Boundary effects on quantum q-breathers in a Bose-Hubbard chain
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Boundary e ff ects on quantum q-breathers in a Bose-Hubbard chain Ricardo A. Pinto a, ∗ , Jean Pierre Nguenang a,b , Sergej Flach a a Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, 01187 Dresden, Germany b Fundamental physics laboratory: Group of Nonlinear physics and Complex systems, Department of Physics, University of Douala, P.O. Box24157, Douala-Cameroon
Abstract
We investigate the spectrum and eigenstates of a Bose-Hubbard chain containing two bosons with fixed boundaryconditions. In the noninteracting case the eigenstates of the system define a two-dimensional normal-mode space. Forthe interacting case weight functions of the eigenstates are computed by perturbation theory and numerical diagonal-ization. We identify paths in the two-dimensional normal-mode space which are rims for the weight functions. Thedecay along and o ff the rims is algebraic. Intersection of two paths (rims) leads to a local enhancement of the weightfunctions. We analyze nonperturbative e ff ects due to the degeneracies and the formation of two-boson bound states. Key words: q-breathers, Bose-Hubbard model, normal modes
PACS: + m, 03.65.Ge
1. Introduction
Localization phenomena due to nonlinearity and spa-tial discreteness in di ff erent physical systems havereceived considerable interest during the past fewdecades. Despite the given translational invariance ofa lattice, nonlinearity may trap initially localized exci-tations. The generic existence and properties of discretebreathers - time-periodic and spatially localized solu-tions of the underlying classical equations of motion- allow us to describe and understand these localiza-tion phenomena [1, 2, 3, 4]. Discrete breathers wereobserved in many di ff erent systems like bond excita-tions in molecules, lattice vibrations and spin excita-tions in solids, electronic currents in coupled Joseph-son junctions, light propagation in interacting opticalwaveguides, cantilever vibrations in micromechanicalarrays, cold atom dynamics in Bose-Einstein conden-sates loaded on optical lattices, among others (for ref-erences see [1, 2]). In many cases quantum e ff ectsare important. Quantum breathers are nearly degener-ate many-quanta bound states which, when superposed,form a spatially localized excitation with a very longtime to tunnel from one lattice site to another (for refer-ences see [1, 2, 4]). ∗ Corresponding author
Email addresses: [email protected] (RicardoA. Pinto), [email protected] (Jean Pierre Nguenang), [email protected] (Sergej Flach)
The application of the above ideas to normal-modespace of a classical nonlinear lattice allowed us to ex-plain many facets of the Fermi-Pasta-Ulam (FPU) para-dox [5], which consists of the nonequipartition of en-ergy among the linear normal modes in a nonlinearchain. There, the energy stays trapped in the initially ex-cited normal mode with only a few other normal modesexcited, leading to localization of energy in normal-mode space. Recent studies showed that, similar to dis-crete breathers, exact time-periodic orbits exist whichare localized in normal-mode space. The properties ofthese q -breathers [6, 7, 8, 9, 10, 11, 12, 13] allow us toquantitatively address the observations of the FPU para-dox. A hallmark of q -breathers is the exponential local-ization of energy in normal-mode space, with exponentsdepending on control parameters of the system.On the quantum side, recently we studied the fateof analogous states (quantum q -breathers) in a one-dimensional lattice with two interacting bosons and pe-riodic boundary conditions [14]. By using perturba-tion theory, supported by numerical diagonalization, wecomputed weight functions of the eigenstates of the sys-tem in the many-body normal-mode space. We didfind localization of the weight function in normal-modespace. However, at variance from the classical case, thedecay is algebraic instead of exponential. The periodicboundary conditions allow us to introduce an irreducibleBloch representation. Since states with di ff erent wave Preprint submitted to Elsevier October 30, 2018 umbers belong to di ff erent Hilbert subspaces, they arenot coupled by a Hubbard interaction term. Therefore,localization along the Bloch wave number is compact.This is also happening for the corresponding classicalnonlinear Schr¨odinger equation with periodic bound-ary conditions [12], when searching for plane-wave-likestates.The classical case however inevitably leads to non-compact distributions in normal-mode space, once fixedboundary conditions are considered. Indeed, also in thequantum case, these conditions violate translational in-variance, and lead to nonzero matrix elements betweenstates with di ff erent Bloch wave numbers, mediated bythe Hubbard interaction. That is the reason for studyingthe properties of quantum q -breathers for finite chainswith fixed boundary conditions. From a technical pointof view, the irreducible normal-mode space dimensionis then increased from one to two.In Sec. 2 we describe the model and introduce thebasis to write down the Hamiltonian matrix. We de-scribe the quantum states of the lattice containing oneand two noninteracting bosons. From the latter case weuse the two-particle states as the basis to write down theHamiltonian matrix in normal-mode space for the in-teracting case, after which the energy spectrum is com-puted. In Sec. 3 we study localization in normal-modespace. We introduce weight functions to describe local-ization in that space, and obtain analytical predictionsusing perturbation theory. We present numerical resultsfrom a diagonalization of the Hamiltonian matrix, andcompare them with analytical estimates. Then we studynonperturbative e ff ects when increasing the interactionparameter. Finally we present our conclusions in Sec.4.
