On the complex behaviour of the density in composite quantum systems
Filiberto Ares, José G. Esteve, Fernando Falceto, Alberto Usón
OOn the complex behaviour of the density in composite quantumsystems.
Filiberto Ares ∗ International Institute of Physics, UFRN, 59078-970, Natal, RN, Brazil
Jos´e G. Esteve † and Fernando Falceto ‡ Departamento de F´ısica Te´orica, Universidad de Zaragoza, 50009 Zaragoza, SpainInstituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI) andCentro de Astropart´ıculas y F´ısica de Altas Energ´ıas (CAPA) 50009 Zaragoza, Spain
Alberto Us´on § Instituto de F´ısica Corpuscular (IFIC),CSIC & Universitat de Val`encia, 46980 Valencia, Spain.
In this paper, we study how the probability of presence of a particle is distributedbetween the two parts of a composite system. We uncover that the difference ofprobability depends on the energy in a striking way and show the pattern of thisdistribution. We discuss the main features of the latter and explain analyticallythose that we understand. For the rest, we formulate some conjectures that we arenot able to prove at present but can be supported by numerical experiments.
I. INTRODUCTION
In spite of its apparent simplicity, many-body quantum systems in one dimension haveturned out to be very useful models in order to understand and unravel many differentphenomena, from entanglement [1–6] and quantum information [7, 8] to new phases ofmatter [9, 10] and quantum chaos [11]. Moreover, the development of experimental tech- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: auson@ific.uv.es a r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r niques in cold atoms, ion traps and polarized molecules has recently allowed to simulatethese systems in the laboratory [12–18].Two of the most studied unidimensional many-body quantum systems are the tightbinding model and the Su-Schrieffer-Heeger (SSH) model. The tight binding model con-sists of a lattice with a fixed number of free fermions which can hop from one site to thenext one with a given probability. In the simplest version of the model, the sites of thelattice (the position of the atoms) are fixed and the hopping probability (the hopping in-tegral) is constant along the chain. Physically, this system can be seen as a toy model fora one-dimensional metal. It can also be mapped into the XX spin chain via the Jordan-Wigner transformation. When the atom vibrations are taken into account, the hoppingprobability depends on the position of the nearest sites and, due to the Peierls theorem,the chain dimerizes. In the Born-Oppenheimer approximation the hopping probabilitiesbetween the even-odd and odd-even sites are different. This is the SSH model, whichdescribes a unidimensional insulator. It was firstly introduced to characterize solitons inthe polyacethylene molecule [19–21]. In last years, the SSH model has attracted muchattention since it displays the essential properties of topological insulators [9, 10].In this paper, we take the union of two different systems of this type. That is, weanalyze systems composed by two different tight binding models coupled by special bondswhich we will call contacts. Physically, this situation corresponds to the junction of twometals with different band structure. We may also combine a tight binding model anda SSH model (metal-insulator) or two SSH models (insulator-insulator). This kind ofjunctions were considered in Ref. [22], in which the ground state entanglement entropybetween the two parts is investigated; see also [23, 24]. Systems with two different criticalparts (such as the tight binding model) or with a critical and a non-critical part (likethe SSH model) have been examined from the perspective of conformal invariance [25–28]. Composite free-fermionic systems are also of interest in quantum transport and nonequilibrium physics [29–43], where a typical problem is the analysis of the evolution of theoverall state of two different chains after being joined together (inhomogeneous quench).Here, we consider the one particle states with a definite energy. Depending on itsenergy, the particle is confined in one of the two parts or, on the contrary, is delocalizedalong the whole chain. In this work, we will analyze how the particle distributes betweenthe two parts. In a way, our problem is not how , but where Schr¨odinger’s cat is. Forthis purpose, we will introduce a quantity that we call leaning , defined as the differencebetween the probabilities of finding the particle in each part of the chain. It happensthat the dependence of the leaning on the energy and on the contact between the twoparts is rather non-trivial [44]. The goal of this paper is to characterize and explain thisbehaviour.The paper is organized as follows. In Section II, we introduce the main system understudy, the union of two tight binding models, and the so-called leaning. In Section III,we will see how to compute analytically the leaning of a one-particle configuration. InSection IV, we calculate the spectral density of the whole chain. Sections V and VI aredevoted respectively to analyze the resonant regions and to determine the boundary ofthe clouds of points that appear in the energy-leaning plot. In Section VII, we conjecturethe existence of a measure in the energy-leaning plane that accounts for the density ofpoints in the thermodynamic limit. Finally, in Section VIII, we present our conclusionsand outlook. The paper is complemented with an Appendix where we show that theaverage of the leaning does not dependent on the value of the contact.
