Multi-fractality of the entanglement Hamiltonian eigen-modes
MMulti-fractality of the entanglement Hamiltonian eigen-modes
Mohammad Pouranvari
Department of Physics, Faculty of Basic Sciences,University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran (Dated: April 30, 2019)We study the fractal properties of single-particle eigen-modes of entanglement Hamiltonian infree fermion models. One of these modes that has the highest entanglement information and thuscalled maximally entangled mode (MEM) is specially considered. In free Fermion models withAnderson localization, fractality of MEM is obtained numerically and compared with the fractalityof Hamiltonian eigen-mode at Fermi level. We show that both eigen-modes have similar fractalproperties: both have same single fractal dimension in delocalized phase which equals the dimensionof the system, and both show multi-fractality at phase transition point. Therefore, we conclude that,fractal behavior of MEM – in addition to the fractal behavior of Hamiltonian eigen-mode – can beused as a quantum phase transition characterization.
I. INTRODUCTION
Quantum phase transition happens at zero tempera-ture, where quantum fluctuations – in contrast to tem-perature dependent fluctuations – are dominant anddrive the phase transition. These fluctuations yield tobroadly distributed observable quantities at the phasetransition point. Anderson transition between delo-calized and localized phases, is one example of quan-tum phase transition which has attracted much atten-tion. The original Anderson phase transition was in-troduced as a three-dimensional (3 d ) tight binding lat-tice model with randomness in on-site energies. For aspecific value of the randomness strength, the state atthe Fermi level becomes localized. In this theory thelower critical dimension is 3. In 1 d and 2 d cases, allstates in the thermodynamic limit are localized for aninfinitesimal amount of disorder and subsequently thereis no Anderson phase transition. Later on, 1 d and 2 d models were proposed with correlated disorder that havedelocalized-localized transition. Anderson transition, asa quantum phase transition, exhibits statistical fluctua-tions at the phase transition point. These fluctuationsare manifested in the anomalous scaling of the inverseparticipation ratio (defined below) of Hamiltonian eigen-mode at the Fermi energy | E F (cid:105) , which lead to fractal be-havior at the phase transition point. Such fractal behav-ior can be used as a tool to distinguish different phases. Multi-fractal analysis has broad applications in differentbranches of science including physiology, geophysics, fluid dynamics, and even in finance.
Some early reports on the fractal behavior of the | E F (cid:105) are Refs. [19–25] where its fractal properties are analyzedin different phases and used as a phase characterization.Some more recent reports are the followings: based on ananalytic calculation, Ref. [26] shows that there is a sym-metry in the multi-fractal spectrum of | E F (cid:105) . The relationbetween single-particle entanglement entropy and fractaldimensions at the phase transition point was found inRefs. [27 and 28]; also Ref. [29] proves analytically andthen shows numerically that multi-fractality of | E F (cid:105) atthe phase transition point can be obtained by using mo- ments of R´eyni entropy. Refs. [30 and 31] calculate thesingularity spectrum of | E F (cid:105) in Anderson 3 d model andcompare the typical average with the ensemble average incalculation of singularity spectrum. Furthermore, in Ref.[32] singularity spectrum is obtained by calculating theprobability distribution of | ψ i | ( ψ i as the wave-functionat site i ) .In this paper, we propose to study the problem ofAnderson transition from the point of view of multi-fractality of entanglement Hamiltonian. Let us recall thedefinition of entanglement Hamiltonian. If the systemis in state | ψ (cid:105) , then the density matrix will be given by ρ = | ψ (cid:105)(cid:104) ψ | . For a bipartite system, reduced density ma-trix for one subsystem is obtained by tracing over de-grees of freedom of other subsystem. As we know, forfree Fermion, we can write the reduced density matrix as e − H ent , where the H ent is a free Fermion Hamiltonian andcalled entanglement Hamiltonian. Entanglement Hamil-tonian eigen-modes of two subsystems are then attachedtogether to make a mode for the entire system. Themode corresponding to smallest magnitude entanglementenergy which has the largest contribution to the entan-glement entropy is distinguished from others, since it hasimportant physical information about the system.
