Many-Body Localization in the Infinite-Interaction Limit
EEigenstate Transition between Constrained and Diagonal Many-Body Localization:Hierarchy of Many-Body-Localized Dynamical Phases of Matter
Chun Chen ∗ and Yan Chen
2, 3, † Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187 Dresden, Germany Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: December 8, 2020)There is a growing consensus that many-body-generalized Anderson insulators can arise in low-dimensional strongly disordered systems if the included interparticle interactions are weak . Then,curiously, can robust localization also persist in the infinite-interaction limit, i.e., when the interac-tion strength is infinitely larger than the randomness strength? If so, is it still many-body Andersonlocalization? To tackle these questions, we study the full many-body localization (MBL) in theRydberg-blockaded atomic quantum simulator with both infinite-strength projection and moderatequasiperiodic modulation. Employing both exact diagonalization (ED) and time-evolving block dec-imation (TEBD) methods, we identify affirmative evidence of a constrained many-body-localizedphase stabilized by a pure quasirandom field transverse to the direction of the projection. Intrigu-ingly, through the lens of quantum dynamics, we find that rotating the modulated field from paralleltowards perpendicular to the projection axis induces an eigenstate transition between the diagonaland the constrained MBL phases. Remarkably, the growth of the entanglement entropy in con-strained MBL follows a double-logarithmic form, whereas it changes to a power law in the diagonallimit. To our knowledge, this is the first fully MBL state exhibiting such a double-logarithmic entan-glement growth. Although the diagonal MBL steered by a strong modulation along the projectiondirection can be understood by extending the phenomenology of local integrals of motion, a thoroughanalysis of the constrained MBL—a genuine infinite-interaction-facilitated localized state—calls forthe new ingredients. As a preliminary first step, we unveil the significance of confined nonlocaleffects in the integrals of motion of the constrained MBL phase, which potentially challenges theestablished framework of the unconstrained MBL and suggests that, crucially, this new insulatingstate realized in the infinite-interaction limit is no longer a many-body Anderson insulator. Sincethe quasiperiodic modulation has been achievable in cold-atom laboratories, the constrained anddiagonal MBL regimes, as well as the eigenstate transition between them, should be within reach ofthe ongoing Rydberg experiments.
I. MOTIVATION
The theoretical framework of many-body localization(MBL) lays its foundation on noninteracting Andersoninsulator [1] and sets from there to address, first per-turbatively [2, 3], the fundamental quest of ergodicitybreaking and instability toward delocalization and eigen-state thermalization [4, 5] under the influence of presum-ably weak albeit ubiquitous many-body interactions inlow spatial dimensions [6, 7].This short-range weak-interaction picture is prevailingand forms the backbone of the conventional MBL. How-ever, it also raises an alternative question of whetherthere can arise the many-body non-Anderson localiza-tion in the circumstances where the interaction strengthsare not weak but infinitely strong, i.e., the many-body-localized phase without an asymptotic Anderson insulat-ing limit. This kind of intrinsic many-body localization,if exists, is distinct in that it cannot be evolved fromthe many-body Anderson insulator if not undergoing ∗ Corresponding [email protected] † Corresponding [email protected] an eigenstate transition from either a static or dynamicviewpoint. Here we restrict to full
