Entanglement scaling in fermion chains with a localization-delocalization transition and inhomogeneous modulations
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r Entanglement scaling in fermion chains with a localization-delocalization transitionand inhomogeneous modulations
Gerg˝o Ro´osz,
1, 2, ∗ Zolt´an Zimbor´as,
3, 4, 5, † and R´obert Juh´asz ‡ Institute of Theoretical Physics, Technische Universit¨at Dresden, 01062 Dresden, Germany Wigner Research Centre for Physics, Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary Wigner Research Centre for Physics, Theoretical Physics Department, H-1525 Budapest, P.O. Box 49, Hungary MTA-BME Lend¨ulet Quantum Information Theory Research Group Mathematical Institute, Budapest University of Technology and Economics, Hungary (Dated: April 7, 2020)We study the scaling of logarithmic negativity between adjacent subsystems in critical fermionchains with various inhomogeneous modulations through numerically calculating its recently estab-lished lower and upper bounds. For random couplings, as well as for a relevant aperiodic modulationof the couplings, which induces an aperiodic singlet state, the bounds are found to increase logarith-mically with the subsystem size, and both prefactors agree with the predicted values characterizingthe corresponding asymptotic singlet state. For the marginal Fibonacci modulation, the prefactorsin front of the logarithm are different for the lower and the upper bound, and vary smoothly withthe strength of the modulation. In the delocalized phase of the quasi-periodic Harper model, thescaling of the bounds of the logarithmic negativity as well as that of the entanglement entropy arecompatible with the logarithmic scaling of the homogeneous chain. At the localization transition,the scaling of the above entanglement characteristics holds to be logarithmic, but the prefactors aresignificantly reduced compared to those of the translationally invariant case, roughly by the samefactor.
I. INTRODUCTION
In the last decade, there has been an increasing inter-est in the the entanglement properties of quantum sys-tems [1–3]. The studies on this subject allowed for adeeper understanding of many-body models, in partic-ular with respect to criticality [4–6], simulability [7–9],and thermalization [10–12]. In many of the early works,the case when the entire quantum system is in a purestate was considered, and one investigated the entangle-ment between a subsystem and the rest of the system. Inthis case, the so-called entanglement entropy is a propermeasure of the quantum entanglement [13]. Consider asystem S in a pure state | Ψ i , and divide the system intotwo complementary parts A and B (i.e., in a way that A ∪ B = S ); calculating the reduced density matrices ρ A = Tr B | Ψ ih Ψ | , ρ B = Tr A | Ψ ih Ψ | , the entanglemententropy is defined as the von Neumann entropy of eitherof them: S = − Tr A ρ A ln ρ A = − Tr B ρ B ln ρ B . (1)However, if one wishes to characterize the entangle-ment between two subsystems whose union does notcover the whole system S , ( A ∪ B = S ), the A ∪ B sub-system is no longer in a pure state, and one has to useanother measure to quantify the entanglement between A and B . One possible candidate is the logarithmic en-tanglement negativity, which is proven to be an entan- ∗ [email protected] † [email protected] ‡ [email protected] glement monotone [14, 15] and is defined as E N = ln || ρ T A A ∪ B || , (2)where ρ A ∪ B is the reduced density matrix of the subsys-tem A ∪ B , T A denotes the partial transpose on subsystem A , and || · || is the trace norm. The partial transpose isdefined as h ϕ A ϕ B | ρ T A A ∪ B | ϕ ′ A ϕ ′ B i = h ϕ ′ A ϕ B | ρ A ∪ B | ϕ A ϕ ′ B i ,with h ϕ A | and h ϕ B | being bases for A and B , respec-tively. It corresponds to the local time reversal, and ifa state is separable (non-entangled) partial transposeddensity matrix is still a valid density matrix, while if itis entangled, the partial transpose of the density