This mode is called maximally entangled mode (MEM).In Ref. [34] it is shown, regarding the localization ofthe mode, that MEM and | E F (cid:105) contain the same physics:both are localized in the localized regime and both areextended in delocalized regime. In addition, their overlapat the phase transition point is larger than their overlapin delocalized or localized phases, although small com-pare to 1. Here, regarding the comparison of two modesfrom indirect point of view of fractality, we ask the fol-lowing questions: does MEM show multi-fractality at thephase transition point? Can we use fractal behavior ofMEM to distinguish different phases?In this paper, we answer the above questions, using two1 d models and the 3 d Anderson model. We obtain | E F (cid:105) using numerically exact diagonalization of the Hamilto-nian which is an N × N matrix ( N is the system size).And to obtain the MEM, we follow the method men-tioned in Ref. [33] where we have to diagonalize another a r X i v : . [ c ond - m a t . s t r- e l ] A p r matrix with dimension N F × N F ( N F is the Fermionnumber). These two diagonalization procedures makethe calculations very time consuming and thus we arelimited in the system size for the case of 3 d Andersonmodel.Our key results are as follows: In the delocalized phase,MEM, like | E F (cid:105) has a single fractal dimension equal todimension of the system d , while in the localized phase,the fractal dimension goes to zero. More importantly, atthe phase transition point, MEM shows multi-fractality;we calculated numerically its multi-fractal spectrum andalso show that MEM obeys the symmetry relation ofanomalous exponents. Furthermore, we can distinguishdifferent phases based on the singular spectrum of theMEM.This paper is organized as follows: in section II weexplain multi-fractality as a mathematical concept andthen apply it to wave-function in lattice systems. Themodels we intend to study are next explained in sectionIII. Section IV contains main results of our numerical cal-culations. Finally, the summary of our work is presentedin section V. II. MULTI-FRACTALITY ANALYSIS
Suppose that we have N numbers, randomly dis-tributed. Dividing this set of numbers into cells withsize (cid:96) , the probability that a number is in the i th cell, p i ( (cid:96) ) is proportional to the numbers included in that cell N i : p i ( (cid:96) ) = N i / N . Scaling behavior of moments of the probability, aver-aged over all cells, tells us the multi-fractal structure ofthese random numbers: (cid:10) p i ( (cid:96) ) q − (cid:11) ∝ (cid:96) τ ( q ) , (1)where the multi-fractal spectrum is defined as below: τ ( q ) = ( q − D ( q ) . (2)If D ( q ) is independent of q , we call D the single-fractaldimension; otherwise, when τ is not a linear function of q , we have multi -fractality. Now, in view of above method of characterizing ran-dom numbers, the fractal behavior of an eigen-functionin a lattice system can be studied, where | ψ i | ’s for the i th lattice sites are the random numbers. We want toobtain the scaling behavior of the so called generalizedinverse participation ratio(GIPR) P q , defined below: P q ( (cid:96) ) = N (cid:96) (cid:88) k µ qk ( (cid:96) ) , (3) µ k ( (cid:96) ) = (cid:96) (cid:88) i | ψ i | , (4)in which we divide the system with size N into N (cid:96) cells,each containing (cid:96) sites and we coarse grain over cells with Eq. (4). For a wave-function P q ∼ λ τ ( q ) , where λ = (cid:96)/N. (5)The behavior of the multi-fractal spectrum τ ( q ) can beused as a characterization for Anderson localization: τ ( q ) ∼ , in localized phase D ( q )( q − , at the phase transition point d ( q − , in delocalized phase (6)i.e. in the localized phase no scaling behavior is seen.In the delocalized phase, the singularity spectrum τ isa linear function of q with a constant slope of d andthus the wave-function is considered to have single-fractal dimension. On the other hand, at the phase transitionpoint, τ ( q ) is a non-linear function of q with a varyingslope of D ( q ) and the wave-function is multi-fractal .In addition, τ ( q ) is written as: τ ( q ) = d ( q −
