notes on bethe ansatz

8/3/2019 Notes on Bethe Ansatz

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Notes on Bethe Ansatz

Mikhail Zvonarev

December 2, 2010

Coordinate and Algebraic Bethe Ansatz techniques are reviewed; the former is applied tothe Lieb-Liniger model, and the latter to the Heisenberg model. Suggestions for the furtherreading are given.

1 Coordinate Bethe Ansatz

Bethe Ansatz means Bethes substitution, the method is named after the work by HansBethe [1]. Bethe found eigenfunctions and spectrum of the one-dimensional spin-1/2 isotropicmagnet (often called one-dimensional spin-1/2 isotropic Heisenberg model). I will demon-strate the usage of the Algebraic Bethe Ansatz on this model in the second part of mylecture.

Next model was solved by coordinate Bethe Ansatz more than 30 years later [2]. This isa model of one-dimensional gas of bosons interacting via the -function potential. Since it

was solved by Lieb and Liniger, it is often called the Lieb-Liniger model. I will demonstratehow coordinate Bethe Ansatz works using this model as an example.

1.1 Hamiltonian of the Lieb-Liniger model in the first and second-quantized form

The Hamiltonian of the one-dimensional Bose gas with zero-range interaction potential is

H = dx [(x)2x(x) + c

(x)(x)(x)(x)], (1.1)

where the units have been chosen such that = 2m = 1. The interaction constant c 0 hasthe dimension of inverse length. The dimensionless coupling strength is given by

=c

. (1.2)

The boson fields and satisfy canonical equal-time commutation relations:

[(x), (y)] = (x y), [(x), (y)] = [(x), (y)] = 0. (1.3)

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The number of particles operator N is

N =

dx (x)(x), (1.4)

and the momentum operator P is

P = i

dx (x)x(x). (1.5)

The number of particles and momentum are integrals of motion:

[H, N] = [H, P] = 0. (1.6)

The equation of motion for the Hamiltonian (1.1)

it = 2x + 2c (1.7)

is called the quantum nonlinear Schrodinger equation.Since the Hamiltonian commutes with N, Eq. (1.6), the number of particles in the model

is conserved and one can look for the eigenfunctions of the Hamiltonian (1.1) in a sectorwith the fixed number of particles. Write

|N(k1, . . . , kN) = 1N!

dNx N(x1, . . . , xN|k1, . . . , kN)(x1) . . . (xN)|0. (1.8)

Here |N(k1, . . . , kN) is an eigenfunction of the second-quantized Hamiltonian (1.1):H

|N(k1, . . . , kN)

= EN(k1, . . . , kN)

|N(k1, . . . , kN)

, (1.9)

the symbol |0 denotes the Fock vacuum state(x)|0 = 0, 0|(x) = 0, 0|0 = 1, (1.10)

the symbol dNx dx1 . . . d xN, and the parameters k1, . . . kN are called quasi-momenta. Thecoordinate wave function N is an eigenfunction of the Hamiltonian (1.1) written in thefirst-quantized form:

HN = Ni=1

2

x2i+ 2c

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when working at finite N. This can be done in several different ways: one can put the systemon a ring of circumference L and impose the periodic boundary conditions, or put it in thehard-walls box of length L, or, more generally, impose twisted boundary conditions. Theinfluence of the boundary conditions should disappear in the thermodynamic limit, but forfinite N they are, of course, affect the results. The exact eigenfunctions of the model areknown for the general case of the twisted boundary conditions. The particular case of the

periodic boundary conditions is, however, enough for our purposes and will be assumedhenceforth.

1.2 Lieb-Liniger model in the c limit: Tonks-Girardeau gasThe limit of infinite repulsion, c , is often called the Tonks-Girardeau limit [3, 4]. Wewill consider this limit in full details since it will give us a lot of intuition about the behaviorof the model at arbitrary c.

1.2.1 Eigenfunctions and spectrum of the Tonks-Girardeau gas

The eigenfunctions of the Hamiltonian (1.11) should go to zero when xi xj for arbitraryi = j :

N(x1, . . . , xi, . . . , xj, . . . , xN) 0, xi xj, i = j, (1.14)and they should solve the free Schrodinger equation when all xj are different. A possiblecandidate is the determinant of an N N matrix with the entries exp{ikjzl} :

det[exp{ikjxl}]. (1.15)

This function should be symmetrized in coordinates:

N(x1, . . . , xN|k1, . . . , kN) = CN!

det[exp{ikjxl}]

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These equations are called Bethe equations; in this particular case they can be solved easily:

kj =2

Lnj , j = 1, . . . , N . (1.20)

Here nj are arbitrary odd numbers for N even and arbitrary even for N odd. Apart fromthis selection rule, the quantization condition for quasimomenta are the same as for freefermions: the allowed values of kj are equidistant in the momentum space and multipleoccupation of the same position in the momentum space is prohibited: the determinantin Eq. (1.19) becomes equal to zero. The fermion-like behavior of this boson system isvery typical for the quantum systems in one spacial dimension; this phenomenon is calledstatistical transmutation. It should be stressed, however, that the Tonks-Girardeau gasis not a system of the non-interacting fermions: the wave function (1.19) is symmetric incoordinates, while the wave functions of the fermion system should be antisymmetric. Thus,correlation properties of the Tonks-Girardeau gas are very different in many aspects fromthose of the free fermion system. An example illustrating this difference, the one-particledensity matrix, will be studied in section 1.5.

The norm of the wave function (1.19) with the periodic boundary conditions can becalculated easily: L

0

dNx|(x1, . . . , xN|k1, . . . , kN)|2 = C2LN, (1.21)

therefore

C =1LN

. (1.22)

It is not difficult to prove that the wave functions (1.16) with the quasi-momenta quantizedby the conditions (1.19) form a complete set of eigenfunctions of the Hamiltonian (1.11) in

the limit c .

1.2.2 The ground state and thermodynamics of the Tonks-Girardeau gas atzero temperature

The ground state of the Tonks-Girardeau gas can be found easily. To minimize EN, Eq. (1.17),one should take the following subset of kj from the set (1.20):

kj =2

L

j N + 1

2

, j = 1, . . . , N . (1.23)

The momentum PN of the ground state is thus equal to zero.

The thermodynamic limit is the limit of infinite number of particles at finite density :

=N

L= const, N, L . (1.24)

One has in this limitN

j=1

L2

dk. (1.25)

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The ground state energy in the canonical ensemble is, therefore,

E0 =L

2

dk k2 =L

323. (1.26)

In the grand canonical ensemble the number of particles is not fixed and depends on the

chemical potential; if the average number of particles in the system is N then

EN =N

j=1

k2j . (1.27)

In the thermodynamic limit the chemical potential is related to the density of particles asfollows:

= ()2. (1.28)

One can see that the thermodynamics of the Tonks-Gigardeau gas at zero temperature is thesame as that of the spinless free-fermion gas, with playing a role of the Fermi momentum

kF : kF = . (1.29)

The pressure of the 1D gas is given by the formula

P =

E0L

,N

=

F

L

T,N

. (1.30)

Here is the entropy of the gas, and F is the Helmholtz free energy, F = E0T . The partialderivatives taken at a fixed T coincide with the partial derivatives taken at a fixed for thezero temperature case. Because of this fact we will omit the corresponding subscripts in thezero temperature case. For example, we will write P =

(E0/L)N. The thermodynamic

definition of the sound velocity is

vS =

L

m

P

L

N

. (1.31)

We set m = 1/2 in the definition of the Lieb-Liniger model (1.1). One thus gets for theTonks-Girardeau gas

P =2

323 (1.32)

and

vS = 2. (1.33)One can see from the expression for kF that the Fermi velocity vF is

vF kFm

= 2 (1.34)

(recall that we set m = 1/2). Comparing Eqs. (1.33) and (1.34) one concludes that the Fermivelocity coinside with the sound velocity in the Tonks-Girardeau gas. This is the expectedresult since the spectrum of the Tonks-Girardeau gas is the same as that of the free-fermiongas. In our studies of the Lieb-Liniger model with finite coupling c we will use the notion ofthe Fermi velocity as given by Eq. (1.34); the sound velocity vS at arbitrary c is always less

than vF, see Fig. 1.