MBL and strong(quasi)randomness to put aside the issues of disorder-free localization and nonthermalization in uniform sys-tems [8–11].Phenomenologically, isolated many-body Anderson in-sulators defined in the weak-interaction limit may bedescribed by the emergent extensive set of local inte-grals of motion (LIOMs or (cid:96) -bits) [12, 13], at least in1D [14]. Then, is it conceivable that localization per-sists but owing to restriction or frustration, the LIOM-based picture breaks down, similar to the inadequacyof Landau’s Fermi-liquid theory in correlated materi-als? Stated differently, the conventional MBL may beapproximated as an extension of the Fermi-liquid the-ory to the entire eigenspectrum. Then, what would bethe counterpart of “non-Fermi liquids” in the context ofMBL? It is known that finite interaction activates moreresonance channels for dephasing, so it is expected tosuppress localization. In this regard, a better route toachieving the unconventional
MBL might be associatedwith the presence of restriction or frustration. Giventhe interaction strength in these locally constrained set-tings can be (effectively) levitated to infinity to blockfractions of the many-body Hilbert space, this consid-eration leaves the door open to the breakdown of the a r X i v : . [ c ond - m a t . d i s - nn ] D ec established MBL framework, for instance, in disorderedRydberg-blockaded chains [15, 16], where two nearest-neighbouring Rydberg atoms cannot be simultaneouslyexcited, thus confining the system’s evolution onto a con-strained Hilbert-space manifold, which can be modelledby a projection action of the infinite strength. Specifi-cally, would there be a singular boundary separating dif-ferent phases of MBL due to abrupt distortion ratherthan a progressive dressing of the (cid:96) -bits? This type ofeigenstate transition does not rely on discrete unitarysymmetries, so it is distinguished from the transition tothe localization-protected symmetry-broken quantum or-der at nonzero energy density [7, 17]. II. THE MODEL
The aforementioned physics might be visible in dis-ordered and locally constrained chain models [16]. Thesimplest of such category takes the following form, H qp = (cid:88) i (cid:16) g i (cid:101) X i + h i (cid:101) Z i (cid:17) , (1)where (cid:101) X i , (cid:101) Z i are projected Pauli matrices, (cid:101) X i := P σ xi P and (cid:101) Z i := P σ zi P . The global operator P prohibits themotifs of ↓↓ -configuration over any adjacent sites, P := (cid:89) i (cid:18) σ zi + σ zi +1 − σ zi σ zi +1 (cid:19) , (2)hence rendering the Hilbert space of the model (1) locallyconstrained.In Ref. [16], we showed that a random version of themodel (1) by quenched disorder exhibits tentative signa-tures of a constrained MBL (cMBL) phase; nevertheless,as being in proximity to the nearby criticality, the Grif-fiths effect therein proliferates, which impedes an identi-fication and a direct investigation of this unconventionalnonergodic state of matter. In current work, we improveour prior construction by conceiving a quasiperiodic con-strained model with open and periodic boundary condi-tions (BCs), i.e., choosing [18–20] g i = g x + W x cos (cid:18) πiφ + φ x (cid:19) , (3) h i = W z cos (cid:18) πiφ + φ z (cid:19) , (4)where the inverse golden ratio 1 /φ = ( √ − / i = 1 , . . . , L , and φ x , φ z ∈ [ − π, π ) are differ-ent sample-dependent random overall phase shifts. SinceHamiltonian (1) is real, time-reversal symmetry T := K is preserved, giving rise to the Gaussian orthogonal en-semble (GOE) in the phase obeying the eigenstate ther-malization hypothesis (ETH) [21]. Additionally, when W z = 0 there is a particle-hole symmetry P := (cid:81) i σ zi W z =0 [ S v N ]/ S P W z [ S v N ]/ S P g x FIG. 1. Static diagnostics with OBCs. (a): Along W z = 0,in a finite shaded range of g x /W x ∈ (0 . , r ] (main) and[ S vN ] /S P (inset) approach r Poi and 0, demonstrating the re-alization of cMBL. This phase survives to finite W z /W x ≈ . g x , the dMBL state at dom-inant W z , and a critical phase at g x /W x ≈ W z /W x ∼ E S vN ] /S P signals a cMBL-thermal transitionaround g x /W x ≈ .
2. (b): At fixed g x /W x = 0 .