1) + ∆ q , (7)where ∆ q are the anomalous exponents that are zero inthe delocalized phase and hold the following symmetryrelation at the phase transition point: ∆ q = ∆ − q . (8)By applying Legendre transformation, one obtains thesingularity spectrum f ( α ): α = dτ ( q ) dq , (9) f ( α ) = q dτdq − τ. (10) f ( α ) is the fractal dimension of points where | ψ i | = N − α , i.e. number of such points that scale as N f ( α ) . III. MODELS
The first model we study is the Aubry-Andre (AA)model. It is a 1 d tight binding model with the Hamil-tonian: H = − t (cid:88) ( c † i c j + c † j c i ) + (cid:88) i (cid:15) i c † i c i , (11)where c † i ( c i ) is the creation (annihilation) operator forthe site i in the second quantization representation and <> indicates nearest neighbor hopping only. Hoppingamplitudes are constant t = 1, and on-site energy (cid:15) i atsite i has an incommensurate period: (cid:15) i = 2 η cos (2 πib ) , (12)where b = √ is the golden ratio. This model hasa phase transition at η = 1. As we change η , we gothrough a phase transition from delocalized states ( η <
1) to localized states ( η > which is a 1 d model with the Hamilto-nian: H = N (cid:88) i,j =1 h ij c † i c j (13)in which on-site energies are zero, and long-range hoppingamplitudes are h ij = w ij / | i − j | a (14)where w ’s are uniformly random numbers distributed be-tween − a = 1between delocalized state ( a <
1) and localized state( a > which is a 1 d long range hop-ping model with the Hamiltonian of Eq. (13): matrix el-ements h ij are random numbers, distributed by a Gaus-sian distribution function that has zero mean and thefollowing variance (with periodic boundary condition): (cid:10) | h ij | (cid:11) = (cid:34) (cid:18) sin π ( i − j ) /Nbπ/N (cid:19) a (cid:35) − , (15)The system is delocalized for a <
1; at the phase tran-sition point a = 1, it undergoes Anderson localizationtransition to localized states for a >
1. This phase tran-sition happens regardless of b , and in our calculation weset b = 1. Specially this model is important for us,since by changing parameter b , we can simulate differ-ent models. Interestingly, it has similar multi-fractalproperties like the Anderson model in three dimensions. And finally, we also use 3 d Anderson model (Eq. (11))with randomly Gaussian distributed on-site energies, (cid:15) i ,and constant nearest-neighbor hopping amplitudes, t =1. The Gaussian distribution has zero mean and variance w . Anderson phase transition happens at w = 6 .
1, withdelocalized behavior for w < . w > . IV. MULTI-FRACTALITY OF MAXIMALLYENTANGLED MODE
Multi-fractal analysis of Hamiltonian eigen-mode atthe Fermi energy | E F (cid:105) has been studied before .Here, fractal properties of MEM is studied and comparedwith the | E F (cid:105) . To do so, in the following we first in-spect profile of MEM in AA model. Then multi-fractalspectrum as well as the singularity spectrum of MEM inPRBA, PRBM, and Anderson 3 d are studied. Then, thesymmetry relation of the anomalous exponents ∆, Eq.(8) for the MEM is verified. FIG. 1. (color online) Left panels are log | ψ i | of MEM forAA model versus site number for a sample with system size N = 4000 in delocalized phase ( η = 0 .
5, top panel), at thephase transition point ( η = 1 .
0, middle panel), and in thelocalized phase ( η = 1 .
5, bottom panel). In each of the rightpanels, a different part of the MEM at the phase transitionpoint is plotted. For each choice, we see the same behavior.Thus MEM at the phase transition point is self-similar. Notethat there is no randomness in AA model and we do not haveto take disorder average.