II. BASIC SET-UP
As we already mentioned in the introduction, the system that we are going to studyconsists of the union of two tight binding fermionic chains of lengths N and N withdifferent hopping parameters t and t respectively. The ends of the two chains areconnected by means of other hoppings t , t (cid:48) which we call contacts, these will be our maintunable parameters.Therefore, the Hamiltonian of the composite system is H = 12 (cid:32) t N − (cid:88) n =1 ( a † n a n +1 + a † n +1 a n ) + t N − (cid:88) m =1 ( b † m b m +1 + b † m +1 b m )+ t ( a † N b N + b † N a N ) + t (cid:48) ( b † a + a † b ) (cid:17) , (1)where a n , b m are the fermionic annihilation operators associated to every piece of thecomposite chain. Note that the sites are enumerated such that the site n = 1 of thesubchain with hopping t is connected with the site m = 1 of the subchain with hopping t by the contact coupling t (cid:48) and, likewise, the site n = N is connected with the site t t
12 2 t−t −t −tt −t E FIG. 1: In the figure we represent the band structure of one particle states for the two separatesubsystems (on the left) and the combined spectrum when we connect them (on the right). In the lattercase and for particular values of the contacts t , t (cid:48) there could be localized states with energy in thediscrete spectrum, outside the band. Here we are not interested in these states and we do not representthem. m = N by the contact t (that is, the sites of the subchain with hopping t are numberedin opposite direction to those of the subchain with hopping t ).This kind of systems have some interesting properties that deserve further research. Forinstance, at certain values of the contacts it possesses a discrete spectrum with localized,topologically protected states. They behave similarly to those of the topological insulators.These features will be studied elsewhere. Here we are rather interested in the continuousspectrum (in the thermodynamic limit) and more precisely in its one particle statesΨ = (cid:32) N (cid:88) n =1 α n a † n + N (cid:88) m =1 β m b † m (cid:33) | (cid:105) , (2)where | (cid:105) represents the vacuum of the Fock space, i.e. a n | (cid:105) = b m | (cid:105) = 0 ∀ m, n .The continuous spectrum in the composite system has a band structure that is obtainedas a superposition of those corresponding to every of its two parts. In fig. 1 we representthis situation.For definiteness and without lose of generality, we shall take t > t >
0. Then, thestates whose energies are such t > | E | > t > t and hardly penetrate, with exponential decay, in the left hand side.On the contrary, those states with energy in the interval [ − t , t ] are distributed alongthe whole chain. Our concern in this work is how the latter split between the two partsof the chain.Therefore, we decompose the one particle Hilbert space H = H ⊕ H , where H contains the wave functions supported on the left hand side ( β m = 0) and H those supported on the right hand side ( α n = 0) and denote by P and P thecorresponding orthogonal projectors; we shall be interested in the expectation value ofthe difference between these projectors L = (cid:104) Ψ | ( P − P ) | Ψ (cid:105) . We will refer to L as the leaning . It measures the difference between the probabilityof finding the particle in the right hand side and that of finding it in the left one.The leaning associated to a one particle stationary state Ψ E will be denoted by L E . Itis immediate to see that, in the thermodynamic limit, L E = 1 for | E | ∈ [ t , t ], however,when E ∈ [ − t , t ] the leaning depends on the energy in a rather complex way, as it isshown in fig. 2.The rest of the paper is devoted to understand this behaviour as fully as possible.In first place, we see a cloud of points that fills a definite region in the E - L plane,apparently bounded by smooth curves. The cloud is symmetric under the exchange of E to − E . We also observe that the points seem to be randomly distributed inside theregion, except for some range of energy where the points group at some definite valuesof L and lie along visible curves. We call these zones resonant. We will study the originof the previous facts and how they depend on the parameters and size of the system.Further properties of these plots will be discussed along this work. III. ANALYTIC APPROACH.
In this section we will show how to determine analytically the leaning of a giveneigenstate. -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =0.0 t =1.0 t =1.5 N =8 000 N =8 000 L e a n i n g ( L ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =0.0 t =1.0 t =1.618 N =8 000 N =8 000 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =2.0 t =1.0 t =1.5 N =9 000 N =8 000 L e a n i n g ( L ) Energy ( E ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =1.272 t ' =0.0 t =1.0 t =1.618 N =8 000 N =8 000 Energy ( E ) FIG. 2: In these plots we show the dependence of the leaning (in the vertical axis) as a function of theenergy (in horizontal axis) for different values of the hoppings t and t , sizes N and N of the twosubsystems, and contacts t , t (cid:48) . The two plots on the left differ in the size of one of the subsystems N .The two on the right in the value of the contact t . The coefficients α n and β n for an eigenfunction of the Hamiltonian with energy E , suchthat | E | < t < t , can be written as α n = A sin( nk + δ ) , β n = A sin( nk + δ ) , where k j ( E ) = arccos Et j , j = 1 , , and the following gluing conditions should be fulfilled A t sin( N k + k + δ ) − A t sin( N k + δ ) = 0 ,A t sin( N k + δ ) − A t sin( N k + k + δ ) = 0 ,A t sin( δ ) − A t (cid:48) sin( k + δ ) = 0 ,A t (cid:48) sin( k + δ ) − A t sin( δ ) = 0 . (3)From the compatibility of these equations we derive the spectral condition and henceforththe allowed values for E .In order to simplify the analysis, we shall take t (cid:48) = 0 and, therefore, the last twoequations imply δ = δ = 0. We shall show along the paper that this assumption doesnot affect, in fact, the generality of our results.Hence the other two equations can be equivalently written C ≡ A /A = t t sin( N k + k )sin( N k ) = t t sin( N k )sin( N k + k ) , (4)and finally L = N − N C N + N C (5)is determined once the value of E is fixed.Equation (4) is our starting point for the study of plots in fig. 2. In the next sectionswe will focus on different aspects or characteristics of these plots and we will show howthey emerge from (4). But before going to that, we shall insert a paragraph to explainthe symmetry under the exchange of E with − E that we observe in the plots.This is due to a chiral transformation in the states that reverses the sign of the Hamil-tonian. Namely, for a one particle state (2), we define its chiral transformed state byΓΨ = (cid:32) N (cid:88) n =1 ( − n α n a † n − N (cid:88) m =1 ( − m β m b † m (cid:33) | (cid:105) . If N + N is even or t t (cid:48) = 0, the Hamiltonian (1) satisfies Γ H Γ = − H while, for theprojectors P , P , we have Γ P i Γ = P i . That is, the chiral symmetry reverses the sign ofthe energy without changing the leaning of the state. This explains the symmetry in theplots of fig. 2. IV. SPECTRAL DENSITY