matrixmay have negative eigenvalues [16]. For one-dimensional,homogeneous critical models, there exist conformal fieldtheory (CFT) results for the entanglement entropy andthe entanglement negativity [17]. The entanglement en-tropy depends on the length ℓ of the subsystem as S ( ℓ ) = c ℓ + const , (3)where c is the so-called central charge, while the entangle-ment negativity of adjacent subsystems of length ℓ scalesas E ( ℓ ) = c ℓ + const . (4)In the recent years, a series of numerical [18–24] andanalytical [17, 25–36] works have been devoted to theentanglement negativity. For non-interacting bosonicsystems, the entanglement negativity is calculable effi-ciently (in the number of nodes) both for the groundstate and for thermal states [15, 37–39]. The reason is,that the partial transpose maps the bosonic Gaussianstates to bosonic Gaussian operators [16, 40]. However,for non-interacting fermionic systems, no similar calcu-lation method is known. The ground and thermal statesare fermionic Gaussian states, but the partial transposeis not a fermionic Gaussian operator [41]. The lack of asimple formula for the entanglement negativity triggereda series of works on upper and lower bounds [36, 42–45].In this work, we will use the upper and lower boundsgiven in [44] to investigate numerically the scaling ofthe entanglement negativity in various fermionic chainswith random or aperiodic inhomogeneities. In the caseof sublattice-symmetry, we will present a simplificationin the calculation of the lower bound. The scaling of en-tanglement entropy will also be studied numerically bythe correlation matrix method [46].The rest of the paper is organized as follows. In Sec. II,the models are defined. In III, we recapitulate the stepsof calculating the negativity upper and lower bounds in-troduced in Ref. [45], and a simplification of the form ofthe lower bound due to sublattice symmetry is presentedin Sec. III C. In Sec. IV, we present our numerical re-sults. These are then discussed in Sec. V. . II. MODELS
In this work, we will consider different variants of thefermionic hopping model having the Hamiltonian of gen-eral form H = − L X l =1 t l ( c † l +1 c l + c † l c l +1 ) + h l c † l c l , (5)where t l and h l are site-dependent hopping amplitudesand on-site energies, respectively, while c † l and c l arefermionic creation and annihilation operators obeyingthe anticommutation rules { c l , c j } = { c † l , c † j } = 0 and { c l , c † j } = δ l,j , for l, j = 1 , , . . . , L . Periodic boundarycondition is considered, so that site L + 1 is identifiedwith site 1. The chemical potential is zero, so the stateswith E n < / A. Off-diagonal inhomogeneity
For this class of models, the on-site energies are all zero h j = 0, j = 1 , . . . , L , while the hopping amplitudes areposition-dependent, either random or follow an aperiodicmodulation. In the first case, which we refer to as randommodel, we assume that the amplitudes t j are indepen-dent, identically distributed quenched random variables,drawn from a uniform distribution in the interval [0 , a and b ) by the substitution rule: a → abb → a . The first few realizations of the Fibonacci sequence are a , ab , aba , abaab .Another two-letter sequence, which we refer to as rel-evant aperiodic modulation (RAM), is obtained by thefollowing inflation rule [47]: a → ababab → a . Generating a sufficiently long aperiodic sequence bythe repeated application of the inflation rule, a modu-lation pattern can be associated with it, in which letter a ( b ) corresponds to amplitude t a ( t b ). The strength ofthe modulation can be characterized by the ratio of twotypes of amplitudes, r = t a /t b .The non-zero energy eigenstates of the randomly dis-ordered model are exponentially localized, and the local-ization length diverges for zero energy as it was shownwith rigorous tools in connection with the off-diagonalAnderson model in [48, 49]. According to the strong-disorder renormalization group (SDRG) method [50, 51],the