Bounds on the entanglement entropy by the number entropy in non-interacting fermionic systems
Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Jesko Sirker, Michael Fleischhauer
SSciPost Physics Submission
Bounds on the entanglement entropy by the number entropyin non-interacting fermionic systems
M. Kiefer-Emmanouilidis , R. Unanyan , J. Sirker , M. Fleischhauer Department of Physics and Research Center OPTIMAS, University of Kaiserslautern,67663 Kaiserslautern, Germany Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2,CanadaMarch 9, 2020
Abstract
Entanglement in a pure state of a many-body system can be characterized by the Rényi en-tropies S ( α ) = ln tr ( ρ α ) / (1 − α ) of the reduced density matrix ρ of a subsystem. These entropiesare, however, difficult to access experimentally and can typically be determined for small sys-tems only. Here we show that for free fermionic systems in a Gaussian state and with particlenumber conservation, ln S (2) can be tightly bound by the much easier accessible Rényi numberentropy S (2) N = − ln (cid:80) n p ( n ) which is a function of the probability distribution p ( n ) of thetotal particle number in the considered subsystem only. A dynamical growth in entanglement,in particular, is therefore always accompanied by a growth—albeit logarithmically slower—ofthe number entropy. We illustrate this relation by presenting numerical results for quenchesin non-interacting one-dimensional lattice models including disorder-free, Anderson-localized,and critical systems with off-diagonal disorder. Contents a r X i v : . [ c ond - m a t . d i s - nn ] M a r ciPost Physics Submission The entanglement between two parts of a many-body system in a pure state can be char-acterized by the Rényi entropies S ( α ) = ln tr ( ρ α ) / (1 − α ) . The von-Neumann entanglemententropy is given by S = S (1) ≡ lim α → S ( α ) = − tr ( ρ ln ρ ) . Their time evolution, for examplefollowing a quantum quench, offers important insights into the dynamics of the many-bodysystem. Except for very small systems, where the reduced density matrix ρ can be obtainedfrom quantum-state tomography [1], these entropies are difficult to access experimentally. Forsystems with particle number conservation, the Rényi entropies can be expressed as S ( α ) = 11 − α ln (cid:32)(cid:88) n p α ( n ) tr ρ α ( n ) (cid:33) = S ( α ) N + S ( α )conf . (1)Here p ( n ) is the probability distribution of the particle number n in the partition and ρ ( n ) isthe block of the reduced density matrix with fixed particle number n , normalized such thattr ρ ( n ) = 1 . The Rényi number entropy S ( α ) N = ln[ (cid:80) n p α ( n )] / (1 − α ) is then the part of theentanglement due to number fluctuations only (i.e., in a system where only one configurationfor each possible particle number n exists we would have S ( α ) = S ( α ) N ) and S ( α )conf describesthe additional entanglement due to the existence of several configurations for a given n . Thisconfigurational entropy takes the particularly simple form, S conf = − (cid:80) n p ( n ) tr [ ρ ( n ) ln ρ ( n )] ,in the limit α → [2]. The corresponding number entropy S N has been measured very recentlyin an experiment on a cold atomic gas [3]. The source of the number entropy are fluctuationsinduced by particle transport. S conf , on the other hand, is determined by the full microscopiccounting statistics. S N is thus much easier measurable in experiments and can also be accessedtheoretically using conformal field theory (CFT) [4].Here we prove that for non-interacting fermions on a lattice where the particle number isconserved, the second Rényi entropy can be bounded from above and below by eπ exp (cid:16) S (2) N (cid:17) − (cid:46) S (2) (cid:46) ln 2 π exp (cid:16) S (2) N (cid:17) , (2)where S (2) N is the second Rényi number entropy. Thus the size and time dependence of theentanglement is directly linked to that of the number entropy. A dynamical growth of