In this section we compute the spectral density λ t ( E ) of the composite system inthe thermodynamic limit (when the size of the chain goes to infinity) while keeping therelative size of the two chains fixed, N = ν N and N = ν N . In order to make thecomputation simpler, we will take t (cid:48) = 0.Call Σ t ,N the spectrum of H for given values of t and N , ( t , t , ν , ν remain fixedand, to simplify the notation, are omitted in the symbol used for the spectrum). We areinterested in the region of the spectrum where the bands of the two pieces of the chainoverlap, i. e. E ∈ ( − t , t ). We define the spectral density as λ t ( E ) = lim δE → + δE lim N →∞ N − (cid:93) (Σ t ,N ∩ [ E − δE, E + δE ]) , (6)where the symbol (cid:93) stands for the cardinality of the set.We will show that the density of states is actually independent of t and can be com-puted by simply adding up the density of the two pieces. The latter can be easily estimatedby going from k space, where the points in the spectrum are regularly spaced by intervals π/ ( N i + 1), to the E space. As a result one gets λ t ( E ) = ν k (cid:48) ( E ) + ν k (cid:48) ( E ) π = ν / (cid:112) t − E + ν / (cid:112) t − E π . (7)One might be tempted to approach the problem by using perturbation theory. Actually,if we decompose the Hamiltonian in (1) as H = H + H I with H the unperturbed pieceand the perturbation given by the contact term H I = 12 (cid:16) t ( a † N b N + b † N a N ) + t (cid:48) ( b † a + a † b ) (cid:17) , one immediately sees that for ϕ k , ϕ k (cid:48) eigenstates of H (cid:104) ϕ k | H I | ϕ k (cid:48) (cid:105) = O (cid:18) N (cid:19) , and, therefore, the interaction between the two pieces decreases when the system getslarger. While this is true, if we try to apply the perturbative expansion we have toface a sort of “small denominators” problem, well known in classical perturbation theory.In fact, when N grows, the gaps E m − E m (cid:48) , which appear in the denominators of theperturbative expansion, can be arbitrarily small (even smaller than O (1 /N ) for certainvalues of m, m (cid:48) ) and the perturbative expansion ceases to make sense.Another indication that we are dealing with a non perturbative phenomenon is thefact that, while for t , t (cid:48) = 0 the only two possible values for the leaning are 1 or −
1, forany value of t (cid:54) = 0 we have (with N large enough) states with an arbitrary value for L in the interval [ − , t (cid:48) = 0 in which the equations (4) apply and the spectrumΣ t ,N is given by the solutions for E of the equation t t t = sin( N k + k ) sin( N k + k )sin( N k ) sin( N k ) . (8)Recall that k i and the energy are related by E = t i cos k i .In the two extreme limits, t = 0 and t → ∞ , it is easy to determine Σ t ,N . Actually,for t = 0, Σ ,N is the union of the spectra of the two parts of the composite system withDirichlet boundary conditions, that isΣ ,N = { E m ; m = 1 , . . . , N + N } == (cid:26) t cos (cid:18) πm N + 1 (cid:19) ; m = 1 , . . . , N (cid:27) ∪ (cid:26) t cos (cid:18) πm N + 1 (cid:19) ; m = 1 , . . . , N (cid:27) . Similarly for t → ∞ the spectrum is given byΣ ∞ ,N = { E ∞ m ; m = 1 , . . . , N + N − } == (cid:26) t cos (cid:18) πm N (cid:19) ; m = 1 , . . . , N − (cid:27) ∪ (cid:26) t cos (cid:18) πm N (cid:19) ; m = 1 , . . . , N − (cid:27) . One may notice that in the latter case the spectrum has two points less than for t = 0.Actually, the missing eigenvalues correspond, for large but finite t , to states localized atthe contact whose energies, close to ± t , lie outside the spectral band. When t goes toinfinity the energy of these states diverges.To understand the spectrum for intermediate values of t and E ∈ ( − t , t ) (recall thatwe assume t > t >
0) it is convenient to write (8) in the form t = f ( E ) f ( E ) , where f i ( E ) = E − (cid:113) t i − E cot( N i k i ( E )) , i = 1 , . The crucial observation now is that f (cid:48) i ( E ) = 1 + N − E (cid:112) t i − E cot( N i k i ( E )) + N cot ( N i k i ( E ))is positive provided t i − E > E N i ( N i + 1) . -5 0 5 10 15 20 0.101 0.103 0.105 0.107 0.109 F ( E ) Energy ( E ) FIG. 3: In this plot we represent the function F ( E ) ≡ f ( E ) f ( E ) whose level set F ( E ) = t is thespectrum. The parameters of the system are t = 1 , t = 1 . , N = 1000 , N = 1700. For clarity of theplot we have selected a short range of energies between 0.1 and 0.11. Notice that, as we prove in thetext, the function is monotonic for intervals in which it is positive. We indicate by and the zeros of f ( E ) and f ( E ) respectively (which correspond to the energies E m ) while the vertical lines are theirasymptotes (associated to the energies E ∞ m ): the dot-dashed ones correspond to f and the dashed to f . This means that for any E in the open interval ( − t , t ) and N i = ν i N , as before, thereexists a K such that f (cid:48) i ( E ) > N > K . As we are interested in the large N limit,we may assume that f (cid:48) i ( E ) is positive for any E ∈ ( − t , t ).A consequence of the previous fact is that F ( E ) ≡ f ( E ) f ( E ) is monotonic in everyinterval in which F ( E ) >
0. Then the picture we get is represented in fig. 3.For a point in Σ ,N , say E m , we have F ( E m ) = 0. Now assume F (cid:48) ( E m ) >
0, then for E slightly larger than E m , F ( E ) > F (cid:48) ( E ) > F ( E ) increases with E until we encounter an eigenvalue of H for t = ∞ , say E ∞ m (cid:48) . At this point F ( E ) diverges. This implies that for each value of t we have one andonly one solution for the equation (8) with energy in the interval [ E m , E ∞ m (cid:48) ]. In the case F (cid:48) ( E m ) < H for each value of t with energy in the interval[ E ∞ m (cid:48) , E m ], where E ∞ m (cid:48) is the point in Σ ∞ ,N immediately smaller than E m . Finally, in the1unlikely instance in which F (cid:48) ( E m ) = 0 one necessarily has f ( E m ) = f ( E m ) = 0 andtherefore F (cid:48)(cid:48) ( E m ) = f (cid:48) ( E m ) f (cid:48) ( E m ) >
0. But this later property means that F ( E ) > E in a punctured neighbourhood of E m , hence the arguments above hold and F ( E ) growsmonotonically to infinity when we separate from E m in both directions and approach theimmediate points of Σ ∞ ,N .Due to this fact, we clearly see that given an interval of energies I ⊂ ( − t , t ) thenumber of stationary states with energies in I varies at most by two with t , namely2 ≥ (cid:93) ( I ∩ Σ t ) − (cid:93) ( I ∩ Σ ) ≥ − . Therefore, the density of states λ t derived from (6) is independent of t and, as weanticipated at the beginning of this section, it can be written as the sum of the densitiesfor the two chains, i. e. λ t ( E ) = ν / (cid:112) t − E + ν / (cid:112) t − E π . (9)For further purposes we also introduce the spectral density normalized in the interval ofenergies ( − t , t ) in which the stationary states extend along the whole chainˆ λ t ( E ) = ν / (cid:112) t − E + ν / (cid:112) t − E πν + 2 ν arcsin( t /t ) . This result has been checked numerically and the results are presented in fig. 4. Therethe histogram for the energy of the states, determined numerically, is plotted againstthe theoretical curve obtained above. It is quite manifest the perfect agreement of bothresults.Notice that we have shown, in passing, that for E ∈ ( − t , t ), t (cid:54) = 0 or ∞ and N largeenough, there is no degeneracy in the spectrum of H . This is very good news becausein the case of a degenerate eigenvalue the leaning of the eigenstates is not well defined;it means that the accidental degeneracy (and the undefinition of the leaning) can onlypossibly happen in the extreme cases t = 0 or ∞ , and this only after fine tunning t , t and N . V. RESONANT REGIONS
We turn now our attention to other feature of the plots in fig. 2: the existence ofresonances.2
Numerical t =2.0 t ' =0.0 t =1.0 t =1.5 N =8 000 N =8 000 F r e q u e n c y Energy ( E ) Theoretical FIG. 4: In the histogram we represent the frequency of the spectrum with bins of width 0 .
01 units ofenergy. The discontinuous line is the theoretical prediction obtained from (9).
We call resonant regions those windows in energy where the points in the plot accu-mulate at a few definite values of the leaning, like for instance in the central zone of theupper-left panel of fig. 2. Outside these regions the points (
E, L E ) spread and the cloudseems to fill the whole allowed band. We have observed that the width of the resonancesdepends on N , in such a way that they shrink when N grows.To understand the reason for these facts let us consider a solution E for (4) C = t t sin(( N + 1) k ( E ))sin( N k ( E )) = t t sin( N k ( E ))sin(( N + 1) k ( E )) , and expand around this value C + ∆ C = t t sin(( N + 1)( k ( E ) + k (cid:48) ( E )∆ E + ... ))sin( N ( k ( E ) + k (cid:48) ( E )∆ E + ... ))= t t sin( N ( k ( E ) + k (cid:48) ( E )∆ E + ... ))sin(( N + 1)( k ( E ) + k (cid:48) ( E )∆ E + ... )) . (10)Now, writting N i = ν i N for some integer N we impose the resonant condition ν k (cid:48) ( E ) ν k (cid:48) ( E ) = m m , m , m ∈ Z . or in other words m = r ν k (cid:48) ( E ) , m = r ν k (cid:48) ( E ) , for some r ∈ R , m and m are relative primes. Then, for large N , it is clearthat if we take ∆ E = πrnN , n ∈ Z , the first subleading terms in the expansion above (for n = O (1)) are N i k (cid:48) i ( E )∆ E = πm i n and one easily checks that E + ∆ E is another solution of (4) upto corrections of order O (1 /N ). The relevant fact is that these solutions give the same value for C , and thereforefor the leaning, upto O (1 /N ) terms. This explains the smooth curves of the Lissajoustype that we observe for certain values of the energy.To determine the width of the window we must consider the subleading corrections.They pose a limit to the validity of our approximation. To be specific let us focus in theargument of the first numerator in (10) and write( N + 1) k ( E + ∆ E ) = ( N + 1) k ( E ) + N k (cid:48) ( E )∆ E + k (cid:48) ( E )∆ E + 12 N k (cid:48)(cid:48) ( E )(∆ E ) + . . . (11)As we discussed before, given our choice of ∆ E = πrn/N the second term of the expansiongives a contribution πm n and hence, inside the sinus function reduces to a global ± πrnN k (cid:48) ( E ) + ν ( πrn ) N k (cid:48)(cid:48) ( E ) . Note that for n = O ( √ N ) the second term above is of order O (1) and our approximationceases to be valid. Hence we conclude that the width of the resonant windows scales like1 / √ N .This is true provided k (cid:48)(cid:48) ( E ) (cid:54) = 0 which does not hold at E = 0. In this case we mustgo one step further in the expansion, so that the first corrections are πrnN k (cid:48) (0) + ν ( πrn ) N k (cid:48)(cid:48)(cid:48) (0) , which are of order O (1) for n = O ( N / ) and the validity of our approach extends as faras ∆ E = O ( N − / ). This explains why the resonances at the center of the plot, whenthey occur, are much wider.4A different question is: how many curves in the E - L plane are there around a resonantvalue E ?, or in other words, how many well separated values for the leaning do we getfor values of the energy near E ? To answer this question we may use the results for thespectral density that we derived in the previous section.First, consider that the separation between two consecutive values for the energy withthe same value of the leaning (upto O (1 /N )) is δE = πr/N. Now, combining this with the spectral density (9) we can obtain the number of statesbetween two consecutive repetitions of the leaning, i. e. the number of curves at theoutset of the resonance. Namely
N λ t ( E ) δE = N ν k (cid:48) ( E ) + ν k (cid:48) ( E ) π πrN = m + m , where for the last equality the conditions for resonance, m i = rν i k (cid:48) i ( E ) have been used.Then, we conclude that the number of curves that we obtain at the resonant value isprecisely m + m . These results are illustrated in fig. 5 where we show the plot for theleaning and we superimpose some resonant values obtained according to our derivation.Note that the number of Lissajous type curves is also correctly predicted.We must add that at some special points, like for instance at E = 0, there may appearsome degeneracy for the leaning which results in a smaller number of different values forit. This is clearly observed in the upper-left panel of fig. (5), where the crossing of thecurves at E = 0 reduces from five to three the number of allowed values for the leaning.Of course, these degeneracy occurs only at E = 0 and is broken in its vicinity, recoveringthere the right number of curves.As it is clear from the plots and from the discussion in this section, the resonancewindows do not depend on the contact t of the two subsystems but they are sensible toits size. In the next section we will discuss a property of the plots that behaves exactly inthe opposite way, i. e. it is independent of the size and varies with the contact coupling t .5 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =0.0 t =1.0 t =1.5 N =8 000 N =8 000 3:2 8:5 5:3 2:1 L e a n i n g ( L ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =0.0 t =1.0 t =1.618 N =8 000 N =8 000 13:8 5:3 7:4 2:1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =2.0 t ' =0.0 t =1.0 t =1.5 N =9 000 N =8 000 27:1622:1317:1012:7 7:49:5 2:1 L e a n i n g ( L ) Energy ( E ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 t =1.272 t ' =0.0 t =1.0 t =1.618 N =8 000 N =8 000 5:3 7:4 2:1 Energy ( E ) FIG. 5: In these figures we show the plots of the leaning versus energy for different composed systemswith the hopping, sizes and contacts that appear in every panel. The vertical lines mark the value ofthe energy for which we expect a resonance according to the discussion in section V. The pair ofnumbers at every vertical line is the theoretical relation between the two resonant frequencies. Noticethat the number of Lissajous type curves at every resonance coincides with the sum of those twonumbers, as it is explained in the text.