ground state of the random model is a random-singlet state [52], which is a product of one-particle states √ ( | i − | i ) on pairs of sites which can be arbitrarilyfar away from each other. The method is approxima-tive but is asymptotically exact, giving the low-energy(large-scale) properties of the system correctly. Analo-gous to this, the ground state of aperiodic models is anaperiodic-singlet state [53], for any r < r → A . The averageentanglement entropy of a subsystem of size ℓ , which ispart of an infinite system increases asymptotically (apartfrom log-periodic oscillations for aperiodic models) as S ( ℓ ) = c eff ℓ + const , (6)where the effective central charge depends on the typeof modulation. For the random model[54], c raneff = ln 2, for the RAM, c RAMeff = λ ( λ − λ − , where [47] λ = (3 + √ r , and its limiting value islim r → c FMeff ( r ) = τ +1) log τ , where τ = √ is thegolden mean [47, 55].The logarithmic negativity in a singlet state is givenby the number of singlets connecting A and B , in unitsof ln 2. As it was shown in Ref. [20] for the random sin-glet state, the average logarithmic negativity of adjacentintervals of size ℓ scales as E ( ℓ ) = κ ln ℓ + const , (7)with the prefactor κ = ln 26 . Recently, a more detailedstudy has appeared about the negativity spectrum of thismodel [56]. The prefactor κ is the half of the prefactorof entanglement entropy, which can be intuitively under-stood since subsystem A borders to B only on one side,while, in the case of entanglement entropy, A borders tothe rest of the system on both sides. According to this,the result in Eq. (7) is expected to hold also for aperiodicsinglet states with a prefactor κ = c eff .The recapitulated SDRG method and its modificationshas a wide range of applications from the description ofentanglement of star-graph like systems [57] and disor-dered surfaces [58, 59] including the dynamics of dis-ordered systems [60–64] and description of their highlyexcited states [65–70] or even a construction of quasi-periodic tensor network model enable us to investigateAdS/CFT correspondence [71]. B. The Harper model
Another model, which we will consider is the Harpermodel. Here, the couplings in the general Hamiltonianin Eq. (5) are constant, t i = 1, while the local potential h j is a quasiperiodic function of j , namely, h j = h cos(2 πj/τ ) , (8)where τ = √ is the golden mean. The irrational-ity of τ makes the model quasiperiodic. This model,when formulated in terms of spin variables is also knownas Aubry-Andr´e model [72]. It is well-known that thismodel shows a delocalization transition, which is exactlyat h = 1 due to its self-dual property [73]. In the region h <
1, all one-particle eigenstates are extended, whereas,for h <
1, they are all localised on a length [72] l loc = 1ln h . (9)At the critical point, h = 1, the one-particle states showan interesting multi-fractal behaviour [74, 75]: they areessentially localised on ∼ L D ( n ) sites, where the dimen-sion D ( n ) of the effective support of the state varies fromstate to state. Its maximal value is found numerically[74]to be D ≈ .
82 .To our knowledge the logarithmic negativity in theground state of the Harper model was not studied sofar in the literature, the entanglement entropy has beenstudied in [76], there the authors focused on the effectof the chemical potential. Here we investigate the sizedependence of the entanglement entropy.
III. NEGATIVITY OF FREE FERMIONS:UPPER AND LOWER BOUNDS
In this section, we briefly summarize the calculationof upper and lower bounds introduced in Ref. [44] and used in the present paper. Instead of the general defi-nitions, here, it is sufficient to restrict ourselves to theformulation valid in the special case of particle num-ber conserving states. The latter means that the den-sity matrix commutes with the particle number operator( N = P Ll =1 c † l c l ), [ ρ, N ] = 0. The ground states of thequadratic fermion Hamiltonian in Eq. (5) are such states. A. Upper bound
