Quantum many-body systems with nearest and next-to-nearest

PHYSICAL REVIEW B 71, 125121 共2005兲

Quantum many-body systems with nearest and next-to-nearest neighbor long-range interactions Meripeni Ezung,1 N. Gurappa,2,* Avinash Khare,3,† and Prasanta K. Panigrahi4,‡ 1School

2Laboratoire

of Physics, University of Hyderabad, Hyderabad 500 046, India de Physique, Theorique et Modeles Statistiques, Bat. 100 Universite Paris-Sud, 91405, Orsay, France 3Institute of Physics, Sachivalaya Marg, Bhubaneswar 751 005, India 4Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India 共Received 1 October 2004; published 22 March 2005兲

The bound and scattering states of the many-body systems, related to the short-range Dyson model, are studied. The Hamiltonian for the full-line problem is connected to decoupled oscillators. The analytically obtainable eigenstates are smaller in number as compared to the Calogero-Sutherland family, indicating the quasiexactly solvable nature of these models. The exactly found scattering states, a smaller set as compared to the Calogero case, can be realized as coherent states. The relation of the scattering Hamiltonian to free particles is also established algebraically. We analyze both AN−1 and BCN models on a circle and construct a part of the excitation spectrum by making use of the symmetry arguments. DOI: 10.1103/PhysRevB.71.125121

PACS number共s兲: 71.10.⫺w, 03.65.Ge

I. INTRODUCTION

Exactly solvable and quantum integrable many-body systems, with long-range interactions, are one of the most active fields of current research. The Calogero-Sutherland model 共CSM兲,1–3 the Sutherland model 共SM兲4 and their variants are the most prominent examples among such systems.5 These models have found application in various branches of physics,6–10 ranging from quantum Hall effect11–13 to gauge theories. It is known that, the CSM is related to random matrix theory2,6,14–17 and enables one to capture the universal aspects of various physical phenomena.6 The Brownian motion model of Dyson connects the random matrix theory 共RMT兲 with exactly solvable models.15 The role of RMT in the description of the level statistics of chaotic systems is well-known.18,19 Although CSM is exactly solvable, the fact that the strength of the pairwise inverse square interaction is independent of distance between the particles, makes it somewhat unrealistic. It is then natural to look for models, where the inverse square interaction is short ranged, which may happen in a physical situation where screening is present. A few years back, a short-range Dyson model was introduced to understand the spectral statistics of systems, which are “nonuniversal with a universal trend.”20 It is known that, there are dynamical systems which are neither chaotic nor integrable, the so-called pseudointegrable systems, which exhibit the above mentioned level statistics.21 Aharanov-Bohm billiards,22 three-dimensional Anderson model at the metal– insulator transition point23 and some polygonal billiards24 fit into the above description. Quite recently, a new class of one dimensional, many-body quantum mechanical models on the line, with nearest and next-to-nearest neighbor interactions, have been introduced,25,26 which are related to the shortrange Dyson model. As we will see the short range nature of the interaction changes the nature of the Hilbert space drastically. Unlike the Calogero model which is exactly solvable, the present one is quasiexactly solvable. Using the symmetrized version of this model, it has been shown earlier that, there exists an off-diagonal long-range order in the system which indicates the presence of different quantum phases.27 1098-0121/2005/71共12兲/125121共8兲/$23.00

An important property of this model is that, it possesses a E0 good thermodynamic limit, i.e., limN→⬁ N is finite. The abovementioned interacting model and its generalizations can also be defined on a circle, analogous to the Sutherland model and can show chaotic behavior in the classical domain.28 The CSM and SM are well studied quantum mechanical systems. Many of their properties like quantum equivalence to non-interacting integrability,29–31 systems,31–33 algebraic construction of eigenfunctions,29,34–36 etc., have been established rigorously. The above mentioned models, with nearest and next-to-nearest neighbor interactions, being of recent origin, need to be analyzed throughly, in order to unravel their properties, as has been done for the CSM, SM and their generalizations. The present paper is devoted to an investigation of these models and deal with both the bound and scattering problems. It is organized as follows: In Sec. II, we show the connection of the model on the line with the decoupled oscillators, which yields some wave functions explicitly. Our analysis reveals that the degeneracy of this model is less as compared to the CSM and this model is of quasiexactly solvable nature. In Sec. III, we proceed to the scattering problem and show that the scattering states, which form a subset of the Calogero case, can be realized as a coherent state. We also explicate the scattering Hamiltonian’s relation with free particles. In Sec. IV, we analyze the nearest and next-to-nearest neighbor AN−1 and BCN models on a circle, and obtain a part of the excitation spectrum by making use of symmetry arguments.