2. Model and spectrum
We consider a one-dimensional periodic lattice with f sites described by the Bose-Hubbard (BH) model.This is a quantum version of the discrete nonlinearSchr¨odinger equation, which has been used to describea great variety of systems [15]. The BH Hamiltonian isˆ H = ˆ H + γ ˆ H [16], withˆ H = − f X j = ˆ a + j (ˆ a j − + ˆ a j + ) , (1)and ˆ H = − f X j = ˆ a + j ˆ a + j ˆ a j ˆ a j . (2) ˆ H describes the nearest-neighbor hopping of particles(bosons) along the lattice, and ˆ H the local interac-tion between them whose strength is controlled by theparameter γ . a + j and a j are the bosonic creation andannihilation operators satisfying the commutation rela-tions [ˆ a j , ˆ a + j ′ ] = δ j , j ′ , [ˆ a j , ˆ a j ′ ] = [ˆ a + j , ˆ a + j ′ ] =
0, and thesystem is subject to fixed boundary conditions. TheHamiltonian (1) commutes with the number operatorˆ N = P fj = ˆ a + j ˆ a j whose eigenvalue is n , the total num-ber of bosons in the lattice. Here n =
2. It is of interestdue to its direct relevance to studies and observation oftwo-vibron bound states in molecules and solids [18, 19,20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Morerecently, two-boson bound states have been observed inBose-Einstein condensates loaded on an optical lattice[33].To describe quantum states, we use a number statebasis | Φ n i = | n n · · · n f i [16], where n i = , , | Φ n i is an eigenstate of the number operator ˆ N witheigenvalue n = P fj = n j . For the case of having only one boson in thelattice ( n =
1) a number state has the form | · · · l · · · i ≡ | l i , where l denotes the latticesite where the boson is. This number state can be alsowritten as | l i = ˆ a + l | i , (3)where the operator ˆ a + l creates a boson at the l -th site ofthe lattice, and | i is the vacuum state.We write down the Hamiltonian matrix in the ba-sis of the above-defined number states. For the single-boson case, the interaction term ˆ H has no contributionto the matrix elements. The eigenstates of ˆ H , for fixedboundary conditions, are standing waves: | Ψ k i = f X l = s f + kl ) | l i ≡ | k i , (4)where k = q π/ ( f + q = , . . . , f . The corre-sponding eigenenergies are ε k = − k ) . (5)We define bosonic operators ˆ a k , ˆ a + k satisfying thecommutation relations [ˆ a k , ˆ a + k ′ ] = δ k , k ′ , [ˆ a k , ˆ a k ′ ] = [ˆ a + k , ˆ a + k ′ ] =
0, such that the state (4) may be written sim-ilar to (3): | k i = ˆ a + k | i , ˆ a + k = f X l = S l , k ˆ a + l , (6)2here the operator ˆ a + k creates a boson in the single-particle state with quantum number (wave number ormomentum) k . The bosonic operators ˆ a k , ˆ a + k are relatedto the operators ˆ a l , ˆ a + l in direct space through the trans-formation matrix S l , k = s f + kl ) . (7) For the two-boson case ( n = | l , l i = r − δ l , l a + l ˆ a + l | i , (8)where l ≥ l because of the indistinguishability of par-ticles. ˆ a + l and ˆ a + l respectively create one boson at thelattice sites l and l . The number of basis states is d = f ( f + /
2. The interaction term ˆ H in (1) con-tributes to the matrix elements of the Hamiltonian in theabove-defined basis.In the noninteracting case ( γ =