First, we consider the upper bound E u . It is formulatedin terms of the covariance matrix γ i − , j = − γ j, i − = 2 C ij − δ ij , (10)where C ij = h c † i c j i is the correlation matrix, with allthe other entries of γ being zero. For every Hamiltonianin the form (5), the covariance matrix can be obtainedfollowing a standard canonical transformation [77]. Fortranslational invariant systems, one can easily obtain aclosed form of the covariance matrix while, for inhomo-geneous systems, it is computable in polynomial time inthe number of fermionic modes L . The covariance matrixcharacterizes all correlations in the system, so it also de-termines the entanglement negativity of any subsystems.However, no simple formula is known for the entangle-ment negativity in terms of the covariance matrix, mak-ing the bounds defined in [44] really valuable. The upperbound we use is given by E u = 12 (cid:2) S / ( ρ × ) − S ( ρ A ∪ B ) (cid:3) , (11)where S α ( ρ ) denotes the R´enyi entropy of state ρ : S α ( ρ ) = 11 − α ln Tr ρ α . (12)The state ρ × is defined by its covariance matrix as γ × = − ( − γ − )( + γ + γ − ) − ( − γ + ) , (13)where γ ± = T ± B γ T ± B , and T ± B = L j ∈ A L j ∈ B ( ± i ) . B. Lower bound
For constructing the lower bound, the matrix Γ = 2 C − " Γ AA Γ AB Γ TAB Γ BB , (14)Using the singular value decomposition of Γ AB , Γ AB = U DV T , where D is a diagonal matrix with non-negativeelements, whereas U and V are orthogonal matrices, onecan transform Γ by U ⊕ V to the following formΓ ′ = " U T Γ AA U DD V T Γ BB V . (15)Denoting the diagonal elements of D , U T Γ AA U , and V T Γ BB V by c i , a i , and b i , respectively, the lower boundis given by E l ( ρ A ∪ B ) = n X j =1 ln h ( a j , b j , c j ) , (16)where h ( a, b, c )= 12 + 12 max { , p ( a + b ) +(2 c ) − ( ab − c ) , | a − b | +( ab − c ) } . (17) C. Simplification of the lower bound by sublatticesymmetry
Here, we show that the expression in Eq. (16) can befurther simplified making use of the sublattice symmetry,which holds in the absence of an on-site potential for aneven L . We also assume, that the lattice is half filled.Due to this, the elements of matrix Γ = 2 C − A and B , the odd (even) indices to the first (second) l/ AB = 2 C AB − AB = " PQ . (18)We are looking for the singular value decompositionΓ AB = U DV T , where the diagonal matrix D contains thenon-negative singular values of Γ AB , while the columnsof U and V are eigenvectors of Γ AB Γ TAB and Γ
TAB Γ AB ,respectively. These matrices are then block diagonal,Γ AB Γ TAB = P P T ⊕ QQ T Γ TAB Γ AB = Q T Q ⊕ P T P. (19)and, as a consequence, U and V are block diagonal, aswell: U = U o ⊕ U e , V = V o ⊕ V e . (20)Transforming Γ by U ⊕ V will bring the diagonal blocksΓ AA = " RR T , Γ BB = " SS T , (21)to the formΓ ′ AA = U T Γ AA U = " U To RU e U Te R T U o , (22)Γ ′ BB = V T Γ BB V = " V To SV e V Te S T V o . (23)The elements a i and b i in Eq. (16) are therefore zero.Using this, the lower bound can be written as E l = X ′ i (cid:18) c i + 1 √ (cid:19) , (24)where the prime means that the summation goes over thesingular values fulfilling c i > √ − IV. NUMERICAL RESULTS
We calculated the lower and upper bounds of the en-tanglement negativity, E l and E u , respectively, whichwere introduced in Ref. [44] and are defined in the pre-vious section, for the models described in Sec. II. It isworth mentioning that the above bounds, apart from anadditive constant for the upper bound, are sharp for thespecial case of singlet states.The numerical calculations were performed accordingto two kinds of schemes. Either the total system size L was kept fixed (and large) and the size ℓ of adjacentsubsystems was varied, or L was varied keeping the ra-tio ℓ/L constant. For a given L and ℓ , E l and E u werecalculated for all possible positions of the subsystems ( L in number) and an averaging was performed in the caseof non-random models. For the random model, 32 differ-ent positions of the subsystems was