125121-1

II. BOUND STATE PROBLEM ON A LINE

The Hamiltonian for the model25 is, 共ប = m = ␻ = 1兲, N

H=−

N

N

1 1 1 ⳵2 + x2 + ␤共␤ − 1兲 2 2 i=1 i 2 1=1 i 共x − i xi+1兲 i=1

兺

N

−␤

2

兺

兺

1

, 兺 i=2 共xi−1 − xi兲共xi − xi+1兲

共1兲

©2005 The American Physical Society

PHYSICAL REVIEW B 71, 125121 共2005兲

EZUNG et al.

where, ⳵i ⬅ ⳵ / ⳵xi and xN+i ⬅ xi. It is worth pointing out that, for three particles, this model is equivalent to the CSM, since the three body term vanishes in this case. The ground-state wave function and the eigenenergy of this system25,26 are, ␺0 = GZ and E0 = 共N / 2 + N␤兲, respectively; here G ⬅ exp兵−兺ix2i 其 and Z ⬅ 兿i共xi − xi+1兲␤. The same model without harmonic confinement describes the scattering problem. In the study of this problem, we make use of our earlier method, which enabled us to map the CSM to decoupled oscillators.31 A series of similarity transformations is used to connect the present interacting system with the oscillator Hilbert space. We now perform a similarity transformation on the Hamiltonian, by its ground-state wave function, to yield H⬘ ⬅

=

x i⳵ i + E 0 − T + , 兺 i=1

冋兺

册

xi⳵i,e−T+/2 = T+e−T+/2 ,

i

N

␺n1,0,0,¯ = ␺0

兺i xi⳵i + E0 .

共3兲

¯ can be connected with the decoupled oscillaExplicitly, H tors: N

He

N 兺i=1 mi=n1

Hmi共xi兲 m i!

,

冉 冊

1 ␺0,n2,0,¯ = ␺0 exp − T+ 共r2兲n2 = ␺0e−1/2T+ Pn22 . 2

兺

共7兲

共8兲

We note that, T+ P2 = 4共2␤ + 1兲, T+2 P2 = 0, and hence, ␺0,1,0,0 = ␺0关P2 − 2共2␤ + 1兲兴. For N particles, it can be verified that, T+r2n = 2n共E0 − 1 + n兲r2共n−1兲 and this gives ␺0,n2,0,¯ as,38

N

n2

1 1 ⳵2 + x2 + E0 − N/2. G =− 2 i=1 i 2 1=1 i −1

共6兲

For the sake of clarity, we give below the explicit derivation for ␺0,1,0,0 for the four particle case and then generalize the result for arbitrary number of particles and levels. For four particles,

␺0,1,0,0 = ␺0e−1/2T+ P2 = ␺0e−1/2T+共x21 + x22 + x23 + x24兲.

¯ ⬅ eT+/2H⬘e−T+/2 = H

Ge

兺 兿 i=1

where, Hmi共xi兲’s are the Hermite polynomials. Similarly, the eigenfunction for the radial degree of freedom, r2 = 兺ix2i , can be obtained from,

it is easy to see that

−T+共␤=0兲/2 ¯ T+共␤=0兲/2

冊

This can be cast in the form,37

共2兲

N 2 N ⳵ / ⳵x2i + ␤兺i=1 1 / 共xi − xi+1兲共⳵i − ⳵i+1兲. Here, where, T+ ⬅ 21 兺i=1 we confine ourselves to a sector of the configuration space given by x1 艌 x2 艌 ¯ 艌 xN−1 艌 xN. Using the identity,

兺

共5兲

⬁

␺−1 0 H␺0

冉

冉 冊

N

1 1 ␺n1,0,0,¯ = ␺0 exp − T+ Rn1 = ␺0 exp − ⳵ 2 R n1 . 2 4 i=1 i

兺

␺0,n2,0,¯ = ␺0 兺

m=0

共E0 − 1 + n2兲! 2 m 共− 1兲m 共r 兲 , m ! 共n2 − m兲! 共E0 − 1 + m兲!

