Equilibration properties of small quantum systems: further examples
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Equilibration properties of small quantum systems:further examples
J M Luck
Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA and CNRS,91191 Gif-sur-Yvette, France
Abstract.
It has been proposed to investigate the equilibration properties of asmall isolated quantum system by means of the matrix of asymptotic transitionprobabilities in some preferential basis. The trace T of this matrix measuresthe degree of equilibration of the system prepared in a typical state of thepreferential basis. This quantity may vary between unity (ideal equilibration) andthe dimension N of the Hilbert space (no equilibration at all). Here we analyzeseveral examples of simple systems where the behavior of T can be investigatedby analytical means. We first study the statistics of T when the Hamiltoniangoverning the dynamics is random and drawn from a distribution invariant underthe group U( N ) or O( N ). We then investigate a quantum spin S in a tiltedmagnetic field making an arbitrary angle with the preferred quantization axis, aswell as a tight-binding particle on a finite electrified chain. The last two casesprovide examples of the interesting situation where varying a system parameter– such as the tilt angle or the electric field – through some scaling regime inducesa continuous crossover from good to bad equilibration properties.E-mail: [email protected] quilibration properties of small quantum systems: further examples
1. Introduction
The last decade has witnessed an immense activity around the themes of therma-lization and equilibration in isolated quantum systems (see [1, 2, 3, 4, 5, 6, 7] forcomprehensive reviews). This renewed interest in an old classic subject of QuantumMechanics was mostly triggered by progress on the experimental side in cold-atomphysics. The physical mechanisms underling equilibration and thermalization are farfrom obvious. On the one hand, an isolated quantum system evolves by some unitarydynamics, and therefore keeps the memory of its initial state. On the other hand, ifthe system is large enough, concepts from Statistical Mechanics can be expected toapply at least approximately. Most recent works have dealt with large many-bodyquantum systems, the key issues including the description of a small subsystem by athermodynamical ensemble (either microcanonical or canonical), the thermalizationproperties of single highly-excited states, and the role of conservation laws and ofintegrability.An alternative and complementary viewpoint, where the main focus is on theunitary evolution of a small quantum system considered as a whole, has beenemphasized in [8]. There, the key quantity is the matrix Q of asymptotic transitionprobabilities in some preferential basis. The trace T of this matrix is also the sumof the inverse participation ratios of all eigenstates of the Hamiltonian in the samebasis. This quantity has been put forward as a measure of the degree of equilibrationof the full system if launched from a typical basis state; the larger T , the poorer theequilibration.The above line of thought has been illustrated by means of a detailed study of asingle tight-binding particle on a finite segment of N sites in a random potential [8].In the absence of disorder, the quantity T exhibits a finite value T = T (0) ≈ / T ≈ N Q grows linearly withthe sample size, with a finite mean asymptotic return probability Q , testifying poorequilibration. The outcome of most physical interest concerns the regime of a weakdisorder, where the disorder strength w is much smaller than the bandwidth. In thisregime, T obeys a universal scaling law of the form T ≈ T (0) + 1 N Φ( x ) , x = N w / . (1.1)This finite-size scaling law interpolates between both above regimes. In the ballisticregime, where x is small, the first correction due to disorder scales as Φ( x ) ≈ x / x is large, the quadratic growth Φ( x ) ≈ Ax , with A ≈ .
21, describes thecrossover to the localized regime. The mean return probability accordingly scales as Q ≈ Aw / at weak disorder. The crossover exponent 2/3 entering the variable x is dictated by the well-known anomalous scaling of the localization length in thevicinity of band edges. Typical equilibration properties of a single particle in a weakrandom potential, as testified by the scaling behavior of T , are therefore governedby the few most localized band-edge states. This is a not an artifact of consideringa global quantity such as T , albeit a genuine physical effect: a typical initial statewill generically have a non-zero overlap with the anomalously localized band-edgestates, and the very slow relaxation of the latter overlap will cause the relatively poorequilibration properties of such a generic initial state.The behavior of T in a clean quantum many-body system has then beenconsidered in [9], where the statistics of inverse participation ratios in the XXZ spinchain with anisotropy parameter ∆ has been investigated numerically. There, the quilibration properties of small quantum systems: further examples T was found to grow linearly with the system size ( T ∼ L ) in the gaplessphase of the model ( | ∆ | < T ∼ exp( aL )) in the gapped phase( | ∆ | > T , even in the simplest of integrablemodels.In this paper we analyse in detail several other examples of simple quantumsystems for which the behavior of T can be investigated by analytical means. Section 2provides a reminder of the transition matrix formalism proposed in [8], emphasizing therole and the meaning of T . In section 3 we investigate the statistics of T correspondingto a random Hamiltonian whose distribution is invariant under the group U( N ) orO( N ), thus keeping in line with the random matrix theory approach to quantumchaos. Section 4 deals with a single quantum spin S submitted to a tilted magnetic fieldmaking an angle θ with respect to the quantization axis defining the preferred basis.Finally, the case of a tight-binding particle on a finite electrified chain is considered insection 5. This situation is somehow a deterministic analogue of the case consideredin [8], where the ladder of Wannier-Stark states replaces Anderson localized states. Itis however richer, as it features two successive scaling regimes in the small-field region.Our findings are summarized and discussed in section 6.
2. A reminder on the transition matrix formalism
This section provides a reminder of the transition matrix formalism which has beenproposed in [8] to describe equilibration in small isolated quantum systems. Considera system whose Hilbert space has dimension N . Let {| a i} ( a = 1 , . . . , N ) be apreferential basis, chosen once for all. In that basis the Hamiltonian H is some N × N matrix. We assume that its eigenvalues E n are non-degenerate. This condition willbe satisfied in all systems considered in this work. The situation of degenerate energyeigenvalues will be considered briefly at the end of this section.If the system is launched from one of the basis states, say | a i , the probability forit to be in state | b i at a subsequent time t reads P ab ( t ) = X m,n e i( E n − E m ) t h b | m ih m | a ih a | n ih n | b i , (2.1)where {| n i} ( n = 1 , . . . , N ) is a basis of normalized eigenstates, such that H| n i = E n | n i . The stationary state reached by the system is characterized by the time-averaged transition probabilities Q ab = lim t →∞ t Z t P ab ( t ′ ) d t ′ . (2.2)In the absence of spectral degeneracies, we have Q ab = X n |h a | n i| |h b | n i| . (2.3)The transition matrix Q is the central object of this formalism. Let us begin witha few general properties. The matrix Q only depends on the eigenstates | n i of theHamiltonian H , and not on its eigenvalues E n . Therefore, any two Hamiltonians with quilibration properties of small quantum systems: further examples Q , irrespective of their spectra. Infull generality, the matrix Q reads Q = RR T , (2.4)where the matrix R is given by R an = |h a | n i| , (2.5)and R T is its transpose. The matrix Q is therefore real symmetric and positivedefinite. The matrix R is real, albeit not symmetric in general; it may therefore havecomplex spectrum. Finally, both matrices Q and R are doubly stochastic: their rowand column sums equal unity: X a R an = X n R an = X a Q ab = X b Q ab = 1 . (2.6)More generally, one can associate real matrices Q and R obeying the above propertieswith any pair of orthonormal bases {| a i} and {| n i} , related to each other by anarbitrary unitary transformation U such that U an = h a | n i . (2.7)Coming back to equilibration properties, the diagonal element Q aa = X n |h a | n i| , (2.8)i.e., the stationary return probability to the basis state | a i , can be recast as Q aa = tr ω a , (2.9)where ω a = X n |h a | n i| | n ih n | (2.10)is the stationary density matrix issued from state | a i . The right-hand side of (2.9) iscalled the purity of ω a . Its reciprocal, i.e., d a = 1 Q aa , (2.11)provides a measure of the dimension of the subspace of Hilbert space where the systemequilibrates if launched from state | a i [11, 12, 13, 14, 15, 16].The equilibration properties of the system launched from a typical basis state areencoded in the trace of the matrix Q : T = X a,n |h a | n i| = X a Q aa = X n I n , (2.12)where I n = X a |h a | n i| (2.13)is the inverse participation ratio (IPR) [17, 18, 19, 20] of the eigenstate | n i of H in thepreferential basis. The return probabilities Q aa and the IPR I n therefore somehowappear as dual to each other.The ratio D = NT (2.14) quilibration properties of small quantum systems: further examples D = 1 N X a d a . (2.15)Hence D provides a measure of the dimension of the subspace which hosts theequilibration dynamics for a typical initial state. The trace T and the effectivedimension D are always between the following extremal values. Ideal equilibration.
If the eigenstates of H are uniformly spread over the preferentialbasis, we have R an = 1 N (2.16)for all a and n . The bases {| a i} and {| n i} are said to be mutually unbiased [21, 22, 23].Hence Q ab = 1 N (2.17)for all a and b , and T min = 1 , D max = N. (2.18)This ideal situation corresponds to a perfectly equilibrated stationary state, with nomemory of the initial state at all. No equilibration.
This situation is the exact opposite of the previous one. If H isdiagonal in the preferential basis, its eigenstates | n i can be ordered so as to have R an = δ an (2.19)for all a and n , and Q ab = δ ab (2.20)for all a and b , δ being the Kronecker symbol, and so T max = N, D min = 1 . (2.21)In such a situation, the system keeps a full memory of its initial state, and so there isno equilibration at all.Finally, let us mention that the above transition matrix formalism can be readilyextended to situations where some of the energy eigenvalues are degenerate [8]. Insuch a situation, the expression (2.12) of the quantity T becomes T = X a,n µ n X i =1 |h a | n, i i| ! , (2.22)where the | n, i i form an orthonormal basis such that H| n, i i = E n | n, i i , for i =1 , . . . , µ n , where µ n is the multiplicity (degeneracy) of the n th energy eigenvalue E n . quilibration properties of small quantum systems: further examples
3. Random matrix theory approach
In this section we investigate the statistics of T obtained by choosing the Hamiltonianmatrix H at random, according to some probability distribution over N × N Hermitianmatrices. We assume that the latter distribution is invariant under the actionof a suitable chosen symmetry group, here U( N ) or O( N ) (see below). Thisassumption keeps in line with the random matrix theory approach to quantum chaos,which has been used extensively to investigate many properties of complex quantumsystems [24, 25, 26, 27], including the energy spectra of closed systems and thescattering and transport properties of open systems. This line of thought impliesin particular that the distribution of the matrix U , introduced in (2.7), encodingthe eigenstates of H in the preferred basis, is given by the flat Haar measure onthe appropriate symmetry group. Moreover, it will be sufficient for our purposeto consider the first two cumulants of T , i.e., the mean value h T i and the variancevar T = h T i − h T i . The unitary case.
We consider first the situation where the Hamiltonian H is an N × N complex Hermitian matrix. The relevant symmetry group is then the unitarygroup U( N ). We have h T i U = N A U , (3.1)(var T ) U = N (cid:0) B U + 2( N − C U + ( N − D U − N A (cid:1) , (3.2)with A U = h| U | i U , B U = h| U | i U ,C U = h| U | | U | i U , D U = h| U | | U | i U , (3.3)where h . . . i U denotes an average with respect to the Haar measure on U( N ).Moments of this kind have attracted much attention in the past, and severalapproaches have been proposed to evaluate them [28, 29, 30, 31, 32, 33, 34, 35, 36, 37].Table 1 gives the values of the moments A U , B U , C U and D U for the unitary groupU( N ) (see (3.3)) and A O , B O , C O and D O for the orthogonal group O( N ) (see (3.8)).U( N ) O( N ) A U = N ( N +1) A O = N ( N +2) B U = N ( N +1)( N +2)( N +3) B O = N ( N +2)( N +4)( N +6) C U = N ( N +1)( N +2)( N +3) C O = N ( N +2)( N +4)( N +6) D U = N ( N − N +3) D O = N +3)( N +5) N ( N − N +2)( N +4)( N +6) Table 1.
Values of the integrals A , B , C and D for both symmetry classes.For the unitary group U( N ) (see (3.3)), the one-vector integrals A U , B U and C U are given in [28, 37] while the two-vector integral D U , given in [35],can be checked using the recursive approach of [37]. For the orthogonalgroup O( N ) (see (3.8)), the one-vector integrals A O , B O and C O are givenin [37], and the two-vector integral D O , given in [36], can again be checkedusing [37]. quilibration properties of small quantum systems: further examples h T i U = 2 NN + 1 , (3.4)(var T ) U = 4( N − N + 1) ( N + 3) . (3.5) The orthogonal case.
We now turn to the situation where the dynamics is invariantunder time reversal, so that the Hamiltonian H is a real symmetric matrix. Therelevant symmetry group is then the orthogonal group O( N ). We have h T i O = N A O , (3.6)(var T ) O = N (cid:0) B O + 2( N − C O + ( N − D O − N A (cid:1) , (3.7)with A O = h Ω i O , B O = h Ω i O ,C O = h Ω Ω i O , D O = h Ω Ω i O , (3.8)where h . . . i O now denotes an average with respect to the Haar measure on O( N ).Using again the results of table 1, we obtain h T i O = 3 NN + 2 , (3.9)(var T ) O = 24 N ( N − N + 1)( N + 2) ( N + 6) . (3.10)For a very large dimension N , T converges to the limits T U = 2 , T O = 3 , (3.11)which only depend on the symmetry class of the Hamiltonian. These finite valuestestify extended eigenstates and good equilibration properties, as could be expectedfor a generic Hamiltonian. The limit values (3.11) are deterministic, in the sense thatfluctuations of individual values of T become negligible for large N , as testified by the1 /N falloff of the variances(var T ) U ≈ N , (var T ) O ≈ N . (3.12)The limit values (3.11) coincide with the outcome of the following naive approach.In the unitary case, consider T as the sum of N independent quantities of theform | u | , modeling the entries u as centered complex Gaussian variables suchthat h| u | i = 1 /N . The identity h| u | i = 2 h| u | i yields h| u | i = 2 /N , and so T U = N h| u | i = 2. Similarly, in the orthogonal case, consider T as the sumof N independent quantities of the form ω , modeling the entries ω as real Gaussianvariables such that h ω i = 1 /N . The identity h ω i = 3 h ω i yields h ω i = 3 /N , andso T O = N h ω i = 3. In the same vein, the 1 /N decay of the variances (3.12) alsoconforms with the above naive approach, viewing T as the sum of N independentrandom variables.For finite values of N (see figure 1), the results (3.4), (3.5) and (3.9), (3.10) have asimple rational dependence on N . The mean values h T i increase monotonically fromunity and saturate to the limits (3.11). The variances vanish both for N = 1 and N → ∞ , and therefore present a maximum. The maximum of (var T ) U , equal to4/45, is reached for N = 2, while the maximum of (var T ) O , equal to 4/25, is reachedfor N = 3 and 4. quilibration properties of small quantum systems: further examples N < T > U(N)O(N) N v a r T Figure 1.
Mean values (left) and variances (right) of T , for a randomHamiltonian with a distribution invariant under the unitary group U( N )(see (3.4), (3.5)) or the orthogonal group O( N ) (see (3.9), (3.10)). Dashedhorizontal lines in left panel: large- N limits T U = 2, T O = 3 (see (3.11)).
4. A spin in a tilted magnetic field
This section is devoted to the dynamics of a single quantum spin in the representationof SU(2) of dimension N = 2 S + 1 . (4.1)The spin S is either an integer (if N is odd) or a half-integer (if N is even). Wechoose the z -axis as preferred quantization axis. The preferred basis thus consists ofthe eigenstates | a i of S z , where a takes the N values a = − S, . . . , S , with unit steps.The spin is subjected to a tilted external magnetic field h making an angle θ with the z -axis, with 0 ≤ θ ≤ π . The corresponding Hamiltonian reads, in dimensionless units H = cos θ S z + sin θ S x . (4.2)The spin S and the tilt angle θ are the two parameters of the model. S = Let us consider first the case of a spin S = , so that N = 2. The Hamiltonian H = 12 (cid:18) cos θ sin θ sin θ − cos θ (cid:19) (4.3)has eigenvalues E ± = ± . The matrix encoding the corresponding eigenstates is U = (cid:18) cos θ sin θ − sin θ cos θ (cid:19) , (4.4)up to arbitrary phases. The matrices R and Q (see (2.3), (2.5)) therefore read R = 12 (cid:18) θ − cos θ − cos θ θ (cid:19) , (4.5) Q = 12 (cid:18) θ − cos θ − cos θ θ (cid:19) . (4.6) quilibration properties of small quantum systems: further examples U and R are respectively orthogonal and symmetric. Wehave finally T = 1 + cos θ. (4.7) In the general case ( S arbitrary), the eigenvalues of H are E n = n = − S, . . . , S ,again with unit steps. The matrix U encoding the corresponding eigenstates in thepreferred basis is nothing but the Wigner ‘small’ rotation matrix d ( S ) ( θ ), describingthe transformation of a spin S under a rotation of angle θ around the y -axis.References [38, 39, 40, 41] give a full account of its properties. The rotation matrix d ( S ) ( θ ) is a real orthogonal matrix, whose entries obey the symmetry property d ( S ) a,n ( θ ) = ( − a − n d ( S ) − a, − n ( θ ) . (4.8)This identity allows us to recast the entries of the matrix R as R an = ( − a − n d ( S ) a,n ( θ ) d ( S ) − a, − n ( θ ) . (4.9)Expanding the product of rotation matrix elements over irreducible representa-tions [38, 39, 40, 41], we obtain R an = N − X j =0 ϕ ( j ) a ϕ ( j ) n P j (cos θ ) . (4.10)In this expression, the integer j takes N = 2 S + 1 values, P j (cos θ ) = d ( j )0 , ( θ ) (4.11)are the Legendre polynomials, and ϕ ( j ) n = ( − n h S, n, S, − n | j, i (4.12)are Clebsch-Gordan coefficients, up to signs.The identity (4.10) has several consequences. First, it shows that R is areal symmetric matrix. The properties that U and R are respectively orthogonaland symmetric, already observed for S = , thus hold for an arbitrary spin S .Furthermore, (4.10) yields the spectral decomposition of the matrix R : r j = P j (cos θ ) are its eigenvalues, with the corresponding normalized eigenvectors ϕ ( j ) n being independent of the angle θ . Finally, the matrix R being symmetric, (2.4) implies Q = R , and so the eigenvalues of Q are q j = r j . We thus obtain the explicit formula T = N − X j =0 P j (cos θ ) . (4.13)Here are a few immediate properties of the above expression. First, it is invariantwhen changing θ to π − θ , as should be. The Legendre polynomials indeed obey thesymmetry property P j ( − x ) = ( − j P j ( x ), so that T is an even function of x = cos θ .For θ = 0 or θ = π , we have exactly T = N . The Hamiltonian indeed reads H = ± S z ,and so its eigenstates coincide with the preferred basis. This is one example of thesituation referred to in section 2 as no equilibration . For S = , i.e., N = 2, theresult (4.7) is recovered, as P ( x ) = 1 and P ( x ) = x . quilibration properties of small quantum systems: further examples N (semi-classical regime) The goal of this section is to estimate the growth law of T in the semi-classical regimeof large values of N . The quantity T is clearly an increasing function of N , at fixedangle θ , as the expression (4.13) is a sum of positive terms.Consider first the mean value h T i , assuming that the magnetic field h actingon the spin is isotropically distributed. This amounts to x = cos θ being uniformlydistributed in the range − ≤ x ≤
1. The identity (see e.g. [42, Vol. II, Ch. 10]) Z − P j ( x ) d x = 22 j + 1 (4.14)yields h T i = N − X j =0 j + 1 = H N − H N , (4.15)where H N are the harmonic numbers, and hence h T i ≈
12 ln (4e γ N ) , (4.16)where γ = 0 .
577 215 . . . is Euler’s constant. The mean value h T i therefore growslogarithmically with N .A similar logarithmic growth holds at any fixed value of the tilt angle θ . This canbe shown by means of the generating series [43, 44, 45] X j ≥ P j (cos θ ) z j = 2 π (1 − z ) K ( k ) , k = − z sin θ (1 − z ) , (4.17)where K ( k ) is the complete elliptic integral of the first kind. We are interested in thesingular behavior of the above expression as z →
1. As the modulus k diverges in thatlimit, it is advantageous to change the square modulus [42, Vol. II, Ch. 13] from k to q = − k − k = 4 z sin θ (1 − z ) + 4 z sin θ . (4.18)We have then K ( k ) = q ′ K ( q ), with q ′ = p − q . Furthermore, as q →
1, i.e., q ′ → K ( q ) ≈ ln (4 /q ′ ), so that X j ≥ P j (cos θ ) z j ≈ π sin θ ln 8 sin θ − z , (4.19)and finally T ≈ π sin θ ln (8e γ N sin θ ) . (4.20)The quantity T thus grows logarithmically in N for any fixed value of the angle θ ,with an angle-dependent amplitude 1 / ( π sin θ ). Hence the product T sin θ , plottedin figure 2 against the reduced angle θ/π , grows to leading order as (1 /π ) ln N ,irrespective of θ . Finally, the growth law (4.16) of the mean value h T i can be recoveredby integrating (4.20) over 0 ≤ θ ≤ π with the measure sin θ d θ . quilibration properties of small quantum systems: further examples θ/π T s i n θ Figure 2.
Product T sin θ against θ/π for N ranging from 2 to 20 (bottomto top). Black curves: integer spin S ( N odd). Red curves: half-integerspin S ( N even). The lowest curve corresponds to S = (see (4.7)). The quantity T obeys a non-trivial scaling behavior in the regime where N is largeand the tilt angle θ is small, interpolating between the growth laws T = T max = N for θ = 0 and T ∼ ln N for any non-zero θ (see (4.20)).This scaling behavior is inherited from the scaling formula obeyed by the Legendrepolynomials for j large and θ small, sometimes referred to as Hilb’s formula (seee.g. [42, Vol. II, Ch. 10]), i.e., P j (cos θ ) ≈ J ( s ) , (4.21)with s = jθ (4.22)and J being the Bessel function. Inserting this into (4.13), we obtain the scaling law T ≈ N F ( u ) , (4.23)or, equivalently, T θ ≈ G ( u ) , (4.24)with u = N θ (4.25)and G ( u ) = uF ( u ) = Z u J ( s ) d s. (4.26)The formula (4.23) is more adapted to small values of the scaling variable u .The function F ( u ) decreases monotonically from F (0) = 1 to zero. For small u , theexpansion F ( u ) = 1 − u · · · (4.27) quilibration properties of small quantum systems: further examples T = N − N ( N − θ + · · · (4.28)of the full expression (4.13). We have F ( u ) = 1 / u ≈ . T drops fromits maximal value T = N at θ = 0 to T = N/ θ ≈ . /N .The formula (4.24) is more adapted to large values of u . The function G ( u ) ismonotonically increasing. It exhibits an infinite sequence of ‘pauses’ at the zeros j n of the Bessel function J , with heights G ( j n ) = G n , around which it varies cubically,as G ( u ) − G n ∼ ( u − j n ) . The asymptotic behavior of the Bessel function J ( s ) ≈ (cid:18) πs (cid:19) / cos (cid:16) s − π (cid:17) ( s → ∞ ) (4.29)implies that G ( u ) grows logarithmically for large u , as G ( u ) ≈ π ln (8e γ u ) , (4.30)where the finite part has been obtained by matching with (4.20). The zeros readapproximately j n ≈ nπ , so that the heights of the pauses also grow logarithmically, as G n ≈ π ln (8 π e γ n ) . (4.31)There are 12 pauses with height less than 2, and 277 pauses with height less than 3.The first of these pauses predict the heights of the ripples which are visible on thedata for finite values of N , shown in figure 2.Figure 3 shows the above scaling functions. The left panel shows the function F ( u )entering (4.23). The right panel shows the function G ( u ) = uF ( u ) entering (4.24).Blue symbols show the first three pauses. u F ( u ) u G ( u ) Figure 3.
The functions F ( u ) (left) and G ( u ) = uF ( u ) (right), respectivelyentering the scaling laws (4.23) and (4.24), against u = Nθ . Bluesymbols on the right panel: first three pauses at heights G ≈ . G ≈ . G ≈ . quilibration properties of small quantum systems: further examples
5. A tight-binding particle on an electrified chain
In this section we investigate a tight-binding particle on a finite chain of N sitessubjected to a uniform electric field F . We assume F > | a i associated with the sitesof the chain ( a = 1 , . . . , N ). In terms of the amplitudes ψ a = h a | ψ i , the eigenvalueequation H ψ = Eψ reads ψ a +1 + ψ a − + (cid:0) a − ( N + 1) (cid:1) F ψ a = Eψ a , (5.1)in reduced units, where the dimensionless electric field F equals the potential dropacross one unit cell, in units of the hopping rate.We shall consider an open chain with Dirichlet boundary conditions ( ψ = ψ N +1 = 0). The choice of the constant offset in the electrostatic potential V a = (cid:0) ( N + 1) − a (cid:1) F (5.2)ensures that the latter vanishes in the middle of the sample. As a consequence, theenergy spectrum is symmetric with respect to E = 0. Indeed, if the wavefunction ψ a describes a state with energy E , then ( − a ψ N +1 − a describes a state with energy − E .Both states have the same IPR. In particular, samples with odd sizes N have a singlenon-degenerate eigenstate at E = 0.The problem of an electron in a uniform electric field is an old classic of QuantumMechanics. The spectrum of a tight-binding particle on the infinite chain has beenknown for long [46, 47, 48]. It consists of an infinite sequence of localized states, thecorresponding energies having a constant spacing (∆ E = F in our reduced units).Such a discrete spectrum has been dubbed a Wannier-Stark ladder [49, 50, 51]. Thefate of these ladders on finite samples has been investigated by several authors in the1970s [52, 53, 54, 55, 56, 57]. In order to investigate the behavior of T in the variousregimes of system size N and applied electric field F , we have to thoroughly revisitthese works. A general approach consists in solving (5.1) on the half-line ( a ≥ E , with boundary conditions ψ = 0 and ψ = 1. This approach isknown under the name of ‘solving the Cauchy problem’. Equation (5.1) readily yields ψ = E + ( N − F,ψ = (cid:0) E + ( N − F (cid:1) (cid:0) E + ( N − F (cid:1) − , (5.3)and so on. The amplitude ψ a = P a − ( E ) thus obtained is a polynomial with degree a − E , dubbed a Lommel polynomial [53]. It also depends parametrically on N and F .On a finite chain of N sites with Dirichlet boundary conditions, the energylevels E n are the N zeros of the polynomial equation ψ N +1 = P N ( E ) = 0 , (5.4)whereas the IPR I of the eigenstate associated with energy E reads I = P Na =1 ψ a (cid:16)P Na =1 ψ a (cid:17) , (5.5) quilibration properties of small quantum systems: further examples ψ a = P a − ( E ).It is useful to begin with a detailed study of the simplest case, namely N = 2.Equation (5.4) giving the two energy levels reads( E + F )( E − F ) − . (5.6)Inserting the expressions (5.3) into (5.5), we obtain I = 1 + ( E + F ) (1 + ( E + F ) ) . (5.7)Eliminating the energy E between (5.6) and (5.7), we obtain AI + BI + C = 0 , (5.8)with A = (4 + F ) , B = − F )(2 + F ) , C = (2 + F ) . (5.9)To sum up, the two energy levels E n are the zeros of (5.6), while the correspondingIPR I n are the zeros of (5.8). The quantity T reads (see (2.12)) T = I + I = − BA = 2(2 + F )4 + F . (5.10)In the present situation, both IPR are equal ( I = I ), and so the polynomialentering (5.8) is a perfect square. This peculiarity however plays no role in theexposition of the method.The pattern for an arbitrary size N is as follows. The energy levels E n are thezeros of the polynomial equation (5.4), while (5.5) allows one to express the IPR I as a rational function of the energy E . As a consequence, the I n are the zeros of apolynomial equation of degree N , of the formΠ N ( I ) = N X k =0 Π N,k I k = 0 , (5.11)where Π N ( I ) is the resultant of the algebraic elimination of E between (5.4) and (5.5).This observation simplifies the calculation of T for all system sizes N . This quantityis indeed the sum of the I n (see (2.12)), and so T = − Π N,N − Π N,N . (5.12)This procedure can be easily automated and implemented in a symbolic algebrasoftware. We thus obtain the following expressions for T . • N = 2 : T = 2(2 + F )4 + F . (5.13) • N = 3 : T = 5 + 4 F + 3 F (2 + F ) . (5.14) • N = 4 : T = 4(24 + 107 F + 91 F + 48 F + 36 F )(1 + 2 F )(5 + 2 F )(16 + 12 F + 9 F ) . (5.15) quilibration properties of small quantum systems: further examples • N = 5 : T = N ( F )(3 + 4 F + 4 F ) (4 + 24 F + 9 F ) , (5.16)with N ( F ) = 2(24 + 206 F + 615 F + 1676 F + 700 F + 528 F + 360 F ) . (5.17)The following observations can be made at this stage. The quantity T is a rationaleven function of the applied field F , whose complexity grows rapidly with N . Its degreein F reads d N = N / N is even, and d N = ( N − / N is odd. We recallthat the degree of a rational function is the larger of the degrees of its numerator anddenominator. In the present situation, these degrees coincide.In the small-field limit, we have T = T (0) + T (2) F + · · · (5.18)The first term is the known value of T for a free tight-binding particle [8], i.e., T (0) = N N + 1) ( N even) , N + 12( N + 1) ( N odd) , (5.19)which saturates to the finite limit T (0) = 32 (5.20)for large system sizes. The coefficient T (2) will be calculated in section 5.2 for all N by means of perturbation theory (see (5.32), (5.33)).In the large-field limit, T obeys a linear-plus-constant growth of the form T ≈ N I ( F ) + J ( F ) , (5.21)asserting the poor equilibration properties characteristic of a localized regime, with I ( F ) = 1 − F + 9 F − F + 1225144 F + · · · , (5.22) J ( F ) = 4 F − F − F + 15943108 F + · · · (5.23)As N increases, more and more terms of the above asymptotic expansions stabilize.This localized regime will be further investigated in sections 5.4 and 5.6. The goal of this section is to determine the coefficient T (2) of the expansion (5.18)by means of second-order perturbation theory. The derivation follows the lines ofsection 5.1 of Reference [8], devoted to the case of a weak diagonal random potential.In the absence of an electric field, the energy eigenvalues of a free tight-bindingparticle [8] read E n = 2 cos nπN + 1 (5.24) quilibration properties of small quantum systems: further examples n = 1 , . . . , N ). The corresponding normalized eigenstates are h a | n i = r N + 1 sin anπN + 1 . (5.25)Their IPR read I n = 12( N + 1) (cid:18) δ n,
12 ( N +1) (cid:19) . (5.26)For odd sizes N , the IPR of the central eigenstate at zero energy, with number n = ( N + 1), is enhanced by a factor 4 /
3. The expression (5.19) can be recoveredby summing (5.26) over n .In the presence of an electric field, the un-normalized n th eigenstate reads | n i F = | n i − F X m B nm | m i + F X m = n C nm | m i + · · · , (5.27)with B nm = h m | W | n i E n − E m , C nm = 1 E n − E m X l h m | W | l ih l | W | n i E n − E l , (5.28)where E n and | n i are given by (5.24) and (5.25), while W is the operator W = X a (cid:0) ( N + 1) − a (cid:1) | a ih a | , (5.29)where the quantity in the parentheses is the ratio V a /F (see (5.2)). The property h n | W | n i = 0 has been used in the derivation of (5.27).The expansion (5.27) yields the following expansions to order F : X a h a | n i F = 1 + F X m B nm + · · · , X a h a | n i F = T (0) + 6 F X a h a | n i (cid:16)X m B nm h a | m i (cid:17) + 4 F X a h a | n i X m = n C nm h a | m i + · · · (5.30)We thus obtain the following expression for the coefficient T (2) : T (2) = 6 X a,n h a | n i (cid:16)X m B nm h a | m i (cid:17) + 4 X a,n h a | n i X m = n C nm h a | m i − X a,n h a | n i X m B nm . (5.31)The final step of the derivation, i.e., the evaluation of the above sums, could notbe done by analytical means. We have instead used the following trick. We knowfrom the general approach of section 5.1 that T (2) is a rational number. A numericalevaluation of (5.31) with very high accuracy ( N = 400 can be reached in one minute ofCPU time) reveals that the denominator of T (2) only grows linearly with the systemsize. It is indeed observed to be a divisor of 140( N + 1) if N is even, and of 105( N + 1)if N is odd. This serendipitous situation allows one to unambiguously reconstructthe numerator of T (2) by means of a polynomial fit. We are thus left with the exactclosed-form expressions: quilibration properties of small quantum systems: further examples • N even: T (2) = N ( N + 2)(2 N + 8 N + 91 N + 166 N + 153)10080( N + 1) . (5.32) • N odd: T (2) = ( N − N + 3)(4 N + 16 N − N − N − N + 1) . (5.33)Up to N = 5, the above results agree with the expansions of the exactexpressions (5.13)–(5.16). The coefficient T (2) turns out to be negative for N = 3and N = 5, resulting in a non-monotonic dependence of T on the applied field F .For large system sizes, the expansion (5.18) becomes T ≈
32 + N F · · · (5.34)We shall come back to this result in section 5.5 (see (5.80)). The problem of a tight-binding particle on the infinite electrified chain has been studiedlong ago [46, 47, 48, 49, 50, 51]. The energy spectrum is an infinite Wannier-Starkladder of the form E k = kF , forgetting about the constant offset ( N + 1) / The central Wannier-Stark statewith energy E = 0 (5.35)is described by the normalized wavefunction ψ a = J a (2 /F ) , (5.36)where J a is the Bessel function. We have ψ − a = ( − a ψ a , so that the probabilities ψ a are symmetric around the origin. All the other states are obtained from the abovecentral one by translations along the chain. The state with energy E k = kF (5.37)is described by the wavefunction ψ a = J a − k (2 /F ) , (5.38)centered at site k .There is a formal identity between the wavefunction (5.36) describing the centralstationary Wannier-Stark state on the electrified chain and the time-dependentwavefunction [58, 59] ψ a ( t ) = i − a J a (2 t ) , (5.39)describing the ballistic spreading of a quantum walker modelled as a free tight-bindingparticle launched from the origin at time t = 0, with time t playing the role of theinverse field 1 /F , in our reduced units.The regime of most physical interest is that of a small electric field. The squarewavefunction of the central Wannier-Stark state is shown in figure 4 for F = 0 .
02. Wesummarize here for future use its behavior in various regions in the small-field regime.The following estimates stemming from the theory of Bessel functions [42, 60] can berecovered by semi-classical techniques based on the saddle-point method, as describedin the physics literature [58, 59]. quilibration properties of small quantum systems: further examples −120−100−80 −60 −40 −20 0 20 40 60 80 100 120 a ψ a Figure 4.
Square wavefunction ψ a of the central Wannier-Stark stateagainst position a , for an electric field F = 0 .
02. Vertical lines show thenominal positions of the peaks in the transition regions ( a = ± Allowed region ( | aF | < ). The wavefunction only takes appreciable values in anallowed region extending up to a distance 2 /F on either side of the origin. The sizeof the allowed region, L ( F ) = 4 /F, (5.40)plays the role of a localization length. More precisely, setting aF = 2 cos θ (0 < θ < π ) , (5.41)we have ψ a ≈ (cid:18) Fπ sin θ (cid:19) / cos (cid:18) F (sin θ − θ cos θ ) − π (cid:19) . (5.42)The allowed region is limited by two transition regions, located symmetrically near a = ± L ( F ) / ± /F , corresponding to θ → θ → π . Transition regions ( | aF | ≈ ). In the transition regions at the endpoints of theallowed region, the wavefunction exhibit sharp peaks. In the right transition region,setting aF = 2 + zF / , (5.43)we have ψ a ≈ F / Ai( z ) , (5.44)where Ai( z ) is the Airy function. The width of the transition regions, λ ( F ) ∼ F − / , (5.45)provides a second length scale which diverges in the small-field regime, albeit muchless rapidly that L ( F ) (see (5.40)). quilibration properties of small quantum systems: further examples Forbidden regions ( | aF | > ). The wavefunction falls off essentially exponentiallyfast in the forbidden regions. In the right forbidden region, setting aF = 2 cosh µ ( µ > , (5.46)we have ψ a ≈ (cid:18) F π sinh µ (cid:19) / exp (cid:18) − F ( µ cosh µ − sinh µ ) (cid:19) . (5.47) All Wannier-Stark states have the common IPR I ( F ) = X a J a (2 /F ) . (5.48)This quantity was already investigated [58] in the context of the time-dependentwavefunction (5.39). It has also attracted some attention in the mathematicalliterature (see [61, 62] and references therein).In the large-field regime ( F ≫ ψ a indeed fall off as ψ a ≈ a ! F a ( a > . (5.49)The expansion (5.22) of I ( F ) can therefore be recovered easily. An all-order asymp-totic expansion can be derived by means of the following expression of the Mellintransform [58]: M ( s ) = Z ∞ F − s − I ( F ) d F = Γ( s )2 s π / cos sπ Γ (cid:0) − s (cid:1) Γ (cid:0) − s (cid:1) ! (0 < Re s < . (5.50)Summing the contributions of the simple poles at s = − , − , . . . , we obtain I ( F ) = X k ≥ ( − k (2 k )! k ! F k . (5.51)The above result can be recovered by using the expression of I ( F ) in terms of ahypergeometric function [61, 62]: I ( F ) = F (cid:18) ,
12 ; 1 , , − F (cid:19) . (5.52)In the small-field regime, we have I ( F ) = F π ln 64e γ F + F / √ π sin(8 /F − π/
4) + · · · (5.53)The first term of the above expression exhibits a logarithmic correction, stemmingfrom the double pole of M ( s ) at s = 1, with respect to the leading scaling I ( F ) ∼ /L ( F ) ∼ F . The presence of this logarithmic correction to scaling was alreadyemphasized in the time-dependent problem [58], where it was put in perspective withthe following phenomenon dubbed bifractality. The generalized IPR I ( q ) ( F ) = X a | J a (2 /F ) | q , (5.54) quilibration properties of small quantum systems: further examples q > F → + , of the form I ( q ) ( F ) ∼ F τ ( q ) , (5.55)with τ ( q ) = q − q < , q −
13 ( q > . (5.56)The usual IPR I ( F ) corresponds to q = 2, i.e., precisely the break point between bothbranches of the above bifractal spectrum, where the generalized IPR is respectivelydominated by the allowed region (for q <
2) and by the transition regions (for q > I ( F ) asa function of the field F . I n and of T In this section we provide a picture of the salient features of the IPR I n of the variouseigenstates, and of their sum T .Figure 5 shows the IPR I n against the energy E n for all eigenstates of finitechains of various lengths ranging from N = 8 to 100. The applied field reads F = 0 . L ( F ) = 40. The horizontal line shows the value I ( F ) = 0 .
035 222 . . . of the IPR of Wannier-Stark states. The IPR I n have beenevaluated by means of a direct numerical diagonalization of the Hamiltonian matrix H .The symmetry of the plot with respect to E = 0 is manifest. As N increases, the IPRof the few first (or last) eigenstates converge very fast to well-defined limiting values.In the small-field regime, these limits will be shown to scale as F / (see (5.68)).They are therefore much larger than the IPR I ( F ) of Wannier-Stark states, whichis proportional to F , with a logarithmic correction (see (5.53)). In the localizedregime, i.e., for sample sizes N larger than the localization length L ( F ), there areapproximately N − L ( F ) bulk states in the central part of the energy spectrum: | E | < N F − . (5.57)These eigenstates are exponentially close to the Wannier-Stark states whose allowedregion is entirely contained in the sample (see e.g. [52, 53, 56]). Their IPR I n istherefore very close to I ( F ). The remaining eigenstates are edge states, living neareither boundary of the system, whose energies lie in the wings of the spectrum: N F − < | E | < N F + 2 . (5.58)Their IPR vary in a broad range between I ( F ) and F / . On shorter samples, whoselength N is smaller than the localization length L ( F ), the distinction between bulkand edge states looses its meaning. All the IPR I n are larger than I ( F ) and manifesta rather irregular dependence on the state number n . This is exemplified by the datafor system sizes N = 8 to 20 in figure 5.The above picture substantiates the asymptotic linear-plus-constant growth of T with the system size N , already advocated in (5.21): T ≈ N I ( F ) + J ( F ) . (5.59)This estimate holds not only for large fields, but all over the localized regime, definedby the inequality N > L ( F ), i.e., N F >
4. The slope I ( F ) is the IPR of Wannier-Stark states (see (5.48)). The offset J ( F ) somehow encodes the contributions of all quilibration properties of small quantum systems: further examples −8 −6 −4 −2 0 2 4 6 8 10 E n I n Figure 5.
IPR I n against energy E n for all eigenstates of finite chainsof various lengths N , for an applied field F = 0 .
1. Horizontal line: I (0 .
1) = 0 .
035 222 . . . edge states. Its exact value is not known for arbitrary values of the field F . It canbe noticed that the coefficients of its large-field expansion (5.23) do not exhibit anyobvious regularity. The quantity J ( F ) will be investigated in section 5.6 in the small-field regime (see (5.104)).Figure 6 shows the ratio T /N against F for several system sizes N . The lowestblack line shows the IPR I ( F ), corresponding to the N → ∞ limit of T /N , accordingto (5.59). The data for finite N converge smoothly to that limit, for any fixed non-zerovalue of the electric field F .The quantitative analysis of the IPR I n and of their sum T in the small-fieldregime will require some special care. Two different characteristic lengths indeeddiverge in the F → + limit, namely L ( F ) ∼ /F (see (5.40)) and λ ( F ) ∼ /F / (see (5.45)). Each of these diverging lengths is responsible for the occurrence of anon-trivial scaling regime (see sections 5.5 and 5.6). ( N ∼ λ ( F ) ∼ /F / )This section is devoted to the regime where the system size N and the size λ ( F ) ofthe transition regions (see (5.45)) are large and comparable. We introduce the scalingvariable x = N F / , (5.60)proportional to the ratio N/λ ( F ). Since the wavefunction of the Wannier-Stark statesin the transition regions is an Airy function (see (5.44)), we dub this regime the Airyscaling regime. Let us first investigate the extremal edge states on avery large sample, with energies near the left edge of the spectrum ( E ≈ − N F − quilibration properties of small quantum systems: further examples F T / N Figure 6.
Ratio
T /N against F for several system sizes N . Lowest blackline: IPR I ( F ), corresponding to N → ∞ . A direct approach goes as follows (see e.g. [54, 55]). Setting χ a = ( − a ψ a , (5.61)and replacing N + 1 by N in the offset of the electrostatic potential, the tight-bindingequation (5.1) becomes χ a +1 + χ a − = (cid:0) aF − N F − E (cid:1) χ a . (5.62)Using a continuum description, this readsd χ d a ≈ ( aF + E ) χ, (5.63)with E = − N F − E − , (5.64)and with the boundary conditions that χ vanishes for a = 0 and a → + ∞ . Then,setting a = ( z + c ) F − / , E = − cF / , (5.65)equation (5.63) boils down to Airy’s equationd χ d z = zχ, (5.66)whose solution decaying as z → + ∞ is χ ( z ) = Ai( z ).The extremal edge states are quantized by the Dirichlet boundary condition at a = 0. The latter requires c = c n , where − c n are the zeros of the Airy function( n = 1 , , . . . ). These zeros are real and negative, and so c n >
0. The extremal edgestates on a very large sample therefore have energies E n ≈ − N F − c n F / . (5.67) quilibration properties of small quantum systems: further examples I n ≈ a n F / , (5.68)with a n = µ ( c n ) µ ( c n ) (5.69)and µ m ( c ) = Z + ∞− c Ai m ( z ) d z. (5.70)The above results can be made more explicit for large n , which turns out to bethe regime of most interest. The asymptotic behavior of the Airy functionAi( z ) ≈ √ π | z | / sin (cid:18) | z | / + π (cid:19) ( z → −∞ ) (5.71)implies c n ≈ (cid:18) πn (cid:19) / . (5.72)Then, for large c , averaging first the powers of the rapidly varying sine function, weobtain µ ( c ) ≈ Z − c d zπ | z | / ≈ √ cπ , (5.73) µ ( c ) ≈ Z − c − c d zπ | z | ≈ π ln cc . (5.74)The finite part c of the logarithm has been evaluated by more sophisticated tech-niques [63], yielding µ ( c ) ≈ π ln(4e γ/ c ) , (5.75)where γ is again Euler’s constant. We thus obtain the estimate a n ≈ (cid:18) πn (cid:19) − / ln(12 π e γ n ) . (5.76) T . The evaluation of T for large but finite systems in theAiry scaling regime goes as follows. The IPR I n of the n th eigenstate scales as I n ≈ N φ n ( x ) , (5.77)with φ n ( x ) ≈ (
32 ( x → ,a n x ( x → ∞ ) . (5.78)The above estimates respectively match (5.26) and (5.68).As a consequence, T obeys a scaling law T ≈ T (0) + 1 N Φ( x ) (5.79)of the very same form as (1.1), corresponding to a particle in a weak disorderedpotential [8], albeit with a different scaling variable x , and a different function Φ( x ). quilibration properties of small quantum systems: further examples T for a free particle, given by (5.19), and so we haveΦ(0) = 0. The following observation [8] also applies to the present situation. Had wechosen the limit value (5.20) for T (0) as the first term in (5.79), instead of the exactexpression (5.19) for finite N , the scaling function Φ( x ) would have acquired a finiteoffset depending on the parity of N , i.e., Φ(0) = − / −
1) for N even(resp. N odd).When the scaling variable x is small, the perturbative result (5.34) impliesΦ( x ) ≈ x x → . (5.80)The asymptotic behavior of the scaling function Φ( x ) at large x can be estimatedas follows. The crossover between both estimates entering (5.78) remains steep, in thesense that it does not broaden out as the state number n gets larger and larger. Thesum T can therefore be estimated as T ≈ N N/ X n =1 max (cid:18) , a n x (cid:19) . (5.81)Using the expression (5.76) for the amplitudes a n , we obtain after some algebraΦ( x ) ≈ n c ( x ) , (5.82)where n c ( x ) is the value of n such that both arguments of the max functionentering (5.81) coincide. To leading order in ln x ≫
1, we have n c ( x ) ≈ ( x ln x ) / π , (5.83)and so Φ( x ) ≈ ( x ln x ) / π ( x → ∞ ) . (5.84)Figure 7 shows a log-log plot of the ratio Φ( x ) /x against the scaling variable x .This ratio has been chosen in order to better reveal the structure of the scalingfunction Φ( x ). Color lines show data pertaining to system sizes 100, 200 and 400. Thestraight line to the left shows the power-law result (5.80) stemming from perturbationtheory. The line to the right shows the estimate (5.84). The data exhibit a verygood convergence to the scaling function Φ( x ), as well as a good agreement with bothanalytical predictions. Small oscillatory corrections are however visible at very high x . ( N ∼ L ( F ) ∼ /F )This section is devoted to the regime where the system size N and the localizationlength L ( F ) (see (5.40)) are large and comparable. We introduce the scaling variable y = N F, (5.85)proportional to the ratio
N/L ( F ), and dub this regime the Bessel scaling regime. quilibration properties of small quantum systems: further examples −1 0 1 2 3 4 ln x −10−9−8−7−6−5 I n ( Φ ( x ) / x ) Figure 7.
Log-log plot of the ratio Φ( x ) /x against the scaling variable x .Color lines: data pertaining to system sizes 100, 200 and 400. Straight lineto the left: power-law result (5.80). Line to the right: estimate (5.84). Let us first investigate the generic edge states on a verylarge sample in the localized regime ( y > ψ a = J a − k (2 /F ) , (5.86)to a part of their allowed region. The allowed region of the above wavefunctionis k − /F < a < k + 2 /F . Left edge states therefore correspond to the range − /F < k < /F , i.e., − y − < E < − y + 2, in agreement with (5.58).The above edge states are quantized by the Dirichlet boundary condition ψ = 0.Setting k = − αF , E = − y − α, (5.87)the expression (5.42) of the wavefunction in the allowed region yields the quantizationcondition sin α n − α n cos α n ≈ πnF . (5.88)As the angle α increases from 0 to π , the edge state number n ranges from 1 to n edge ,where n edge = 2 F = L ( F )2 (5.89)is our estimate for the number of edge states in each wing of the spectrum in thesmall-field regime. The quantization condition (5.88) yields the density of states ρ ( α ) = 2 πF α sin α, (5.90) quilibration properties of small quantum systems: further examples Z π ρ ( α ) d α = n edge . (5.91)The IPR of the above edge states read I n = S ,n S ,n , (5.92)with S m,n = X a ≥ ψ ma,n , (5.93)and ψ a,n is given by (5.86), together with the quantization condition (5.88). Thesums S m,n can be estimated by using the expression (5.42) of the wavefunction in theallowed region, averaging first the powers of the sine function, and transforming thesums over a into integrals over θ in the range 0 < θ < α n , according to (5.41). Wethus obtain S ,n ≈ · π Z α n d θ = α n π , (5.94) S ,n ≈ · Fπ Z α n ε d θ sin θ = 3 F π ln (cid:18) ε tan α n (cid:19) , (5.95)and so I n ≈ F α n ln (cid:18) ε tan α n (cid:19) . (5.96)The cutoff ε is independent of the state number n . It can therefore be determined bymatching the Airy and Bessel regimes. For n ≪ n edge , i.e., nF ≪
1, (5.88) reads α n ≈ (cid:18) πnF (cid:19) / . (5.97)Identifying the expressions (5.68), (5.76) in the Airy regime and (5.96), (5.97) in theBessel regime yields ε = (cid:18) F γ (cid:19) / . (5.98)We are thus left with the estimate I n ≈ I ( F, α n ) = F α n ln (cid:18) γ F tan α n (cid:19) . (5.99) T . The evaluation of T for finite systems in the Besselscaling regime goes as follows. The cases y > y < quilibration properties of small quantum systems: further examples • y > . Consider first the simple situation of samples in the localized regime,such that
N > L ( F ), i.e., y >
4. As already discussed in section 5.4, there are N − n edge = N − L ( F ) = ( y − /F bulk states, whose IPR essentially coincide with I ( F ), and 2 n edge = L ( F ) = 4 /F edge states, whose IPR vary according to (5.99). Wethus obtain T ≈ ( y − I ( F ) F + T edge , (5.100)where the contribution of all edge states reads T edge ≈ Z π ρ ( α ) I ( F, α ) d α = A ln 64e γ F + 3 B, (5.101)with A = 1 π Z π sin αα d α = Si( π ) π = 0 .
589 489 . . . , (5.102) B = 1 π Z π sin αα ln tan α α = − .
462 288 . . . (5.103)The result (5.100) has the expected linear-plus-constant form (5.21), (5.59). Ityields the following estimate for the offset J ( F ) in the small-field regime: J ( F ) ≈ (cid:18) A − π (cid:19) ln 64e γ F + 3 B. (5.104)Finally, (5.100) can be recast as T ≈ U ( y ) ln 1 F + V ( y ) , (5.105)with U ( y ) = y − π + A, (5.106) V ( y ) = (cid:18) y − π + A (cid:19) ln(64e γ ) + 3 B. (5.107) • y < . Consider now shorter samples, such that
N < L ( F ), i.e., y <
4. Asdiscussed in section 5.4, there is no clear-cut distinction between bulk and edge statesin this situation. As a consequence, the formula (5.99) for the IPR of edge statesis not sufficient to derive an estimate of T for y <
4. We nevertheless hypothesizethat T still obeys a logarithmic scaling law in F of the form (5.105), even thoughthe amplitudes U ( y ) and V ( y ) cannot be predicted analytically. Figure 8 presentsnumerical results for these amplitudes. For each value of y , T has been calculatedfor 15 roughly equidistant system sizes N on a logarithmic scale between N = 10and N = 5000. These values of T obey accurately the logarithmic dependenceon F , i.e., on N = y/F , hypothesized in the scaling form (5.105). The resultingestimates for U ( y ) and V ( y ) are shown as symbols in both panels of figure 8.Black lines show fits of the form U ( y ) = ( u ln y + u ) √ y + ( u ln y + u ) y and V ( y ) = 3 / v ln y + v ) √ y + ( v ln y + v ) y . The form of these fits is freely inspiredby the expected crossover at small y with the estimate (5.84) for large x in the Airyregime, i.e., T ≈
32 + √ y π (cid:16) ln yF / (cid:17) / . (5.108)In any case, the proposed fits describe the data accurately. Finally, the blue straightlines show the analytical results (5.106), (5.107) for y >
4. The data suggest that thefunctions U ( y ) and V ( y ) are continuous at y = 4, as well as their first derivatives. quilibration properties of small quantum systems: further examples y U ( y ) y V ( y ) Figure 8.
Amplitudes U ( y ) (left) and V ( y ) (right) entering the scalingformula (5.105) for T . Red symbols: numerical data for y <
4. Blacklines: fits to these data (see text). Blue straight lines: analyticalresults (5.106), (5.107) for y >
4. Blue symbols: values U (0) = 0 and V (0) = 3 /
6. Discussion
A novel way of investigating equilibration – or lack of equilibration – in small isolatedquantum systems has been put forward recently [8]. The key quantity of that approachis the trace T of the matrix Q of asymptotic transition probabilities in some preferentialbasis chosen once for all. The quantity T is also the sum of the inverse participationratios of all eigenstates of the Hamiltonian in the same basis. This transition matrixformalism therefore requires the knowledge of all energy eigenvectors, so that itsscope is limited to small systems. This is however consistent with the purpose of theapproach, which is precisely to quantify the degree of equilibration of small quantumsystems. The quantity T provides a measure of the degree of equilibration of thesystem if launched from a typical basis state of the preferential basis. It may varybetween T min = 1 and T max = N , the dimension of the Hilbert space; the larger T ,the poorer the equilibration.The models investigated in [8] and in the present work embrace the mostsignificant physical systems with a finite-dimensional Hilbert space. The outcomesof both papers allow one to get a full understanding of the behavior of the quantity T .Quantum systems whose dynamics is governed by a generic Hamiltonian exhibit goodtypical equilibration properties, so that T saturates to a non-trivial limit for large N .This limit is however always found to be larger than T min = 1. It equals T (0) = 3 / T U = 2 or T O = 3for a random Hamiltonian whose distribution is invariant under the groups U( N ) orO( N ) (section 3).An interesting situation is met when varying a system parameter throughsome scaling regime induces a continuous crossover from good to bad equilibrationproperties. Three systems have been shown to exhibit such a crossover phenomenon,where T is found to obey non-trivial scaling behavior. Table 2 summarizes thefollowing comparative discussion of these three models. For a spin S in a tilted quilibration properties of small quantum systems: further examples θ with the preferred quantization axis, wehave T = T max = N for θ = 0, while T ∼ ln N for any fixed non-zero tilt angle θ . Bothregimes are connected by the scaling laws (4.23), (4.24), with scaling variable u = N θ .For a tight-binding particle on a finite open chain in a weak random potential [8], T obeys the scaling law recalled in the Introduction (see (1.1)), with scaling variable x = N w / , which describes the crossover between the finite value T = T (0) = 3 / T ≈ N Q in the localizedregime, testifying poor equilibration, with Q ≈ Aw / for a weak disorder. For a tight-binding particle on a finite electrified chain (section 5), T obeys a similar but richerbehavior in the situation where the applied field F is small, so that the localizationlength L ( F ) = 4 /F is large. The quantity T indeed exhibits two successive scaling lawsdescribing the crossover between the ballistic regime and the insulating one, namelya first scaling law of the form (5.79) with scaling variable x = N F / in the so-calledAiry scaling regime, followed by a second scaling law of the form (5.105) with scalingvariable y = N F in the so-called Bessel scaling regime.model N control parameter scaling variable Ref.spin in tiltedmagnetic field 2 S + 1 tilt angle θ u = N θ
Sec. 4particle ondisordered chain numberof sites disorder strength w x = N w / [8]particle onelectrified chain numberof sites electric field F (cid:26) x = N F / (Airy) y = N F (Bessel) Sec. 5
Table 2.
The three models known to exhibit a continuous crossover fromgood to bad equilibration properties in the regime where N is large whereasthe control parameter is small: definition of the control parameters and ofthe variables entering the scaling behavior of T . Numerical results on the XXZ spin chain [9] demonstrate that a transition fromgood to bad equilibration properties, in the sense of a change in the growth law of T ,can be observed in a clean quantum many-body system which does not exhibit many-body localization. There, the transition is found to coincide with a thermodynamicalphase transition, with T ∼ L in the gapless phase of the model ( | ∆ | < T ∼ exp( aL ) in the gapped phase ( | ∆ | > T and various typesof phase transitions, involving or not the appearance of many-body localized phases. Acknowledgments
It is a pleasure to thank P Krapivsky, K Mallick and G Misguich for fruitful discussions,and especially V Pasquier for his interest in this work and for a careful reading of themanuscript. quilibration properties of small quantum systems: further examples References [1] Cazalilla M A and Rigol M 2010
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