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A generalized Lieb–Liniger model

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2016 J. Phys. A: Math. Theor. 49 085205

(http://iopscience.iop.org/1751-8121/49/8/085205)

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A generalized Lieb–Liniger model

Hagar Veksler1 and Shmuel Fishman

Physics Department, Technion- Israel Institute of Technology, Haifa 32000, Israel

E-mail: hagarb@tx.technion.ac.il

Received 25 August 2015, revised 15 December 2015Accepted for publication 18 December 2015Published 20 January 2016

AbstractIn 1963, Lieb and Liniger solved exactly a one-dimensional model of bosonsinteracting by a repulsive δ-potential and calculated the ground state in thethermodynamic limit. In the present work, we extend this model to a potentialof three δ-functions—one of them is repulsive and the other two are attractive—modeling some aspects of the interaction between atoms, and we present anapproximate solution for a dilute gas. In this limit, for low energy states, theresults are found to be reduced to the ones of an effective Lieb–Liniger modelwith an effective δ-function of strength ceff and the regime of stability isidentified. This may shed light on some aspects of interacting bosons.

Keywords: Bethe ansatz, Lieb–Liniger model, interacting bosons

(Some figures may appear in colour only in the online journal)

1. introduction

The physics of Bose gases is a fascinating and complicated field of research. Since it involvesa many-body problem, analytical results are rare and in some parameter regimes, one can useapproximations to describe experimental systems with very good accuracy. For example, for aweakly interacting Bose gas, the mean-field approximation can be used to reduce the many-body Hamiltonian into a one-body nonlinear Schrödinger equation, the Gross–Pitaevskiiequation [1–3]. In the opposite limit, a strongly interacting one-dimensional Bose gas can bemapped into a gas of free fermions (Tonks–Girardeau gas, see, for example, [4–7]). Exactsolutions in other regimes are highly desired.

Simple models like the Lieb–Liniger (LL) model, that may not have direct experimentalrealization, may alert us to unexpected physical phenomena that are overlooked when ‘rea-sonable approximations’ are made and motivate experiments [6–8]. The model introduced inthe present work is of this type.

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor. 49 (2016) 085205 (12pp) doi:10.1088/1751-8113/49/8/085205

1 Author to whom any correspondence should be addressed.

1751-8113/16/085205+12$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1

In their seminal work from 1963 [9], Lieb and Liniger managed to solve exactly a one-dimensional model for interacting bosons. They considered the Schrödinger equation for Nparticles interacting via a δ-function potential

m xc x x E

21

j

N

j j sj s

N

j s

2

1

2

2, 1

( ) ( )å å d y y-

¶¶

+ - == =

>

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥where xj is the coordinate of the jth particle and c is the amplitude of the δ-function. Making aBethe ansatz1

x x x k k kmc

x x, , 1 exp ii

sign , 2NP

P

n

N

n Pj s

P P j s11

2n j s( ) ( ) ( ) ( )[ ]

å å y ¼ µ - - - -= >

⎧⎨⎩⎫⎬⎭

⎡⎣⎢

⎤⎦⎥

where kPnare the k vectors obtained by the permutation P (where P[ ] is its parity) of the set

k k, , N1 ¼ . Lieb and Liniger wrote Bethe ansatz equations for the ks by imposing periodicboundary conditions on a ring of length L [10, 11],

k Lk k mc

k k mc

mc

k kmc

k k

exp ii

i

1i

1i

. 3jh j

Nj h

j h h j

Nj h

j h

2

2

2

2

{ }( )( )

( )

( )

( )

=- +

- -=

+-

--

¹ ¹

These N coupled equations are solved numerically and the energy

Em

k2

4j

N

j

2

1

2 ( )å==

is calculated for the ground state and the excitations [9, 12, 13].In the present work, we study a simple model that takes into account the range of inter-

particle interactions without giving up mathematical simplicity. It is a generalization of the LLmodel [9] where in addition to the repulsion there is also attraction. It is defined by theSchrödinger equation for N interacting particles of mass m,