共4兲 From the above diagonalized form, it is evident that, a part of the spectrum of H is like that of N uncoupled oscillators and is linear in the coupling parameter ␤. The eigenfunctions of H, which can be connected by the above mapping with the oscillator eigenspace are captured in our approach. For these states, one can write down the raising and lowering operators for H, akin to the CSM. However, the eigenfunctions of H can be constructed straightforwardly by making use of Eq. 共3兲; since the eigenfunctions of 兺ixi⳵i are homogeneous polynomials of degree n in the particle coordinates, n being any integer. Although, the similarity transformation formally maps the Hilbert space of the interacting problem to that of the free oscillators, it needs a careful study, because of the singular interactions present in the Hamiltonian. Below, we clarify this point. Below, we present some eigenfunctions for the N-particle N n xi , case, computed using the power-sum basis, Pn共x兲 = 兺i=1 −T+/2 Pn, and the corresponding eni.e., ␺n = ␺0Sn; here, Sn ⬅ e ergy eigenvalue is En = 共n + E0兲. The wave function 共unnormalized兲 corresponding to the N xi, is found to center-of-mass degree of freedom, R = 1 / N兺i=1 be 关we use the notation ␺n1,n2,¯,nN = ␺0 exp共− 21 T+兲兿l Pnl l兴,

=c␺0LnE0−1共r2兲, 2

共9兲

where, LnE0−1共r2兲 is the Lagurre polynomial. 2 Furthermore, for four particles, S3 = P3 − 23 共2␤ + 1兲P1. However, it can be checked that, S4 = e−T+/2 P4, does not terminate as a polynomial and results in a function with negative powers of the particle coordinates, which is not normalizable with respect to the ground-state wave function as a measure. This indicates that, the model is quasiexactly solvable. Finding the other wave functions explicitly, for an arbitrary number of particles, and also the exact degeneracy structure of this model remains an open problem. In the above, we have concentrated in finding the wave functions in the Cartesian basis; the interested readers can see Ref. 25 for some wave functions in the angular basis. The power sum basis is not found to be convenient for obtaining explicit form of wave functions; apart from the center of mass and radial degree of freedom. We now list a few eigenfunctions constructed by using the elementary symmetric functions 共we follow the notations of Ref. 39 for the symmetric polynomials兲:

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QUANTUM MANY-BODY SYSTEMS WITH NEAREST AND…

e1 =

兺

兺

x i, e 2 =

1艋i艋N

兺

x ix j , e 3 =

1艋i⬍j艋N

1艋i⬍j⬍k艋N

x ix j x k, ¯ , 共10兲

N

eN =

xi . 兿 i=1

bound-state Hamiltonian. To be precise, we show that the polynomial parts of the bound-state wave functions enter into the construction of scattering states. For this purpose, we first identify a SU共1 , 1兲 algebra containing the bound and scattering Hamiltonians, after suitable similarity transformations, as elements. It is easy to see that, the following generators 1 1 ⳵2 共⳵i − ⳵i+1兲 ⬅ T+ , 2 +␤ 2 i=1 ⳵xi i=1 共xi − xi+1兲

兺

兺

1 Sˆ−1共− H/2兲Sˆ = − 2

冉兺

Z−1共− Hsca兲Z =

1 B2,0,0,0 = e21 − 2, B1,1,0,0 = e1e2 − 共3 − 4␤兲e1, B3,0,0,0 2

B2,1,0,0 = e21e2 − 共3 − 2␤兲e21 − 2e2 + 共3 − 4␤兲, B4,0,0,0 =

−

12e21

1 2

+ 12,

B5,0,0,0 =

e51

−

20e31

+ 60e1 ,

− 12共3 − 2␤兲.

III. SCATTERING PROBLEM

By finding a canonical conjugate of T+

Now, we proceed to study a subset of the scattering states of the Hamiltonian,

兺

1

共12兲

共16兲

one can construct the coherent state 具x 兩 m , k典, the eigenstate of T+,45 which are nothing but the scattering states, 具x兩m,k典 = U−1 Pm共x兲 = e−1/2k with

冉兺 N

T+ Pm共x兲 ⬅

2˜ T

−

Pm共x兲,

N

共17兲

冊

1 1 ⳵2 共⳵i − ⳵i+1兲 Pm共x兲 = 0. +␤ 2 i=1 ⳵x2i 共x − i xi+1兲 i=1

兺

共18兲 It is known that this equation admits homogeneous solutions26 i.e., T0 Pm共x兲 = −关共m + E0兲 / 2兴Pm共x兲. Here m refers to the degree of homogeneity of Pm共x兲. It is easy to see that, U−1 Pm共x兲 is the eigenstate of T+. Starting from T+ Pm共x兲 = 0, one gets, U−1T+UU−1 Pm共x兲 = 0,

共19兲

1 T+U−1 Pm共x兲 = − k2U−1 Pm共x兲. 2

共20兲

i.e.,

N

1 1 ⳵2 Hsca = − 2 + ␤共␤ − 1兲 2 2 i=1 ⳵xi i=1 共xi − xi+1兲

共15兲

:42–44

˜ 兴 = 1, 关T+,T −

A. Realization of the scattering states as coherent states

, 兺 i=2 共xi−1 − xi兲共xi − xi+1兲

关T0,T±兴 = ± T± .