entan-glement in any non-interacting fermion system, in particular, implies that the number entropygrows as well, albeit logarithmically slower. Vice versa, in a fully localized phase, where thefluctuations of the particle number of a partition are expected to saturate, the number ofaccessible configurations and thus entanglement can no longer increase either.Our paper is organized as follows. In Sec. 2 we present the proof for the lower and upperbounds in Eq. (2). In Sec. 3 we exemplify the usefulness of these bounds based on numeri-cal data for the time evolution of the entropies after quantum quenches in one-dimensionalfermionic lattice models with and without disorder. This includes, in particular, the interest-ing case of off-diagonal disorder (bond disorder), where the von Neumann entropy shows avery slow ln ln t increase in time [5–8], while the number entropy scales as ln ln ln t . Extensionsto fermionic models with interactions will be discussed elsewhere [9].2 ciPost Physics Submission In the following, we will establish a relation between the second Rényi entropy S (2) = − ln tr ( ρ ) of a quantum state ρ , which we will refer to as purity entropy, and the corresponding numberentropy S (2) N = − ln (cid:80) n p n . Specificially, we consider models of non-interacting fermions withparticle number conservation. Since the number entropy does not account for the different configurations of particles in theconsidered subsystem, a trivial lower bound for the purity entropy is given by S (2) ≥ S (2) N . (3)This is, however, in most cases only a very weak bound. An alternative and often muchbetter lower bound can be obtained using the relation between S (2) and the particle numberfluctuations ∆ n derived in [10] n ≥ S (2) ≥ n . (4)From the right hand side of Eq. (4) together with the modified version of Shannon’s inequalityfor discrete variables [11]
12 ln (cid:20) πe (cid:18) ∆ n + 112 (cid:19)(cid:21) ≥ S N ≥ S (2) N , (5)we find the alternative lower bound for the purity entropy S (2) ≥ eπ exp (2 S N ) − ≥ eπ exp (cid:16) S (2) N (cid:17) − . (6)For S (2) N > ln( eπ/ / ≈ . this bound is positive and for S (2) N (cid:38) . it is a stricter lowerbound than the trivial relation (3). In order to derive an upper bound for the purity entropy we make use of the fact that thequantum state ρ for a non-interacting fermionic system in any dimension is completely deter-mined by its single-particle correlations and has a Gaussian form. This applies in particular toall eigenstates of free-fermion Hamiltonians and to all time-evolved states under such Hamil-tonians if the initial state is Gaussian. Since we assume, furthermore, total particle numberconservation, ρ can be represented as [12–14] ρ = 1 Z exp (cid:32) − (cid:88) mn c † m C mn c n (cid:33) , (7)where c m ( c † m ) are the fermionic annihilation (creation) operators at lattice site m . Here C isa Hermitian matrix which is determined entirely by the matrix f of (normal) single-particlecorrelations 3 ciPost Physics Submission f mn = (cid:68) c † m c n (cid:69) = tr (cid:16) ρ c † m c n (cid:17) = (cid:20)
11 + e C (cid:21) mn , (8)and Z = tr (cid:110) exp (cid:0) − (cid:80) nm c † n C nm c m (cid:1)(cid:111) .We will now show that the particle number fluctuations ∆ n in a partition are boundedfrom above by the Rényi number entropy S (2) N . Making use again of relation (4) this will thenresult in an upper bound on the purity entropy in terms of S (2) N . To do so, it is useful tointroduce the moment generating function of the total particle number ˆ N in the partition [15] χ ( θ ) ≡ (cid:68) e i θ ˆ N (cid:69) = tr (cid:110) ρ e i θ ˆ N (cid:111) = (cid:88) n p ( n ) e i nθ . (9)For Gaussian fermionic states, the generating function can be written as a determinant [18] χ ( θ ) = det (cid:104) + ( e i θ − + e C (cid:105) . Making use of Parsevals theorem one then finds (cid:88) n p n = 12 π (cid:90) π dθ (cid:12)(cid:12) χ ( θ ) (cid:12)(cid:12) = 12 π (cid:90) π dθ (cid:12)(cid:12)(cid:12) det (cid:104) + ( e i θ − + e C (cid:105)(cid:12)(cid:12)(cid:12) (10) = 12 π (cid:90) π dθ det (cid:0) + 2 e C cos θ + e C (cid:1) det ( + e C ) = 12 π (cid:90) π dθ det ( − G + G cos θ ) . In the last equation we introduced the matrix G = 2 e C ( + e C ) = 2 f ( − f ) ≤ . (11)We see that the argument in the last line of Eq. (10) is a positive-definite matrix. Thus we canapply the arithmetic-geometric inequality to get an upper bound on det ( − G + G cos θ ) .Denoting the lattice size as M we find (cid:88) n p n ≤ π (cid:90) π dθ (cid:20) θ − tr ( G ) M (cid:21) M → π (cid:90) π dθ exp (cid:104) (cos θ − tr ( G ) (cid:105) , (12)where the second line holds in the thermodynamic limit M → ∞ . The integral can becalculated elementary in terms of the modified Bessel function of the first kind I ( x ) resultingin (cid:88) n p n ≤ exp (cid:0) − tr ( G ) (cid:1) I (cid:0) tr ( G ) (cid:1) . (13)Furthermore, we see from Eq. (11) that the trace of the matrix G gives the fluctuations of thetotal particle number tr ( G ) = 2 tr (cid:0) f ( − f ) (cid:1) = 2 ∆ n . (14)Combined with Eq. (13) we therefore find for the number entropy S (2) N = − ln (cid:88) n p n ≥ n − ln (cid:16) I (cid:0) n (cid:1)(cid:17) . (15)4 ciPost Physics Submission Using the asymptotic expansion of the modified Bessel function in the limit of large ∆ n , thisexpression can be simplified to S (2) N ≥
12 ln (cid:0) π ∆ n (cid:1) . (16)In the opposite limit of small ∆ n the contribution of the modified Bessel function in Eq. (15)can be neglected and we find instead S (2) N ≥ n . (17)Now making use of the left hand side of the inequality in Eq. (4), we eventually arrive at anupper bound on the purity entropy in terms of the Rényi number entropy. This bound can bewritten explicitly in the two limiting cases of either small values of ∆ n S (2) (cid:46) (2 ln 2) S (2) N , (18)or large values of ∆ n S (2) (cid:46) ln 2 π exp (cid:16) S (2) N (cid:17) . (19)Eqs. (18,19) and (6) are the main results of our paper. They show that the entanglementquantified by the logarithm of the purity entropy S (2) is bounded both from below and aboveby the number entropy. As a consequence, a growth of entanglement in free fermionic systemsis always accompanied by a logarithmically slower growth of the number entropy. Next, we illustrate our results by considering one-dimensional tight-binding models of non-interacting fermions. By applying a Jordan-Wigner transformation, these models can alter-natively also be seen as spin- / XX chains. We discuss free fermions without disorder inSec. 3.1, with potential disorder leading to Anderson localization in Sec. 3.2, and with bond(off-diagonal) disorder resulting in a critical system in Sec. 3.3.The Hamiltonian for all these systems has the same structure H = L − (cid:88) j =1 J j (ˆ c † j ˆ c j +1 + h.c. ) + L (cid:88) j =1 D j ˆ c † j ˆ c j , (20)with hopping amplitudes J j , onsite potentials D j , system size L , and open boundary condi-tions. In the following, we always consider the Rényi and number entropies for a partition ofsize l = L/ . We are interested in the time evolution of the entanglement entropies follow-ing a quantum quench starting from an initial state | Ψ ini (cid:105) which is not an eigenstate of theHamiltonian (20).We use exact diagonalization (ED) methods to obtain the eigenvalues of the reduced densitymatrix ρ ( t ) , see Eq. (7), which is calculated from the time-evolved state | Ψ( t ) (cid:105) = e − i ˆ Ht | Ψ ini (cid:105) following Refs. [12–14]. From the eigenvalues f m of the correlation matrix f in Eq. (8), we candirectly obtain all quantities of interest efficiently. This includes the von-Neumann and purityentropies 5 ciPost Physics Submission S = − (cid:88) m f m ln( f m ) − (cid:88) m (1 − f m ) ln(1 − f m ) ,S (2) = − (cid:88) m ln(1 − f m (1 − f m )) , (21)as well as the number distribution which can be calculated from the corresponding character-istic