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1.3 Lieb-Liniger model at arbitrary repulsion strength

We will repeat, for arbitrary repulsion strength c, the steps performed in section 1.2: weconstruct the eigenfunctions and the spectrum of the Lieb-Liniger model (1.11) at arbitrarycoupling strength c, identify the ground state of the system and take the thermodynamiclimit. We then calculate the velocity of sound vS, the compressibility , and the Luttinger

parameter K from the thermodynamics of the Lieb-Liniger model. We discuss the weak(Bogoliubov) and strong (Tonks-Girardeau) coupling limits in the solution of the Lieb-Linigermodel.

1.3.1 Eigenfunctions and spectrum of the Lieb-Liniger model

The structure of the eigenfunctions at general c is much more complicated as compared tothe eigenfunctions at c = , considered in section 1.2.1. That particular case, however,shows a way one should proceed: The eigenfunctions N(x1, . . . , xN) in the N-particle sectorare symmetric under any interchange of coordinates, therefore it is sufficient to consider the

following domain T in the coordinate space:

T : x1 < x2 < < xN. (1.35)In this domain N is an eigenfunction of the free Hamiltonian

H0N = N

j=1

2

x2j(1.36)

H0NN = EN(k1, . . . , kN)N (1.37)The following boundary condition should be satisfied:

xj+1

xj c

N = 0, xj+1 = xj + 0. (1.38)

This boundary condition obviously reduces to Eq. (1.14) in the Tonks-Girardeau limit. Toprove Eq. (1.38) use the variables z = xj+1 xj and Z = (xj+1 + xj)/2. One has

x2

x1= 2

z,

2

x21+

2

x22= 2

2

z2+

1

2

2

Z2. (1.39)

Integrating Eq. (1.12) over z in the small domain |z| < 0 one can see that the eigenfunc-tion of the free-particle Hamiltonian (1.36) obeying the boundary condition (1.38) is indeed

an eigenfunction of the Hamiltonian (1.11).

To get an explicit expression for the eigenfunctions (1.12) we start from the trial expres-sion (1.15). We suggest the following form of the eigenfunctions:

N = const

Nj>l1

xj

xl+ c


One should show that this expression satisfies Eqs. (1.36) and (1.38). Check, for instance,that

x2

x1 cN = 0, x2 = x1 + 0. (1.41)

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Do do this, rewrite Eq. (1.40) as follows:

N =

x2

x1+ c

N, (1.42)

where

N = constN

j=3

xj

x1+ c

xj

x2+ c

Nj>l3

xj

xl+ c


The function N is antisymmetric with respect to the interchange of x1 and x2,

N(x1, x2) = N(x2, x1). (1.44)Writing the left hand side of the Eq. (1.41) in the form

x2

x1

2 c2

N (1.45)

one sees that it is antisymmetric with respect to the interchange ofx1 and x2, and, therefore,equal to zero when x2 x1. The boundary condition (1.38) for the other xj can be checkedsimilarly. We thus showed that the wave function (1.40) is indeed an eigenfunction of theHamiltonian (1.11). Using the definition of the determinant

det[exp{ikjxl}] =P

(1)[P] exp{

iNn=1

xnkPn

}(1.46)

one gets for (1.40)

N = const

{N!

j>l

[(kj kl)2 + c2]}1/2

P (1)[P] exp{i

N

n=1

xnkPn}j>l

[kPj

kPl

ic sign(xj

xl)]. (1.47)

This is the desired expression for the eigenfunctions of the Hamiltonian (1.11). We wroteEq. (1.47) in such a form that it is valid for arbitrary values of x1, . . . , xN, thus removingthe restriction (1.35). One can see that the wave functions given by Eq. (1.47) reduce to thewave functions given by Eq. (1.16) in the c limit.

The eigenvalues of the Hamiltonian (1.11) and the momentum operator (1.13) on theeigenfunctions (1.47) are

EN =N

j=1

k2j , PN =N

j=1

kj, (1.48)

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respectively. These formulas look very similar to that of a free system and because of thissimilarity the parameters kj are called quasi-momenta.

No boundary conditions were imposed so far. Let us do it now and use the periodicboundary conditions. The condition (1.18) imposed on the function (1.47) gives

exp {ikjL} =Nl=j

kj

kl + ic

kj kl ic, j = 1, . . . , N . (1.49)

These coupled nonlinear equations are called Bethe equations. In the c limit theseequations reduce to Eqs. (1.19). It can be shown that all their solutions are real (see, forexample, Ref. [5]). It is often more convenient to use the logarithmic form of Bethe equations:

Lkj +N

m=1

(kj km) = 2

nj N + 12

, j = 1, . . . , N . (1.50)

Here {nj} is a set of integers, and the function is

(k) = i ln

ic + kic k

, () = . (1.51)

An important property of the representation (1.50) is that the sets ofnj and the sets of quasi-momenta kj are in one to one correspondence. It is often more convenient to parameterizea quantum state by the set nj rather than by the set of quasi-momenta kj.

Let us sum up Bethe equations (1.50) over j. The function (k) defined by Eq. (1.51) isantisymmetric, therefore

N

j,m=1(kj km) = 0 (1.52)

and one gets

LN

j=1

kj = 2N

j=1

nj N + 1

2

. (1.53)

Comparing this equation with Eq. (1.48) one gets a relation between the momentum of anarbitrary state and the set of quantum numbers nj :

PN =2

L

Nj=1

nj N + 1

2

. (1.54)

The calculation of the normalization constant in Eq. (1.47) is a nontrivial task (recallthat in the Tonks-Girardeau limit this constant is given by Eqs. (1.21) and (1.22)). A closedexpression for this constant was suggested by Gaudin (see [6] and references therein) andproved in Ref. [7]; a detailed discussion can be found in Ref. [5].

1.3.2 The ground state and thermodynamics at zero temperature

The ground state of the model in the N-particle sector corresponds to the following choiceof the quantum numbers nj in Eq. (1.50):

nj = j, j = 1, . . . , N , (1.55)

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that is the Bethe equations (1.50) take the form

Lkj +N

m=1

(kj km) = 2

j N + 12

, j = 1, . . . , N . (1.56)

in the ground state of the model. The momentum of the ground state is equal to zero.

Consider now the thermodynamic limit of the model, defined by Eq. (1.24). Write

Nj=1

=N

j=1

kj+1 kjkj+1 kj = L

Nj=1

(kj)(kj+1 kj) (1.57)

where we have defined (kj) as follows

(kj) =1

L(kj+1 kj) . (1.58)

In the thermodynamic limit one has for an arbitrary function f(kj) :N

j=1

f(kj) L

dk (k)f(k). (1.59)

The parameter plays a role of the Fermi momentum: all the states with |k| < are occu-pied, and all the states with |k| > are empty. The value of is given by the normalizationcondition

dk (k) =

N

L= . (1.60)

To get the thermodynamic limit of the Bethe equations (1.56), take the difference of theequations with the indices j + 1 and j :

L(kj+1 kj) +N

j=1

[(kj+1 km) (kj km)] = 2. (1.61)

The difference kj+1 kj is small in the thermodynamic limit, therefore Eq. (1.61) can bewritten as follows

L(kj+1 kj) + (kj+1 kj)N

m=1

(kj km) = 2, N, L , (1.62)

where (k) (k)/k. Using Eqs. (1.58) and (1.59) one gets after some algebra

(k) 12

dq (q)K(k, q) =1

2, k , (1.63)

where

K(k, q) (k q) = 2cc2 + (k q)2 . (1.64)