ˆ = T T − T 共T + 1兲 = T T − T 共T − 1兲. C − + 0 0 + − 0 0

At this point, it is worth recollecting the StanleyMacdonald conjecture,39 which states that, the coefficients of the interaction parameter ␤ are positive integers, when the Jack polynomials39 are expressed in terms of the monomial symmetric functions with a suitable normalization. This conjecture was later proved by Sahi.40 A similar feature appears in the case of the Hi-Jack polynomials,41 which are the polynomial part of the wave functions of the CSM, but with an exception that the coefficient ␤ can also be negative integer. Remarkably, from the above explicit computations of the polynomials, we also find that the coefficients of the interacting parameter ␤ are integers 共both positive and negative兲, though we have used elementary symmetric functions. It will be interesting to check whether the modified StanleyMacdonald conjecture also holds in the present case for N particles.

兺

共14兲

The quadratic Casimir for the above algebra is given by,

共11兲

N

兺i x2i ⬅ T− ,

关T+,T−兴 = − 2T0,

B4,1,0,0 = e41e2 − 2共3 − ␤兲e41 − 12e21e2 + 12e2 + 6共9 − 4␤兲e21

N

共13兲

one can easily check that T± and T0 satisfy the usual SU共1 , 1兲 algebra:

1 B3,1,0,0 = e31e2 − 共9 − 4␤兲e31 − 6e1e2 + 6共3 − 2␤兲e1 , 2

− ␤2

冊

x i⳵ i + E 0 ⬅ T 0 ,

i

N 兩xi − xi+1兩␤ and Sˆ ⬅ exp共− 21 兺ix2i 兲Z exp共− 21 T+兲. where, Z ⬅ 兿i=1 Defining,

= e31 − 6e1 ,

e41

N

N

In this case, ␺兵mi其 = ␺0B兵mi其, B兵mi其 = e−T+/2兿i共ei兲mi, and the corresponding eigenvalues are, E兵mi其 = 兺iimi + E0. Some of the B兵mi其’s for the four particle case are listed below:

The scattering state is given by, ␺sca = ZU−1 Pm共x兲, since Hsca = −ZT+Z−1, therefore, Hsca␺sca = k2 / 2␺sca. To find 具x 兩 m , k典 explicitly, we have to determine ˜T−. By choosing ˜T = T F共T 兲, Eq. 共16兲 becomes −

and realize them as the SU共1 , 1兲 coherent states of the 125121-3

−

0

关T+,T−F共T0兲兴 = F共T0兲T+T− − F共T0 + 1兲T−T+ = 1

PHYSICAL REVIEW B 71, 125121 共2005兲

EZUNG et al.

N

ˆ + T 共T − 1兲兴 − F共T + 1兲关C ˆ + T 共T + 1兲兴 = 1, F共T0兲关C 0 0 0 0 0

1 ˜ , K− = x2 = K − 2 i=1 i

兺

共21兲 yielding,

and F共T0兲 =

− T0 + a ˆ + T 共T − 1兲 C 0 0

共22兲

.

Here, a is a parameter to be fixed along with the value of the ˆ , by demanding that the above commuquadratic Casimir C tator is valid in the eigenspace of T0. Equation 共16兲 when used on Pm共x兲, yields, a = 1 − 共E0 + m兲 / 2. Similarly, ˆ P 共x兲 = 关T T − T 共T + 1兲兴P 共x兲 = CP 共x兲, C m − + 0 0 m m where,

C = 21 共m + E0兲关1 − 共m + E0兲 / 2兴. F共T0兲Pm共x兲 =

具x兩m,k典 = e

e

⬁

= e−1/4k

2

Pm共x兲 = e 共k2/2兲n

e

e

共26兲

˜

˜

connects the Hsca, with interactions, to a interaction-free system, i.e.,

1 Pm共x兲. T0 − 共m + E0兲/2 −1/4k2 −T+ −1/2k2˜T−

兺

U = ei␲/2共K++K−兲e−i␲/2共K++K−兲 ,

One then finds,

UHscaU† = Hsca共␤ = 0兲 = −

共24兲

Explicitly, we have, −1/2k2˜T− −T+

i ˜ , 共2xi⳵i + 1兲 = K 0 4 i=1

˜ 兲兴 = ± iK 共K ˜ 兲, ˜ 兲 , K 共K ˜ 兲兴 satisfy 关K0 , K±共K 关K−共K ± ± ± − + + ˜ 兲. It can be verified that, the following unitary op= 2iK0共K 0 erator,

共23兲

− T0 + a Pm共x兲 C + T0共T0 − 1兲

=−

N

K0 = −

1 2

兺i

⳵2 . ⳵x2i

共27兲

The above mapping motivates one to analyze, explicitly, the precise correspondence of the respective Hilbert spaces of the interacting and noninteracting systems. This analysis for the present case, as well as for the CSM, needs further investigation.

Pm共x兲

−1+m兲 2 L共E 共r /2兲Pm共x兲 兺 n n=0 共E0 + m + n兲! 0

2

= e−k /4共k/2兲−共E0−1+m兲共r兲−共E0−1+m兲JE0−1+m共kr兲Pm共x兲,

IV. MODEL ON A CIRCLE

共25兲 where, r2 = 兺ix2i and JE0−1+m共kr兲 is the Bessel function. Note that, there is an additional factor of e−T+ in the above equation, this has been introduced for calculational convenience and does not alter our results, since e−T+ Pm = Pm. In order to arrive at the above result, we have made use of the following:

A. AN−1 model

Recently, Jain et al.,25,26 have studied a model with nearest and next to nearest neighbor interactions and with periodic boundary conditions as given by H=−

T+关r2n Pm共x兲兴 = 2n共E0 − 1 + m + n兲r2共n−1兲 Pm共x兲, and also the identity,46 ⬁

J␣共2冑xz兲ez共xz兲−␣/2 = 兺

n=0

zn L␣共x兲. 共n + ␣ + 1兲! n

sin2

冋

冋

1 ␲ 共x j − x j+1兲 L

册 冋

册

册

␲2 ␲ ␲ 共x j−1 − x j兲 cot 共x j − x j+1兲 , 2 兺 cot L j L L 共28兲

with x j+N = x j. They have shown that the ground state energy eigenvalue and the eigenfunction for this model are given by N j

The fact that as in the Calogero case, a part of the spectrum of the scattering Hamiltonian matches with that of the free particles suggests a possible connection of this system with free particles.47 We now show the same by making use of the algebraic structures already introduced. The following generators,

冋

␺0 = 兿 sin

B. Connection to free particles

˜ = H 共␤ = 0兲, K + sca

␲2

兺j ⳵2j + ␤共␤ − 1兲 L2 兺j

− ␤2

Note that, the above wave function can also be obtained by solving the scattering Hamiltonian explicitly. However, we have chosen this algebraic method, since it will be of subsequent use.

K+ = Hsca,

1 2

␲ 共x j − x j+1兲 L

册

␤

,E0 = N␤2

␲2 . L2

共29兲

The purpose of this section is to obtain a part of the excitation spectrum of this model. To that end, we substitute

␺ = ␺ 0␾ ,

共30兲

in the eigenvalue equation for the Hamiltonian Eq. 共28兲. It is then easily shown that ␾ satisfies the equation

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PHYSICAL REVIEW B 71, 125121 共2005兲

QUANTUM MANY-BODY SYSTEMS WITH NEAREST AND…

再

N

N

冋

册

1 ␲ ␲ ␲ − ⳵2j − ␤ cot 共x j − x j+1兲 − cot 共x j−1 − x j兲 ⳵ j 2 j=1 L j=1 L L

兺

冎

兺

+ E0 − E ␾ = 0.

N

兺

共31兲 + 2

sin

Introducing, z j = exp共2i␲x j / L兲, Eq. 共31兲 reduces to H 1␾ = 共 ⑀ − ⑀ 0兲 ␾

共32兲

H1 =

N

D2j + ␤ 兺 兺 j=1 j=1

冋

册

z j + z j+1 共D j − D j+1兲, z j − z j+1

+ g1 共33兲

with D j ⬅ z j⳵ / ⳵z j and ⑀ − ⑀0 = 共E − E0兲L2 / 2␲2. It is worth pointing out that the Eqs. 共32兲 and 共33兲 are structurally similar to those in the SM. It may be noted that H1 commutes with the momentum N zi⳵ / ⳵zi. Hence ␾ is also an eigenstate operator P = 2␲ / L兺i=1 of the momentum operator, i.e., P␾ = ␬␾ .

冋

1

␲ 共xi + xi+1兲 L

␲2 兺 L2 i

i=1

N

− ␤2

cot

冉

␲ 共xi − xi+1兲 L

册

␲2 ␲ 共xi−1 − xi兲 2 兺 cot L i=1 L

␲ ␲ 共xi − xi+1兲 + cot 共xi + xi+1兲 L L

1 ␲2 + g2 2 ␲ L 共xi兲 L

冋 册

兺i

sin2

冊

1 . 共37兲 2␲ 共xi兲 L

冋 册

N

N

N

i=1

i=1

i=1

N

− ␪i+1兲兴␤/2

关sin2共␪i + ␪i+1兲兴␤/2 , 兿 i=1

共38兲

where g1 , g2 are related to ␥ , ␥1 by

␥ g1 = 关␥ + 2␥1 − 1兴, 2

共35兲

is also an eigenstate of H⬘ and P with eigenvalues ⑀ − ⑀0 + Nq2 + 2q␬ and ␬ + Nq, respectively. Here q is any integer 共both positive and negative兲. Note that the multiplication by G implements Galilei boost. It may be noted that the Hamiltonian and hence the ␾ 2i␲x j/L equation is invariant under z j → z−1 , hence j . Since, z j = e −1 −2i␲x j/L zj = e , thereby indicating the presence of left and right moving modes with momentum ␬ and −␬. Hence, it follows that, if one obtains a solution with momentum ␬, then by changing z j → z−1 j , one can get another solution with the same energy but with the opposite momentum 共−␬兲. Thus all the excited states with nonzero momentum are 共at least兲 doubly degenerate. Finally, let us discuss the solutions to the ␾ equation. So far we have been able to obtain the following solutions.

g2 = 2␥1共␥1 − 1兲.

共39兲

The corresponding ground state energy turns out to be E0 =

N␲2 共␥ + 2␥1 + 2␤兲2 . 2L2

共40兲

By setting one or both of the coupling constants ␥ , ␥1 to zero we get the other root systems, i.e., BN:␥1 = 0, CN:␥ = 0, DN:␥ = ␥1 = 0.

共41兲

The purpose of this subsection is to obtain a part of the excitation spectrum of the BCN , BN , CN , DN models. To that end, we substitute

␺ = ␺ 0␾ ,

共42兲

in the eigenvalue equation for the above Hamiltonian where ␺0 is as given by Eq. 共38兲. It is easy to show that in that case ␾ satisfies the equation:

冋兺

共i兲 ␾ = e1, ⑀ − ⑀0 = 1 + 2␤ ,

N

j=1

共ii兲 ␾ = eN−1, ⑀ − ⑀0 = N − 1 + 2␤ , 共iii兲 ␾ = eN, ⑀ − ⑀0 = N.

冊冉

sin2

␺0 = 兿 sin␥ ␪i 兿 共sin2 2␪i兲␥1/2 兿 关sin2共␪i

N

G = ⌸ zi ,

sin2

册冣

冢 冋

1

Following them, we restrict the coordinates xi to the sector L 艌 x1 艌 x2 艌 . . . 艌 xN 艌 0. As shown by Auberson et al.,26 the ground state eigenfunction is given by

共34兲

Furthermore, if ␾ is an eigenstate of H1 and P then

␾ ⬘ = G q␾ ,

兺

␲ 共xi−1 + xi兲 L

− cot

where N

N

1 ⳵2 ␲2 H=− + ␤ 共 ␤ − 1兲 2 i=1 ⳵x2i L2 i=1

⳵2 ⳵␪2j N

共36兲

+ 2␤

兺 j=1

cot共␪ j − ␪ j+1兲

冉

冊

冉

冊

N

⳵ ⳵ − + 2␤ 兺 cot共␪ j ⳵␪ j ⳵␪ j+1 j=1 N

⳵ ⳵ ⳵ + ␪ j+1兲 + + 2␥ 兺 cot ␪ j ⳵ ␪j ⳵␪ j ⳵␪ j+1 j=1

B. BCN model

Recently Auberson et al.,26 have studied a BCN model with nearest and next-to-nearest neighbor interactions and with periodic boundary conditions as given by 125121-5

N

+ 4␥1

⳵

2L2

cot 2␪ j + 共E − E0兲 2 兺 ⳵␪ j ␲ j=1

册

␾ = 0,

共43兲

PHYSICAL REVIEW B 71, 125121 共2005兲

EZUNG et al.

where ␪ j = ␲x j / L. Introducing, z j = exp共2i␪ j兲, Eq. 共43兲 reduces to H 1␾ = 共 ⑀ − ⑀ 0兲 ␾

共44兲

where N

N

H1 =

z +1

j Dj + D2j + ␥ 兺 兺 z j=1 j − 1 j=1 N

+␤

兺 j=1

冋

册

N

z2j 2␥1 2 j=1 z j

+1

兺

−1 N

Dj

冉

冊

z j + z j+1 z jz j+1 + 1 共D j − D j+1兲 + ␤ 共D j z j − z j+1 z jz j+1 − 1 j=1

兺

+ D j+1兲.