function p ( n ) = 1 l + 1 l (cid:88) k =0 exp (cid:18) − i πknl + 1 (cid:19) χ ( k ) , χ ( k ) = (cid:68) e i πkl +1 ˆ N (cid:69) = (cid:89) m (cid:16) (cid:16) e i πkl +1 − (cid:17) f m (cid:17) . (22) The case of fermions on a one-dimensional lattice has been studied extensively in the past,both analytically using conformal field theory [19] as well as numerically, see e.g. Ref. [6].The conformal field theory results show that the von-Neumann entropy as well as all Rényientropies increase linearly in time in the thermodynamic limit although with a slope which isnon-universal. Here we choose a density-wave state with a fermion on every second site, | Ψ ini (cid:105) = l (cid:89) j =1 ˆ c † j | (cid:105) , (23)as initial state. Note, however, that the results are qualitatively the same for any genericinitial product state. The numerical results in Fig. 1(a) show that S (2) ( t ) increases linearly intime until the particle-hole pairs created by the quench reach the boundaries of the partitionof size l = L/ . This happens for times t ∼ l/v ≈ l/ [6, 19] where v ≈ is the velocity of theexcitations. For times t > l/v boundary effects dominate the dynamics and S (2) is oscillatingaround an average value which depends on the size of the partition. Fig. 1(a) confirms thatthe lower and upper bounds obtained here are valid for all times, including long times whereboundary effects dominate. The upper bound (19), in particular, is a very tight bound for alltimes in this case, see the inset of Fig. 1(a).Based on the bounds and verified by the numerical results above we find that the Rényinumber entropy grows as S (2) N ( t ) ∼ ln t for free fermions on a one-dimensional lattice withoutdisorder. This logarithmic growth can be understood as follows: Consider a quench in a half-filled system where at long times each arrangement of particles has approximately the sameprobability. Then for a system of size L we have L particles. If we cut the system in twohalfs of size L , the probability to find k particles in one half is given by p ( k, L ) = (cid:0) Lk (cid:1)(cid:0) LL − k (cid:1)(cid:0) LL (cid:1) . (24)For (cid:28) k < L we can approximate this distribution by a normal distribution ˜ p ( k, L ) = 2 √ Lπ exp (cid:34) − L (cid:18) k − L (cid:19) (cid:35) . (25)We can now obtain the Rényi number entropies by integrating over the continuous distribution6 ciPost Physics Submission t S ( ) ( a ) ub S (2) lb-1lb-2 t S ( α ) N ( b ) S N S (2)N S (3)N S (4)N S (5)N
275 300 325 t S ( ) Figure 1: (a) S (2) ( t ) for a quench from the initial state (23) using the Hamiltonian (20) with J j = 1 , D j = 0 , and system size L = 1024 . The purity entropy S (2) grows linearly until itreaches its maximum at t ≈ l/v . The upper bound (ub), Eq. (19), and the two lower boundsfrom Eq. (6), (lb-1) exp(2 S N ) / ( eπ ) − / , and (lb-2) exp (cid:16) S (2) N (cid:17) / ( eπ ) − / , encapsule S (2) .Note that the upper bound is tight, see inset. (b) Rényi number entropies S ( α ) N ( t ) for Gaussian waves who are initially spaced at equal distances and whose width increases linearlyin time, see Eq. (29), for ν = 1 . We find S ( α ) N ( t ) = const + ln t . The saturation at long timesis a finite-size effect. S ( α ) N ( L ) ≈ − α ln (cid:18)(cid:90) ∞−∞ dk ˜ p α ( k, L ) (cid:19) = ln (cid:20) √ π α / (2 − α ) (cid:21) + 12 ln L . (26)The von-Neumann number entropy can be obtained by S N ( L ) = lim α → S ( α ) N ( L ) ≈ (cid:18) (cid:20) Lπ (cid:21)(cid:19) . (27)If we now consider excitations which spread ballistically ∼ vt then we have regions of size L ∼ vt in which each arrangement of particles has approximately equal probability. Puttingthis into the results for the number entropy and the Rényi number entropies we obtain thefinal result S ( α ) N ( t ) = const + 12 ln t (28)which includes the von-Neumann case ( α → ). Note that the constants do depend on themicroscopic details of the model but are monotonically decreasing with