0) the eigenstates of ˆ H in terms of bosonic operators in the normal-mode spaceread [see Eq. (6)]: | k , k i = r − δ q , q a + k ˆ a + k | i , q ≥ q . (9)ˆ a + k and ˆ a + k respectively create one boson in the single-particle states k and k of the form (4). Using Eqs. (6)and (7), the relation between the basis states in normal-mode space (9) and the basis states in direct space (8)reads: | k , k i = p − δ q , q √ × " f X l = f X l > l ( S l , k S l , k + S l , k S l , k ) | l , l i + √ f X l = S l , k S l , k | l , l i . (10)In the interacting case ( γ > d × d matrix [ d = f ( f + /
2] whose elements H ( i , j )( i , j = , . . . , d ) are H ( i , j ) = h k ′ , k ′ | ˆ H | k , k i ≡ h q ′ , q ′ | ˆ H | q , q i . (11)The integer j that labels the column of the matrix ele-ment (11) is related to the mode numbers q and q by j q , q = ( q − f + − ( q − q + + q . (12) The same relation holds for the integer i q ′ , q ′ labeling therow of the matrix element (11).The matrix elements (11) are H ( i , j ) = H ( i , j ) + γ H ( i , j ) , (13)where H ( i , j ) = ( ε k + ε k ) δ i , j , (14)and H ( i , j ) = f q , q , q ′ , q ′ f X l = S l , k S l , k S l , k ′ S l , k ′ . (15) ε k is the single-particle energy given by Eq. (5), and thecoe ffi cients f q , q , q ′ , q ′ are f q , q , q ′ , q ′ = − q (2 − δ q , q )(2 − δ q ′ , q ′ )( f + . (16)In Fig. 1 we show the energy spectrum of the Hamil-tonian matrix (13) obtained by numerical diagonaliza-tion for di ff erent values of the interaction parameter γ .In all calculations by numerical diagonalization we used f =
40, which leads to a matrix dimension d = E ν ( ν = , . . . , d ). At γ =
0, the spectrum consists ofthe two-boson continuum, whose eigenstates | k , k i aregiven by (10). The eigenenergies are the sum of the twosingle-particle energies: E k , k = − k ) + cos( k )] . (17) ν -25-20-15 -10 -505 E ν γ = 0γ = 2γ = 4γ = 6γ = 8γ = 10 Figure 1: Energy spectrum of the two-boson BH chain with fixedboundary conditions for di ff erent values of the interaction strength γ .The eigenvalues are plotted as a function of the eigenvalue label (seetext). Here f = γ >
0, eigenvalues in the lower part of the spec-trum are pushed down, and beyond γ ≈ f states splits o ff from the two-boson continuum. Theseare the two-boson bound states, with a high probabilityof finding the two bosons on the same lattice site, whilethe probability of them being separated by a distance r decreases exponentially with increasing r [16, 15, 14].The critical value γ b = ff from the continuum maybe explained as follows. In the limit f → ∞ the un-normalized bound state with highest energy E = − γ isgiven by [14, 17]: | Ψ i = f X l = ( − l | l , l i . (18)For γ b = E ∈ [ − ,
3. Localization in normal-mode space
We recall that the normal-mode space is spanned byboth momenta k and k . The conditions 0 < k , < π and k ≤ k reduce the normal-mode space to a trianglethat we call the irreducible triangle , as sketched in Fig.2. For finite f and γ the eigenstates | Ψ i will spread inthe basis of the γ = {| k , k i} . We measuresuch a spreading by computing the weight function innormal-mode space C ( k , k ) = |h k , k | Ψ i| . We use perturbation theory to calculate the weightfunctions, where γ is the perturbation. We fix the mo-mentum k and k , and choose an eigenstate | ˜ k , ˜ k i ofthe unperturbed case γ =
0. The wave numbers ˜ k and˜ k define a seed point P = (˜ k , ˜ k ) in the irreducible tri-angle (see Fig. 2). Upon increase of γ , the chosen eigen-state transforms into a new eigenstate | Ψ ˜ k ˜ k i , which willhave overlap with several eigenstates of the γ = γ : | Ψ ˜ k ˜ k i = | ˜ k , ˜ k i + γ | Ψ (1)˜ k , ˜ k i , (19)where | Ψ (1)˜ k , ˜ k i = X k ′ , ˜ k X k ′ , ˜ k k ′ ≥ k ′ h k ′ , k ′ | ˆ H | ˜ k , ˜ k i E k ˜ k − E k ′ k ′ | k ′ , k ′ i . (20) Thus for k , ˜ k and k , ˜ k the weight function C ( k , k ; ˜ k , ˜ k ) = |h k , k | Ψ ˜ k ˜ k i| is C ( k , k ; ˜ k , ˜ k ) = γ |h k , k | ˆ H | ˜ k , ˜ k i| | E k ˜ k − E k k | , (21)where E k k and E k ˜ k are eigenenergies of the unper-turbed system given by (17). For convenience we usenew variables in normal-mode space k ± = k ± k , (22)which are the total (Bloch) and relative wave numbersrespectively. They have values 0 < k + < π and 0 < k − < π . Since we are interested in the behavior of theweight function around the core at (˜ k , ˜ k ), we define thecoordinates relative to that point: ∆ ± = k ± − ˜ k ± . (23)Thus, (21) becomes C ( k , k ; ˜ k , ˜ k ) = γ f q , q , ˜ q , ˜ q [16( E k ˜ k − E k k )] × R k + , k − ;˜ k + , ˜ k − , (24)where f q , q , ˜ q , ˜ q is given by Eq. (16).The coe ffi cient R k + , k − ;˜ k + , ˜ k − consists of a sum of eightterms of the form g ( ζ ) = sin h (2 f + ζ i sin (cid:16) ζ (cid:17) , (25)with pairwise opposite signs (see appendix A). Foreach term, the argument ζ is a certain combination ofthe wave numbers k + , k − and ˜ k + , ˜ k − (see appendix Afor details). Unless the argument of any of the eightterms g ( ζ ) vanishes, all of them cancel each other and R k + , k − ;˜ k + , ˜ k − =
0. Thus the condition ζ = R k + , k − ;˜ k + , ˜ k − , together with the relations (22) and(23), defines lines k = k ( k ) in the normal-mode spacewhere the weight function C ( k , k ; ˜ k , ˜ k ) is nonzero.These lines are schematically shown in Fig. 2 (the an-alytical derivation of these lines is given in appendixA). Note that these lines are specularly reflected at theboundaries k = k = π of the irreducible triangle.To study the localization in normal-mode space awayfrom the core using the formula (24), we consider thetwo cases ∆ − = ∆ + > ∆ + and ∆ − (Fig. 2). For4 = (k , k ) ~ k / π k / π ~ ∆ + ∆ − P P P = ( π− k , π− k ) ~ ~ Figure 2: Sketch of the di ff erent lines in the two-dimensional normal-mode space along which the weight function (21) is nonzero. The seedpoint P = (˜ k , ˜ k ) corresponding to the unperturbed eigenstate | ˜ k , ˜ k i is represented by the black spot. Its conjugate point ¯ P = ( π − ˜ k , π − ˜ k )is represented by the grey spot. The axes defining the coordinates ∆ + and ∆ − are indicated by the arrows emerging from P . each case we obtain, with | ∆ ± | ≪ π , C ± ( k , k ; ˜ k , ˜ k ) = γ f + ! (2 − δ q , q )(2 − δ ˜ q , ˜ q ) × ∆ − ± ( h cos(˜ k ) + cos(˜ k ) i ∆ ± + sin(˜ k ) ± sin(˜ k ) ) − . (26)The e ff ective interaction strength is γ/ ( f + γ → f → ∞ we have compactification ofthe eigenstates. The formula (26) shows localization innormal-mode space. Depending on the seed (˜ k , ˜ k ) wefind algebraic decay within the irreducible triangle, C ∼ ∆ − α , with α = ,
4. If sin ˜ k ± sin ˜ k , α =
2. Ifsin ˜ k ± sin ˜ k = α =
4. E.g. for ˜ k = ˜ k C − ∼ γ f + ! (˜ k ) ∆ − . (27)Note that along the ∆ + direction in the irreducible tri-angle, R k + , k − ;˜ k + , ˜ k − = f +