5m x

c x x c x x l c x x l E2 j

N

j j sj s

N

j s lj sj s

N

j s lj sj s

N

j s

2

1

2

2 0, 1 , 1 , 1

( )( ) ( ) ( ) å å å åd d d y y-¶¶

+ - + - - + - + == =

>=>

=>

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

where the inter-particle interaction is modeled as a sum of three δ-functions: the central one isrepulsive c 00( )> while the peripheral ones are attractive c 0l( )< . This model is inspired bythe van der Waals potential, which has repulsive and attractive regimes. By adjusting theparameters c c l, ,l0 of (5), one can model scattering from many inter-particle potentials [14–16]. By writing (5) in terms of rescaled coordinates y x lj j= ,

6

ml y

c

ly y

c

ly y

c

ly y E

21 1 ,

j

N

j j sj s

N

j sl

j sj s

N

j sl

j sj s

N

j s

2

21

2

20

, 1 , 1 , 1

( )

( ) ( ) ( ) å å å åd d d y y-¶¶

+ - + - - + - + == =

>=>

=>

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1 For those who are not familiar with Lieb–Liniger solution, we recommend lecture notes by Mikhail Zvonarev,http://cmt.harvard.edu/demler/TEACHING/Physics284/LectureZvonarev.pdf and the books [10, 11].

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

2

it is possible to identify that the important parameters of this model are mlc02 and mlcl

2 .Increasing l is equivalent to decreasing ÿ. Therefore, quantum effects are stronger where l issmall. This will be proved by detailed calculations in the following sections.

In section 2, we present Bethe ansatz equations for two bosons interacting via three δ-functions interaction potential and in section 3 an approximation is introduced that allows usto extend the LL Bethe ansatz equations to an arbitrary number of particles. The ground statesolution for the approximate equations is found in section 4. Section 5 specifies the para-meters of the regime where the gas is stable. The results and their experimental relevance arediscussed in section 6.

2. Bethe ansatz equations for two bosons interacting via three δ-functionsinteraction potential

We start by writing Bethe ansatz equations for a simple case where there are only two bosons.In this case, the equations are intuitive.

Consider two bosons of mass m trapped on a ring of length L and interact according to(5). It is convenient to write the wave function ψ in terms of center of mass coordinate,r x x 21 1 2( )= + and relative motion coordinate, r x x 22 1 2( )= - ,

r rL

r,1

e 7ik r1 2 21 1( ) ( ) ( )˜y f=

where k n L21 p= and n is an integer so that periodic boundary conditions are satisfied. Atthe center of mass frame of reference, k 01 = and the wave function of the relative motion,

r2( )f , satisfies the Schrödinger equation

m rc r c r l c r l r E r

4

1

2

1

22

1

22 8l l

2 2

22 0 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )

d d d f f-¶¶

+ + - + + =⎡⎣⎢

⎤⎦⎥

which can be written also as

m rc r c r l c r l r E r

22 2 2 . 9l l

2 2

22 0 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ( )

d d d f f-¶¶

+ + - + + =⎡⎣⎢

⎤⎦⎥

As usual in such cases, the wave function takes a different functional form in each of the fourintervals ,L l

4 2- -⎡⎣ ⎤⎦, , 0l

2-⎡⎣ ⎤⎦, 0, l

2⎡⎣ ⎤⎦ and ,l L

2 4⎡⎣ ⎤⎦. The result for r2( )f is

r C k r iC r Q k r

Q r l k r l Q r l k r l

cos sign sin

sign 2 sin 2 sign 2 sin 2

10l l

2 2 2 2 0 2 2

2 2 2 2 2 2

( ) ( ˜ ) { ( ) ( ˜ )

( ) [ ˜ ( )] ( ) [ ˜ ( )]}( )

f = +

+ - - + + +

where C is a normalization constant while Q0 and Ql should be determined. They are easilydetermined for the δ-function interaction since the jump of the derivative at the locations ofthe δ-function satisfies

rr

r

r

r

mcr

d

d

d

d

211

r r2

2

2 0

2

2 02 2

2 2

( ) ( ) ( ) ( ) ( )* * ** *

ff f

fD ¢ º - =+ ++ -

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

3

where r c c0,2 0* *= = or r l c c2, l2* *= = . This results in two equations for Q0 and Ql

k Q kmc

Q k li 1 2i sin 2 12l2 0 20

2 2˜ ( ˜ ) { ( ˜ )} ( )