共45兲

Here D j ⬅ z j⳵ / ⳵z j while ⑀ − ⑀0 = 共E − E0兲L2 / 2␲2. Note that apart from the cyclic symmetry, the Hamiltonian and hence the ␾ equation is also invariant under z j → z−1 j . As a consequence, as in the BCN Sutherland model,48,49 it turns out that even in our case the polynomial eigenfunctions of H1 with BCN symmetry as given by Eq. 共45兲 are symmetric polynomials in 共z j + 1 / z j兲, i.e., in cos共2␲x j / L兲. Finally, let us discuss the solutions to the ␾ equation. So far we have been able to obtain only one solution in the BCN case but are able to obtain several solutions in the DN case and also a few in the BN and CN cases. C. Exact solution for the BCN model

It is easily checked that the exact solution is

␾ ⬅ ␾BCN = ␾1 + ␣,

N

H+1

= H1共z j ; ␤, ␥, ␥1 = 0兲 +

z −1

j Dj 兺 j=1 z j + 1

= H1共z j ; ␤, ␥ − 1, ␥1 = 1兲.

共51兲

We thus see that ␺+ essentially satisfies the same equation as satisfied by the BCN Hamiltonian but with the value of ␥ and ␥1 shifted to ␥ − 1 and 1, respectively. Thus it follows that once we obtain solutions of the BCN problem, all these will give us new solutions of the spinorial type for the BN model as given by Eqs. 共49兲–共51兲. So far we have been able to obtain only one solution in the BCN case as given by Eqs. 共46兲 and 共47兲. Using that solution, it then follows that the new spinorial solution for the BN model is as given by Eq. 共49兲 with energy

⑀ − ⑀0 = 2 + ␥ + 4␤ +

N 关1 + 2␥ + 4␤兴, 4

共52兲

while ␺+ ⬅ ␾BCN共z j ; ␤ , ␥ − 1 , ␥1 = 1兲 with ␾BCN being given by Eqs. 共46兲 and 共47兲. E. Exact solutions for the DN model „␥ = ␥1 = 0…

Unlike BN, there are two distinct classes of spinor representations for DN. Besides, there are also some additional solutions in this case. It may be noted that in this case the Hamiltonian H1 acting on ␾ is H1共z j ; ␤兲 = H1共z j ; ␤, ␥ = 0, ␥1 = 0兲.

⑀ − ⑀0 = 1 + ␥ + 2␥1 + 4␤ , 共46兲

共53兲

The first new solution that we have is given by

where N

␾1 = 兺 j=1

冉 冊

2N␥ ␣= . 1 + ␥ + 2␥1 + 4␤

1 zj + , zj

N

共47兲

We will see that this solution 共and in fact other solutions, if any, in the BCN case兲 will play important roles in our construction of solutions for other systems like BN, CN, and DN. D. Exact solutions for the BN model „␥1 = 0…

Apart from the obvious solution ␾共z j ; ␤ , ␥ , ␥1 = 0兲 as given by Eq. 共46兲 it turns out that there are other solutions corresponding to the spinorial representation for the BN model. These are N

␾ ⬅ ␾+ = ⌸

j=1

冉冑

zj +

冑 冊 1

zj

,

⑀+ − ⑀0 =

␾ ⬅ ␾− = ⌸

j=1

共48兲

where ⑀ is as given by Eq. 共48兲 while

冊

,

⑀− − ⑀0 =

N 关1 + 4␤兴. 共54兲 4

␾ = ␾ −␺ − ,

共55兲

where ␾− is as given by Eq. 共54兲 and consider the equation H1␾ = 共⑀ − ⑀0兲␾. It is easily seen that in that case ␺− satisfies

共50兲

共56兲

where ⑀− is as given by Eq. 共54兲 while N

H−1

= H1共z j ; ␤, ␥ = 0, ␥1 = 0兲 + = H1共z j ; ␤, ␥ = 1, ␥1 = 0兲.

共49兲

where ␾+ is as given by Eq. 共48兲. It is easily seen that in that case ␺+ satisfies

+

1

冑z j

Note that the two solutions Eq. 共48兲 共with ␥ = 0兲 and Eq. 共54兲 which correspond to the two different spinorial representations are degenerate in energy. In order to obtain the other solution, we start with the ansatz

In order to obtain the other solution, we start with the ansatz

H+1 共z j ; ␤, ␥, ␥1 = 0兲␺+ = 共⑀ − ⑀+兲␺+ ,

zj −

H−1 共z j ; ␤, ␥ = 0, ␥1 = 0兲␺− = 共⑀ − ⑀−兲␺− ,

N 关1 + 2␥ + 4␤兴. 4

␾ = ␾ +␺ + ,

冉冑

z +1

j Dj 兺 j=1 z j − 1

共57兲

We thus see that ␺− essentially satisfies the same equation as satisfied by the BN Hamiltonian but with the value of ␥ being fixed at 1. Using the solution for the BN case, it then follows that the new spinorial solution for the DN model is as given by Eq. 共55兲 with energy