1) at all points but ¯ P = (¯ k = π − ˜ k , ¯ k = π − ˜ k ). This is the conjugate point of theseed P (Fig. 2), where two lines intersect. At this point R k + , k − ;˜ k + , ˜ k − = f + | ˜ k , ˜ k i and | ¯ k , ¯ k i have energies E k , ¯ k = − E k , ˜ k . In Fig. 3 we show the weight function in the two-dimensional normal-mode space obtained by numerical / π k / π (a) −14−12−10−8−6−4−20.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 k / π k / π (b) −14−12−10−8−6−4−2 Figure 3: 3-D plot of the logarithm of the weight function in thenormal-mode space for the eigenstate ν = f =
40 and γ = . diagonalization and the formula (24) respectively, withcharacteristic localization profiles. We find agreementof the numerical data with the results from perturbationtheory. The largest value is at the point P = ( π, π ) ∼ (0 . π, . π ), and it decays mainly along the lines de-scribed in the previous section (Fig. 2). Note also thepresence of the local maximum at the conjugate point¯ P ∼ (0 . π, . π ) in both cases.In Figs. 4 and 5 we plot the weight function of theeigenstate shown in Fig. 3 along the directions ∆ + and ∆ − respectively for di ff erent values of the interaction pa-rameter γ . The state becomes less localized with in-creasing γ , as expected from the above analysis. Thedecay of the weight function is well described by per-turbation theory (dashed lines). The peak of the weightfunction at the conjugate point is clearly seen in Fig. 4.In Fig. 6 we plot the weight function of di ff erentstates along the ∆ + direction. It decays as a power lawthat ranges from ∆ − for states near the lower corner ofthe irreducible triangle (see Fig. 8) to ∆ − for states ful-filling ˜ k ≈ π − ˜ k . In Fig. 7 we plot the decay of theweight function along the ∆ − direction, where we seethe power-law decay that ranges from ∆ − for states ful-filling ˜ k ≈ ˜ k (see Fig. 8), to ∆ − for states fulfilling5 ∆ + / π -12 -10 -8 -6 -4 -2 C + ( k , k ; k , k ) γ = 0.0001γ = 0.001γ = 0.01γ = 0.1 Figure 4: Weight function for di ff erent values of the interactionstrength γ of the eigenstate ν =
145 along the ∆ + direction. Thedashed lines are results from formula (24). Here f = -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 ∆ − / π -10 -8 -6 -4 -2 C - ( k , k ; k , k ) γ = 0.0001γ = 0.001γ = 0.01γ = 0.1 Figure 5: Weight function for di ff erent values of the interactionstrength γ of the eigenstate ν =
145 along the ∆ − direction. Thedashed lines are results from formula (24). Here f = -1 ∆ + / π -7 -6 -5 -4 -3 -2 -1 C + ( k , k ; k , k ) ν = 1ν = 3ν = 25ν = 241ν = 328 y ~ x -4 y ~ x -2 Figure 6: Weight function of di ff erent eigenstates (labeled by the in-dex ν ) along the ∆ + direction. Here γ = . f = ˜ k ≈ π − ˜ k . The results from numerical diagonalizationagree very well with those from the perturbation theoryanalysis. -1 ∆ − / π -5 -4 -3 -2 -1 C - ( k , k ; k , k ) ν = 145ν = 160ν = 201ν = 241ν = 328 y ~ x -4 y ~ x -2 Figure 7: Weight function of di ff erent eigenstates (labeled by the in-dex ν ) along the ∆ − direction. Here γ = . f = k / π k / π ν = 1ν = 3ν = 25ν = 145ν = 160ν = 201ν = 241ν = 328 Figure 8: Location P = (ˆ k , ˆ k ) of the eigenstates, shown in Figs. 6and 7, in the irreducible triangle. ff ects The results in the previous section were obtainedfor small values of the interaction parameter γ up to γ = .
