinson bound [4, 55] in this case to obtain (cid:107) A ( t ) − A ( t, R ) (cid:107) ≤ (cid:107) A (cid:107) O ( e et − R ) . In fact, in the limit of α → ∞ , Theorem 2 has already been proven in Ref. [42].For a finite α the situation is much less clear. Using thedirect generalization [38, 56] of the Lieb-Robinson bound forthe /r α Hamiltonian in Eq. (1) leads to (cid:107) A ( t ) − A ( t, R ) (cid:107) ≤(cid:107) A (cid:107) | X |O ( e vt /R α − D ) , which only guarantees that A ( t ) willbe well approximated by A ( t, R ) when t (cid:28) log( R ) , thus re-quiring exponentially larger R to maintain the level of approx-imation in the α → ∞ case. As shown later, this requirement t (cid:28) log( R ) prohibits a proof of Theorem 2 using the strategyof Ref. [42]. However, recent improvements to the long-rangeLieb-Robinson bound [45] significantly improve the situation.The improved bound enables the following Lemma to be de-rived (see [57]), which together with additional techniques de-scribed below leads to a proof of Theorem 2. Lemma 2.
There exists a constant v = O (1) such that for α > D , γ = D +1 α − D , and < t < t R ≡ ( R v ) γ [58], (cid:107) A ( t ) − A ( t, R ) (cid:107) ≤ (cid:107) A (cid:107)| X | [ O ( e vt − Rtγ ) + O ( t α (1+ γ ) R α − D )] . (7)A crucial consequence of Lemma 2 is that we must onlychoose R polynomially large in t in order to ensure that A ( t ) iswell approximated by A ( t, R ) . The quantity t R characterizesthe edge of the “light cone”, meaning that (cid:107) A ( t ) − A ( t, R ) (cid:107) is only parametrically small in R for t < t R .The locality structure of D ij ( s ) can be understood with thehelp of Lemma 2 and the decomposition, D ij ( s ) = ∞ (cid:88) R =1 G ij ( s, R ) ≡ − i ∞ (cid:88) R =1 (cid:90) ∞−∞ f (∆ t ) g ( s ) ij ( t, R ) dt. (8)Here, g ( s ) ij ( t, R ) ≡ ˜ h ( s ) ij ( t, R ) − ˜ h ( s ) ij ( t, R − and ˜ h ( s ) ij ( t, R ) ≡ (cid:82) dµ ( U R ) U R ˜ h ( s ) ij ( t ) U † R ; Eq. (8) follows by bringing thesummation inside the integral, and using ˜ h ( s ) ij ( t, ∞ ) =˜ h ( s ) ij ( t ) and ˜ h ( s ) ij ( t,
0) = 0 to collapse the summation to (cid:80) ∞ R =1 g ( s ) ij ( t, R ) = ˜ h ( s ) ij ( t ) . We emphasize that G ij ( s, R ) acts only on sites within a distance R from i or j (Fig. 1), andin this sense is “local”. In order to bound how (cid:107)G ij ( s, R ) (cid:107) decays with R and r ij , we must first derive a bound for (cid:107) g ( s ) ij ( t, R ) (cid:107) . It is useful to tackle the short-time and long-time behavior separately. Short-time behavior : For < t < t R we can apply Lemma2. Using a triangle inequality (cid:107) g ( s ) ij ( t, R ) (cid:107) ≤ (cid:107) ˜ h ( s ) ij ( t ) − ˜ h ( s ) ij ( t, R ) (cid:107) + (cid:107) ˜ h ( s ) ij ( t ) − ˜ h ( s ) ij ( t, R − (cid:107) and the inequalities vt − R/t γ < −O ( R / (1+ γ ) ) and (cid:107) ˜ h ij (cid:107) ≤ r − αij , Lemma 2gives (cid:107) g ( s ) ij ( t, R ) (cid:107) ≤ [ O ( e −O ( R γ ) ) + O ( t α (1+ γ ) R α − D )] r − αij . Long-time behavior : When t > t R , for reasons that willbecome clear soon it suffices to bound (cid:107) g ( s ) ij ( t, R ) (cid:107) directlyby (cid:107) ˜ h ij (cid:107) ≤ r − αij , which follows because (cid:107) A ( t, R ) (cid:107) = (cid:107) A (cid:107) for any A , t and R . i j R V Figure 1: Illustration of the locality structure of D ( s ) . Each G ij ( s, R ) is an interaction between a ball of sites centered on i witha radius R and a ball of sites centered on j with a radius R . The in-teraction strength (cid:107)G ij ( s, R ) (cid:107) decays as /r αij and also as /R D − α for large R , represented by the fading color of the balls. For a givensubregion V with boundary ∂V (blue square), the maximum rateof entanglement entropy change only involves interactions G ij ( s, R ) that act on both sites in V and sites outside V . Performing the integration over t in the definition of G ij ( s, R ) [see Eq. (8)], we find [59] (cid:107)G ij ( s, R ) (cid:107) ≤ O ( e −O ( R γ ) ) + O ( R D − α ) + O ( F [ O ( t R )]) r αij , where F ( x ) = (cid:82) ∞ x f ( t ) dt also decays sub-exponentially. Im-portantly, because Lemma 2 states that t R = O ( R / (1+ γ ) ) , (cid:107)G ij ( s, R ) (cid:107) is dominated by O ( R D − α ) for large R . Note thatthe directly generalized Lieb-Robinson bound in Refs. [38,56] gives t R ∼ log( R ) ; in this case, the term O ( F [ O ( t R )]) above would not decay in R for large R .To summarize what we have obtained so far, D ( s ) = (cid:88) ij ∞ (cid:88) R =1 G ij ( s, R ) , (cid:107)G ij ( s, R ) (cid:107) ≤ O ( R D − α ) r αij . (9)Eq. (9) reveals the locality structure hidden in D ( s ) (seeFig. 1 for an illustration); Theorem 2 can now be proved us-ing Lemma 1 by summing over all (cid:107)G ij ( s, R ) (cid:107) whose sup-port overlap with V and ¯ V simultaneously. Our summa-tion strategy is to first sum over all i and j that contributeto | dS V ( | ψ ( s ) (cid:105) ) /ds | for a given R , and sum over R next.The first step involves two scenarios: (1) For i with r i ≤ R we need to sum j over the entire lattice because G ij ( s, R ) will always cross the boundary, leading to the summation (cid:80) i,r i
In this supplemental material, we prove Lemma 2 in the main text by generalizing the following result of Ref. [45]: