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Logarithmic current fluctuations in non-equilibrium quantum spin chains

Logarithmic current ?uctuations in non-equilibrium quantum spin chains
Tibor Antal,1 P. L. Krapivsky,2, 3 and A. R? akos4
Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138, USA 2 Department of Physics, Boston University, Boston, MA 02215, USA 3 Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 4 Research Group for Condensed Matter Physics of the Hungarian Academy of Sciences, H-1111 Budapest, Hungary We study zero-temperature quantum spin chains which are characterized by a non-vanishing current. For the XX model starting from the initial state | · · · ↑↑↑↓↓↓ · · · we derive an exact expression for the variance of the total spin current. We show that asymptotically the variance exhibits an anomalously slow logarithmic growth; we also extract the sub-leading constant term. We then argue that the logarithmic growth remains valid for the XXZ model in the critical region.
PACS numbers: 05.60.Gg, 75.10.Jm, 05.70.Ln

arXiv:0808.3514v1 [cond-mat.stat-mech] 26 Aug 2008



eral XXZ chain in section V. A summary of our results and relation to other work is given in section VI.

A distinguishing feature of non-equilibrium states is the presence of currents [1, 2, 3]. Fluctuations of currents often exhibit universal behavior [4] and shed light on the nature of non-equilibrium systems. Current ?uctuations in classical systems have been extensively investigated, see e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and a review [15]. The total current grows linearly with time and current ?uctuations usually exhibit an algebraic growth. Quantum ?uctuations in general, and spin current ?uctuations in particular, are much less understood; even in the simplest systems quantum ?uctuations often behave very di?erently from standard statistical ?uctuations (see e.g. [16]). In this paper we study current ?uctuations in quantum spin chains. How to impose currents in spin chains? Perhaps the simplest way is to start with a spin chain in the following inhomogeneous product state | · · · ↑↑↑↓↓↓ · · · (1)



Most generally, we consider the quantum XXZ Heisenberg spin chain with Hamiltonian H=?
x y y z z sx n sn+1 + sn sn+1 + ?sn sn+1 n


This choice [17] allows one to avoid complications and arbitrariness of coupling the chain to spin reservoirs. State (1) evolves according to the Heisenberg equations of motion. The average magnetization pro?le has been computed for the simplest quantum chains, e.g. for the XX model where the perturbed region was found to grow ballistically [17]. Numerical works [18] suggest that the growth is also ballistic for the XXZ chain in the critical region (described by Hamiltonian (2) with |?| < 1). The main goal of this paper is to study the ?uctuations of spin current in quantum chains. After a brief description of the model in section II, in section III we present a simple derivation of the asymptotic properties of non-equilibrium states in free fermion systems with special initial conditions. In section IV we probe ?uctuations of the current speci?cally in the XX spin chain. First we present a back-of-the-envelope calculation for the variance of the time integrated current; then we establish an exact result from which we extract the long time asymptotical behavior. We discuss the more gen-

Here we set the coupling constants to unity in the x and y directions. The coupling constant ? in the z direction is called the anisotropy parameter. The z component of z the total magnetization M z = ∞ n=?∞ sn is a conserved quantity in this model, and our aim is to study the corresponding current. In the following we shall focus on the time evolution of this spin chain starting from an inhomogeneous initial state, whereby the left and right halves of the in?nite chain are set to di?erent quantum states and are joined at time zero. Such initial states provide a particularly convenient framework to study currents and their ?uctuations in quantum spin chains. We mainly consider the special case of ? = 0 where the model reduces to free fermions. In this system, known as the XX model, the time evolution can be written in a compact form, which enables us to perform exact calculations. In particular, it is possible to evaluate the scaling limit of the magnetization pro?le and other physical quantities [17, 19]. Corrections to this scaling behavior were considered in [20, 21]. Interesting non-equilibrium behavior was found in disordered spin chains [22, 23] and chains at ?nite temperatures [19, 24, 25]. An alternative method has been proposed to generate stationary currents in spin chains using a Lagrange multiplier [26, 27, 28, 29, 30]. Using this method ?uctuations of basic quantities have been studied in non-equilibrium steady states in [31]. The relaxation of a large class of initial conditions towards equilibrium was considered in numerous studies starting from late sixties, see [32, 33, 34] and [35] for a

2 review of more recent work. Our focus, however, is on non-equilibrium states with a non-vanishing current.
π 2

Here we show a simple method to obtain the long time asymptotic behavior of a free fermion system by a continuous hydrodynamic description. This allows one to avoid a lengthy exact calculation and yet gives asymptotically exact results. The Hamiltonian of the free fermion system can be written in the form H=
k ? ηk ? ?(k )ηk ηk

0 ?π 2 ?π -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x = n/t


where and ηk are creation and annihilation operators of fermions with momentum k and ?(k ) is the energy of an excitation with wave number k . In the simplest situation, the system is initially divided into two half in?nite chains, each of them being in a homogeneous pure state. In this case, the elementary excitations can be considered initially homogeneously distributed in each half chains. At time zero, each mode starts moving with velocity v (k ) = ?′ (k ). As the excitations are entirely independent, they do not interact and keep moving with their initial velocities. This argument suggests that whether an excitations is present at a space-time point (n, t) depends only on the ratio x = n/t. Moreover, keeping x ?xed, a ?nite neighborhood of site n becomes asymptotically homogeneous for t → ∞. This physical picture is not exact due to the ?nite lattice spacing. However, we believe that the above description becomes asymptotically exact for any free fermion system in the scaling limit: n → ∞, t → ∞, n/t = const. (See [25] for a rigorous derivation of this scaling limit for the XX model.) Whether an excitation is present at position n at time t can be decided by noting that for n > 0, the modes which are present were initially on the left side of the chain with v (k ) > n/t and on the right side of the chain with v (k ) < n/t. Similar argument applies for n < 0. This method can easily be extended to cases where the two half-in?nite chains are initially in mixed states, e.g., one can consider cases where the two half-in?nite chains are set to di?erent temperatures [25]. As an illustration, let us calculate the magnetization pro?le in the XX model with the simplest initial condition (1). The spectrum of the model is ?(k ) = ? cos(k ), that is, v (k ) = sin(k ). Initially, all the modes are ?lled on the left, while the right side of the chain is in the vacuum state. At time t, around site n > 0, the modes with sin(k ) > n/t are ?lled, that is, all the modes with k0 < k < π ? k0 , where k0 = arcsin(n/t). Similarly, for n < 0, only the modes with sin(k ) > n/t are ?lled, that is, the modes with ?π < k < ?π ? k0 , and k0 < k < π . For an illustration see Fig. 1. As each mode carries a unit magnetization, the average z magnetization can be obtained by simply integrating through the

FIG. 1: Hydrodynamic description for the XX chain started from the | · · · ↑↑↑↓↓↓ · · · initial condition. We consider the scaling limit where t → ∞ and x = n/t = const. The shaded region shows the elementary excitations that are present in the scaling points indexed by x.

?lled modes m(x = n/t) = 1/2 + dk/2π . This results 1 arcsin(x) in the well known pro?le m(x = n/t) = ? π for ?1 < x < 1, and the magnetization keeps its initial values outside this region [17]. Other applications are given in appendix A.



The local magnetization current operator for a quantum spin chain can be obtained through a continuity equation for the local magnetization [27]. For the XX model this gives
x x y jn = sy n sn+1 ? sn sn+1


for the current between spin n and n + 1. (We measure time in units of ). The time integrated current C0 , i.e., the net transported magnetization up to time t through the bond between spin 0 and spin 1, is a quantity which is less obvious to de?ne for a quantum system in general. However, in the case of the setup (1), C0 can simply be expressed as C0 = (sz n + 1/2).