The linear integral equation (1.63) is called Lieb equation. Its solution, together with the

normalization condition (1.60), gives the quasi-momentum distribution function (k). All

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thermodynamic parameters can be calculated from (k) and . The ground state energy inthe canonical ensemble is

E0 =N

j=1

k2j L

dk (k)k2. (1.65)

Following [2], change the variables as follows:

k = z, c = , (z) = g(z). (1.66)

Written in these variables, Eqs. (1.60), (1.63), and (1.65) are, respectively,

11

dz g(z) = , (1.67)

g(z) 12

11

dy2g(y)

2 + (y z)2 =1

2, (1.68)

and

e() E0

N2 =3

311 dy g(y)y

2. (1.69)

Recall that is given by Eq. (1.2). The system of equations (1.67) and (1.68) is much moresuitable for the numerical analysis than the system (1.60) and (1.63). To solve the system(1.67) and (1.68) the following two steps should be done:

(i) Solve Eq. (1.68) for a given ;

(ii) use Eq. (1.67) to determine as a function of ;

Having solved the system (1.67) and (1.68), one can easily go back to the equations in theform (1.60) and (1.63), if necessary. The Fermi momentum can be found by combining

Eqs. (1.2) and (1.66): =

=

[11

dz g(z)

]1. (1.70)

The quasi-momentum density (k) is related to g(z) by Eq. (1.66).

1.3.3 Velocity of sound, compressibility, and Luttinger parameter

Discuss first the thermodynamic definition of the sound velocity, Eq. (1.31). Taking E0 fromthe Eq. (1.69), and using Eq. (1.30) one gets

vS = 2[

3e() 2e() + 12

2e()]1/2

= 2[

() 12

()]1/2

, (1.71)

where prime denotes differentiation with respect to and is the chemical potential givenby

=

E0N

L

= 2[3e() e()]. (1.72)

Note an important technical point: since P and depend on N and L only via = N/L,the following identities are valid:

P

N

= P

L

,

N

=

L

. (1.73)

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These identities are very useful in getting various thermodynamic relations.

Another, microscopic definition of the sound velocity is

vS =(k)

P(k)

k=

, (1.74)

where (k) is the energy change of the system when a particle with the quasi-momentum kis added; P(k) is the momentum change of the system. The equivalence of the definitions(1.31) and (1.74) in the Lieb-Liniger was proven in Ref. [2]; for more modern presentationone can consult Ref. [5]. We will omit this proof since it is quite long; we only present hereseveral equivalent representations for vS obtained in course of this proof:

vS =

2

=

=

2()2. (1.75)

Using the compressibility

1

LL

PN

=1

2

L

(1.76)

we can write the velocity of sound as

vS =

2

. (1.77)

The Luttinger Liquid Hamiltonian contains two parameters: the velocity of sound, vS,and the Luttinger parameter, K. These parameters are related to the compressibility of thesystem [8]:

K

vS= 2 any model (1.78)

In case the microscopic theory has some extra symmetry, there could exist some other re-lations. For example, if the system is Galilean-invariant, that is, the kinetic energy of anyparticle it consists of is p2/2m, one can show [9] that

KvS = 2 Galilean-invariant model (1.79)

One can see that Eqs. (1.78) and (1.79) are consistent with Eq. (1.77), as it should be sincethe Lieb-Liniger model is Galilean-invariant. Note that 2 = vF, where vF is the soundvelocity of the Tonks-Girardeau gas, Eqs. (1.33) and (1.34).

1.3.4 Weak and strong coupling limits and numerics

The large limit is easy to work out since Eq. (1.68) admits a regular perturbative expansionwith 1/ being a small parameter. One has

g(z) =1

2+

1

2+

2

32+ , , (1.80)

Equation (1.67) then gives

=

2 +4

32 32

154 16

455+

,

. (1.81)

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Inverting this expression, one gets

=

+

2

4

32+

16

33+

22

15 1

16

4+ , . (1.82)

For the ground state energy per particle defined by Eq. (1.69) one gets

e() =2

3

[1 4

+

12

2+ (2 15) 32

153+

](1.83)

From the exact expression (1.71) for the sound velocity in the Lieb-Liniger model, one getsthe following large expansion:

vlarge = vF

1 4

+

12

2+

, (error < 1% for > 10). (1.84)

The leading terms in all the expansions written in the present section correspond to the

Tonks-Girardeau limit and were obtained in the section 1.2.2.The opposite limit, 0, is much more difficult to analyze since the kernel in Eq. (1.63)

becomes singular when 0. The leading term is obtained in Ref. [2]:

g(z)

1 z22

, 0. (1.85)

For the ground state energy per particle defined by Eq. (1.69) one gets

e() 1 4

3

, 0. (1.86)

Let us substitute the expansion (1.86) into Eq. (1.71):

vsmall = vF1

1

23/2

1/2, 0 (error < 1% for < 10). (1.87)

Since only two terms are written explicitly in the expansion (1.86), one should expandEq. (1.87) and drop all the terms except the leading and the first subleading. However,the numerical analysis shows that Eq. (1.87) gives much better approximation to the exactexpression (1.71) if not expanded. We plot vS/vF, vlarge/vF and vsmall/vF versus in Fig. 1.Recall that vF is the velocity of sound in the Tonks-Girardeau gas, Eq. (1.34), and the ratiovS/vF is the inverse Luttinger parameter, Eq. (1.79),

K1 =vSvF

. (1.88)

1.4 Excitations in the Lieb-Liniger model

We construct elementary (single quasi-particle and single quasi-hole) excitations in theLieb-Liniger model and construct more complex excitations out of them.

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0.01 0.1 1 10 100

0.2

0.4

0.6

0.8

1

vvF

Figure 1: Shown is the log-linear plot of the inverse Luttinger parameter in the Lieb-Linigermodel, Eq. (1.88), versus . Red curve corresponds to the exact (numerical) result for vS,black dotted curve is obtained from Eq. (1.84), and black dashed curve from Eq. (1.87).

1.4.1 One-particle excitations in the Tonks-Girardeau gas

The ground state quasimomentum distribution in the N-particle sector of the Tonks-Girargeaugas is given by Eq. (1.23):

Lkj = 2 j N + 1

2 , j = 1, . . . , N , (1.89)

while in the N + 1-particle sector it takes a form

Lkj = 2

j N + 2

2

, j = 1, . . . , N + 1. (1.90)

Therefore, every kj is shifted by /L with respect to kj. A way to avoid dealing with thisshift is to impose anti-periodic boundary conditions in the N + 1-particle sector. Then theground state quasimomentum distribution is

Lkj = 2 j N + 1

2 , j = 1, . . . , N + 1. (1.91)

Note that the ground state is 2-fold degenerate, with another configuration correspondingto j = 0, 1, . . . , N . While the above choice of the boundary conditions eliminates the /Lshift, its influence onto the ground state energy is negligible small (vanishes as 1/N) in thelarge N limit.

The structure of the one-particle excitations is now evident. They can be decomposed intothe topological part resulting in the boundary conditions change and truly one-particleexcitation. The corresponding Bethe equations are

Lkj = 2 j N + 1

2 , j = 1, . . . , N . (1.92)

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andLkN+1 = 2M, |M| > N/2. (1.93)

One can see by comparing Eqs. (1.89) and (1.92) that kj = kj for j = 1, . . . , N . This resultis specific to the Tonks-Girardeau gas and does not hold true for the finite repulsion case.However, the separation of the topological part of the one-particle excitation by using the

above trick with the boundary conditions will prove useful in the finite repulsion case as well.