= +

and

ik Qmc

k l i Q k l Q k lcos 2 sin 2 sin , 13ll

l2 2 2 0 2 2˜ { ( ˜ ) [ ( ˜ ) ( ˜ )]} ( )

= + +

leading to

Q k

ik mc k lmc

kk l

k m c c k l k mc k l

cos 2 sin 2

2 sin 2 sin. 14l

l

l l

2

22

20

22 2

22 4 2

02

2 22

2

( ˜ )

˜ ( ˜ ) ˜ ( ˜ )

˜ ( ˜ ) ˜ ( ˜ )( )

= -

+

- -

⎡⎣⎢

⎤⎦⎥

k2˜ should be determined to ensure periodic boundary conditions r r L 22 2( ) ( )f f¢ = ¢ + . Inaddition, r r2 2( ) ( )f f= - so that r r2 2( ) ( )f f¢ = - ¢ - . Therefore, in particular, r r L2 42( )∣f¢ =must vanish, leading to

eQ k Q k k l

Q k Q k k l

1 2 cos 2

1 2 cos 2. 15ik L l

l

2 0 2 2 2

0 2 2 2

2( ˜ ) ( ˜ ) ( ˜ )( ˜ ) ( ˜ ) ( ˜ )

( )˜ =- -+ +

Now, we return to coordinates x x,1 2. For this purpose, we use the relations:k k k 21 1 2( ˜ ˜ )= + , k k k 22 1 2( ˜ ˜ )= - , r x x 21 1 2( )= + and r x x 22 1 2( )= - resulting in

k r k r k x k x 161 1 2 2 1 1 2 2˜ ˜ ( )+ = +

and

k r k r k x k x . 171 1 2 2 2 1 1 2˜ ˜ ( )- = +

The function ψ of (7) takes the form

x x C e e

C x x Q k k e e

CQ k k x x l e e

e e

CQ k k x x l e e

e e

,

sign

sign

sign

. 18

i k x k x i k x k x

i k x k x i k x k x

li k x k x i k k l

i k x k x i k k l

li k x k x i k k l

i k x k x i k k l

1 2

1 2 0 1 2

1 2 1 22

2

1 2 1 22

2

1 1 2 2 2 1 1 2

1 1 2 2 2 1 1 2

1 1 2 2 1 2

2 1 1 2 1 2

1 1 2 2 1 2

2 1 1 2 1 2

( ) [ ]( ) ( )( )

( ) ( )[]

( ) ( )[] ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

y = ++ - - -+ - - --+ - - +-

+ +

+ +

+ - -

+ -

+ -

+ - -

The periodic boundary condition xL

x xL

x2

,2

,2 2 2 2y y+ = -⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠ results in

eQ k k Q k k k k l

Q k k Q k k k k l

1 2 cos 2

1 2 cos 219ik L l

l

0 1 2 1 2 1 2

0 1 2 1 2 1 2

1( ) ( ) (( ) )( ) ( ) (( ) )

( )=- - - - -+ - + - -

and

eQ k k Q k k k k l

Q k k Q k k k k l

1 2 cos 2

1 2 cos 220ik L l

l

0 1 2 1 2 1 2

0 1 2 1 2 1 2

2( ) ( ) (( ) )( ) ( ) (( ) )

( )=+ - + - -- - - - -

which are identical to each other and to (15) (under the assumption k 01 = , namely, in thecenter of mass frame of reference). In the derivation we used the fact thatx x x x L1 2 1 2- - + involves rotation around the circle and consequently all the signsare changed.