The average of the integrated current C0 through the central bond grows asymptotically as π ?1 t [17], which can also be seen from the hydrodynamic picture described above. The hydrodynamic approach of Sec. III, however, does not allow to probe ?uctuations. Therefore we must return to the microscopic description. Below we shall focus 2 on the variance of the total current D(t) ≡ C0 ? C0 2 .

3 Let us de?ne the left, right and total magnetization as follows: ML =

sz n,

MR =

sz n, (6)

M = ML + MR .

The variance of the integrated current is equal to the variance of the left (right) magnetization:
2 D(t) = MR t

? MR

2 t

2 = ML


? ML 2 t.


Since the total magnetization is conserved and the initial state is an M eigenstate, the ?uctuation of M remains zero for any time t (ML + MR )2

chain the fermions are ?lled up to the Fermi energy ?L (?R ). In this case, the asymptotic state — which builds up near the origin — includes fermions from momentum ?kR to kL , where kL (kR ) are the Fermi momenta corresponding to ?L (?R ). For an illustration see Fig. 3, and more details are given in appendix A 1. The correlation function ρz (n) can easily be calculated for these asymptotic states [27]. One ?nds that in general it behaves 2 as ρz (n) = ? π21 n2 sin (n?), where ? depends on kL and kR . As the φ dependence of the asymptotic form averages out we conclude that the result (13) is unchanged for this class of initial states. The exact evaluation of D(t) is based on (9). Following the strategy of [17] we write sz i in terms of the local fermionic creation and annihilation operators c? , c: 1 ? sz n = cn cn ? . 2 (14)

? ML + MR

2 t

= 0.


By exploiting this property we can rewrite (7) as D(t) = ML

MR =


? ML MR ( sz l


sz m



z ? sz l sm t ) . (9)

In Heisenberg picture the time dependence of these operators under the dynamics of the XX chain take the simple form

Before presenting an exact calculation we provide a back-of-the-envelope derivation of our main result. The idea is to evaluate D(t) by substituting correlations in (9) with their stationary values in the local state which builds up at the origin for t → ∞ (see [17] and our Fig. 1). The reason is that the main contribution comes from “(j ? i) not too large”, and for t ? 1 these “points” are located at the origin. This leads to D(t) = ? where
z z z ρz (n) = sz k s k +n ? s k s k +n .

cn (t) =
j =?∞

ij ?n Jj ?n (t)cj ,


where Jn (t) are the Bessel functions. Inserting this into (9) one gets D(t) =
l≤0,m≥1 α,β,γ,δ

i?α+β ?γ +δ × Jα?l (t)Jβ ?l (t)Jγ ?m (t)Jδ?m (t) ×

nρ (n),



? ? ? c? α cβ cγ cδ ? cα cβ cγ cδ



The expectation value in the above formula is taken in the initial state. One ?nds (11)
? ? ? c? α cβ cγ cδ ? cα cβ cγ cδ =

In the (homogeneous) “maximal current” stationary state of the XX model ρz (n) takes the same form as in the ground state, where it is given by the well-known expression: ρz (n) = ? π21 n2 0 r = odd . r = even (12)

?δα,δ δβ,γ 0

if α ↑, β ↓ otherwise


for initial states which are product states of individual spins pointing either up or down (like (1)). Here, α ↑, β ↓ are shorthand notations for α, β with sα =↑, sβ =↓. Using identities J?k (t) = (?1)k Jk (t) and [43] Jk+p (t)Jk+q (t) = t
k ≥1

This gives a logarithmical divergence for D(t), which one can regularize by truncating the sum in (10). For a ?nite but large t the volume of the region around the origin, which can be described by this maximal current state, grows linearly with t. Hence we choose the upper limit in the sum in (10) to be proportional to t and obtain

Jp (t)Jq+1 (t) ? Jp+1 (t)Jq (t) 2(p ? q )

for the sums over l and m, one obtains D(t) = t2 4 Jα?1 (t)Jβ (t) ? Jα (t)Jβ ?1 (t) α?β . (18)

α↑,β ↓

D(t) ? ?

nρz (n) =

1 ln(t). 2π 2


Specially, for our initial condition (1) we get D(t) = t2 4 Jl?1 (t)Jm?1 (t) + Jl (t)Jm (t) l+m?1

This argument remains valid for a more general class of initial conditions, where on the left (right) half of the

. (19)


4 This is an exact expression for the ?uctuations of the current at any time. From this formula we obtain the long time asymptotic behavior 1 (ln t + C ) (20) 2π 2 with constant C = 2.963510026 . . .. The leading term coincides with our heuristic argument (13). The derivation of (20) is relegated to appendix B. We also evaluated expression (19) numerically and plotted it in Fig. 2 for t < 50. We observe a logarithmic increase in time, and our numerical estimate for the constant is C ≈ 2.9633, which is in good agreement with the exact result. On top of this logarithmic growth one observes oscillations with decreasing amplitude. We found that the formula D(t) = D(t) = 1 cos 2t ln t + C ? (ln t + C ′ ) , 2π 2 t (21) time evolution in this general case. Due to the symmetry in the XY plane, however, the z component Mz of the total magnetization is still conserved, hence the magnetization current can be studied. It has been investigated numerically by Gobert et. al. [18]. Our interest lies in the critical region (?1 < ? < 1), and in the isotropic ferromagnetic (? = 1) and antiferromagnetic (? = ?1) case. The XX model belongs to the critical region, and the behavior in the entire critical region is believed to be similar to the behavior of the XX model. In particular, the magnetization pro?le plausibly scales linearly with time: m(n, t) → M(n/t). On the other hand, the magnetization pro?le is expected to become frozen when |?| > 1. Algebraic scaling seems to emerge for the XXX model (|?| = 1), viz. m(n, t) → M(n/ta ) in the scaling regime n → ∞, t → ∞ with n/ta kept ?nite. Numerically the exponent is a = 0.6 ± 0.1 [18], so the non-trivial part of the pro?le is sub-ballistic. For the XX chain we obtained the correct current ?uctuations in the leading order from the simple formula (10) when the upper limit of the sum was chosen to grow linearly with time. The reason for this choice of upper limit is that the front and the whole pro?le “moves” linearly with time, hence the cuto? must behave similarly. We shall use (10) also for the XXZ chain, and we assume that the upper bound moves again linearly vt in the critical region (?1 < ? < 1). The actual value of the ‘velocity’ v is unknown, but it does not a?ect the leading order term anyway. The next problem is that we should use the spin correlations ρz (n) in the presence of current in (10). For the XX chain we know [17, 27] that the current does not a?ect the z component of the correlations signi?cantly (may introduce a modulation), only the x and y components. Here we boldly assume the same for the XXZ model, at least for its large distance behavior, and we use the equilibrium correlations in (10). The asymptotic formulae for ρz (n) are [36, 37] ? ? ?δn?2 0<?<1 ? ? ??[1 + (?1)n ](2π 2 )?1 n?2 ? = 0 ρz (n) = (22) 2 ? (?1)n A n?4π δ ? δn?2 ?1 < ? < 0 ? ? √ ? (?1)n B n?1 ln n ? δn?2 ? = ?1 where we used the shorthand notation 1 δ= 4π arccos(?) (23)

with C ′ ≈ 1.95 (and C given exactly in (B23)) gives a very good ?t to the numerical data even for relatively short times. Since the mean current is 1/π , in average one Fermion crosses the origin in time π . Hence the cos 2t oscillations can be interpreted as a consequence of the quantum nature of the magnetization: each passing Fermion causes a bump in the ?uctuations. Similar arguments were used to explain oscillations in the magnetization pro?le in the same system [21].