1.4.2 One-particle shift function

We consider the N-particle ground state, Eq. (1.56). We then add one article and imposeanti-periodic boundary conditions, like in the Tonks-Girardeau limit. The resulting Betheequations are

Lkj +Nk=1

(kj kk) + (kj kN+1) = 2

j N + 12

, j = 1, . . . , N , (1.94)

and

LkN+1 +Nk=1

(kN+1 kk) = 2M, |M| > N2

. (1.95)

The difference of Eqs. (1.56) and (1.94) is

L(kj kj) +Nk=1

[(kj kk) (kj kk)] (kj kN+1) = 0. (1.96)

Using that kj+1 kj 1/L in the thermodynamic limit one getsNk=1

[(kj kk) (kj kk)]

= (kj kj)Nk=1

(kj kk) Nk=1

(kk kk)(kj kk), N, L (1.97)

and it follows from Eq. (1.56) that

(kj+1 kj)

L +Nk=1

(kj kk)

= 2, N, L . (1.98)

Let us define the so-called shift function F(kj|kN+1) as follows (we write kp instead of kN+1in order to lighten the notations)

F(kj|kp) = kj kjkj+1 kj . (1.99)

Using Eqs. (1.97)(1.99) one gets from Eq. (1.96) an integral equation (often called dressingequation) onto the shift function:

F(k|kp) 12

dq K(k, q)F(q|kp) = 12

(kp k). (1.100)

This dressing equation is of crucial importance for the analysis of one-particle excitations.

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1.4.3 Momentum of the excited state

We introduce the so-called dressed momentum k() by the formula

p(k) = k +

dq (q)(k q). (1.101)

An important property of k(k), valid for any is

p() = kF. (1.102)It can be proven by integrating Eq. (1.63) with respect to k. Note that in the infinite repulsionlimit the integral in Eq. (1.101) vanishes:

p(k) = k as = . (1.103)

The importance of the definition (1.101) becomes evident when one calculates the mo-

mentum difference P(kp) between the N-particle ground state and N + 1-particle excitedstate. One has

P(kp) =Nk=1

(kk kk) + kp = N

j=1

F(kj|kp)(kj+1 kj) + kp

dk F(k|kp) + kp, N . (1.104)

Multiplying both parts of Eq. (1.100) by (k), integrating over k, and taking into account

Eq. (1.63) one gets an identity

dk F(k|kp) =

dk (k)(kp k) (1.105)

which impliesP(kp) = p(kp), |kp| . (1.106)

1.4.4 Energy of the excited state

Let us introduce the integral equation onto the so-called dressed energy () :

(k) 12

dq (q)K(k, q) = k2 (1.107)

with an extra condition() = 0. (1.108)

It can be shown [5] that (k) possesses the following properties

(k) = (k) (1.109)(k) > 0, k > 0 (1.110)

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and

(k) < 0, |k| < (1.111)(k) > 0, |k| > (1.112)

Differentiating Eq. (1.107) and taking the integral by parts one gets

(k) 12

dq (q)K(k, q) = 2k. (1.113)

Let us now calculate the energy difference E(kp) between the N-particle ground stateand N + 1-particle excited state. One has

E(kp) = k2p +

N

j=1 k2j k2j

= k2p N

j=1

2kj(kj kj) k2p

dk 2kF(k|kp). (1.114)

Multiplying Eq. (1.113) by F(k|kp), integrating over k, and using Eq. (1.100), one arrives atthe identity

dk 2kF(k|kp) = 1

2

dq (q)K(q, kp). (1.115)

Substituting this identity into Eq. (1.114) we finally get

E(kp) = (kp), |kp| . (1.116)This formula clarifies a physical meaning of (k) introduced earlier by Eq. (1.107): it can beinterpreted as the energy of an excitation with the momentum p(k).

1.4.5 One-hole excitations

One-hole type of excitations can be generated by removing one particle from the N-particleground state. An analysis similar to the one carried out for the one-particle excitation gives

P(kh) = p(kh), kh (1.117)for the dressed momentum of the excitation, and

E(kh) = (kh), kh (1.118)

for the dressed energy.

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Figure 2: Particle-hole excitation corresponding to min E for a given momentum 0 P 2kF.

1.4.6 Multiple particle-hole excitations

An arbitrary excitation in the Lieb-Liniger model can be created from one-particle and one-hole excitations. The momentum difference between the excited state and the ground stateis

P =

particles

p(kp) holes

p(kh) (1.119)

and the energy difference is

E =

particles

(kp) holes

(kh). (1.120)

1.4.7 Dispersion curves and their exact lower bound

By the dispersion curve we call the curve of E plotted as a function of P. What is theminimal value of E for a given P lying in the region 0 P 2kF? The answer tothis question (first given in Ref. [10]) is very elegant and intuitively clear: the correspondingexcitation is shown in Fig. 2. It consists of a single particle-hole pair, with the particle lyingat the right Fermi point and

kF p(kh) = P kF kF. (1.121)

We plot E as a function of P for several values of , see Fig. 3. Note that the explicit

analytic expression for min E can be found easily for = :E

k2F=

P

kF

2 P

kF

, 0 P 2kF as = . (1.122)

1.5 Correlation functions of the Tonks-Girardeau gas

The knowledge of the exact eigenfunctions and spectrum for the Lieb-Liniger model makes itpossible to develop the microscopic approach to the calculation of its correlation functions.

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0 0.5 1 1.5 2

PkF

0

0.2

0.4

0.6

0.8

1

E

kF

2

Figure 3: Shown are the plots of min E as a function of P for = , 100, 10, 1, 0.1 (fromupper to lower).

The procedure is straightforward: calculate the form-factors of local field operators in thebasis formed by the eigenfunctions of the Hamiltonian (1.1) in the sector with given numberof particles N, then sum up over all intermediate states, and, finally, take the thermodynamiclimit. It is very easy to formulate such a program, and very difficult to implement it becauseof the complicated structure of the Bethe wave function (1.47) and Bethe equations (1.49).The book [5] is mostly devoted to the development of this program and is, probably, the mostcomprehensive monograph in this field. Among the results not included in the cited book oneshould mention the papers [11, 12, 13], where the local field operators were represented viathe fundamental operators of the Quantum Inverse Scattering Problem (this approach workfor spin chains, where the quantum space is finite-dimensional, but not for the Lieb-Linigermodel; there is, however a trick (unpublished) solving the inverse scattering problem for theLieb-Liniger model as well).

Unfortunately, the final expressions for the correlation functions obtained within thisstraightforward approach are, in general, too complicated to be successfully analyzed eitherby analytic methods or numerically. There are, however, several important exceptions. Oneof them, the Tonks-Girardeau gas, is discussed here.

The relatively simple form of the eigenfunctions of the Tonks-Girardeau gas make itpossible to calculate the correlation functions explicitly for arbitrary number of particles N.

The first non-trivial correlation function which was calculated is the one-particle densitymatrix. The calculations were done by Lenard [14] and the result is now known as theLenards formula. We present this formula in section 1.5.1. Lenards formula contains thedeterminant of an NN matrix. The N limit of the determinant of an NN matrix iscalled the Fredholm determinant. It is discussed in section 1.5.2. The thermodynamic limitof the Lenards formula is studied in sections 1.5.3 and 1.5.4. The momentum distributionfunction for the Tonks-Girardeau gas is calculated in section 1.5.5.

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1.5.1 One-particle density matrix: Lenards formula

The one-particle density matrix in the N-particle sector is defined as follows

N(x) = (x)(0), (1.123)

where the average is taken over the ground state of the system in the N-particle sector.The expression for the N(x) in the first-quantized representation can be obtained easily(remember the commutation relations (1.3) and the representation (1.8)):

N(x) = N(k1, . . . , kN)|(x)(0)|N(k1, . . . , kN) = 1N!

L0

dNz dNz

N(z1, . . . , zN)N(z1, . . . , zN)0|(z1) . . . (zN)|(x)(0)|(z1) . . . (zN)|0

= NL

0

dz2 . . . d zNN(x, z2, . . . , zN)N(0, z2, . . . , zN) (1.124)

In the Tonks-Girardeau case N is given by Eqs. (1.16), the normalization constant C for theperiodic boundary conditions is given by Eq. (1.22), and the ground state quasi-momentumdistribution is given by Eq. (1.23). Therefore

N(x) =1

(N 1)!LNL0

dz2 . . . d zN

P P

(1)[P]+[P] exp{ixkP1}N

a=2exp{iza(kPa kPa)}sgn(za x). (1.125)

All the integrals in this expression can be taken explicitly and give

1

L

L0

dza exp{iza(kPa kPa)}sgn(za x) = f(kPa kPa), a = 2, 3, . . . , N , (1.126)

where

f(ka kb) = kakb 2

L

exp{ix(ka kb)} 1i(ka kb) . (1.127)

One gets, therefore,

N(x) =1

(N 1)!LP

P

(1)[P]+[P] exp{ixkP1}f(kP1 kP1)

Na=1

f(kPa kPa). (1.128)

Represent the permutation P in the internal sum as a sequence of two permutations

P = PP (1.129)

where P is the running index of the external sum. Further, introduce the notation

kPa

P[ka]. (1.130)

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Written in these notations, Eq. (1.128) reads

N(x) =1

(N 1)!LP

P

(1)[P] exp{ixP[k1]}f(P[P[k1]] P[k1])

Na=1

f(P[P[ka]] P[ka]). (1.131)

One can easily notice the following identity:

Na=1

f(P[P[ka]] P[ka]) =Na=1

f(P[ka] ka), (1.132)

therefore

N(x) =1

(N 1)!LP

(1)[P]Na=1

f(P[ka] ka)P

exp{ixP[k1]}f(P[P[k1]] P[k1])

= 1LP

(1)[P]Na=1

f(P[ka] ka)N

j=1

exp{ixkj}f(P[kj] kj)

=1

L

Nj=1

exp{ixkj}P

(1)[P]a=j

f(P[ka] ka). (1.133)

Consider the determinant of an N N matrix with the entries f(kj kl) :

f(k1 k1) f(k1 kN)...

. . ....

f(kN k1) f(kN kN)

= P

(1)[P]N

a=1

f(P[ka] ka). (1.134)

To reproduce the last line of the Eq. (1.133) one should replace the l-th column of the matrixf(kj kl) with the expression exp{iz1kl}/L :

N(x) =

detN(I+ V1 + V2)

=0

= detN(I + V1 + V2) detN(I+ V1), (1.135)

where

(V1)ab =

2

L

exp{ix(ka kb)} 1i(ka kb)

, (V2)ab =1

L

exp

{ixkb

}, a, b = 1, . . . , N .

(1.136)Equation (1.135) can be rewritten in the following form

N(x) =

detN(I+ V1 + V2)

=0

= detN(I + V1 + V2) detN(I+ V1), (1.137)

where

(V1)ab = 2 2L

sin[x2

(ka kb)](ka kb) , (V2)ab =

1

Lexp

i x

2(ka + kb)

, a, b = 1, . . . , N .

(1.138)

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Equations (1.137) and (1.138) are the desired Lenards formula for the one-particle densitymatrix of the Tonks-Girardeau gas at zero temperature. It can be easily generalized to afinite temperature case.

It is proper place to stress again that the Tonks-Girardeau gas is not equivalent to thefree-fermion gas: though the spectrum of the Tonks-Girardeau gas is the same as that of the

free fermion gas, the wave functions have different symmetry. This changes some observablesdrastically: for example, the one-particle density matrix for the free fermion gas

ffN (x) =1

L

Nj=1

exp{ixkj} (1.139)

is totally different from that of the Tonks-Girardeau gas, Eqs. (1.137) and (1.138).

1.5.2 The Fredholm determinant

We have seen that in the N-particle sector the density matrix (1.123) is expressed via thedeterminants of N N matrices, Eqs. (1.137) and (1.138). In the thermodynamic limit(1.24) these matrices becomes infinite-dimensional. Thus, an important object comes intoplay: the determinant of an infinite-dimensional matrix (it is called Fredholm determinant).This object appears regularly in the calculations of the correlation functions of the integrablemodels.

We give in this section two alternative definitions of the Fredholm determinant. The firstone is: Let V be an N N matrix with the entries Vab = V(ka, kb). Let

ka = 2aN 1

1 , a = 0, 1, . . . , N 1. (1.140)Then the Fredholm determinant det(I + V) is defined as follows:

det(I + V) = limN

detN

I+

2

N 1V

. (1.141)

The matrix I on the right hand side of Eq. (1.141) is the N N identity matrix. We seethat with increasing number of divisions, N, the possible values of ka, Eq. (1.140), fill theinterval 1 ka 1 densely. The objects I and V on the left hand side of the Eq. (1.141)should thus be some integral operators with the kernels defined on [

1, 1]

[

1, 1]. This can

be readily seen from an alternative definition of the Fredholm determinant:

det(I+ V) =

N=0

1

N!

11

dk1 . . .

11

dkNdet

V(k1, k1) V(k1, kN)... . . . ...

V(kN, k1) V(kN, kN)

, (1.142)

where V is a linear integral operator with the kernel V(k, p) defined on [1, 1] [1, 1]. Theequivalence of the definitions (1.141) and (1.142) is proven, for example, in Ref. [15]. Thesedefinitions are somehow complementary: the former one is more suitable for a numericalanalysis, while the latter one reveals better the analytic structure of the problem.

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1.5.3 One-particle density matrix: thermodynamic limit

Let us set up a convention first. The one-dimensional density is controlled by the chemicalpotential in the infinite limit as given by Eq. (1.28):

= . (1.143)

It follows from this equation that

has a dimension of inverse length. In the rest of section1.5 we measure distances in units of 1/

. This convention implies that the momentum is

measured in units of the Fermi momentum kF, Eq. (1.29).

The ground state and zero temperature thermodynamics of the Tonks-Girardeau gas aredescribed in section 1.2.2. Using formulas from that section, write the thermodynamic limitof the Lenards formula (1.137) in the form prescribed by the definition (1.141):

(x)

(0)= det(I+ V1 + V2) det(I + V1). (1.144)

The kernels of the operators V1 and V2 in Eq. (1.144) are given by the thermodynamic limitof Eq. (1.138):

V1(k, p) = sin x

2(k p)

(k p) (1.145)

and

V2(k, p) =1

2exp{ix

2(k + p)}. (1.146)

These kernels are defined on [1, 1] [1, 1], in accordance with our convention formulatedbelow Eq. (1.143). The parameter is

= 2. (1.147)

Note that (0) = ; one has = 1/ if measured in units of

.

1.5.4 One-particle density matrix: asymptotics and numerics

Since (x) = (x), we assume x 0 in this section. For short distances (x 1) one caneasily find from Eqs. (1.144) and (1.142) that

(x)

(0) = 1 x2

6

x3

18 +

x4

120 +

11x5

2700 + . . . , x 0. (1.148)For = 0 this is simply the expansion of sin x/x. Notice that higher order terms that arenot exhibited on the r.h.s. of Eq. (1.148), depend on nonlinearly.

Next, consider the long-distance expansion of Eq. (1.144). Compared to the short-distance analysis, this is a more sophisticated task. The analysis carried out in Refs. [16, 17,18] shows the power-law decay of the density matrix:

(x)

(0)=

Cx

[1 +

1

8x2

cos2x 1

4

+

3

16x3sin2x

], x (1.149)

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0 5 10 15 20 25 30

x

0

0.2

0.4

0.6

0.8

1

x0

Figure 4: Shown is the function (x)/(0) versus x. The density matrix (x) is given byEq. (1.144); in getting the plot the Fredholm determinants entering Eq. (1.144) were ap-proximated by finite-dimensional matrices, Eq. (1.141), with N = 800.

with relative corrections of the order of x1. The constant C is given by

C = e1/221/3A6 0.924, (1.150)with A = 1.2824271 . . . being Glaishers constant. The result (1.149) confirms the Luttingerliquid theory predictions.

For the curve in Fig. 4 we took N = 800 in using the representation (1.141) for theFredholm determinants entering Eq. (1.144). For x large than 30 it is safe to use the asymp-

totic expansion (1.149) instead of the exact expression (1.144): the relative difference (x)between the exact expression and the leading term of the large x expansion

(x) =(x) C/x

(x)(1.151)

is very small for x = 30 :(30) 6 104. (1.152)

so the subleading terms in expansion (1.149) are not relevant for most applications. On theother hand, the function 1/

x converges to zero very slowly and the knowledge of constant C

entering the expansion (1.149) is required with a good precision. Thus, the exact expression(1.150) is very much appreciated.

1.5.5 The momentum distribution

The momentum distribution function n(k) is given by

n(k) =

dx eikx(x), (1.153)

where (x) is the one-particle density matrix (1.123). Recall that x is measured in units of

1/ as defined below Eq. (1.143), and that (x) = (x). The momentum distribution is23


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0 0.5 1 1.5 2

k

0

0.5

1

1.5

2

n

k

Figure 5: Shown is the momentum distribution for the Tonks-Girardeau gas (solid curve) andfor the free fermion gas (dashed curve). Both distributions are normalized by the condition(1.154). Dotted curve represents the large k asymptotics of the momentum distribution,Eq. (1.155).

also symmetric, n(k) = n(k). It is normalized as follows0

dk n(k) = 1. (1.154)

We plot n(k) in Fig. 5. The density matrix (x) for x 30 was calculated from the Fredholmdeterminant representation; for x 30 we have used the asymptotic expansion (1.149). Thismomentum distribution is compared with the momentum distribution of the free fermion gas.

For the large momentum tail of n(k) one gets

n(k) 432

1

k4, k . (1.155)

The 1/k4 decay originates from the cusp of (x) at x = 0 seen in the third term on the right-hand side of Eq. (1.148). The coefficient 3/42 in Eq. (1.155) is taken from the Ref. [19].It is useful to note that Ref. [19] contains more general result: the large momentum tail ofn(k) decays as 1/k4 at any coupling strength; the coefficient stands at 1/k4 term was alsofound in the cited reference.

It follows from the large x expansion (1.149) that the momentum distribution is divergentat small momenta,

n(k) C

2

k, k 0. (1.156)

where the coefficient C is given by Eq. (1.150). This singularity is predicted by the Luttingerliquid theory for any coupling strength , and is the manifestation of quasi-long-range order,that is, a quasicondensate. The singularity is integrable for any = 0, therefore zeromomentum state is not macroscopically occupied. Note, however, that a significant fractionof particles lies in low-momentum states: for the Tongs-Girardeau gas about 15% of particleshave a momentum lower than 0.01 (in units of kF, as discussed below Eq. (1.143)). When

lowers this fraction is increased.

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2 Algebraic Bethe Ansatz

The Quantum Inverse Scattering Method, and its descendant, Algebraic Bethe Ansatz, wereproposed in late 70s (several crucial works appeared in the years 1978 and 1979). One canconsult chapters VI and VII of the book [5] for the list of references. Various relativistic andnon-relativistic quantum field theory models in 1 + 1 (one space one time) dimension wereanalyzed. Given the time limitations of a single lecture, we will learn the method by usingit to solve the spin-1/2 Heisenberg chain. This is, probably, the simplest model which canbe solved by the Algebraic Bethe Ansatz: it is non-relativistic (remember how careful oneshould treat infinite Dirac sea in the Luttinger model) and has finite-dimensional quantumspace.

2.1 Coordinate Bethe Ansatz for the spin-1/2 XXZ Heisenbergchain

The coordinate Bethe Ansatz for the spin-1/2 XXZ Heisenberg spin chain is reviewed. It issimilar to the coordinate Bethe Ansatz for the Lieb-Liniger model presented in section 1.

2.1.1 Hamiltonian, eigenfunctions and spectrum

Recall some basic notations. The Pauli spin matrices are

(x) =

0 11 0

(y) =

0 ii 0

(z) =

1 00 1

. (2.1)

It is often convenient to use instead of (x)

and (y)

:

(x) = (+) + (), (y) = i((+) ()) (2.2)

Useful multiplication formula

(p)(q) = ipqr(r) + pqI (2.3)

The tensor product for 2 2 matrices a and b is defined as follows1:

a b = a11 a12a21 a22

b11 b12b21 b22

=

a11b11 a11b12 a12b11 a12b12

a11b21 a11b22 a12b21 a12b22a21b11 a21b12 a22b11 a22b12a21b21 a21b22 a22b21 a22b22

. (2.4)

We will work with the operators living in the tensor product of M 2 2 matrices. Periodicboundary conditions are imposed. The Hamiltonian of the XXZ chain is

H = H0 + M 2hSz, (2.5)1When the type of the operators product, ordinary or tensorial, is clear from the context, the symbol

is usually omitted to lighten the formulas. Expression (2.6) is an example, there mm+1 means mm+1.

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H0 = M

m=1

(x)m

(x)m+1 +

(y)m

(y)m+1 +

(z)m

(z)m+1

, (2.6)

Sz =1

2

Mm=1

(z)m . (2.7)

Eigenfunctions of this Hamiltonian are

|N(1, . . . , N) = 1N!

Mm1,...,mN=1

N(m1, . . . , mN|1, . . . , N)()m1 ()mN|0 (2.8)

where |0 is the ferromagnetic vacuum (all spins up), |0 = Mj=1| . There exists an operatorVperforming an important unitary transformation

VH(h)V1 = H(h), V=M

m=1(z)m . (2.9)

Since (x)| = | and (x)()| = (+)| , the wave function V|N(1, . . . , N) is

V|N(1, . . . , N) = 1N!

Mm1,...,mN=1

N(m1, . . . , mN|1, . . . , N)(+)m1 (+)mN|0, (2.10)

where |0 is the dual ferromagnetic vacuum (all spins down), |0 = Mj=1| . ComparingEqs. (2.8) and (2.10) we see that the first-quantized wave function is the same in theseexpressions. Therefore, it is sufficient to consider h > 0, since all the observables will havea definite parity with respect to h in our problem. In particular, the ground state energy

is an even function of h, and the magnetization is an odd one. Another useful similaritytransformation, which exists for even number of sites only, is

UH(, h = 0)U1 = H(, h = 0), U=M/2m=1

(z)2m. (2.11)

Due to the existence of this transformation, some authors choose the + sign in front ofthe Hamiltonian (2.5). We, however, will keep the sign throughout this text. In ourfurther calculations M is assumed to be even.

The function N is the Bethe ansatz wave function:

N =

Nb>a1

sgn(mb ma)

Q(1)Q exp

i

Na=1

map0(Qa)

exp i

2

Nb>a1

(Qb Qa)sgn(mb ma)

(2.12)

The summation runs through all the permutations Q of the integers 1, 2, . . . , N . The anisotropyparameter is convenient to represent as follows

= cos(2). (2.13)

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Since the model is defined on a lattice, the momentum operator should be defined via theshift operator. The eigenfunctions of the shift operator Q in the N-particle sector have theform

QN =N=1

p0() (2.14)

Thus, one calls sometimes p0 by a momentum density. For the parameters the wordrapidities us used. Unfortunately, the word quasi-momentum distribution is used for as well, so one should be careful. Momentum density p0 and scattering phase are expressedvia rapidities as follows:

p0() = i lncosh( i)cosh( + i)

, p0(0) = 0, (2.15)

and

(1 2) = i ln sinh(2i + 1 2)sinh(2i 1 + 2) , (0) = 0. (2.16)

Note that p0() and () are antisymmetric, and

() 4 as (2.17)

Periodic boundary conditions imposed onto the eigenfunctions of the Hamiltonian (2.5)imply a set of nonlinear equations onto rapidities. These equations are called Bethe equa-tions: [

cosh( i)cosh( + i)

]M= (1)N1

N=

sinh(2i + )sinh(2i + ) . (2.18)

They can be written in the logarithmic form

M p0() +

N=1

( ) = 2n, = 1, . . . , N . (2.19)

Here n are integers (half-integers) when N is odd (even). We stress that these equationsdo not contain magnetic field. The eigenvalues of the Hamiltonian (2.5) are, of course,field-dependent:

EN =N=1

0() hM, (2.20)

where the one-particle bare dispersion 0 is the followin function of the quasi-momentum p0:

0() = 2 sin(2) dp0()d

+ 2h = 2sin2

(2)cosh( + i) cosh( i) + 2h

= 4sin2(2)

cosh(2) + cos(2)+ 2h. (2.21)

Heredp0()

d=

2 sin(2)

cosh(2) + cos(2). (2.22)

We see that the state N contains N particles with quasi-momenta p0() ( = 1, . . . , N )and energies 0(). These excitations are excitations over the ferromagnetic vacuum |0 andthey are usually called spin waves.