A. Profile of MEM
First, we look at the profile of MEM in different phasesfor AA model, which is a disorder-free model and we donot have to take disorder average. We plot MEM in thedelocalized phase, localized phase and at the phase tran-sition point in Fig. 1. As we can see, in the delocalizedphase, MEM is spread over sites, while it is localized atone site in the localized phase. On the other hand, itshows self-similarity at the phase transition point. i.e.behavior of any part of the MEM is similar to that of theentire mode. Because of such self-similarity, MEM showsmulti-fractality.Moreover, Since fractal properties of eigen-modes areextracted from the GIPR, we compare the GIPR of MEMand | E F (cid:105) according to Eq. (3). GIPR of AA model fordifferent q ’s are plotted in Fig. 2. For each q , althoughthe behavior is not identical, similar trend is observed.From this simple calculation, we can deduce that MEMhas much the same fractal properties as | E F (cid:105) . In addi-tion, similar to the GIPR of the Hamiltonian eigen-modeat the Fermi level, GIPR of the MEM distinguishes dif-ferent phases and can be used as a phase detection pa-rameter. B. Multi-fractal Spectrum
In this subsection, we consider the behavior of multi-fractal spectrum τ ( q ) as a function of q for PRBA, PRBM P q AA | E F q =0.4| MEM q =0.4| E F q =0.5| MEM q =0.5 AA | E F q =3| MEM q =3| E F q =4| MEM q =4 FIG. 2. (color online) Generalized participation ratio, Eq.(3) of both | E F (cid:105) and MEM, for the AA model. Left panel: q = 0 . , . q = 3 ,
4. Both modes havesimilar behavior in delocalized phase ( η <
1) and in localizedphase ( η > N = 3000. Since there is norandomness in AA model, we do not have to take disorderaverage. and Anderson 3 d models. In models with randomness, touse Eqs. (5), (9), and (10) we need to take the average ofquantities over (quenched) random sample realizations.To do so we can take either ensemble average or typicalaverage. As verified by Ref. [30], typical average yieldsmore accurate results (since very small numbers in | ψ i | are also take into account). Thus, we only present theresults obtained using typical averages. To obtain typicalaverage of GIPR for models with disorder, we rewrite Eq.(5) as: e (cid:104) ln P q ( λ ) (cid:105) ∝ λ τ ( q ) typ , (16)where (cid:104)· · · (cid:105) stands for arithmetic average over disorderrealization. Thus, τ ( q ) typ = lim λ → (cid:104) ln P q ( λ ) (cid:105) ln λ . (17)In taking the limit, we are free to either fix (cid:96) andchoose a sequence of system sizes N , or we can fix N and choose a sequence of smaller values of cell size:1 (cid:28) (cid:96) < N . Here we choose the former; we choose (cid:96) ∼ d models), and (cid:96) ∼
10 (for 3 d model) for q < (cid:96) = 1 for q > N . The reason that we choose (cid:96) > q is the following: numerical inaccuracies that are thecalculated eigen-mode (either for | E F (cid:105) or MEM) becomeexaggerated for negative q and thus to avoid them, wecoarse grain over a cell with size (cid:96) . Then, the slope ofthe straight line fitting (cid:104) ln P q (cid:105) versus − ln( N ) gives usthe τ ( q ). − ln( N ) τ ( q ) typ = (cid:104) ln N (cid:96) (cid:88) k =1 µ qk (cid:105) , (18)with similar calculations based on Eqs. (9) and (10) we q ( q ) t y p PRBA | E F , a = 0.5| MEM , a = 0.5| E F , a = 1| MEM , a = 1| E F , a = 1.5| MEM , a = 1.5 FIG. 3. (color online) Multi-fractality spectrum, τ ( q ) forPRBA. System sizes are between 1000 and 5000 in step of500. For each data point we averaged over 1000 samples. obtain α and f ( α ): − ln( N ) α ( q ) typ = (cid:104) (cid:80) N (cid:96) k =1 µ qk ln µ k P q (cid:105) (19) − ln( N ) f ( q ) typ = (cid:104) (cid:80) N (cid:96) k =1 µ qk ln µ qk P q − ln P q (cid:105) (20)We know that, multi-fractal spectrum behavior ofHamiltonian eigen-mode | E F (cid:105) depends on the phase ofthe system: in the delocalized phase, τ ( q ) is a straightline with a constant slope equal to dimension of the sys-tem. In the localized phase, the spectrum goes to zerofor q >
0, and at the phase transition point, the slope ofthe spectrum is not constant, yielding to multi-fractality.The multi-fractal spectrum of MEM and | E F (cid:105) for PRBA,PRBM, and Anderson 3 d model are plotted in Fig. 3,Fig. 4, and Fig. 5 respectively. In Fig. 6 disorder av-eraged (cid:104) ln P q (cid:105) versus − ln( N ) plotted and fitted with astraight line for MEM at the phase transition point forPRBA, PRBM, and Anderson 3 d models. The slope ofthis line which is τ ( q ) and the accuracy of the fitted lineby R-squared measure are calculated.In PRBA and PRBM models, the behavior of τ in thedelocalized phase for both | E F (cid:105) and | M EM (cid:105) are iden-tical, both are straight lines; although we see a slightdiscrepancy behavior for Anderson 3 d model. In the lo-calized phase, τ ( q ) goes to zero. And, more importantlyat the phase transition point, τ ( q ) is a non-linear functionof q .We also calculate the fractal dimension of MEM andplot them in Fig. 7. The single fractal dimension ofMEM in delocalized phase equals to 1 = d for PRBAand PRBM models, and for Anderson 3 d model, it isaround 3 = d . For the localized phase, fractal dimensiongoes to zero as it should. The fractal dimension of MEMat the phase transition point is also plotted, which as we q ( q ) t y p PRBM | E F , a = 0.5| MEM , a = 0.5| E F , a = 1| MEM , a = 1| E F , a = 1.5| MEM , a = 1.5 FIG. 4. (color online) Multi-fractality spectrum, τ ( q ) forPRBM. System sizes are between 1000 and 5000 in step of500. For each data point we averaged over 1000 samples. q ( q ) t y p Anderson 3D | E F , w = 1| MEM , w = 1| E F , w = 6.1| MEM , w = 6.1| E F , w = 12| MEM , w = 12 FIG. 5. (color online) Multi-fractality spectrum, τ ( q ) for An-derson 3 d model with Gaussian distribution. System sizes arebetween 4 × × × ×
30, with 300 samples for smallsizes and 50 samples for large sizes. can see, is not a constant and thus MEM is multi-fractalat the phase transition point.