1, for which perturbation theory gives a gooddescription of the results obtained by numerical diag-onalization. However, when increasing γ several non-perturbative e ff ects occur. These are: Split o ff of the two-boson bound state band : This ef-fect was discussed in Sec. 2.2 (Figs. 1). When γ > ff from the two-boson continuum, and the corresponding eigenstates arecorrelated in direct space, i.e. with large probability thetwo bosons are occupying identical lattice sites. Thus,in normal-mode space these eigenstates become delo-calized as shown in Fig. 9. Degenerate levels in the noninteracting case : Theanalysis using perturbation theory is valid as long as theeigenstate which is continued from the noninteractingcase is not degenerate. Because of the finiteness of thelattice the momenta ˜ k and ˜ k are restricted to discrete6 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 k / π k / π (a) −14−12−10−8−6−4−20.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 k / π k / π (b) −14−12−10−8−6−4−2 Figure 9: 3-D plot of the logarithm of the weight function in normal-mode space for the eigenstate ν =
25, that belongs to the two-bosonbound state band. Results were obtained by numerical diagonaliza-tion. (a) γ = .
5. The two-boson bound state band did not split o ff and the eigenstate is localized in normal-mode space. (b) γ =
2. Atthis interaction value the two-boson bound state band splits o ff and theeigenstate becomes delocalized in normal-mode space. Here f = values and define a grid in the two-dimensional normal-mode space. A grid point (˜ k , ˜ k ) defines a line of con-stant energy in normal-mode space through Eq. (17),with E k , k = E k , ˜ k (Fig. 10-a). The nondegeneracycondition implies that this line should not pass throughany other grid point. It is easy to see from Eq. (17) thatall states | ˜ k , π − ˜ k i are degenerate, with E ˜ k ,π − ˜ k = k = π − k (thick line in Fig.10-a). In Fig. 10-b we show the weight function of aneigenstate that is located on that diagonal in the non-interacting case. As expected, even for small values of γ , the state completely delocalizes along the degeneracydiagonal. Avoided crossings : Upon increase of the interac-tion parameter γ , the energies of continued eigenstateschange, and will resonate with eigenvalues of otherstates.The first possible avoided level crossing defines acritical value of the interaction parameter γ up to whichfirst-order perturbation theory is applicable. To estimate k / π k / π (a) / π k / π (b) −14−12−10−8−6−4−2 Figure 10: (a) Lines of constant energy given by (17). The red thickline is the E k , k = | k , k = π − k i are degenerate. (b) 3-D plot of the logarithm of the weight function innormal-mode space for the eigenstate ν = γ = . f =
40. In the noninteracting casethis state corresponds to | k , k = π − k i . this value, γ c , we assume that before the first avoidedcrossing is encountered, the eigenenergies depend lin-early on γ . This dependence may be estimated usingfirst-order perturbation theory in γ . The result is, forlarge f , E ˜ k , ˜ k ( γ ) ≈ E k , ˜ k + b (˜ k , ˜ k ) f γ, (28)where b (˜ k , ˜ k ) = k = , − k = ˜ k > , − k > , ˜ k > ˜ k . (29)Let us consider two levels E and E that interact inthe first avoided level crossing. At γ = δ E . For nonzero γ the energies linearly changein γ : E = E k , ˜ k + | b (˜ k , ˜ k ) | f γ, (30) E = E k , ˜ k + δ E − | b (˜ k , ˜ k ) | f γ. (31)7y equating E and E at γ = γ c we obtain γ c (˜ k , ˜ k ) = f δ E | b (˜ k , ˜ k ) | + | b (˜ k , ˜ k ) | . (32)The first avoided crossing of the level E will happenwith its nearest neighbor in the spectrum of the γ = δ E (˜ k , ˜ k ). Using Eq. (17)with ˜ k , ˜ k ≪ π/ δ E ≈ √ π/ ( f + . There-fore γ c ∼ / f .The coe ffi cient b (˜ k , ˜ k ) depends on the state | ˜ k , ˜ k i under consideration through (29). The coe ffi cient b (˜ k , ˜ k ) must have opposite sign as compared to b (˜ k , ˜ k ) for the avoided crossing to take place. Forthe states ν =
145 and ν =
41 located at (˜ k , ˜ k ) ≈ (0 . π, . π ) and (˜ k , ˜ k ) ≈ (0 . π, . π ) respectively(see Fig. 8), b (˜ k , ˜ k ) = − b (˜ k , ˜ k ) =