0.3 D (t )



0 0 10 20 t FIG. 2: Shown is the variance D(t) vs. time t. Red curve is a result of the exact numerical evaluation of (19), blue curve shows the result (20) of our asymptotical analysis. On top of the logarithmic growth we ?nd a subleading oscillating term. Eq. (21) gives a ?t, which is almost indistinguishable from the numerical data (red line). 30 40 50



In a general XXZ chain (2) the coupling is nonzero in the z direction. There is no explicit solution for the

We now insert (22) into equation (10). The amplitude A = A(?) was ?rst guessed in [36]. Yet we do not need this result. Indeed, the leading oscillating term in ρz (n) has the exponent a = 4π 2 δ varying in the range 1 < a < 2 when the anisotropy parameter varies in the ?1 < ? < 0 range. Because this leading term oscillates, we form pairs and ?nd that the oscillating terms in (10) yield the contribution A n?a+1 ? A (n + 1)?a+1 ? aA n?a (24)

5 that decays faster than n?1 . Hence the oscillating term provides merely a constant contribution to the variance D(t) while the sub-leading n?2 term results in the leading logarithmically diverging contribution. Therefore D(t) = δ ln t (25) region). Our arguments are heuristic and a more rigorous derivation is a key challenge for future work. Intriguingly, ?uctuations seem more tractable than e.g. the average magnetization pro?le in the XXZ chain, which is completely unknown. For free fermion systems, current ?uctuations were found to be asymptotically Gaussian [40] and therefore the variance provides a complete characterization of the full current statistics. Moreover, according to [41] this indicates that the entanglement (between the left and right halves) is simply proportional to the variance of the current (with a factor 3/π 2 ). In order to check this relation we compared our results for D(t) to numerical results for the time dependent entanglement entropy (for the same model and initial condition) presented in Fig. 19 of [18], and we found a good agreement in leading order. Little is known about higher moments of current ?uctuations for interacting fermions, and there is no reason to believe that they are Gaussian. The full current statistics is very di?cult to probe (both theoretically and experimentally) for interesting interacting fermion systems [42]. (Even for classical interacting particles, the derivation of the full counting statistics is usually a formidable challenge, see [6, 8, 9, 11, 12, 13, 14].) For the XXZ model, however, it is perhaps possible to compute higher order cumulants with the same approach we applied to obtain the second cumulant (the variance).

This prediction implies that the logarithmic behavior of the variance is universal. The amplitude diverges at ? = 1; analytically δ → (4π )?1 [2(1 ? ?)]?1/2 as ? ↑ 1. This divergence is not very surprising since the isotropic Heisenberg ferromagnet apparently exhibits a truly different behavior. In the other extreme ? ↓ ?1, the amplitude δ approaches the ?nite value (4π 2 )?1 . However, as indicated by the last √ formula in (22), the oscillating asymptotic [38]. A is Bn?1 ln n for the isotropic anti-ferromagnet √ calculation similar to (24) gives n?2 ln n after canceling the oscillations. This leads to dn √ ln n ? (ln t)3/2 (26) D? n Note that for the isotropic anti-ferromagnet the magnetization is non-trivial in the interval that grows slower than√ linearly with time. (The natural guess is the di?usive t growth.) However, the upper limit in the integral in (26) would a?ect only the pre-factor. Thus our tentative predictions for the variance are: enhanced logarithmic growth D ? (ln t)3/2 for ? = ?1; and universal logarithmic behavior D(t) = δ ln t with known pre-factor (23) in the critical region ?1 < ? < 1.



We have studied the ?uctuations of the time integrated magnetization current in the quantum XX chain started from an inhomogeneous non-stationary initial state (1). We derived an exact formula (19) for the variance D(t) of the current. We have shown that D(t) increases logarithmically in the long time limit (20), which is consistent with numerical evaluation of the exact formula (see Fig. 2) [39]. On top of this logarithmic growth we observed oscillations with decreasing amplitude. We argue that this logarithmic leading order behavior remains unchanged for a more general class of initial conditions (where the magnetization on the two half-chains is not saturated). These small logarithmic ?uctuations re?ect the ideal conductor nature of the integrable XX quantum chain. The current simply “slides” through the system ballistically with no disturbance, hence the tiny ?uctuations. Conversely, if an impurity is present at the origin, the variance grows linearly with time [40]. Similarly, in stochastic particle systems [4, 5, 6, 7, 8, 9] the noise – which is intrinsically present in these models – generates algebraic ?uctuations. We have argued that current ?uctuations in the homogeneous XXZ model are also logarithmic (in the critical

We thank Alexander Abanov, Deepak Dhar, Viktor Eisler, R? obert Juh? asz, Eduardo Novais, Pierre Pujol, Zolt? an R? acz and Gunter M. Sch¨ utz for illuminating discussions. We also gratefully acknowledge ?nancial support from NIH grant R01GM078986 (T. A.), the Hungarian Scienti?c Research Fund OTKA PD-72607 (R. A.), and Je?rey Epstein for support of the Program for Evolutionary Dynamics at Harvard University.


In section III we demonstrate how the hydrodynamic approach works for the XX model in the simplest case, i.e., when all the spins point up (down) on the left (right) side at t = 0. One can easily recover the exact results for a slightly more complicated initial state as well. Consider a product state composed of a ground state (in an appropriate external ?eld) with mL (mR ) magnetization on the left (right) side of the chain. That is, the modes ?kL(R) < k < kL(R) with kL(R) = π (1/2 + mL(R) ) are ?lled on the left (right) side. Without any restriction, in the following we assume that mL > mR .

π kL
π 2

8 7 6 5 M+ 4 3 2 1 0 -1 ?vR ?vL 0 vL vR 1 0 5 10 15 20 t FIG. 4: P Shown is the average transferred total magnetization z M+ = n≥0 (Sn + 1/2) as a function of time started from the | · · · ↑↑↑↓↓↓ · · · initial condition in the dimerized XX model. The data points show the result of a time-dependent DMRG calculation published in [18], solid lines indicate the prediction of the hydrodynamic description. The coupling constants for the weak link are 1, 0.8, 0.6, 0.4, 0.2 (from above to below). 25 30 35 40 y = n/t

kR 0 ? kR ?π 2 ? kL ?π k

FIG. 3: Hydrodynamic description for the XX chain started from an initial condition where the left (right) side of the chain is in the ground-state with magnetization mL (mR ). We consider the scaling limit where t → ∞ and y = n/t = const. The shaded region shows the elementary excitations that are present in the scaling points indexed by y .

As an illustration see ?gure 3. Here we assumed vL < vR , as the opposite case is entirely similar. In all cases m(x) has the symmetry property m(?x) = mL + mR ? m(x). We mention that the special case of mR = ?mL was considered in detail in [17, 25].
2. Dimerized XX model

with vL ≡ v (kL ) = cos(πmL ) and vR ≡ v (kR ) = cos(πmR ). and m(?x) = mL + mR ? m(x). (2) A qualitatively di?erent case is when mL > 0 and mR < 0. The argument of section III leads to the following asymptotic pro?le: ? m L +m R 0 < x < vL ? ? ? mR 2 1 arcsin(x) + ? , vL < x < vR 2 4 2π m(x) = (A2) arcsin(x) 1 ? vR < x < 1 ? π ? mR + 2 ? mR vR < x,