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

4

3. Approximate Bethe ansatz equations for an arbitrary number of bosons

The two particle solution cannot be simply generalized to an arbitrary number of particlessince for small inter-particle distances,

x x l, 21j s∣ ∣ ( )- <

the sign function in equation corresponding to (18) varies substantially. For small l, the effectof the regime (21) may be negligible as demonstrated in what follows. This is reasonable for adilute gas where l L N . In such a situation, the LL solution is valid with the replacementmc k k Q Q k k li 2 cos 2P P l P P

20j s j s

( ) (( ) ) - = + - , leading to

x x C i x k

Q k k Q k k

k k l x x

, , exp

1 2

cos 2 sign 22

NP n

N

n P

j sP P l P P

P P j s

11

0

n

j s j s

j s

( )

{ ( ( ) ( )

(( ) )) ( )}] ( )

å å

y ¼ =

+ - + -

´ - -

=

>

⎡⎣⎢⎢

⎧⎨⎩⎫⎬⎭

and

eQ k k Q k k k k l

Q k k Q k k k k l

1 2 cos 2

1 2 cos 2. 23ik L

s j

j s l j s j s

j s l j s j s

0

0

j( ) ( ) (( ) )( ) ( ) (( ) )

( )=- - - - -

+ - + - -¹

The kj are distinct, namely, the wave function vanishes if k kj s= for s j¹ as was shown inthe original work of LL [9].

In the region where inequalities (21) are not satisfied for any of the particle pairs, the signfunctions in (10) are all equal. Therefore, in this regime, (22) is a solution with the spectrum(23). There is a Hamiltonian that is different from the original one, for which (22) and (23) areeigenfunctions and eigenvalues even if some of the inequalities (21) are satisfied. It is justdefined by the eigenfunctions and eigenvalues. For l = 0, this Hamiltonian and the originalone are identical. If the spectrum and the L2-norm of the eigenfunctions are continuous in l,the relative difference in the spectrum and the wave functions (in the L2-norm) goes to zero inthe limit l 0 . If they are also differentiable as a function of l, then the relative differencebehaves as Nl/L.

We show that for the low energy states, the three δ-function system can be replaced by asystem with one δ-function of strength ceff.

We assume

k k l 1 24j s( ) ( )-

for all wave vectors kj. In section 5, we show that this limit is relevant for the ground state andlow-lying excitations of a dilute gas since k const N

Lmax ( ) is small. In the leading order ink lj ,

Q Q k k l

im

kc c

mc lc c

mc c l mc l

m c c l mc l

2 cos 2

2

2 22

12

, 25

l j h

l

ll

l

l l

0

22

0

2 002

02

2

20

2

4 2

(( ) )

˜ ( )

+ -

» - + +

+ + +

- -

⎧

⎨⎪⎪

⎩⎪⎪

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎫

⎬⎪⎪

⎭⎪⎪

the error is of the order Nl/L. Comparing (23) with (3), one finds that for small k lj , thebehavior of the present problem is indeed similar to the one found for one δ-function potential

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

5

of strength

c c c

mc lc c

mc c l mc l

m c c l mc l2

2 22

12

, 26eff l

ll

l

l l0

2 002

02

2

20

2

4 2

( )

= + +

+ + +

- -

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

in the leading order in k lj . Equation (26) is the main result of the present work, and it enablesone to understand the physics of the three δ-functions’ interaction in terms of the one δ-function interaction. Of particular interest are situations where ceff is very different fromc c2 l0 + (the total strength of interactions). In order to find such situations, we define theparameters r c cl 0= and x mc l0

2= and rewrite (26) as

c

c c

rx r rx x

r rx rx

rx r

r rx rx

21

2 2

1 2 1

1 2

1 2 127

eff

l0

12

12

2

12

2

( )( )

( )

( )

( )( )

+= +

+ + +

+ - -

=+ +

+ - -

For weak interactions, (x 1 and rx 1 ),c

c c21. 28

eff

l0( )

+»

However, for very strong interactions (x ¥),

c

c c r x2

2

1 20 . 29

eff

l0 ( )( )

+»

-+

-

This is a surprising result. It is instructive to analyze the behavior of c c c2eff l0( )+ ,equation (27), as a function of x. We are interested in the regime x 0> and r0.5 0- < < . Atx = 0, the derivative of (27) is negative and therefore the function decreases. At

xr

r

1 2300

( ) ( )= -+

it turns out that c 0eff = (even though c c2 0l0 + ¹ ). Higher values of x result in negativevalues of ceff, namely, the effective interaction is attractive (even though c c2 0l0 + > ). Aschematic description of c c c2eff l0( )+ is given in figure 1.

The result c 0eff = at x x0= is verified numerically (see figure 2) and will be discussedin what follows. In the two particle case it is exact. For a related result see [14, 15, 19].

4. Ground state energy

In the previous section, we derived the approximate Bethe ansatz equations (23) for N bosonsinteracting by a three δ-function potential (5). The solution for these N coupled equations,k k k, , , N1 2( )¼ , can be used to calculate the energy of the gas

Em

k2

. 31j

N

j

2

1

2 ( )å==

In the ground state, kj∣ ∣ are minimal (but yet kj are different, as in the original work of LL [9]).

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

6

Lieb and Liniger [9] managed to calculate the ground state energy in the thermodynamiclimit (N ¥) by solving only two coupled integral equations (36) and (37) (instead of Nequations of the form (23)). Here, we obtain similar equations by using the logarithmic formof (23),

G k k L k k nN

21

232j j

s jj s j( ) ( ) ( )åq pº + - = -

+

¹

⎜ ⎟⎛⎝

⎞⎠

where

k iQ k Q k kl

Q k Q k klln

2 cos 2 1

2 cos 2 1. 33l

l

0

0( ) ( ) ( ) ( )

( ) ( ) ( )( )q =

+ -+ +

⎡⎣⎢

⎤⎦⎥

We see that if l = 0, the ground state corresponds to the choice n jj = , j N1, ,( )= ¼ . Thisis true also for l 0¹ , as long as θ is a monotonic increasing function of k. To see this, assumek kj m> , then, by monotonicity of θ, k k k kj s m s( ) ( )q q- > - for all s, thereforeG k G kj m( ) ( )> and G kj( ) is monotonic. Since θ is an odd function, G k 0 0j( )= = . Thekj for the ground state are the smallest possible in absolute value, hence, we choose n jj = forthe ground state. Therefore, in the monotonic regime,

L k k k k k k 2 34j j j js j

j s1 1( ) ( ) ( ) ( )åq p- + - ¢ - =+ +¹

where k k k( ) ( )q q¢ º ¶ ¶ and kj and kj 1+ are adjacent wave numbers. Typically, θ ismonotonic and (34) is justified at the regime where (24) holds (see section 5 for more details).The density of states per unit length in k space, is defined as

kL k k

135j

j j1( )

( )( )r =

-+

Figure 1. Schematic description of c c c2eff l0( )+ as a function of x forr c c 0.25l 0= = - . The inset expands the region where ceff changes its sign and thered dot is x , 00( ).

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

7

and satisfies

k kN

Ld . 36( ) ( )ò r =

-L

L

It is used to write (34) in the form

k q q k q1

2d

1

2. 37( ) ( ) ( ) ( )òr

pr q

p- ¢ - =

-L

L

Here, Λ is the Fermi momentum (this should not be confused with fermionic systems!). Theground state energy (31) is

EL

mk k k

2d . 380

22( ) ( )

ò r=-L

L

In order to solve equations (36) and (37), we change into dimensionless variables:

zk c m c m c mL

N

c mL

Nd l, , , , , . 39l

ll

l0

02 2 0

02 2

( )

a a g g=L

=L

=L

= = = L

In these variables, z z( ) ( )h q= L , the density of states is g z z( ) ( )r= L , and equations (36)–(38) are, respectively [9],

Figure 2. The dimensionless energy e of (42) as a function of dimensionless interactionstrengths for c c 4l 0= - (namely, r 0.25= - ) and 0 300a< < . (a) e as a function of

2 l0g g+ . Different lines represent different choices of d of (39), from top to bottom:d = 0 (blue), d = 0.02 (green), d = 0.04 (red), d = 0.06 (turquoise), d = 0.08(purple). Points where the effective interaction is attractive were excluded from thefigure (these were supposed to appear in the bottom purple curve in the regimex x 20> = , see equation (30)), so that the highest value of x which does appear in thefigure is x = 1.94 and it corresponds to the purple point 7.7, 0.114( ). (b) The energy eof (a), plotted as a function of effg (equation (43)).