VI. THE BOUNDARY.
In this section, we will find an analytical expression for the curves that limit thedistribution of points in the E - L plane.We are interested in the boundaries of the cloud in the E - L plane that are valid in thethermodynamic limit, when N , N → ∞ . To achieve this goal we first look for an upperand lower bound of C at a given value of the energy. It is clear that, given the monotonicdecreasing character of the leaning in (5) with C , the latter leads respectively to lowerand upper bounds for L .It also happens that, as we show below, it is possible to obtain bounds for C which6are valid for any N , N and are optimal, in the sense that they can be approached asmuch as we want by varying the size of the two parts of the system.As we look for bounds for C independent of N , N , it makes sense to replace N k ( E )and N k ( E ) inside the trigonometric functions of (4) by two continuous variables ξ , ξ ∈ [0 , π ) independent, in principle, of E .To justify this replacement consider, on the one hand side, that we are looking forbounds for C , then if we relax the conditions for the equation (4) we are sure that thebounds for the modified equation are still valid for the original one. On the other hand,we may argue that by considering N i large enough we may approach any value ξ i as muchas we want which implies that our bounds, valid for any N i , are optimal.To proceed, we replace the equation (4) by t t sin( ξ + k ( E ))sin ξ = t t sin ξ sin( ξ + k ( E )) , (12)and, consequently, C = t t sin ( ξ + k ( E ))sin ξ . (13)If we replace ξ i by the new variables z i = cos k i ( E ) + sin k i ( E ) cot ξ i , i = 1 , , then equation (12) is easily solved z = t ¯ t z − , where for later convenience we have introduced the dual contact ¯ t = t t /t . Now we usethe previous relation to express C in terms of the single variable z to obtain C = sin k t sin k ¯ t z − t ¯ t cos k i + t z − ¯ z − k i + z − . (14)Then, we simply have to determine the maximum and minimum of (14) as a functionof z , for every value of the energy. The task is, of course, straightforward but somehowpainful. The final expressions are rather cumbersome and of little interest to us. Insteadof writting down the analytic expression for the upper and lower bound of C and theleaning, we prefer to plot it for some special cases.Notice that, as the bounds for C are independent of N , N , those of L only dependon ν and ν , namely L = ν C − ν ν C + ν . -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Boundary t =2.0 t ' =0.0 t =1.0 t =1.5 t =0.75 t ' =0.0 N =9 000 N =8 000 L e a n i n g ( L ) Energy ( E ) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 Boundary t =1.27 t ' =1.27 t =1.0 t =1.62 t =1.27 t ' =0.0 N =8 000 N =8 000 Energy (E)
FIG. 6: In these plots we show the distribution for the leaning versus the energy of different chains. Inthe left panel the two systems are related by duality t ↔ t t /t . It is clear that although the twodistributions of points (represented in different colors) are not identical they fill the same region. In thepanel on the right we represent the selfdual case t = √ t t and the two distributions differ by the largest contact t (cid:48) . The situation is similar to the previous one, different distribution but the sameboundaries. The discontinuous lines represent the theoretical boundaries obtained as explained in thetext. The comparison between the analytic and numerical results for different relative sizesand values of the contact are collected in the plots of fig. 6.An interesting fact that we would like to emphasize is the duality between higher andlower contact. In fact we can show that the bounds are unchanged if we replace t by¯ t = t t /t . A duality that maps zero to infinity or, as we mentioned above, higher to lower couplingconstant.The duality is easily proven starting from (14). There we see that the value of C isunchanged if we replace t by ¯ t and simultaneously z by z − . Therefore the maximumand minimum for C are unchanged under the duality.In the left panel of fig. 6, we plot the leaning for dual values of the contact and we seethat the respective allowed regions perfectly match. On the right side of this figure, theleaning for a self dual value of the contact t = √ t t is plotted.Let us discuss now which is the effect of taking t (cid:48) (cid:54) = 0. First notice that if we replacein (3) δ i by δ i − N i k i ( E ) / , i = 1 ,
2, which is a simple redefinition of the unknowns,8we obtain a more symmetric equation. Actually, replacing now δ i + N i k i / ξ i and N i k i / − δ i − k i by η i we obtain the equivalent equations: C = t t sin( ξ + k ( E ))sin ξ = t t sin ξ sin( ξ + k ( E )) ,C = t t (cid:48) sin( η + k ( E ))sin η = t (cid:48) t sin η sin( η + k ( E )) . (15)Notice that the second line is like the first one by simply replacing ξ by η and t by t (cid:48) .Then we have two equations for obtaining bounds on C , one with t and another with t (cid:48) . It happens that the smaller t + t is (its minimum value is attained for the selfdualcase t = √ t t ) the more restrictive the bounds are, and this applies both for the upperand the lower bound.Consequently, only one of the contacts matters for determining the boundaries of theallowed region in E - L plane, namely the one closer to the selfdual value or equivalentlythe one with the least value for t + t .More formally, if we introduce an order relation defined by: t ≺ t (cid:48) if and only if t + t < t (cid:48) + t (cid:48) , the smallest of the two contacts determines the shape of the cloud.This can be checked in the numerical experiments where it is apparent that a modifi-cation of the larger contact does not alter the shape of the cloud, as can actually be seenin the right plot of fig. 6. VII. THE PROBABILITY MEASURE
If we examine the different plots of previous sections, one observes that for most ofthe allowed region the points that represent (