considered in eachrandom sample, while the number of samples was 10 forsmaller systems, gradually decreasing down to 5000 forthe largest system with L = 2048.In order to make corrections to an expected large- L dependence E l,u ( L ) = κ l,u ln L + const (25)more visible in the second scheme, we also calculatedeffective, size-dependent prefactors from consecutive datapoints at L and L ′ > L as κ l,u ( L ) = E l,u ( L ′ ) − E l,u ( L )ln( L ′ /L ) . (26)For the Harper model, we also considered the averageentanglement entropy of a subsystem of size ℓ , the scalingof which is so far unexplored, and which can be numer-ically calculated from the eigenvalues of the correlationmatrix of fermions restricted to the subsystem [4, 46]. A. Homogeneous system
First, we investigated the behavior of the lower and up-per bounds in the homogeneous chain. Here, we assumethat the system is infinite ( L → ∞ ), which allows us touse the exact expressions of the correlations C i,j = h c † i c j i [78], while keep the subsystem size ℓ finite. The numeri-cal results are shown in Fig. 1. The upper bound followsthe scaling E u ( ℓ ) ∼ κ u ln ℓ + const (27)with κ u = 0 . ± . Ε l ( l ) , Ε u ( l ) ln l Ε u (l) Ε l (l) FIG. 1. Upper and lower bounds for the entanglement neg-ativity in the homogeneous fermion chain. The subsystemsizes are up to ℓ = 1024 in the case of the upper bound, and ℓ = 500000 in the case of the lower bound. The lower boundshows log-periodic oscillations; the boarders of the periods areindicated by arrows. are the consequences of the definition of the lower limit[see Eq. (24)], in which only the singular values greaterthan √ − n th singular value ( n = 1 , , . . . ) slowly increases with in-creasing ℓ . When a singular value crosses the threshold √ −
1, so that the number of terms in the sum increasesby one, a new period of the oscillations starts.To investigate the overall logarithmic trend, one hasto fit to identical parts of the periods, like the break-ing points indicated by arrows in Fig. 1. To do this, atleast a few period long data set is needed. The periodof these oscillations is about 3 . . . . . Fortunately, in this size range,only the largest 1 . . . QQ T and P T P . Since we have a closed formula forthe matrix elements of Q and P , we can use the powermethod and multiplicate with QQ T ( P T P ) without stor-ing the matrix elements in the memory. The ratios of theconsecutive eigenvalues (in decreasing order) of the ma-trix QQ T ( P T P ) are between 0 .
25 and 0 .
1, so the powermethod converges rapidly.Using the data for the lower bounds at the four break-ing points shown in Fig. 1, we calculated effective pref-actors from neighboring data points, which are in or-der, 0 . . . / ln ℓ , an extrapolation to ℓ → ∞ results in κ l = 0 . . κ l = 0 . / Ε u ( L ) , Ε l ( L ) ln LUB l=L/8UB l=L/4LB l=L/8LB l=L/40.10 0.12 0.14 0.16 0.180.20 3 4 5 6 7 κ u ( L ) , κ l ( L ) ln L ln(2)/6UB l=L/8UB l=L/4LB l=L/8LB l=L/4 FIG. 2. Left. Lower (LB) and upper (UB) bound of theentanglement negativity in the random model, as a functionof the system size L , for fixed ratios ℓ/L = 1 / ℓ/L =1 /
4. Right. The corresponding effective prefactors defined inEq. (26). The horizontal line indicates the asymptotic valueln(2) / . . . . predicted by the SDRG method. B. Off-diagonal inhomogeneity
The dependence of the lower and upper bounds on L for fixed ratios ℓ/L , as well as the corresponding effec-tive prefactors for the random model are shown in Fig.2. As can be seen, both the lower and upper boundsscale logarithmically with L for large L . The effectiveprefactor of the upper bound displays a slow crossoverfrom the clean system’s value 1 / / L , and, atthe system sizes available by the numerical method, itis still considerably far from it. We note that, underthe same circumstances, the prefactor of the entangle-ment entropy has similar deviations from the asymptoticvalue (not shown). In the case of the lower bound, theprefactor in the homogeneous system (0 .