2. Thisleads to a critical value of the interaction parameter γ c ≈ .
28, which is in reasonable agreement with thenumerical results: γ c ≈ . ν = γ c ≈ . ν =
4. Conclusions
In this work we studied the properties of quan-tum q-breathers in a one-dimensional lattice contain-ing two bosons modeled by the BH Hamiltonian withfixed boundary conditions. Because of the lack oftranslational invariance, the normal-mode space is two-dimensional and reduces to a triangle when working inthe irreducible representation of the product basis states(the irreducible triangle). To explore localization phe-nomena in this system we computed appropriate weightfunctions of the eigenstates in the normal-mode spaceusing both perturbation theory and numerical diagonal-ization. We find that the weight function is sizable onlyalong the mutually perpendicular directions defined bythe total and relative momentum, thus it defines lines inthe irreducible triangle that show specular reflections atthe boundaries of the irreducible triangle. We observelocalization of the weight function along these lines.The localization is stronger when the size of the sys-tem increases or the interaction parameter is weaker, theformer because the e ff ective interaction drops in the di-lute limit of large chains. We found algebraic localiza-tion. The power of the decay is di ff erent for each eigen-state depending on which seed wave numbers have inthe noninteracting case, ranging from two to four.An interesting e ff ect is the local maximum of theweight function at the symmetry-related (conjugate)point of the eigenstate core in normal-mode space, due to a crossing between di ff erent paths described by thelines along which the weight function is nonzero withinperturbation theory.In addition to the existence of degeneracies betweeneigenstates in the noninteracting case, we analyzedother nonperturbative e ff ects as the interaction param-eter increases, which limit the applicability of perturba-tion theory to describe the system: The splitting o ff ofthe two-boson bound states from the two-boson contin-uum, and the occurrence of avoided level crossings. Thefirst e ff ect manifests as a delocalization of the weightfunction of the bound states due to the two-boson cor-relation in direct space. The second e ff ect manifests asa sudden change of the location of an eigenstate in thenormal-mode space due to resonant interaction with an-other eigenstate. Both e ff ects define critical values ofthe interaction parameter below which one may analyzethe system by perturbation theory. The occurrence of anavoided level crossing gives the smallest critical value.Although we considered a system with fixed bound-ary conditions, we still obtain algebraic decay as inthe case with periodic boundary conditions [14]. Thequestion how to restore exponential localization of clas-sical q-breathers from algebraic decay of quantum q-breathers in the limit of large numbers of particles is stillopen. When going to that limit, one may use a Hartreeapproximation and describe the system with a productstate wavefunction, or use a coherent state representa-tion. Both ways lead to the nonlinear Schr¨odinger equa-tion where classical q-breathers are known to exist [12]. Acknowledgements
J.P.N. acknowledges the warm hospitality of the MaxPlanck Institute for the Physics of Complex Systems inDresden. This work was supported by the DFG (grantno. FL200 /
8) and by the ESF network-programmeAQDJJ.