E, L ) pairs form a cloud more dense nearthe boundaries and more sparse at the middle. Of course, the previous is not true atthe resonance windows, where the points form definite curves. But, as we discussed insection V, one can prove that the resonant regions shrink with the size of the system andeventually disappear in the thermodynamic limit.Then the question that might have sense and we will study is whether there is a measurein the E - L plane that represents the density of points in the thermodynamic limit andhow it depends on the parameters of the system. We believe that such a measure existsand for t , t (cid:48) (cid:54) = 0 is absolutely continuous with respect to the Lebesgue measure.9To be more precise, take N = ν N and N = ν N , let the hopping parameters be asusual t > t > t , t (cid:48) . Now we define the following probability measure onthe Borelians S ⊂ X = [ − t , t ] × [ − , µ N ( S ) = K N (cid:16) (cid:93) { Ψ E | ( E, L E ) ∈ S } (cid:17) , where by (cid:93) we denote the cardinality of the set, Ψ E is an eigenfunction of the Hamiltonianof the composite system with chains of length ν N and ν N , and L E is the leaning ofΨ E . K N is the appropriate normalization constant to obtain a probability measure, i. e. µ N ( X ) = 1.We assert that these measures converge when N → ∞ , to a probability measure µ onthe Borelians of X .We do not have a proof for the existence of µ , only numerical evidences based on thegood behaviour of different expectation values and its apparent convergence with N asillustrated in fig. 7. There we show the plot of the leaning for different values of N and N with the same relative sizes N /N and the running average (cid:104) L n (cid:105) for n = 1 thatcorresponds to the lowest (blurry) curve and going upwards n = 7 , ,
2. We see that thecurves for different sizes of the system agree to a large extent and they seem to havea smooth large N limit. The well defined limit for the different moments is a strongnumerical indication of the existence of a Borelian measure when N → ∞ .Sometimes it will be important to emphasize the dependence of the limiting measureon some of the parameters of the theory, in that case we will write those parameters assubindices. In the following, we will focus mainly on the dependence of µ N or µ on thecontacts, so we will write it µ N,t ,t (cid:48) or µ t ,t (cid:48) . The first observation is that due to the parityinvariance of the Hamiltonian µ N,t ,t (cid:48) = µ N,t (cid:48) ,t , therefore if the limit exists we must have µ t ,t (cid:48) = µ t (cid:48) ,t .In general, we do not know how to determine µ t ,t (cid:48) , only when t = t (cid:48) = 0 (or ∞ )because, in this case, the chain splits into two independent homogeneous systems withwell defined leaning. Therefore, µ , can be obtained as the normalized sum of the corre-sponding spectral measures. Hence, we haved µ , = (cid:0) ρ ( E ) δ ( L + 1) + ρ ( E ) δ ( L − (cid:1) d E d L, -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 t =1.0 t =2.0 t =1.5 t ' =2.0 N =2 250 N =2 000 N =4 500 N =4 000 N =9 000 N =8 000 L e a n i n g ( L ) Energy ( E ) FIG. 7: In this plot we represent the distribution of points in the L − E plane for chains that differ insize, while sharing the proportion between the two subsystems. Every size is represented in a differentcolor. The curves are the running average (over 200 points) of L n for n = 1 , , , N → ∞ , seems to exist. where ρ i ( E ) = ν i / (cid:112) t i − E πν + 2 ν arcsin( t /t ) , and δ represents the Dirac delta function. That is, in this case the measure is supportedin the upper and lower boundary of X , with L = ± µ . However, basedon numerical experiments and some analytical hints we can establish some conjecturesthat we introduce in the following. According to the discussion of section VI, the support of the measure is in the regionbetween the curves L max ( E ) , L min ( E ). Moreover, we proved in that section that thelimiting curves depend only on the smallest contact i. e. they depend only on t provided t ≺ t (cid:48) .The stronger conjecture that is supported in different numerical experiments shown1in fig. 8, is that not only the support of the measure depends only on the lowest contact, but the measure itself has the same property i. e. we conjecture µ t ,t (cid:48) = µ t ,t (cid:48)(cid:48) , if t ≺ t (cid:48) , t (cid:48)(cid:48) . For this reason, and in order to simplify the notation, from now on we will refer tothe measure writing only the smallest contact µ t . Another property of the limiting curves that we showed in section VI is its invarianceunder duality t (cid:55)→ ¯ t = t t /t . Based again in numerical evidences, see fig. 8 foran example, we conjecture µ t = µ ¯ t . That is, we assert that not only the support of the measure is left invariant underthe duality transformation, but also the measure itself.We do not have any analytic argument to substantiate these two last properties. Butif we compute the running moments in L of the distributions, when varying t (cid:48) or whenreplacing t by t , we find that they are as close as they possibly could be. This is shownin fig. 8. From the dependence of the clouds with t , illustrated in fig. 9, it seems reasonableto conjecture that, except for t = t (cid:48) = 0 (or ∞ ), the measure is absolutely continu-ous with respect to the Lebesgue measure. That is, there is a function σ t ∈ L ( E, L )such that d µ t = σ t ( E, L ) d E d L. Although we can not determine µ t , based on the arguments of section IV we assertthat the marginal distribution for E does not depend on t . Therefore, we can writeˆ λ t ( E ) ≡ (cid:90) − σ t ( E, L )d L = ν / (cid:112) t − E + ν / (cid:112) t − E ν arcsin( t /t ) + ν π , where the right hand side has been computed using the spectral density obtainedin section IV or, alternatively, the measure that we determined before for t = 0. Finally, we can prove that the expected value of L at fixed value of the energy isagain independent of t . Indeed, using again the probability measure at t = 0, we2 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 t =1.0 t =1.5 t =2.0 t ' =2.0 t =2.0 t ' =0.0 t =0.75 t ' =0.0 N =9 000 N =8 000 L e a n i n g ( L ) Energy ( E ) FIG. 8: In this plot we show the distribution of points and running average of the momenta L n ( n = 1 , , , t (cid:48) = 2 . . t = 2 by its dual value t t /t = 0 .