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2.1.2 Particular case: XXX model

The limiting case = 1 corresponds to the isotropic ferromagnetic. The ground state is theferromagnetic vacuum, so the magnetization takes maximal possible value. For > 1 theground state remains to be ferromagnetic. The point = 1 corresponds to the isotropicantiferromagnetic.

(I) Consider = 1 (antiferromagnetic). To take this limit we perform the rescalingprocedure

, 2

2

, 0. (2.23)The Bethe equations (2.18) become (recall that M is even)

+ i/2

i/2M

=N

=

+ i i , = 1, . . . , N . (2.24)

The bare dispersion (2.21) becomes

0() = 22 + 1

4

+ 2h. (2.25)

(II) Consider = 1 (ferromagnetic). To take this limit we perform the rescaling proce-dure

+ i 2

, 2

, 0. (2.26)The Bethe equations (2.18) become the same as for the antiferromagnetic case, while thebare dispersion is

0() = 22 + 1

4

+ 2h. (2.27)

2.2 Thermodynamics of the XXZ model (optional reading)

The thermodynamic limit of the Bethe equations is taken. The procedure is similar to theone carried out for the Lieb-Liniger model, section 1.

2.2.1 Lieb equation

To begin with, note some important properties of the ground state. It can be proven thatthe parameters n fill the symmetric interval around zero and the rapidities are real inthe ground state. It will be convenient to work with the density of particles in the rapidityspace (we will sometimes call it the distribution function of s):

() =1

M(+1 ) , (2.28)

that isN

=1

f() = MN

=1

(+1

)()f(), (2.29)

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where f() is an arbitrary function.

We consider now the limit of large number of sites M and large number of particles N :

D =N

M const, N, M . (2.30)

This limit will be called thermodynamic limit. One has in this limitN=1

(+1 )()f()

d ()f(). (2.31)

The integration limits are defined by the normalization condition

D =

d (). (2.32)

The Bethe equations (2.19) become in the thermodynamic limit

2()

d K(, )() =dp0()

d. (2.33)

where

K(, ) =( )

=

sin(4)

sinh( + 2i) sinh( 2i)

=2 sin(4)

cosh[2(

)]

cos(4)

. (2.34)

The magnetization is = (z)n = 1 2D. (2.35)

Having calculated () one gets all the thermodynamics of the model. The ground stateenergy density

=E

M(2.36)

can be found from Eq. (2.20):

= h +

d ()0() = h(1 2D)

d ()

4sin2(2)

cosh(2) + cos(2) . (2.37)

2.2.2 Interaction with magnetic field

We need to know how to calculate the ground state density at a given value of a chemicalpotential. We write

() 12

d K(, )() = 0(), (2.38)

with the condition

() = 0. (2.39)29


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0 0.25 0.5 0.75 1 1.25 1.5

2.5

2

1.5

1

0.5

0

0

Figure 6: Shown is the ground state energy density () as a function of anisotropy forh = 0.

When h = 0 one has = 0 and D = 1/2. The system of equations (2.32) and (2.33) canbe solved analytically and gives

() =

[2( 2)cosh

2]1

. (2.40)

The ground state at zero field is antiferromagnetic. The ground state has nontrivial structurein the region 0 h < hc, where

hc = 4 sin2

= 2(1 ). (2.41)Above the critical field hc the ground state is ferromagnetic. The parameter decays frominfinity to zero when h is changing from 0 to hc. At h = 0 one can calculate an energy persite, , from Eqs. (2.37) and (2.40). For several values of it is given in the table.

0 6

4

3

2

= cos 2 1 12

0 12

1

0 2

33

2

4

2

2.77

(2.42)

A result for an arbitrary value of is shown in Fig. 6

An important characteristics of the system is the magnetic susceptibility

=

h

D

. (2.43)

We obtain the curves of and at = 1/2 solving the system of Eqs. (2.32) and (2.33)numerically. The technique and the results are discussed in a separate section.

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2.2.3 Zero temperature thermodynamics at = 0

Bethe equations can be solved explicitly in this case. One has from Eq. (2.33)

() =1

1

cosh(2). (2.44)

The dressing equation for (), Eq. (2.38) becomes

() = 4cosh(2)

+ 2h (2.45)

and the condition (2.39) gives us an explicit relation between and h :

cosh(2) =2

h. (2.46)

Equation (2.32) gives us, in turn, the relation between and D :

tan

2D

= tanh(). (2.47)

Using these relations and the identities

tanh2() =cosh(2) 1cosh(2) + 1

=2 h2 + h

, tanh(2) =2tanh

1 + tanh2 =

1

2

4 h2 (2.48)

one gets the relation between D and h :

h = 2cos(D), 0 D 1

2 . (2.49)

For the energy per site, Eq. (2.37), one has

= h(1 2D) 4

tanh(2)

= h

1 4

arctan

2 h2 + h

4

1 h

2

4, 0 h 2

=

2cos(D)(1

2D)

4

sin(D), 0

D

1

2. (2.50)

For the magnetic susceptibility, Eq. (2.43), one has

=1

1 h

2

4

12

, 0 h 2. (2.51)

The magnetization is plotted in Fig. 7. The behavior of susceptibility is obvious from thisplot.

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-2 -1 1 2

-1

-0.5

0.5

1

Figure 7: Shown is the magnetization as a function of h for = 0 and T = 0.

2.2.4 Finite-temperature thermodynamics at = 0

We will follow the notations of Ref. [20] in this section. Namely,

H =M

j=1

(Sxj S

xj+1 + S

yj S

yj+1

) 2h Mj=1

Szj

=1

2

Mj=1

(+j

j+1 +

j

+j+1

) h Mj=1

(2+j j 1). (2.52)

Perform the Jordan-Wigner transformation

cm = ei

m1

j=1

+

j

j m, cm = +mei

m1

j=1

+

j

j . (2.53)

The operators c and c are Fermi operators

{cm, cn} = mn (2.54)Since cmcm =

+m

m, the inverse transform is

m = eim1

j=1cjcjcm,

+m = c

me

im1

j=1cjcj . (2.55)

The Hamiltonian takes the form (we have omitted c-number term)

H =1

2

Mj=1

(cjcj+1 + cj+1cj) 2h

Mj=1

cjcj. (2.56)

Take the Fourier transform

ck =1

M

Mj=1

eikjcj , cj =

k

eikjck, k =2

Mq, q = 1, 2, . . . , M . (2.57)

ThereforeH =

k

(cos k

2h)ckck. (2.58)

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We calculate now the partition function of the model. It seems that the closed expressionfor it does not exist on a finite lattice. However, in the thermodynamic limit the closedexpression can be obtained. Indeed,

Z= tr eH = tr e

k kckck =

k (1 + ek

). (2.59)

Introducing the free energy,F = T log Z (2.60)

and taking the limit of large system size,

k

M2

dk, (2.61)

one getsF

M= T

2 dk log (1 + ek) . (2.62)Finally, taking into account the definition of the Fermi distribution function,

fk =1

ek + 1, 1 fk = 1

1 + ek(2.63)

one represents F as followsF

M=

T

2

dk log (1 fk) . (2.64)

It is useful to calculate the entropy from this expression. Indeed,

SM

= 1M

FT

M

= 12

dk log(1 fk) T

2

dk

Tlog(1 fk). (2.65)

We get then the internal energy U of the fermion gas,

U

M=

F

M+ T

S

M= T

2

2

dk

Tlog(1 fk) = 1

2

dk kfk. (2.66)

The density of the particles, D = N/M, is

D =1

2dkfk. (2.67)

We consider the dispersion k of the form

k = cos k 2h, k . (2.68)

Note that the sign in front of cos k is immaterial, it depends on the range of k. At T = 0 onehas D = 1 when h = 1/2 and D = 0 when h = 1/2. So, h = 1/2 are the critical valuesof the magnetic field, and we define the magnetization as follows

= 2D 1. (2.69)

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0 0.5 1 1.5 2

T

0

0.25

0.5

0.75

1

1.25

1.5

T

Figure 8: Shown is the zero field susceptibility (T, h = 0) as a function of T for = 0.