9, [ r ] ([ S vN ] /S P )stays to be r Poi ( ∼
0) under the increase of W z toward dMBL.(d): The [∆ E S vN ] /S P in this case becomes smooth. that anticommutes with H qp . To our knowledge, no dis-crete Abelian symmetry is present in the Hamiltonian(1), so the possibility of a localization-protected sponta-neous symmetry breaking [17] is excluded.It is worth stressing that the kinetic constraint hasbeen realized in the Rydberg-blockaded chain [15] andthe quasiperiodic modulation has played a vital role inexperiments [22–24] to achieve the signature of MBL inunconstrained systems. Accordingly, the actual value ofthe model (1) resides right in its high experimental rele-vance.Throughout this paper, W x = 1 sets the energy scale,i.e., the system is quasirandom at least along x direction.We will provide evidence that this quasiperiodicitymodification facilitates the realization of a stable cMBLphase in the vicinity of W z = 0. More importantly, we dis-cover an eigenstate transition between the cMBL phasenear W z = 0 and the diagonal MBL (dMBL) phase at W z (cid:29) W x through the lens of real-time quantum dy-namics, which indicates that cMBL and dMBL are dis-tinctive dynamical phases of matter that both displaylocalization-induced nonergodicity but their underlyingemergent integrability differs in nature. III. STATIC DIAGNOSTICS
The configuration-averaged level-spacing ratio [ r ] andbipartite entanglement entropy [ S vN ] are single-valuequantities routinely adopted to characterize dynamicalstates of matter. One defining feature of robust localiza-tion is the vanishing repulsion between contiguous gapsand the resulting Poisson distribution of r n := min { δ n , δ n − } max { δ n , δ n − } (5)with mean [ r ] = r Poi ≈ .
386 where δ n := E n − E n − assuming { E n } an ascending list [25]. The half-chainvon Neumann entropy is defined by S vN := − Tr [ ρ R log ρ R ] (6)where ρ R is the reduced density matrix of the right half.Figure 1(a) shows the evolution of [ r ] (main) and [ S vN ](inset) as a function of g x along the W z = 0 axis. Within0 . (cid:46) g x (cid:46)
1, [ r ] and [ S vN ] /S P converge to r Poi and 0 underthe increase of system size L , verifying the stabilizationof a cMBL phase. This new phase forms a dome in thephase diagram of model (1) and expands up to finite W z ≈ . g x = 0 . W z from cMBL to dMBL.Differing in entanglement structure, transition betweencMBL and ETH phase can be probed via [∆ E S vN ] /S P theintrasample deviation of S vN [16, 26, 27]. In accord to thechange of [ r ] and [ S vN ] in (a), Fig. 1(c) illustrates the sep-aration of cMBL and thermal phase through the indica-tion of a sharpening peak of [∆ E S vN ] /S P at the transitionpoint ( g x ≈ . , W z = 0). By contrast, the entanglement-deviation curve in Fig. 1(d) suggests that the increaseof W z at fixed g x = 0 . IV. EIGENSTATE TRANSITION FROMENTANGLEMENT GROWTH
Instead, we demonstrate here the qualitative differencebetween cMBL and dMBL from the angle of the real-timeevolution of entanglement. Notably, we find a clear eigen-state transition between these two dynamical regimes inthe numerical quantum quench experiments.We use two quantities, the bipartite entanglement en-tropy and the quantum Fisher information (QFI). Theinitial state is randomly selected from the complete basisof nonentangled product states of σ zi -spins that respectsthe local constraint. For each system size, we generatemore than 1000 random pairs of ( φ x , φ z ) for the Hamilto-nian, and for each quasiperiodic arrangement, we let thechain evolve and calculate S vN , QFI by ED (quadrupleprecision) and TEBD [28] before averaging.Figure 2(a) compiles time evolutions of [ S vN ] along thecut g x = 0 . W z in a log-log format. Thesalient feature there is the qualitative functional changein the time-evolution profiles. This eigenstate transitionis elaborated in Figs. 2(c) and (e) where we focus on theentanglement growth deeply inside cMBL and dMBL,respectively. For concreteness, after a transient period dMBLW z =8 -1 [f Q ] Y ) t -1 -4 -3 [ S v N ] t [ S v N ] t [f Q ] t FIG. 2. Transition in dynamics from cMBL to dMBL withOBCs and fixed g x = 0 .
9. The top row summarizes functionalchanges of the growth of [ S vN ] and [ f Q ] as a function of W z .Fits in the middle row suggest that for cMBL at W z = 0, theentanglement (QFI) growth follows a double (triple) logarith-mic form. The bottom row targets the dynamics of dMBL atlarge W z : consistent with the logarithmic rise of [ f Q ], [ S vN ]grows as a power law of t in dMBL. The four insets in (c)-(f)present the corresponding TEBD results of L = 28. t (cid:46) S vN ] in dMBL growssteadily as a power law of t [with an exponent ( ≈ . t ≈ ) butits saturated value is far less than the thermal entropy S T ≈ log ( F L/ ) − / (2 ln 2) − .