75. We seethat in this case the running average of the three different chains coincide. This coincidence is morestriking if we look at the two consecutive magnifying insets. have (cid:104) L (cid:105) E ≡ ˆ λ t ( E ) − (cid:90) − σ t ( E, L ) L d L = ν / (cid:112) t − E − ν / (cid:112) t − E ν / (cid:112) t − E + ν / (cid:112) t − E . (16)As it is explained in the appendix, this result can be proven by estimating therunning average of L in the large N limit. Numerical experiments also support ourresult. These are shown in fig. 9 where we present the running average of the leaningthat we obtained numerically for different values of t (the thick curve transversingthe cloud in its lower part), and we check that it is independent of t and agreesextremely well with the conjectured predictions. If we look at the two magnifyinginsets it is clear that curves for different t , represented in different colors, agreeperfectly. They also coincide with the theoretical value of (16) that we plot in whiteand is the line that cuts right in the middle the numerical curves. In contrast, thenumerical curves for (cid:104) L (cid:105) for different values of t (in different colors, in the upperpart of the plot) are well separated.3 -1-0.5 0 0.5 1 -1 -0.5 0 0.5 1 t =1.0 t =1.5 t =3.0 t ' =0.0 t =2.0 t ' =0.0 t =1.225 t ' =0.00 N =9 000 N =8 000 L e a n i n g ( L ) Energy ( E ) FIG. 9: In this plot we show the distribution of points and running average (cid:104) L (cid:105) and (cid:104) L (cid:105) , over 200points, for chains that differ in the contact, t , and are represented in different colors. While the threecurves corresponding to (cid:104) L (cid:105) are clearly different, those for (cid:104) L (cid:105) coincide as it is made manifest in theinsets. The curve in white (visible in the insets) represent the predicted value. The explanation for thisfact is presented in the appendix. VIII. CONCLUSIONS, GENERALIZATIONS AND EXTENSIONS
We have shown that the probability of presence for one particle in a composite system,characterized here with the leaning, follows a rather intricate pattern.We have been able to unravel many of its properties, like the nature of the resonantregions, the boundaries of the allowed region, its independence of the largest contact, theduality between large and small contact coupling or the universal properties of the averageleaning.It is interesting to remark that the leaning, in the thermodynamic limit, is definitely anon perturbative property. This can be argued in several ways: first, the duality discussedin sections VI and VII and mentioned above allows to identify the small and large couplingconstant region t ↔ t t /t ; second, we observe that for no matter how little t (cid:54) = 0 wemay find (for N large enough) states with a leaning arbitrary close to 0, very far from theunperturbed system where | L | = 1 for any state.4Suggestively enough, the non perturbative character of the leaning can be traced backto the existence (for large N ) of small denominators in the quantum perturbative expan-sion. This is reminiscent of the same phenomena in the canonical perturbation expansionin classical mechanics, which is one of the essential ingredients for the existence of nonintegrable systems and chaotic dynamics.Here, of course, we may not have sensitive dependence of initial conditions for theevolution, as the dynamics is linear; but, instead, the expected position of one particlestationary states depends sensitively of its energy.While, as we just stated, the leaning depends critically on the energy of the stationarystate, we may obtain a predictable result if we consider the average over a range ofenergy. This is observed numerically and can be rigorously proven. For the latter proofwe have to get rid of the small denominators problem and it is interesting to remark thatthe way we proceed is very much reminiscent of the analogous strategy for the KAMtheorem in classical perturbation theory: we fix initially a cut-off that suppresses thesmall denominators, then we can proceed with the different estimates before removingthe cut-off.An open problem is to compute the density in the E - L plane for the stationary statesin the thermodynamic limit. There are strong numerical indications that such a Borelianmeasure exists, it is absolutely continuous with respect to the Lebesgue measure (exceptfor t and t (cid:48) equal to 0 or ∞ ) and varies continuously with t in the total variationtopology (the convergence for t , t (cid:48) → , ∞ should occur only in the weak topology).Let us remark that the previous results were obtained for the simplest kind of systemscomposed of two homogeneous tight-binding chains joined at every end by links withdifferent hopping parameter. However, the behaviour that we have described in the paperseems rather universal and it has been observed for the SSH chain (alternating hopping),for the Ising chain and, more generally, for the XY spin chain. One can consider alsodifferent types of contacts (of finite range) without affecting the essential properties of theleaning. Particular examples beyond tight binding models will be presented elsewhere.Finally, we would like to comment that all the systems mentioned in the previousparagraph can be mapped to free fermionic chains and therefore can be analyzed withrelatively ease. It would be interesting to go beyond that and explore the behaviour ofthe leaning for systems composed of interacting chains like the Hubbard model or others.5We plan to approach these problems in our future research. Acknowledgments:
Research partially supported by grants E21 17R, DGIID- DGAand PGC2018-095328-B-100, MINECO (Spain). FA is supported by Brazilian MinistriesMEC and MCTIC and acknowledges the warm hospitality and support of Departamentode F´ısica Te´orica, Universidad de Zaragoza, during several stages of this work.
Appendix A