α , see Eq. (26). Thuswe conclude that all Rényi number entropies behave the same qualitatively and S ( α ) N > S ( α +1) N .An alternative perspective to understand the logarithmic spreading—more closely relatedto the numerical simulations— can be obtained by considering Gaussian waves | Ψ i ( x, t ) | = 1 √ πνt exp (cid:18) − ( x − x i ) νt (cid:19) (29)with initial positions x i spread evenly along a line. Here ν is a constant and the width of theGaussian wave is increasing linearly in time. The probability to find the particle i at x > isthen given by 7 ciPost Physics Submission P ( x i , t ) = (cid:90) ∞ | Ψ i ( x, t ) | = 12 (cid:18) erf (cid:18) x i √ νt (cid:19)(cid:19) . (30)If we have N particles in total then the probability to find k at x > is p ( k, t ) = (cid:88) n i ∈{ , }(cid:48) N (cid:89) i =1 [ P ( x i , t )] n i [1 − P ( x i , t )] − n i (31)The sum (cid:80) (cid:48) is over all permutations of the { n i } and has to be evaluated with the constraint (cid:80) Ni =1 n i = k . It can be directly evaluated if N is not too large. Results for N = 20 particlesare shown in Fig. 1 (b) and confirm Eq. (28). Static potential disorder in an isolated quantum system of non-interacting particles can induceAnderson localization (AL), defined as the absence of particle diffusion [20]. For one and twodimensions, Anderson localization occurs for any strength of disorder D [21]. For a one-dimensional system we can extract the localization length ξ in dependence of energy (cid:15) anddisorder strength D using a transfer matrix approach as described in [22]. If we quench aone-dimensional system with potential disorder we thus expect that for times t (cid:29) ξ/v boththe number and configurational entropies will stop increasing.To study the Anderson case numerically, we set the hopping in Eq. (20) uniformly to J j = 1 and draw random values for the potential from a box distribution D j ∈ [ − D/ , D/ . We nowquench the system from initial random product states at half filling | Ψ ini (cid:105) = l (cid:89) j =1 η j ˆ c † j | (cid:105) , (32)where η j ∈ { , } is random, and half-filling is imposed by requiring (cid:80) j η j = l/ . The randomproduct state will on average yield a state with energy (cid:15) = 0 and we consider both a weaklydisordered case, D = 2 , and a strongly disordered case, D = 20 . Since we consider systemslarge compared to the localization length we do not expect a qualitative difference in the timedependence of the purity entropies in the two cases: there will be an increase of entropy intime until a constant value is reached that does depend on the localization length ξ but isindependent of system size L if L (cid:29) ξ . Our numerical simulations, shown in Fig. 2(a-b),verify this behavior. Furthermore, they also verify the lower and upper bounds in terms of thenumber entropy. Note that S (2) is bounded quite tightly from below by S (2) N for small valuesof S (2) , see Fig. 2(b). Here the trivial lower bound, Eq. (3), is the better bound since both thepurity entropy and the corresponding number entropy are quite small in the localized regime. As the third example, we consider the Hamiltonian (20) with bond disorder, J j ∈ (0 , .It is known that this system in the thermodynamic limit is at an infinite randomness fixedpoint [23–26]. The mean localization length scales as ξ loc ( (cid:15) ) ∼ | ln( (cid:15) ) | Ψ with Ψ being thecritical exponent. The system therefore shows a localization-delocalization transition as a8 ciPost Physics Submission