C. Singularity Spectrum
Next, we consider the behavior of the singularity spec-trum f ( α ) versus α . In the delocalized phase, f ( α )should be narrow around α = d : when the system is delo-calized, we expect the eigen-mode (either the eigen-modeof the Hamiltonian or the MEM of entanglement Hamil-tonian) to spread over all sites and by the normaliza-tion condition (cid:80) L d i =1 | ψ i | = 1, we find that | ψ i | ∼ L − d .Thus, according to Eq. (9), f ( α ) should be narrowed around α ∼ d with the value of f ( α ) ∼ d (i.e. the frac-tal dimension of points with | ψ i | ∼ L − d is very closeto the dimension of system d ). For PRBA and PRBM f ( α ) at α ∼ d , f ( α )at α ∼ f ( α ) has parabolic behavior which is the sign ofthe multi-fractality of the mode. In the localized phase,the eigen-mode is localized at a few number of sites andhas a very small value at many other sites, thus f ( α )broadens toward larger α , i.e. plot is shifted to the right.Our calculation of singularity spectrum of MEM forPRBA, PRBM, and Anderson 3d is plotted in Fig. 8.According to our calculation, the singularity spectrum ofMEM in the delocalized phase is centered around α = d = 1 for PRBA and PRBM models and around α = d =3 for Anderson 3 d model, although it spreads a bit around3 for the Anderson 3d model. At the phase transitionpoint, we see a parabolic behavior as it is predicted andcalculated for Hamiltonian eigen-mode at the Fermi level.And finally, in the localized phase, f ( α ) is broadenedtoward larger α . We note that f ( α ) versus α is morebroadened in the case of Anderson 3 d model than in the1 d cases. Beside some inaccuracies that come from twoexact diagonalizations to obtain MEM (as we explainedin Introduction), we expect more broad behavior for theAnderson 3 d case. Since the linear size of the system, N is the reference in the calculation of α and f ( α ) (seeEqs. (9, 10)), and dimension of the system is three timeslarger than the 1 d cases, thus α goes to larger values.As we can see, the behavior of the singularity spectrumof MEM like | E F (cid:105) depends on the phase considered, andthus it can be used as a phase detection parameter.By looking at the Fig. 8, we see that singularity spec-trum at the phase transition point for the three stud-ied models are symmetric (in contrast to the results ob-tained in the Refs. [26 and 46] for PRBM and Ref. [32]for Anderson 3 d models, where the reason come fromchoosing (cid:96) > f ( α ) versus α which could indicateuniformities (non-uniformities) in the hierarchical orga-nization of mode was pointed out in Ref. [48]. D. Symmetry Relation of ∆ q The symmetry relation of anomalous exponents, Eq.(8) is proved analytically and numerically in Refs. [26and 30] for | E F (cid:105) . Here we present numerical verificationof the symmetry relation for the MEM in PRBA, PRBM,and Anderson 3 d models in the main panels of Fig. 9. Aswe can see the symmetry relation of Eq. (8) is respectedfor the MEM of the entanglement Hamiltonian. On theother hand, we fit the ∆ q for the | E F (cid:105) and MEM withthe parabolic equation of Aq ( B − q ) and find the A and B constants. The values of A and B are reported in theTable I. For three models considered, and for both | E F (cid:105) and MEM, B ∼ τ (1) = 0 → ∆ =0, and so B = 1). Moreover, A MEM is approximately ln( N ) l n P q q = 2.5, m = 5.07, R =1.00 q = 1.5, m = 3.33, R =1.00 q = 0.5, m = 1.73, R =1.00 q =0.5, m = 0.35, R =0.99 q =1.5, m =0.19, R =0.95 q =2.5, m =0.47, R =0.94 8.4 8.2 8.0 7.8 7.6 7.4 7.2 7.0 ln( N ) l n P q q = 2.5, m = 4.86, R =1.00 q = 1.5, m = 3.21, R =1.00 q = 0.5, m = 1.69, R =1.00 