A. Lines of nonzero weight function
For fixed ˜ k , ˜ k , the coe ffi cient R ( k , k ; ˜ k , ˜ k ) in Eq.(24) is given by R ( k , k ; ˜ k , ˜ k ) = g ( k − + ˜ k − ) + g ( ∆ − ) − g ( k − + ˜ k + ) − g ( k − − ˜ k + ) − g ( k + + ˜ k − ) − g ( k + − ˜ k − ) + g ( ∆ + ) + g ( k + + ˜ k + ) , (33)where g ( ζ ) = sin[(2 f + ζ ]sin( ζ ) . (34)8he lines k = k ( k ) in the normal-mode space (irre-ducible triangle) along which R ( k , k ; ˜ k , ˜ k ) , g ( ζ ) = f + . (35)Let us analyze each of the arguments in Eq. (33): • k − + ˜ k − =
0: This implies that k − k = − (˜ k − ˜ k ) . (36)Since k ≥ k the above condition is possible onlyfor points (˜ k , ˜ k ) , ( k , k ) on the diagonal k = k . • ∆ − =
0: This condition leads to k = (˜ k − ˜ k ) + k , (37)which is the equation of the line along the ∆ + di-rection that cuts the k axis at k (0) = ˜ k − ˜ k . • k − + ˜ k + =
0: This implies that k − k = − (˜ k + ˜ k ) , (38)which is possible only if ˜ k = ˜ k = k , k ) ison the diagonal k = k . • k − − ˜ k + =
0: This leads to the equation k = (˜ k + ˜ k ) + k , (39)which describes a line parallel to the ∆ + directionthat cuts the k axis at k (0) = ˜ k + ˜ k . • k + + ˜ k − =
0: This implies that k = − (˜ k + ˜ k ) − k , (40)which is valid only if k = k = ˜ k = ˜ k = • k + − ˜ k − =
0: This leads to the equation k = (˜ k − ˜ k ) − k , (41)which is the equation of a line parallel to the ∆ − direction that cuts the k axis at k (0) = ˜ k − ˜ k . • ∆ + =
0: This leads to the equation k = (˜ k + ˜ k ) − k , (42)which describes the line along to the ∆ − directionthat cuts the k axis at k (0) = ˜ k + ˜ k . k / π k / π k / π k / π P d ∆ k Figure 11: Sketch of the discrete normal-mode space (irreducibletriangle) with a line of constant energy in the circular approxima-tion passing through the grid point P = (˜ k , ˜ k ) and the grid point(˜ k + ∆ k , ˜ k + ∆ k ). The strip of width d = √ ∆ k and area A s con-tains N g = A s / ∆ k grid points through which lines of constant en-ergy pass. The typical line separation within the strip is δ k ≈ d / N g . ∆ k = π/ ( f +
1) is the grid spacing.
B. Energy separation between nearest-neighbor lev-els in the noninteracting case
The finite size of the lattice leads to discrete values ofthe momenta k and k , and thus to a grid in the normal-mode space (Fig. 11). Let us consider a line of constantenergy which passes through the seed point P = (˜ k , ˜ k )given by E k , k = E k , ˜ k with Eq. (17). For small valuesof k and k , the energy in Eq. (17) may be approxi-mated to E k , k ≈ − + k + k , (43)which is the equation for a circle ( circular approxima-tion ). So the equation for the line of constant energypassing through the point P is k ( k ; ˜ k , ˜ k ) = q ˜ k + ˜ k − k . (44)Through another grid point at (˜ k + ∆ k , ˜ k + ∆ k ), sep-arated from P by a distance d = √ ∆ k ≈ ∆ k ( ∆ k is thegrid spacing), another line of constant energy with theform (44) passes (Fig. 11), defining a strip of area A s in the irreducible triangle. The strip contains N g gridpoints through which lines of constant energy pass. Theaverage line separation within the strip is δ k ≈ d / N g .The number of grid points in the strip is N g = A s ∆ k . (45)9he area of the strip is A s = π k + ∆ k ) + (˜ k + ∆ k ) − ˜ k − ˜ k ] = π ∆ k + (˜ k + ˜ k ) ∆ k ] . (46)Therefore N g = π ∆ k ( ∆ k + ˜ k + ˜ k ) , (47)and hence δ k (˜ k , ˜ k ) = ∆ k π (˜ k + ˜ k + ∆ k ) . (48)The corresponding energy separation is δ E (˜ k , ˜ k ) = E k + δ k , ˜ k + δ k − E k , ˜ k , (49)with δ k = δ k = δ k / √
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