There are two qualitatively di?erent cases. (1) If the sign of mL and mR are equal the scaling pro?le consists of three segments. Here we assume that mL > 0, mR > 0 but the case mL < 0, mR < 0 is entirely analogous. Applying the argument of section III here the magnetization pro?le takes the following scaling form: ? m L +m R 0 < x < vL ? 2 arcsin(x) mR 1 m(x) = (A1) + 4 ? 2π , vL < x < vR ? 2 mR vR < x,

with Jm = 1 for odd m, and Jm = δ < 1 for even m. The spectrum of the model is ?) = ± ω (k 1 2 ? 1 + δ 2 + 2δ cos k, (A4)

? is a sublattice wave number going from ?π to where k ? has moπ (an excitation with sublattice wave number k ? mentum k = k/2). The two signs in (A4) indicate two branches with k ∈ (?π/2, π/2). For the simplest initial condition with mL = ?mR = 1/2, the fastest mode ? = ±(π ? arccos δ )] has speed present on the left [at k δ . That is, the front still moves ballisticly, but it can go ?) = x/2 arbitrarily slowly. By inverting the function ω ′ (k k1,2 = arccos ?x2 ± (1 ? x2 )(δ 2 ? x2 ) δ (A5)

with 0 < x < δ , after some elementary calculations, one obtains the magnetization pro?le as well 1 1 x 1 . (k2 ? k1 ) = ? arcsin m(x) = ? + 2 2π π δ (A6)

In all the above example the speed of the front was one, but it is not necessary. Consider, as an example, the dimerized XX model [35] H=?
x y y Jm (sx m sm+1 + sm sm+1 )


Surprisingly, the dimerization only rescaled the asymptotic magnetization pro?le, and the result agrees with a homogeneous chain with coupling δ at each bond. That means that the weaker links behave as bottlenecks and they govern the dynamics of the chain. The average timeintegrated current through the origin is tδ/π , which was also found to be linear in time in [18]. The value of their current agrees with our result (see ?gure 4).


Now in (19) we write l,m≥1 = n≥2 l+m=n , and divide the summation through n into four parts
[t1/2?? ] [t1/2+? ] [t] ∞

Here we study the large time behavior of (19). As the summation in (19) goes to in?nity, we use a general large t asymptotic formula [43] of Bessel functions which is valid for all m < t: π 2 cos tf (?) ? 4 1 , (B1) Jm (t) = +O π t (1 ? ?2 )1/4 t where f (?) = 1 ? ?2 ? ? cos?1 ? , ?= m . t (B2)

n≥2 n=2

n=[t1/2?? ]+1

n=[t1/2+? ]+1



where [·] denotes the integer part. The corresponding terms in (19) will be referred to as D(t) = D1 (t) + D2 (t) + D3 (t) + D4 (t) (B8)

We will also use the scaled variable λ = l/t. For m > t the Bessel function Jm (t) becomes exponentially small thus we can safely neglect those terms in (19). Hence in the double sum we consider only the range λ, ? ≤ 1. Based on (B1) the asymptotic form of a Bessel function with a shifted index can be obtained by simply Taylor expanding f (?) in the argument of the cosine (the term coming from the expansion of the denominator is negligible), which gives Jm?1 (t) = 2 ? cos a ? 1 ? sin a +O πt (1 ? ?2 )1/4 ?2 1 t (B3)

respectively. We choose the exponent in (B7) in the range 0 < ? < 1/2; later we shall take the ? → 0 limit. Since t1/2?? ? t1/2 as t → ∞, in D1 (t) we can use the simple formula (B6), and immediately perform one summation. Thus the ?rst sum becomes
[t1/2?? ]

D1 (t) = π

?2 n even

1 , n?1


which in the t → ∞ limit is simply D1 (t) = (1/2 ? ?) ln(t) + γE + ln(2) , 2π 2 (B10)

where we introduce the shorthand notation π π a = tf (λ) ? , b = tf (?) ? . 4 4 Now using the asymptotic formulas (B1) and (B3) in (19) we obtain Jl?1 (t)Jm?1 (t) + Jl (t)Jm (t) = where T = (1 + λ?) cos a cos b + ? λ 1 ? λ2 1 ? ?2 sin a sin b (2/πt)T , (1 ? λ2 )1/4 (1 ? ?2 )1/4 (B4)

where γE = 0.5772 . . . is the Euler constant. Now we consider the contribution from the region t1/2?? ≤ n ≤ t1/2?? . Using cos2 (·) ≤ 1 we ?nd that
[t1/2+? ]