q =0.5, m = 0.38, R =1.00 q =1.5, m =0.21, R =0.99 q =2.5, m =0.50, R =0.98 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 ln( N ) l n P q q = 2.5, m = 24.39, R =1.00 q = 1.5, m = 15.09, R =1.00 q = 0.5, m = 6.56, R =1.00 q =0.5, m = 0.73, R =0.98 q =1.5, m =0.34, R =0.93 q =2.5, m =0.86, R =0.92 FIG. 6. (color online) Plot of disorder averaged ln P q vs − ln N for PRBA(left), PRBM(middle), and Anderson 3 d (right)models for the MEM at the phase transition point. This calculation is done for some selected values of q and for each q theslope of the fitted line is indicated by m . The R-squared which is the sign of how close data points are to the fitted line is alsocalculated (the closer to 1, the better fitted line). q D ( q ) t y p PRBA | MEM , a =0.5| MEM , a =1| MEM , a =1.5 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 q D ( q ) t y p PRBM | MEM , a =0.5| MEM , a =1| MEM , a =1.5 q D ( q ) t y p Anderson | MEM , w =1| MEM , w =6.1| MEM , w =12 FIG. 7. (color online) Fractal dimension of the MEM of the entanglement Hamiltonian for PRBA (left panel), PRBM (middlepanel), and Anderson 3 d (right panel) models. For each model, fractal dimension is calculated in delocalized phase, at thephase transition point, and in the localized phase. three times larger than A | E F (cid:105) in each model. TABLE I. Constants A and B when we fit ∆ q for | E F (cid:105) andMEM with the equation Aq ( B − q ) (see sub-panels of Fig. 9).PRBA PRBM Anderson 3 dA | E F (cid:105) A MEM B | E F (cid:105) B MEM
V. CONCLUDING REMARKS
It has been shown that Anderson transition as a quan-tum phase transition exhibits multi-fractal behavior atthe critical point. In fact, the generalized participa-tion ratio of Hamiltonian eigen-mode at the Fermi levelis a measure that shows multi-fractality of the system.Recently, entanglement Hamiltonian and its associatedmaximally entangled mode has attracted attention as atool to characterize systems behavior particularly at thecritical point. We note that obtaining an explicit relationfor eigenvectors of the entanglement Hamiltonian (EH)based on the eigenvectors of Hamiltonian is not trivial and they are not directly related. In a study , peoplefound the explicit expression for the EH matrix elementsin the ground state of free fermion models. People alsofound that at the extreme limit of strong coupling be-tween two chosen subsystems, EH of a subsystem and itsHamiltonian are proportional . In this paper, we haveshown that multi-fractality of Anderson transition car-ries over to MEM much in the same way as Hamiltonianeigen-mode at the Fermi level.Based on numerical calculations for PRBA and PRBM1 d models, and also Anderson 3 d model, we showed thatsingle particle MEM of the entanglement Hamiltonian,has the same fractal properties as the Hamiltonian eigen-mode at the Fermi level; although for Anderson 3 d modelwe see a little deviations, since we could not reach verylarge system sizes. For MEM, in the delocalized phase, τ ( q ) has a slope equal to the dimension of the system,while in the localized phase, it goes to zero. Interestingly,at the phase transition point, MEM is multi-fractal andits multi-fractality is similar to that of the | E F (cid:105) . MEMalso follows the symmetry relation of anomalous expo-nents. Moreover, singularity spectrum f ( α ) of MEM,is similar to f ( α ) of | E F (cid:105) : in the delocalized phase itis around α ∼ d ; at the phase transition point it hasparabolic shape with the maximum value d , and it broad- f () t y p PRBA | MEM , a = 1 0 1 2 30.00.20.40.60.81.0 PRBA | MEM , a = 0.5| MEM , a = 1.50.5 1.0 1.5 2.00.00.20.40.60.81.0 