06 where F is the Fi-bonacci number [16]. In stark comparison, the growthof [ S vN ] in cMBL as displayed by Fig. 2(c) follows a dif-ferent functional form: within 10 (cid:46) t (cid:46) , the double-logarithmic function fits the entropy data reasonably well(see also the Appendix). Moreover, the equilibrated [ S vN ]reaches a subthermal value in cMBL and obeys a volumescaling law.Experimentally, a closely-related quantity, the QFI,which sets the lower bound of entanglement, was mea-sured in trapped-ion chain [29] to witness the entangle-ment growth under the interplay between MBL and long-range interactions. Following [29], we start from a N´eelstate in an even chain, | ψ ( t = 0) (cid:105) = |↓↑ . . . ↓↑(cid:105) , character-ized by a staggered Z spin-imbalance operator, I := 1 L L (cid:88) i =1 ( − i σ zi , (7)then the associated QFI density reduces to the connectedcorrelation function of I , f Q ( t ) = 4 L (cid:0) (cid:104) ψ ( t ) | I | ψ ( t ) (cid:105)−(cid:104) ψ ( t ) | I | ψ ( t ) (cid:105) (cid:1) , (8)which links multipartite entanglement to the fluctuationsencoded in measurable quantum correlators. Figure 2(b)is a semi-log plot of the averaged [ f Q ] along the line g x = 0 . W z color-coded the same way as inFig. 2(a). Likewise, the notable change in the functionalform of [ f Q ] echoes again the same eigenstate transitionbetween cMBL and dMBL. Specifically, Fig. 2(d) showsthat the long-time growth of [ f Q ] in cMBL matches atriple-log form, which reinforces that the double-log func-tion in (c) is the appropriate fit for the growth of [ S vN ].Parallel relation between [ S vN ] and [ f Q ] carries over tothe dMBL where the power-law growth of [ S vN ] in (e)transforms into a logarithmic growth of [ f Q ] in (f). Ta-ble I recaps the distinction between cMBL and dMBL inthe fundamental dynamical aspects of entanglement andits witness.To supplement the ED simulation in the main pan-els, we employ the TEBD and matrix-product-operatortechniques to verify the cMBL-dMBL transition in largersystem sizes. A fourth-order Suzuki-Trotter decomposi-tion is implemented, and the truncation error per stepis kept lower than 10 − . The corresponding results andthe fits are consistently presented in the insets of Fig. 2.However, due to the continual entanglement accumula-tion, matrix-product-state algorithms of this type retainefficiency only within limited time scales ( t (cid:47) ). V. EIGENSTATE TRANSITION FROMTRANSPORT
Additionally, there are marked differences betweencMBL and dMBL, as reflected through the chain’s re-laxation from the prepared N´eel state and the spreadof initialized local energy inhomogeneity. In accordancewith the time evolution of [ S vN ] and [ f Q ], the decay of I ( t ) := (cid:104) ψ ( t ) | I | ψ ( t ) (cid:105) is examined in Fig. 3(a). Apart froma quick suppression during t (cid:47)
1, both cMBL and dMBLrelax to a steady state with finite magnetization. Theythus retain remnants of the initial spin configuration incontrast to the thermal phase where [ I ( t )] vanishes irre-vocably. Notice that under the increase of W z , the frozenmoment [ I ∞ ] at infinite t develops monotonously from ∼ . ∼ . I ( t )] is also dampedmore severely in dMBL than in cMBL.Following [30], the energy transport of the constrainedmodel is investigated by monitoring the spread of a localenergy inhomogeneity initialized on the central site of an TABLE I. Hierarchies of dynamic characteristics encompass-ing constrained, unconstrained, and diagonal MBL phases.