125121-6

PHYSICAL REVIEW B 71, 125121 共2005兲

QUANTUM MANY-BODY SYSTEMS WITH NEAREST AND…

⑀ − ⑀0 = 2 + 4␤ +

N 关1 + 4␤兴, 4

共58兲

while ␺− ⬅ ␾BCN共z j ; ␤ , ␥ = 1 , ␥1 = 0兲 with ␾BCN being given by Eqs. 共46兲 and 共47兲. Notice that this solution is degenerate in energy with the solution Eq. 共49兲 共with ␥ = 0兲. In addition, we find that the product of the two “spinorial solutions” is also a solution of the DN model, i.e., N

冉 冊

␾ ⬅ ␾ +␾ − = ⌸ z j − j=1

1 , zj

⑀+− − ⑀0 = N关1 + 2␤兴. 共59兲

In order to obtain another solution, as above we start with the ansatz

␾=␾ ␾ ␺ +

共60兲

− +−

where ␾+ , ␾− are as given by Eqs. 共48兲 and 共53兲, respectively, and consider the equation H1␾ = 共⑀ − ⑀0兲␾. It is easily seen that in that case ␺+− satisfies +− +− +− H+− 1 共z j ; ␤, ␥ = 0, ␥1 = 0兲␺ = 共⑀ − ⑀ 兲␺ ,

共61兲

␾ ⬅ ␾31 = A

1 z1

z2 +

1 z2

z2j 2 2 j=1 z j

兺

+1 −1

⑀ − ⑀0 = 3 + 6␥1 + 8␤,

册

+ B␾1 ,

B=

8A␤ . 1 + 2␥1 + 2␤

共65兲

G. Exact solutions for the DN=4 model „␥ = ␥1 = 0…

Clearly, one obvious solution in the DN=4 case is obtained from the solution Eq. 共64兲 by putting ␥1 = 0. The other solution is obtained by making use of the ansatz as given by Eq. 共60兲, i.e., let

␾ = ␾31␾+− ,

共66兲

with ␾ being given by Eq. 共64兲. Using Eqs. 共60兲–共62兲 it then follows that this is a solution for the DN=4 model with 31

B=

8A␤ . 3 + 2␤

共67兲

V. CONCLUSIONS

共62兲

We thus see that ␺+− essentially satisfies the same equation as satisfied by the BCN Hamiltonian but with the value of ␥1 being fixed at 1. Using the solution for the BCN case as given by Eqs. 共46兲 and 共47兲 共with ␥ = 0 , ␥1 = 1兲, it then follows that the new solution for the DN model is as given by Eq. 共60兲 with energy

⑀ − ⑀0 = 3 + 4␤ + N关1 + 2␤兴,

1 +C.P. z3

Here, by C.P. one means cyclic permutations and ␾1 is as given by Eq. 共46兲.

Dj

= H1共z j ; ␤, ␥ = 0, ␥1 = 1兲.

z3 +

where

⑀ − ⑀0 = 13 + 16␤, N

共63兲

while ␺+− ⬅ ␾BCN共z j ; ␤ , ␥ = 0 , ␥1 = 1兲 with ␾BCN being given by Eqs. 共46兲 and 共47兲. F. Exact solution for the CN=4 model „␥ = 0…

So far we have discussed all the solutions which are valid for any N. In addition, in the special case of N = 4, we have been able to obtain a solution in the CN=4 共and two solutions in the DN=4兲 case. The solution for the CN=4 case is given by

In conclusion, we have carried out a systematic study of the many-body Hamiltonians, related to the short range Dyson model. The interacting system was connected with decoupled oscillators and by explicitly computing some of the bound-state eigenfunctions, it was shown that, the present model is quasiexactly solvable, as compared to the Calogero–Sutherland model. A subset of the scattering states of this model is obtained and is shown to be a coherent state. The scattering Hamiltonian is related to free particles by a unitary transformation. Finally, we have also studied both AN−1 and BCN models on a circle and obtained a part of the excitation spectrum in both the cases. Finding other exact eigenstates, if present, and the nature of the degeneracy structure still remain open problems. ACKNOWLEDGMENTS

M.E., N.G., and P.K.P. would like to thank Professor V. Srinivasan and Professor S. Chaturvedi for useful discussions.

6 B.

*Electronic address: [email protected] †Electronic

z1 +

共64兲

where ⑀+− is as given by Eq. 共59兲 while H+− 1 = H1共z j ; ␤, ␥ = 0, ␥1 = 0兲 +

冋冉 冊冉 冊冉 冊

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