In this appendix, we present a proof of the invariance of the leaning average under achange of the contact.More precisely, given a fermionic chain like the one described in section II (0 < t < t and, for simplicity, t (cid:48) = 0) we compute the running average of the leaning in an intervalof energy [ E − ∆ E, E + ∆ E ] ⊂ ( − t , t ).For that, let us denote the spectra for contact t and 0 respectively by Σ t ,N = { E m ; m ∈ M } and Σ ,N = { E (cid:101) m ; (cid:101) m ∈ M } as before, and the eigenstates by ψ m and ϕ (cid:101) m respectively.Introduce R ⊂ M such that { E m ; m ∈ R } = Σ t ,N ∩ [ E − ∆ E, E + ∆ E ] , and similarly R ⊂ M for Σ ,N ∩ [ E − ∆ E, E + ∆ E ]. With this data we define the densitymatrices ρ = 1 (cid:93) ( R ) (cid:88) m ∈ R | ψ m (cid:105) (cid:104) ψ m | , and ρ = 1 (cid:93) ( R ) (cid:88) (cid:101) m ∈ R | ϕ (cid:101) m (cid:105) (cid:104) ϕ (cid:101) m | . We will prove that in the thermodynamic limitlim N →∞ Tr (( ρ − ρ )( P − P )) = 0 . Or in other words, the leaning averaged over a range of energy does not depend on thecontact.To show it we express ρ in terms of the unperturbed basis, ρ = 1 (cid:93) ( R ) (cid:88) r ∈ R (cid:88) (cid:101) m, (cid:101) m (cid:48) ∈ M U (cid:101) m,r U (cid:101) m (cid:48) ,r | ϕ (cid:101) m (cid:105) (cid:104) ϕ (cid:101) m (cid:48) | , U is the unitary matrix corresponding to the change of basis, that is U (cid:101) m,r = (cid:104) ϕ (cid:101) m | ψ r (cid:105) .Now we decompose the sets of indices into two disjoint sets, M = R ∪ P and M = R ∪ P , and write ρ = 1 (cid:93) ( R ) (cid:88) r ∈ R (cid:88) (cid:101) r, (cid:101) r (cid:48) ∈ R U (cid:101) r,r U (cid:101) r (cid:48) ,r | ϕ (cid:101) r (cid:105) (cid:104) ϕ (cid:101) r (cid:48) | + (cid:88) (cid:101) r ∈ R , (cid:101) p ∈ P U (cid:101) r,r U (cid:101) p,r | ϕ (cid:101) r (cid:105) (cid:104) ϕ (cid:101) p | + (cid:88) (cid:101) r ∈ R , (cid:101) p ∈ P U (cid:101) p,r U (cid:101) r,r | ϕ (cid:101) p (cid:105) (cid:104) ϕ (cid:101) r | + (cid:88) (cid:101) p, (cid:101) p (cid:48) ∈ P U (cid:101) p,r U (cid:101) p (cid:48) ,r | ϕ (cid:101) p (cid:105) (cid:104) ϕ (cid:101) p (cid:48) | . (A1)Due to the orthonormality properties of U we can replace the first term above (cid:88) r ∈ R (cid:88) (cid:101) r, (cid:101) r (cid:48) ∈ R U (cid:101) r,r U (cid:101) r (cid:48) ,r | ϕ (cid:101) r (cid:105) (cid:104) ϕ (cid:101) r (cid:48) | = (cid:88) (cid:101) r ∈ R | ϕ (cid:101) r (cid:105) (cid:104) ϕ (cid:101) r | − (cid:88) p ∈ P (cid:88) (cid:101) r, (cid:101) r (cid:48) ∈ R U (cid:101) r,p U (cid:101) r (cid:48) ,p | ϕ (cid:101) r (cid:105) (cid:104) ϕ (cid:101) r (cid:48) | . With the above replacement and taking into account that the stationary states of theunperturbed Hamiltonian are also eigenstates of the leaning operator:( P − P ) ϕ (cid:101) m = (cid:15) (cid:101) m ϕ (cid:101) m with (cid:15) (cid:101) m = ± , we can writeTr( ρ ( P − P )) = (cid:93) ( R ) (cid:93) ( R ) Tr( ρ ( P − P ))+ 1 (cid:93) ( R ) (cid:88) (cid:101) p ∈ P ,r ∈ R (cid:15) (cid:101) p | U (cid:101) pr | − (cid:88) (cid:101) r ∈ R ,p ∈ P (cid:15) (cid:101) r | U (cid:101) rp | . (A2)Now we must estimate the matrix elements of U . This can be done by means of theidentity | U (cid:101) mm | = | (cid:104) ϕ (cid:101) m | H I | ψ m (cid:105) | ( E (cid:101) m − E m ) , (A3)where we can take advantage of the fact that ϕ (cid:101) m and ψ m are extended wave functionsand H I acts only locally at the interface of the two components of the chain. Indeed, onehas | (cid:104) ϕ (cid:101) m | H I | ψ m (cid:105) | = O ( N − ) . The problem, however, is that the denominator ( E (cid:101) m − E m ) for large N and particularvalues of m and (cid:101) m can be very small (even smaller than 1 /N ). This is an instance of thesmall denominator problem in quantum mechanics.7To avoid this potential divergence we must introduce a cut-off. A similar strategy(although much simpler in this case) to the one followed for proving the KAM theoremin classical mechanics.For instance, to estimate the second term in (A2), | F | ≡ (cid:93) ( R ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:101) p ∈ P ,r ∈ R (cid:15) (cid:101) p | U (cid:101) pr | (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:93) ( R ) (cid:88) (cid:101) p ∈ P ,r ∈ R | U (cid:101) pr | , we fix a range of energy δE < ∆ E and decompose the set R into two disjoint subsets R = R < ∪ R > with R < = { r ∈ R s. t. | E r − E | < ∆ E − δE } , which implies that | E (cid:101) p − E r < | > δE , for any (cid:101) p ∈ P , r < ∈ R < , and the small denominatorproblem is relegated to the set of indices R > .Thus we have | F | ≤ (cid:93) ( R ) (cid:88) (cid:101) p ∈ P ,r > ∈ R > | U (cid:101) pr > | + (cid:88) (cid:101) p ∈ P ,r < ∈ R < | U (cid:101) pr < | ≤ (cid:93) ( R > ) (cid:93) ( R ) + 1 (cid:93) ( R ) (cid:88) (cid:101) p ∈ P ,r < ∈ R < | (cid:104) ϕ (cid:101) p | H I | ψ r < (cid:105) | ( δE ) , (A4)where we have used the normalization condition for the rows of U to estimate the firstterm and (A3) together with the bound for the energy difference for the second.Now (cid:88) (cid:101) p ∈ P | (cid:104) ϕ (cid:101) p | H I | ψ r < (cid:105) | ≤ (cid:104) ψ r < | H I | ψ r < (cid:105) ≤ t M , with M = min (cid:40) ν N − t (cid:112) t − ( | E | + ∆ E ) , ν N − t (cid:112) t − ( | E | + ∆ E ) (cid:41) . The important fact here is that for fixed | E | + ∆ E < t < t we have M = O ( N ) for large N . Inserting this into (A4) and performing the sum, we obtain | F | ≤ (cid:93) ( R > ) (cid:93) ( R ) + 2 t M ( δE ) . The same estimate can be used for the third term on the right hand side of (A2) to get | Tr(( ρ − ρ )( P − P )) | ≤ | (cid:93) ( R ) − (cid:93) ( R ) | (cid:93) ( R ) | Tr( ρ ( P − P )) | + 2 (cid:93) ( R > ) (cid:93) ( R ) + 4 t M ( δE ) . (A5)8From the spectrum of H we derive (cid:93) ( R ) ≥ Nπ (cid:18) ν t + ν t (cid:19) ∆ E − , and, from the results in section IV, we have | (cid:93) ( R ) − (cid:93) ( R ) | ≤ . Likewise we can obtain the upper bound (cid:93) ( R > ) ≤ Nπ (cid:32) ν (cid:112) t − ( | E | + ∆ E ) + ν (cid:112) t − ( | E | + ∆ E ) (cid:33) δE + 2 . Using the previous estimates and choosing the cutoff such that δE → , but N ( δE ) → ∞ , when N → ∞ , e. g. δE = N − / , we obtain from (A5)lim N →∞ Tr(( ρ − ρ )( P − P )) = 0 , as stated before. [1] G. 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