[ S vN ] [Quantum Fisher Info.]cMBL log log ( t ) log log log ( t )uMBL log ( t ) log log ( t )dMBL t α log ( t ) odd chain at infinite temperature, i.e., the system’s initialdensity matrix assumes ρ ( t = 0) = 1dim H (cid:16) + ε (cid:101) X L +12 (cid:17) , (9)where dim H the dimension of the projected Hilbert spaceand ε the disturbance of energy on site i c := ( L + 1) / ε travels isgiven by R ( t ) := 1 Tr [ (cid:101) ρ ( t ) H qp ] L (cid:88) i =1 {| i − i c | Tr [ (cid:101) ρ ( t ) H i ] } , (10)where H i := g i (cid:101) X i + h i (cid:101) Z i and the time-independentbackground has been subtracted via using (cid:101) ρ ( t = 0) := H ε (cid:101) X L +12 . As per ETH, the inhomogeneity ε is even-tually smeared uniformly over the entire chain by unitarytime evolution and in that circumstance [ R ( t = ∞ )] ≈ L .Figure 3(b) contrasts the behaviour of [ R ( t )] betweencMBL and dMBL. Concretely, for dMBL, [ R ] stays van-ishingly small, thereby ε remains confined to i c and showsno diffusion toward infinite t . In comparison, as the con-sequence of a fast expansion within t (cid:47) R ∞ ] after an oscillatory relaxation remains sub-thermal . Taken together, the failure of energy and spintransport indicates the violation of ETH and restrength-ens the observation that no thermalization is establishedacross the system in either cMBL or dMBL phase. VI. INTEGRALS OF MOTION ANDDYNAMICAL ORDER PARAMETERS
Key distinction between cMBL and dMBL can be fur-ther resolved from studying the long-time limit of thespatial distribution of the energy-inhomogeneity propa-gation. We utilize 3 quantities to access this informationcomplementarily. (i) For each quasirandom realization,we parse the definition of R ( t ) as per the site index, ε i ( t ) := Tr [ (cid:101) ρ ( t ) H i ] Tr [ (cid:101) ρ ( t ) H qp ] , (11)which measures in percentage the extra energy on posi-tion i with respect to the total conserved perturbation ε . Observing that ε i normally approaches a constant ε i, ∞ at infinite t , one might implement the trick [31],lim T →∞ T (cid:82) T O ( t ) dt ≈ (cid:80) n (cid:104) n | O | n (cid:105)| n (cid:105)(cid:104) n | , to extract its value, ε i, ∞ := ε i ( t → ∞ ) ≈ (cid:80) n (cid:104) n | (cid:101) X L +12 | n (cid:105)(cid:104) n | H i | n (cid:105) (cid:80) n E n (cid:104) n | (cid:101) X L +12 | n (cid:105) , (12)where {| n (cid:105)} comprises an eigenbasis satisfying H qp | n (cid:105) = E n | n (cid:105) . Evidently, the profile of { ε i, ∞ } bears important cMBLg x =0.9W z =0 W z =0 [ e i , ¥ ] |i-i c | (c) peakdip hump W z =2 W z =3 [ e i , ¥ ] |i-i c | (e) [ e r e s ] -8 -6 -4 -2 | [ e i , ¥ ] | |i-i c | FIG. 3. cMBL-dMBL transition in transport with PBCs andfixed g x = 0 . , L = 19. (a),(b): Time evolution of the Z anti-ferromagnetic imbalance [ I ( t )] and the energy spread [ R ( t )] asa function of W z . (c) exemplifies the peak-dip-hump lineshapeof [ ε i, ∞ ] in cMBL. The time-profiles of [ ε i c ± , ( t )] that char-acterize the nonmonotonicity of the dip-hump structure aregiven by (d). (e) shows the lineshape of [ ε i, ∞ ] in dMBL; theexponential decay can be seen from the semi-log inset whereinthe cMBL data (black dots) are overlaid for comparison. (f):The changes in dynamic “order parameters” [ R ∞ ] and [ ε res ]as tuning W z signal the transition between cMBL and dMBL.Light to solid colours in (d),(f) correspond to L = 15 , , information pertaining to the local structure of integralsof motion (IOMs). (ii) The summation of ε i, ∞ weightedby the separation returns the equilibrated value of theeffective traveling distance, R ∞ = L (cid:88) i =1 ( | i − i c | · ε i, ∞ ) . (13)(iii) In view of the fact that the contribution from i c ismissing from R ∞ , one can define ε i c , ∞ as the residualenergy density at the release place, ε res := ε L +12 , ∞ . (14)All the three quantities defined above can be used todistinguish ETH and MBL. Here we point out that theyalso serve as a set of dynamical “order parameters” tohelp differentiate between the cMBL and dMBL regimesand identify the transition point therein. VII. LIOMS AND POSITIVE DEFINITENESSOF DMBL
Before proceeding to the numerics, let’s gain some un-derstanding on the dMBL limit within the conventional LIOM framework. The first crucial step forward is tointroduce ˘ Z i := P i +1 (cid:101) Z i P i − (15)where P i := ( + Z i ) as the new building blocks of theconstrained (cid:96) -bits. The convenience of ˘ Z i stems fromthe relation Tr ˘ Z i = 0, which should be contrasted to Tr (cid:101) Z i >