13) is closer tothe asymptotic limit ln(2) /
6, and, accordingly, it showsa more rapid crossover.In the case of the relevant aperiodic modulation, thelower and upper bounds, as well as the effective prefactorsas a function of L are exemplified in Fig. 3 for r = 0 . L = 17 , , , , Ε u ( L ) , Ε l ( L ) ln L Ε u (L) Ε l (L) 0.9 1 1.1 0 0.5 κ u , l (r) / κ R A M r FIG. 3. Left. Lower and upper bound of the entanglementnegativity for RAM, as a function of the system size L . In-set. The ratios of the prefactors κ u,l ( r ) obtained for differentvalues of r and κ RAM = c RAMeff / . . . . predicted bythe SDRG method. are the lengths of words obtained by the repeated appli-cation of the inflation rule starting with letter a , whilethe subsystem size was ℓ = [ L/ · ] stands forthe integer part. By this choice of L , log-periodic os-cillations in the data can be avoided [47]. As can beseen, the bounds follow a logarithmic scaling and theirasymptotic prefactors shown in the inset for different dis-order strengths r are in agreement with the expectation κ l,u = κ = c eff / r . Here, thesystem sizes and the subsystem sizes were chosen to be L = F n and ℓ = F n − , respectively, where F n denotes the n th term of the Fibonacci sequence. As can be seen inFig. 4, the bounds plotted against ln L show log-periodicoscillations with a period of three data points. This is inaccordance with that the self-similarity of the aperiodicsinglet state is achieved by applying the third power ofthe inflation transformation [47]. Therefore, in estimat-ing the prefactors, we used every third data points. Forboth bounds, we find a logarithmic dependence on L asgiven in Eq. (25). The prefactor of the upper boundshows a variation with r qualitatively similar to that ofthe prefactor of the entanglement entropy [47]: For mod-erate strengths of the modulation, it has a slight devia-tion from the clean system’s value, then it tends rapidlyto the singlet-state limit. In the case of the lower bound,the prefactor in the clean system (0 . . . . . ) hardly differ, therefore thevariation with r is very weak. κ l , κ u r 0.5 1 1.5 4 6 E u ( L ) , Ε l ( L ) ln Lr=0.05r=0.45 FIG. 4. Prefactors of the lower and upper bounds as afunction of the modulation strength r for the Fibonacci mod-ulation. The horizontal line indicates the singlet-state limit,lim r → c FMeff ( r ) / . · · · . In the inset, the dependenceof the upper (open symbols) and lower (full symbols) boundon L for two different values of the modulation strength areshown. The straight line has a slope characteristic for thesinglet-state limit. C. The Harper model
In the Harper model, we investigated first the scalingof the entanglement entropy in the ground state. In thesecalculations, system sizes were chosen to be twice of Fi-bonacci numbers L = 2 F n , and the size of the subsystemwas F n − .The results are shown in Fig. 5. We find that, in theextended phase, the entanglement entropy scales loga-rithmically as S = 0 .
33 ln L + const, and the measuredprefactor (0 .
33) is compatible with that of the homoge-neous system 1 / S ( L ) = c eff L + const , (28)however, the effective central charge c eff = 0 .
78 differssignificantly from the central charge of the homogeneoussystem.In the localized phase, the entanglement entropy satu-rates to finite values in the limit L → ∞ . As it is demon-strated in Fig. 5, in large systems L ≫ l loc , in which thelength scale is set by the localization length l loc ratherthan the system size, the entanglement entropy followsthe law S ( l loc ) = c eff l loc + const. (29) S ln L (a)L=144233377610987159741816765 2 4 0.75 1 1.25 S h (b)L=2*144L=2*233L=2*377L=2*610 0.5 11 4 S ln l loc S~0.26ln l loc
FIG. 5. Entanglement entropy of the Harper model. a) En-tanglement entropy in the extended phase h = 0 . h = 1 . L . The straight lines have slopes 0 .
33 and 0 . h for various sys-tem sizes. The inset shows the entanglement entropy in thelocalised phase as a function of the logarithm of the localiza-tion length l loc = 1 / ln h . The green line has a slope 0 . with the same prefactor as found in Eq. (28). Approach-ing the critical point, this law is, however, deteriorateswhen the diverging localisation length becomes compa-rable with the system size.Numerical results on the entanglement negativity areshown in Fig. 6. In the delocalized phase and in thecritical point, the upper bound increases logarithmicallywith the system size. In the delocalized phase, just likefor the entranglement entropy, the prefactor is found toagree with the that of the conformally invariant homoge-neous system. In the critical point, the prefactor of theupper bound is reduced to κ u = 0 . h → ∞ , and the ratio is 1 .
05 alreadyfor h = 1 .
5. This makes possible to investigate the be- Ε u ( L ) , Ε l ( L ) ln L (a)h=0.5h=1.0 0.2 0.6 1 0 2 4 Ε u , Ε l ln l loc (b) L=2*377L=2*6100.21 ln(l loc )+const.