f () t y p PRBM | MEM , a = 1 0 1 2 30.00.20.40.60.81.0 PRBM | MEM , a = 0.5| MEM , a = 1.50 2 4 6 80.51.01.52.02.53.0 f () t y p Anderson | MEM , w = 6.1 0 10 20123 Anderson | MEM , w = 1| MEM , w = 12 FIG. 8. (color online) Singularity spectrum f ( α ) for PRBA,PRBM, and Anderson 3 d models. The left panels show thesingularity spectrum of MEM at the phase transition point.The right panels show the singularity spectrum of MEM fordelocalized and localized phases. For each model, the rangeof q is between − ens in the localized phase. And thus, by looking at multi-fractal spectrum or singularity spectrum of MEM, we candistinguish different phases.Multi-fractality of an observable at the quantum phasetransition means that this observable is self-similar; andfinite-size scaling of observable is a legitimate method ofobtaining critical exponents. Here we saw that entan-glement Hamiltonian shows multi-fractality; which indi-rectly verifies that reduced density matrix and even en-tanglement entropy should exhibit self-similarity at thephase transition point, and thus their finite size scalingcan be used as a method to calculate the critical expo-nents, as it has been done in Refs. [51–53].Multi-fractality of entanglement Hamiltonian wasstudied in this paper through its eigen-modes. This studycould also be done by inspecting the multi-fractality ofelements of entanglement Hamiltonian. It might also beinteresting to consider entanglement entropy, or a mea-sure of eigenvalues of reduced density matrix, to see ifthey also carry signatures of multi-fractality. VI. ACKNOWLEDGMENTS
This research is supported by University of Mazan-daran, National Merit Foundation of Iran, and Institutefor Research in Fundamental Sciences (IPM). Part of thiswork was done while I was working at Shiraz University.I would like to thank Dr. Abbas Ali Saberi and AfshinMontakhab for useful discussions and their constructivecriticism on the manuscript. P. W. Anderson, Phys. Rev. , 1492 (1958). F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008). F. Wegner, Z. Phys. B, , 206 (1980). H. E. Stanley, L. A. N. Amaral, A. L. Goldberger, S.Havlin, P. Ch. Ivanov, C.-K. Pengb, Physica A: StatisticalMechanics and its Applications, , 309, (1999). P. Ch. Ivanov, L. A. N. Amaral, A. L. Goldberger, S.Havlin, M. G. Rosenblum, H. E. Stanley, Z. R. Struzik,Chaos: An Interdisciplinary Journal of Nonlinear Science, , 3,(2001). A. Davis, A. Marshak, W. Wiscombe, R. Cahalan, Journalof Geophysical Research: Atmospheres, , 8055 (1994). Y. Tessier, S. Lovejoy, P. Hubert, D. Schertzer, S. Pec-knold, Journal of Geophysical Research: Atmospheres, , 26427, (1996). S. Lovejoy, D. Schertzer, Journal ofGeophysical Research: Atmospheres, , 2021 (1990). L. Biferale, G. Boffetta, A. Celani, B. J. Devenish, A. Lan-otte, and F. Toschi, Phys. Rev. Lett. , 064502 (2004). C. Meneveau, K. R. Sreenivasan, Nuclear Physics B, ,0920, (1987). D. Schertzer et al, Fractals , 427 (1997). R. R. Prasad, C. Meneveau, and K. R. Sreenivasan, Phys.Rev. Lett. , 74 (1988). C. Meneveau, K. R. Sreenivasan, P. Kailasnath, M. S. Fan,Phys. Rev. A , 894 (1990). R. Benzi, G. Paladin, G. Parisi, A. Vulpiani , Journal ofPhysics A: Mathematical and General, , 3521 (1984). L . Zunino, B. M. Tabak, A. Figliola, D. G. Perez, M.Garavaglia, O. A. Rosso, Physica A: Statistical Mechanicsand its Applications, , 6558 (2008). K. Matia, Y. Ashkenazy, H. E. Stanley, Europhysics Let-ters, , 422 (2003). q PRBA, a = 1 | MEM , q | MEM , q q q | E F fit | E F | MEM fit |
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MEM
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