0, thereby ˘ Z i behaves like a normal spin free ofrestrictions. Following [16], it can then be proved thatas long as W z (cid:29) g x + W x , the set of tensor-product oper-ators I L := {Z i ⊗· · ·⊗Z i k } fulfilling 1 (cid:54) i (cid:54) i (cid:54) · · · i k (cid:54) L, i a +1 (cid:54) = i a , (cid:54) k (cid:54) L +12 may be constructed as a com-plete, mutually commuting, and linearly-independent ba-sis to express any nontrivial operators that commute with H qp . In terms of quasilocal unitaries, Z i a ≈ U ˘ Z i a U † .This is because the set of states {|Z i Z i · · ·Z i k (cid:105)} derivedfrom I L reproduces faithfully the effective eigenbasis ofthe projected Hilbert space for dMBL. Accordingly, theIOM in Eq. (12) may be recast intodMBL: (cid:88) n (cid:104) n | (cid:101) X i | n (cid:105)| n (cid:105)(cid:104) n | ≈ L − (cid:88) m =0 (cid:88) r V [ i ] r,m (cid:98) O [ i ] r,m , (16)where (cid:98) O [ i ] r,m denotes the element of I L that possesses thesupport on site i (i.e., contains Z i ) and whose furthestboundary from i is of distance m . The nonidentical indi-viduals comprising this specified subset are then labelledby r . Besides the finite support of Z i , the other keyproperty that promotes (cid:80) n (cid:104) n | (cid:101) X i | n (cid:105)| n (cid:105)(cid:104) n | to the LIOMof dMBL is the locality condition of the real coefficients,i.e., V [ i ] r,m ∼ e − m/ξ . In addition, the universal Hamilto-nian governing the dynamics of dMBL may assume thefollowing form in the LIOM representation, H dMBLqp = (cid:88) i (cid:101) h i Z i + (cid:88) k (cid:88) i ...i k J i ...i k Z i Z i · · · Z i k , (17)where from Figs. 2(e),(f) it is feasible to infer that J i ...i k ∼ | i k − i | − /α · φ −| i k − i | (18)decays as an exponentially-suppressed power law of theLIOMs’ separation.Being the trace of the product of two IOMs, one imme-diate consequence of Eq. (16) is the positive definiteness of [ ε i, ∞ ] featured by a monotonically exponential decayin space. From Fig. 3(e) we find that this is indeed thecase even when W z ≈ g x + W x . VIII. PEAK, DIP, HUMP IN CMBL
Now we are in the position to highlight the occurrenceof negativity and the resulting peak-dip-hump structure in[ ε i, ∞ ] [see Figs. 3(c),(d)] as the peculiar characteristics ofcMBL that distinguish it from both dMBL and uncon-strained MBL (uMBL) by the presence of pronouncednonlocal correlations. The unambiguous negativity of[ ε i c ± ] in Fig. 3(c) and the nonmonotonicity of [ ε i c ± , ]in Fig. 3(d) cleanly point to the insufficiency of Eq. (16)when addressing the cMBL from the dMBL side. Toremedy the inconsistency, we propose as a scenario thatthe missing pieces could come from the terms in I L thatare nonlocal with respect to i , i.e., their support on i vanishes: For cMBL, (cid:88) n (cid:104) n | (cid:101) X i | n (cid:105)| n (cid:105)(cid:104) n | ≈ L − (cid:88) m =0 (cid:88) r,r (cid:16) V [ i ] r,m (cid:98) O [ i ] r,m + V [ i ] r,m (cid:98) O [ i ] r,m (cid:17) . (19)The superscript [ i ] signifies the absence of Z i in the asso-ciated expansion. Under the successive decrease of W z ,it can be anticipated that the weights V [ i ] r,m of small m grow significantly such that a finite-size core centred at i is forming wherein the nonlocal contributions, albeit con-fined, become predominant. On the contrary, for those m beyond the core, the importance of V [ i ] r,m has to dimin-ish abruptly so that the rapid decay tail and the overallsignatures of localization can be well maintained.Alternatively, the core formation may be monitored by[ R ∞ ] and [ ε res ]. Figure 3(f) illustrates that the duo con-stitutes the desired “order parameters” from quantumdynamics that take values zero and unity in dMBL andsaturate to the nontrivial plateaus in cMBL. The criti-cal W z of the transition is hence estimated to be 0 . g x = 0 .