FIG. 6. Upper and lower bounds of the entanglement nega-tivity of the Harper model. a) Dependence on the system sizein the delocalized phase ( h = 0 .
5) and in the critical point( h = 1). The slopes of fitted lines from top to bottom are0 .
25, 0 . .
16, and 0 . .
21 is drawn to guide theeye. havior of the negativity in the localized phase. Althoughthe difference of the upper and lower bounds becomeslarger as one approaches the transition point, there isan intermediate regime, where the transition point is nottoo far, but the difference of the bounds is still moderate.Similarly to the behavior of the entanglement entropy inthe localized phase, the upper bound of entanglementnegativity increases according to E u ( l loc ) = κ u ln l loc + const. (30)in the vicinity of the transition point, where κ u = 0 . V. CONCLUSION AND OUTLOOK
In this paper, we investigated numerically the scalingof the entanglement negativity in fermionic chains withdifferent aperiodic and random modulations. In general,we found a logarithmic increase of both bounds with thesubsystem size, provided the model is critical. Some ofthe off-diagonal modulations, like the random, the rele-vant aperiodic, and the extremal Fibonacci modulationasymptotically induce a state composed of singlets. Inthese cases, we have confirmed that the prefactors of thelower and upper bounds coincide and agree with the halfof the prefactor of the entanglement entropy, κ = c eff .In the case of the Fibonacci modulation of finitestrength r , the ground state is no longer a singlet state,and the prefactor of the entanglement entropy is knownto vary continuously and monotonically with r . We havefound a similar behavior of the prefactors of both bounds,although for the lower bound the difference between thetwo extremal values is very small. The prefactor of theupper bound, which correctly gives the prefactor of theentanglement negativity in the extremal cases r → r = 1, is presumably holds to be correct in the interme-diate regime 0 < r <
1, as well. However, κ ( r ) is notexpected to be simply related to c eff ( r ), as their ratio isdifferent in the two extremal cases (1 / / f ≈ .
78 for the entanglement entropy and f ≈ .
85 for the negativityupper bound). This may suggest that the scaling canbe described by the prefactors of the homogeneous sys-tem, but using an effective length L ′ ∼ L f . Indeed, theone-particle states of the model in the critical point areknown to be fractals, with an effective number of sitesparticipating in them scaling as N ∼ L D . The fractaldimension D varies from state to state and its maximalvalue[74] D ≈ .
82 is close to f . The clarification ofa possible relationship between the reduction factor andthe fractal dimensions of eigenstates is an open questionwhich is deferred to future research.As a possible future direction, the present investiga-tions could be extended to various generalized Harpermodels. For instance, one could add paring terms ( c l c l +1 )to the Hamiltonian and investigate the so called quasi-periodic Ising model [80] with the free-fermion methodsapplied here. We also mention here recent generalizationsof the Harper model with additional long-range couplings[81, 82]. ACKNOWLEDGMENTS
This work was supported by the by the DeutscheForschungsgemeinschaft through the Cluster of Excel-lence on Complexity and Topology in Quantum Matterct.qmat (EXC 2147) and the National Research, Devel-opment and Innovation Office NKFIH under grants No.K128989, K124152, K124176, KH129601 and throughthe Hungarian Quantum Technology National ExcellenceProgram (Project No. 2017-1.2.1-NKP-2017-00001). ZZwas also partially funded by the Janos Bolyai and theBolyai+ Scholarships. [1] J. Eisert and M. Cramer, M. B. Plenio, Rev. Mod. Phys. , 277 (2010).[2] P. Calabrese, J. Cardy, and B. Doyon (Eds.), Entangle-ment entropy in extended quantum systems (special is-sue), J. Phys. A , 500301 (2009).[3] N. Laflorencie, Phys. Rep. , 1 (2016).[4] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.Rev. Lett. , 48 (2004).[5] P. Calabrese and J. Cardy, J. Stat. Mech. P06002 (2004).[6] B. Hsu, M. Mulligan, E. Fradkin, and E-A. Kim, Phys.Rev. B , 115421 (2009).[7] F. Verstraete and J. I. Cirac, Phys. Rev. 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