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

8

zg zd , 4001

1

0( ) ( )òg a=-

g z y y z g y1

2d

1

241

1

1( ) ( ) ( ) ( )òp

hp

- ¢ - =-

and

emE L

Nyg y y

2d . 420

2

2 303

03 1

12( ) ( )

òg

aº =

-

How does one solve equations (40) and (41)? First, it is necessary to choose values for , l0a aand d. These values are related to the parameters of the Hamiltonian via the Fermi momentumΛ which is unknown at this stage. One should only keep in mind that c cl l0 0a a = andtherefore the ratio l 0a a does reflect the ratio between attraction and repulsion in theHamiltonian. The integral equation (41) (with parameters , l0a a and d) can be solvednumerically. The solution, g z( ), should be substituted in (40) in order to find 0g . By repeatingthe above scheme for different parameters, it is possible to plot the dimensionless energy e asa function of the dimensionless interaction strengths 0g and lg and the dimensionless length d.for small values of d, the energy e depends only on the effective strength of interaction, that is(26) in dimensionless units,

d d d

d d2

2 2

1. 43eff l

l l l

l l0

0 012 0 0

12 0

2

( )( )g g g

a g g a g a g

a a a= + +

+ + +

- -

The significance of this effective strength of interaction is demonstrated in figure 2. In thisfigure, we present the solutions e d, ,l0( )a a that were calculated by solving (40)–(42). Infigure 2(a), the energy is plotted as a function of the total interaction strength 2 l0g g+ anddifferent choices of d are represented by different colors. It is clear that the effect of d is notnegligible. Even at the regime d 1 , it is evident that the value of l has a strong effect on theground state energy. Furthermore, even for a given value of d, the total interaction strength

2 l0g g+ (which is proportional to c c2 l0 + ) is not in one to one correspondence with theenergy and therefore cannot be used to characterize the gas. Figure 2(b) shows that in theregime d 1 , the energy indeed depends only on effg of (43). The results are consistentwith (27).

5. Regime of stability and definition of dilute gas

The Bethe ansatz equations (23) and the effective interaction (26), are valid only where (24) issatisfied. Therefore, it is important to identify the regime where k k l 1j s( )- . In the ori-ginal LL model, the ground state energy and the values of ks are maximal for strong inter-actions, c ¥, where k nn L

2= p and ns are integers n , ,N N

2 2= - ¼ . Then, the maximal

absolute value of k is k N

Lmax = p and for all j s, ,

k k lNl

L

2. 44j s( ) ( )p

- <

For dilute gas, the inter-particle separation L/N is much larger then the interaction range l and(24) is satisfied.

The same argument can be written for the three δ-functions’ interaction potential (5). If θof (33) is a monotonic increasing function of k, the ground state is given by n jj = ,j N1, ,( )= ¼ and k kNmax = .

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

9

Let us analyze the function k( )q and identify the regime of parameters where it ismonotonic. first, note that k( )q is monotonically increasing if and only iff k Q Q kl2 cos 2

i l1

0( ) [ ( )]º + is monotonically increasing. For k 0 , f k cm

k eff2( )

= -and therefore it is monotonically increasing as long as c 0eff > . Hence, if c 0eff > , there existsome k* (which depends on the parameters c c l, ,l0 and does not depend on L and N since θ isindependent of these variables) such that for all k k*< , k( )q is monotonically increasing. Forthe ground state, the smallest quasi momenta kj∣ ∣ are occupied, namely, n jj = withj N1, , ,= ¼ and

G k N. 45j( ) ( )p<

θ is an angle variable and therefore it is bounded (actually, for very small k, q p= - ). Hence

kN

Lconst 46j∣ ∣ ( ) ( )<

and can be made arbitrary small. Now, by increasing L (or decreasing N), one may tune thevalue of kmax such that the conditions

k kmax *<

and

k l 1max

are satisfied simultaneously, the regime (24) of dilute gas is achieved and our solution iscorrect up to a term of order Nl/L.

For a dilute gas there is a range of parameters where c 0eff > and the solution is stable.There is also a range of parameters where c 0eff < and the system is unstable.