9. Furthermore, from Fig. 3(c) the core where sub-stantial nonlocal effects take place spans roughly 5 to 7lattice sites which, as per the saturated value of [ R ∞ ] inFig. 3(f), is comparable to a thermal segment of approx-imately 3 lattice-spacing long. IX. SUMMARY AND OUTLOOK
To conclude, we discover a cMBL regime in the quasir-andom Rydberg-blockaded chain. The orthogonality be-tween the field strength and the projection direction ren-ders this new MBL phase fundamentally different fromdMBL and uMBL. In particular, the entanglement en-tropy in cMBL grows as an unusual double-logarithmicfunction of time, as opposed to the power-law growth indMBL and the single-logarithmic growth in uMBL.Even though LIOMs capture the phenomenology ofdMBL, the cMBL-dMBL transition triggered by the ro-tation of the field orientation accentuates the impor-tance of the nonlocal components in the IOMs of cMBL,which, together with the double-logarithmic entangle-ment growth, raises doubts about how to define themeaningful LIOMs (if exist) and the universal (fixed-point) Hamiltonian that underpin the cMBL. The contin-ual theoretical and experimental investigations on theseopen questions may further our understanding of the un-conventional MBL beyond the current scope and lead toa paradigm shift in MBL from the weak-interaction do-main to the infinite-interaction realm. [S vN ] loglog(t) log(t) [log(t)] g log(t) log(t) FIG. 4. A replot of the same entanglement entropydata [ S vN ] of the cMBL regime (stabilized by g x /W x =0 . , W z /W x = 0 on an open chain of L = 18) as is given byFig. 2(c) of the main text but now being fitted by three dif-ferent types of functions: (i) the double-logarithmic functionlog log( t ) and (ii) the single-logarithmic function log( t ) as wellas (iii) the single logarithm up to some power [log( t )] γ where γ ≈ . t gives the best fit among the three functions. ACKNOWLEDGMENTS
The discussion with M. Heyl was acknowledged. Thiswork is supported by the SKP of China (Grant Nos.2016YFA0300504 and 2017YFA0304204) and the NSFCGrant No. 11625416.
Appendix A: Additional Curve Fitting
In this appendix, we analyze and compare in somedetail three different types of fitting functions for thecMBL data points of entanglement entropy [ S vN ] withinthe range of evolution time W x t/ (cid:126) ∈ (10 , ) [see alsoFig. 2(c) in the main text].As shown by Fig. 4, it is manifest that for the cMBLphase, the double-logarithmic fitting function log log( t ) matches the [ S vN ] data curve significantly better than ei-ther the single-logarithmic fitting function log( t ), whichis widely recognized as one of the defining characteris-tics of the unconstrained MBL systems, or alternativelythe single-logarithmic function up to some power [log( t )] γ (this form of growth was argued to occur right at thecritical point between the unconstrained MBL phase andthe delocalized thermal phase by a dynamical real-spacerenormalization group approach). [1] P. W. Anderson, Absence of diffusion in certain randomlattices, Phys. Rev. , 1492 (1958).[2] D. Basko, I. Aleiner, and B. 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