10 20 lnt S ( ) ( a ) ub S (2) lb lnt ( b ) ub S (2) lb lnlnt ( c ) ub S (2) lb S (2)N Figure 2: (a) In the weakly disordered case D = 2 , the upper bound (ub), (ln 2 /π ) exp (cid:16) S (2) N (cid:17) is quite tight for the regime in which S (2) ( t ) grows. The lower bound (lb) shown is exp(2 S N ) / ( eπ ) − / . (b) For strong disorder, D = 20 , the entanglement remains very smalland Eq. (18), S (2)N , is the better upper bound (ub). For the same reason S (2) N (lb) is abetter lower bound. In both cases averages over 2000 disorder realizations and initial statesare shown. (c) S (2) ( t ) for bond disorder after a quench starting from the half-filled randomproduct state (32) for L = 1024 sites and disorder realizations. (ub) corresponds to (ln 2 /π ) exp (cid:16) S (2) N (cid:17) , (lb) to the maximum of S (2) N and exp(2 S N ) / ( πe ) − / . We confirm that S (2) ∼ ln ln t . At long times, S (2) and the upper bound (ub) (19) based on the number entropyonly differ by a constant shift. The grey dotted line signals the point in time where doubleprecision is no longer sufficient to obtain reliable results, see also Ref. [6].function of energy for (cid:15) → . The entanglement dynamics of this model has been investigatedpreviously in Ref. [6] and of the related transverse Ising chain in Ref. [5].Starting from the random half-filled state in Eq. (32) we show in Fig. 2(c) the time evolutionof S (2) ( t ) obtained from exact diagonalizations of a system with L = 1024 lattice sites oververy long times. One notices an extremely slow, but monotonic double-logarithmic increase of S (2) in time consistent with the results in Ref. [6]. The data provide, furthermore, verificationof the upper and lower bounds for S (2) in terms of the number entropy derived in Sec. 2. Weconclude that in the critical bond disordered case the number entropy in the thermodynamiclimit also grows without bounds but extremely slowly, S (2) N ∼ ln ln ln t . In this paper we have considered the entanglement properties of Gaussian states of non-interacting fermions with particle number conservation. We have proven that for any suchsystem—with and without disorder, on arbitrary lattice geometries, and in arbitrary dimensions—the second Rényi entanglement entropy S (2) can be bounded from above and below by the cor-responding number entropy S (2) N . Our result implies an asymptotic scaling S (2) ∝ exp (cid:16) S (2) N (cid:17) ,i.e., a growth in the entanglement entropy always implies a growth, albeit logarithmically9 ciPost Physics Submission slower, of the number entropy and vice versa. While the precise upper and lower bounds havebeen derived for S (2) , all Rényi entropies are expected to show the same asymptotic scalingwith time or length. The connection between a growth in the entanglement entropy and alogarithmic slower growth in the corresponding number entropy is thus expected to hold forall Rényi entanglement entropies including the von-Neumann entanglement entropy.Apart from being of fundamental importance for our understanding of entanglement infermionic systems with particle number conservation, the bounds derived here are also usefulfor experiments on cold atomic gases. In such systems a measurement of the particle-numberdistribution function p ( n, t ) is possible [3] allowing to obtain any Rényi number entropy. De-termining the entire configurational entropy and thus the full entanglement entropy experi-mentally, on the other hand, remains an open issue. Here our results provide an avenue toobtain the asymptotic scaling of the entanglement entropy from p ( n, t ) alone.An interesting question is if similar relations between entanglement and number entropiesalso exist for interacting fermionic systems with particle number conservation. This questionwill be studied in a forthcoming publication [9]. Acknowledgements
J.S. acknowledges support by the Natural Sciences and Engineering Research Council (NSERC,Canada) and by the Deutsche Forschungsgemeinschaft (DFG) via Research Unit FOR 2316.We are grateful for the computing resources and support provided by Compute Canada andWestgrid. M.K., R.U., and M.F. acknowledge financial support from the Deutsche Forschungs-gemeinschaft (DFG) via SFB TR 185, project number 277625399. M. K. would like to thankJ. Léonard and M. Greiner for hospitality and fruitful discussions.
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