D2 (t) ≤

n=[t1/2?? ]+1

1 2? ≈ 2 ln(t) n?1 π


1 ? ?2 cos a sin b ? ? 1 ? λ2 sin a cos b. (B5) √ From hereafter we omit noting the relative O(1/ t) corrections. When λ, ? ? 1, we can replace (1 ? λ2 )1/4 (1 ? ?2 )1/4 by one in (B4), and the expression for T simpli?es to T = cos(a ? b) = cos[tf (λ) ? tf (?)]. Using (B2) we expand f (λ) and f (?) to give tf (λ) ? tf (ν ? λ) = ? π (l ? m) l2 ? m2 + + tO(λ4 , ?4 ). 2 2t

for ?xed positive ? and large t. Hence one can see that for small ? the contribution from D2 (t) becomes negligible (as compared to D1 (t). In D3 (t) and D4 (t) we can replace summation by integration, since t1/2+? → ∞ as t → ∞. We take the square of (B5) and drop rapidly oscillating terms (like sin a cos a) while sin2 and cos2 are replaced by 1/2; this results in replacement of T 2 by (1 + λ?)/2. Overall, we ?nd that D3 (t) = 1 2π 2

n=[t1/2+? ]+1

1 (n ? 1)2 1 + λ? (1 ? λ2 )(1 ? ?2 ) (B12)

× and D4 (t) = 1 2π 2

l+ m = n

√ When l, m ? t, we can keep only the ?rst leading term of this sum. In this limit (B4) reads 2 π (l ? m) Jl?1 (t)Jm?1 (t)+ Jl (t)Jm (t) = . (B6) cos πt 2


1 (n ? 1)2
l+ m = n


1 + λ? . (B13) (1 ? λ2 )(1 ? ?2 )

8 Note that there is only an exponentially small contribution from n > 2t terms. Notice that the second sum in (B12-B13) can be replaced by 1 n


C3 =

dν ν

1 0

dx [F (ν, x) ? 1] = 0.34929294 . . . (B19)

F (ν, x)dx,


where we used the shorthand notation F (ν, x) = [1 ? ν 2 x2 ][1 ? ν 2 (1 ? x)2 ] 1 + ν 2 x(1 ? x) , (B15)

By a similar argument the contribution (B13) from the t ≤ n ≤ 2t region remains ?nite in the t → ∞ limit: D4 (t) = with

C4 2π 2


and introduced ν = n/t = λ + ? and x = λ/ν . For t → ∞ the ?rst sum in (B12-B13) can also be replaced by an integral, which leads to D3 (t) = 1 2π 2
1 t??1/2

dν ν


C4 = dx F (ν, x) (B16)

dν ν

ν ?1

dx F (ν, x) = 1.34385423 . . . (B21)
1?ν ?1


It is useful to rewrite (B16) as D3 (t) = 1 2π 2
1 t??1/2

dν ν

1 0

dx [F (ν, x) ? 1] + (1/2 ? ?) ln t , (B17) 2π 2

Combining the contributions (B10), (B11), (B18), and (B20) from the three regions of (B7), and taking now the ? → 0 limit, we obtain D(t) = where C = γE + ln 2 + C3 + C4 = 2.963510026 . . . (B23) 1 (ln t + C ) 2π 2 (B22)

where the ?rst integral is convergent in the limit t → ∞, ? → 0. Therefore the contribution gathered in the t1/2+? ≤ n ≤ t region is (1/2 ? ?) ln t + C3 D3 (t) = 2π 2 (B18)

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