6. Summary and discussion

In this paper, we analyzed a one-dimensional dilute Bose gas for an extension of the LLmodel defined by (5). By dilute gas, we mean that l L N , that is, the effective size of aparticle l (for example, the van der Waals radius of an atom) is much smaller than the inter-particle distance. Using this assumption and the Bethe ansatz, we derived the approximateequations for the spectrum (19) , (20), (23). In principle, these can be solved numerically. Forlow energies in this situation k l 1j∣ ∣ and the model can be approximated by a LL modelwith one δ-function of strength ceff given by (26) and in dimensionless units by (43). Theerror of this approximation is of order Nl/L. This is a good approximation for the dilute gas.The effective interaction ceff depends on cl and c0 via the parameters mlc0

2 and mlcl2

that reflect the interaction strength in rescaled coordinates (see equation (6)). These are alsothe ratios between the characteristic potential energy scales, c ll and c l0 , and the kineticenergy scale, ml2 2 , of a particle trapped in a well of length l.

Naively one would expect that for small kj, c c c2eff l0» + . It turns out to be correct forrelatively weak interaction energy. For stronger interactions, ceff becomes very small and evenchanges its sign (see figure 1). Note that this result holds also for the two particle case where itis exact. It is a surprising result, verified numerically in figure 2 and its experimental ver-ification should be considered a challenge. The knowledge of ceff enables to calculate theground state and the low excited states if the conditions for stability are satisfied. In section 4,the ground state is calculated in the thermodynamic limit for a dilute gas. In particular, it isdemonstrated to depend on all parameters via ceff. We have shown that for a dilute gas there isa regime of parameters where c 0eff > and therefore the system is stable. For other

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

10

parameters, c 0eff < and the dilute gas is unstable. In this regime, the results of [17, 18]regarding dynamics of attractive gas might be realized. If the gas is not dilute, we cannotdetermine the stability of the system. This theoretical model enables us to predict qualitativefeatures of interacting bosons for realistic systems.

The potential (5) can be realized, for example, in optical lattices [6] with tight harmonictrapping along two perpendicular directions (E w ) and almost flat potential along thethird direction. The inter-particle interactions are in three dimensions and can be modeled by a‘delta shell’ potential

V r

c

r mr r

c

r mr r r

3

4for

2for

0 otherwise

, 47l

0

in3 in

out2

outout out out

( ) ( )

w

e we

=

<

< < +

^

^

⎧

⎨⎪⎪

⎩⎪⎪

where r , 0in oute . In a previous work [16], we calculated the scattering length a and theeffective range re of the potential (47) (see appendix A of [16]), wrote a three-dimensionalSchödinger equation and integrated it over two axes to obtain a one-dimensional equation ofthe form (5). This leads to the relations

r l3 2, 48out ( )=

a c c1

42 49l0( ) ( )

w= +

^

and

rc l

a c c

a2

2

2

3. 50e

l

l

2

0( )( )=

++

As seen from (26), for l = 0, ceff is proportional to the scattering length. However, for l 0¹ ,ceff cannot be expressed in terms of a and re. Therefore, it motivates introducing an effectivescattering length that dominates the spectrum.

From an experimental point of view, it looks that ceff is the only quantity that one canmeasure in order to characterize the inter-particle interactions (because it determines thespectrum). Hence, it makes sense to define an effective scattering length

ac

4. 51eff

eff ( )w

=^

This scattering length, which includes corrections originating in the non-vanishing interactionrange, is unique for one-dimensional bosonic systems.

Acknowledgments

We thank Eliot Lieb and Avy Soffer for suspecting an error in the original version of the workand Nimrod Moiseyev, Daniel Podolsky, Yoav Sagi and Efrat Shimshoni for illuminating andinformative discussions. The work was supported in part by the Israel Science Foundation(ISF) grant number 1028/12, by the US-Israel Binational Science Foundation (BSF) grantnumber 2010132 and by the Shlomo Kaplansky academic chair. SF thanks the Kavli Institutefor Theoretical Physics (KITP) in Santa Barbara for its hospitality, where this research wassupported in part by the National Science Foundation under Grant No. NSF PHY11-25915.

J. Phys. A: Math. Theor. 49 (2016) 085205 H Veksler and S Fishman

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