For T = 0 it is shown in Fig. 7. Note that the sign of the above equation is different fromthat of Eq. (2.35).

Let us calculate the magnetic susceptibility of the system, Eq. (2.43), focusing on thecase of zero field. One can easily show that

(T, h = 0) =1

dk/2

cosh2(2

cos k), (T = 0, h = 0) =

4

. (2.70)

For arbitrary temperature this function is plotted in Fig. 8. Note non-monotonous behaviorof this function.

2.3 Algebraic Bethe Ansatz for the Heisenberg chain

The eigenfunctions (2.8) are given in the form of non-trivial combinations of the local spinoperators acting onto the vacuum (sometimes called pseudovacuum) state. At present,there exist an alternative formulation of the Bethe-ansatz. In this formulation the operatorsacting onto the vacuum state are not local spin operators: each of them acts non-trivially atall lattice sites. This is the so-called Algebraic Bethe Ansatz, while the constructions alongthe line of Bethes work are often referred to as Coordinate Bethe Ansatz. The ABA makesexplicit the fact that the model (2.5) is integrable, that is it have infinitely many mutually

commuting integrals of motion. All these integrals of motion can be calculated explicitlyusing ABA.

2.3.1 Generating functional for the integrals of motion

We will stick to the notations used in the book [5]. We consider a lattice with L sites andimpose periodic boundary conditions. The ABA solution of the model (2.5) starts withdefining the so-called L-operator

L(n

|) =

i cosh( i

zn)

n sin2

+n sin2 cosh( + i

zn) , (2.71)

34


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where are Pauli matrices, Eq. (2.1), and is a free (complex-valued) parameter, customarilycalled a spectral parameter. The entries ofL(n|) are quantum operators acting nontriviallyon the n-th lattice site. The monodromy matrix T() is defined as a product of the L-operators taken over all lattice sites:

T() A() B()

C() D()

= L(L|) L(2|)L(1|). (2.72)The entries of T() are quantum operators acting nontrivially on the whole lattice. Theyare polynomials of the local spin operators, but the complicated structure of these poly-nomials makes them hard to be used for calculations of any observable. In the contrast,local spin operator jm can be expressed through A, B, C, and D in a closed and relativelysimple form [11, 12, 13, 21]. This representation (called the solution of the quantum inversescattering problem), combined with numerics, was used quite successfully for the analysis ofthe dynamical properties (dynamical correlation functions) of the Heisenberg chain [22, 23].However, how to extract the analytic (and numerical) values of the dynamical exponents

near the threshold is an open problem within this approach; its discussion leave aside thescope of the present lecture.

The commutation relations between the entries of the monodromy matrix (2.72) arenontrivial. To get them we introduce an object which plays a crutial role in the ABAformalism. It is the so-called R-matrix

R(, ) =

f(, ) 0 0 00 g(, ) 1 00 1 g(, ) 00 0 0 f(, )

, (2.73)

wheref(, ) =

sinh( + 2i)sinh( ) , g(, ) = i

sin2

sinh( ) (2.74)

One can check that the R-matrix (2.73) and the L-operator (2.71) obey the so-called inter-twining relation

R(, ) (L(n|) L(n|)) = (L(n|) L(n|)) R(, ). (2.75)

We recall that the tensor product for 2 2 matrices a and b is defined by Eq. (2.4). Notethat

(a

b)(c

d) = (ac)

(bd). (2.76)

Equation (2.75) implies the intertwining relation for the monodromy matrix:

R(, ) (T() T()) = (T() T()) R(, ). (2.77)

The proof of Eq. (2.77) is by induction. Equation (2.75) is the basis of the induction. Byforming the product L(m|) L(2|)L(1|) and increasing m from 1 to L we get Eq. (2.77).

Since Eq. (2.77) is the relation between 4 4 matrices, it generates 16 relations betweentheir entries. In particular, the operators of the same sort commute:

[A(), A()] = [B(), B()] = [C(), C()] = [D(), D()] = 0. (2.78)

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However, the commutation relations between different operators are nontrivial:

[A(), D()] = g(, ) (C()B() C()B()) (2.79)[A(), D()] = g(, ) {B()C() B()C()} (2.80)[B(), C()] = g(, ) (D()A()

D()A()) (2.81)

[B(), C()] = g(, ) {A()D() A()D()} (2.82)A()B() = f(, )B()A() + g(, )B()A() (2.83)

B()A() = f(, )A()B() + g(, )A()B() (2.84)

A()C() = f(, )C()A() + g(, )C()A() (2.85)

C()A() = f(, )A()C() + g(, )A()C() (2.86)

B()D() = f(, )D()B() + g(, )D()B() (2.87)

D()B() = f(, )B()D() + g(, )B()D() (2.88)

C()D() = f(, )D()C() + g(, )D()C() (2.89)

D()C() = f(, )C()D() + g(, )C()D() (2.90)

Next important object of the Algebraic Bethe Ansatz is the transfer matrix (), whichis defined as the trace of the monodromy matrix

() = Tr T() = A() + D(). (2.91)

It follows from Eq. (2.77) and the fact that the trace of the tensor product of matrices is

equal to the product of their traces that

[(), ()] = 0. (2.92)

Equation (2.92) implies that is a generating function of the integrals of motion of theproblem: expanding () in one gets a set of commuting integrals of motion Im :

[Im, In] = 0, n, m = 1, 2, 3, . . . . (2.93)

This set can be chosen in many different ways since any analytic function of () can playthe role of the generating functional. In practice, one works with local integrals of motion.Locality means that I

mshould be written in the following form:

Im =M

j=1

Im(j), (2.94)

where the operators Im(j) act non-trivially in m neighboring lattice sites only. In themodel (2.6) a set of Im satisfying locality condition is generated as follows

2

Im+1 = idm

dmln ()

=0

, 0 =i

2 i, m = 1, 2, 3, . . . . (2.95)

2

Note the analogy with the generating functional of the irreducible diagrams in Quantum Field Theory.

36


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In particular, the Hamiltonian (2.6) can be written as

H = 2I2 sin2 + L cos2. (2.96)

To understand why 0 chosen as indicated in Eq. (2.95) implies the locality of Im, andto prove Eq. (2.96), let us analyze the

L-operator (2.71):

L(n|0) = i2

(1 + n)sin2. (2.97)

Given two lattice sites, i and j, the operator

ij =1

2(1 + i j) (2.98)

is the permutation operator:

ijij = 1, ij(ai 1j)ij = 1i aj , a = x,y,z. (2.99)

Here by 1j we denote the identity matrix acting at the site j. Therefore, the operator (2.97)can be written as

L(n|0) = i sin20n. (2.100)The subscript 0 indicate the auxiliary (matrix) space, in which the operators act, andthe subscript n indicate the quantum space, in which the operators n act in Eq. (2.97).Substituting the L-operator (2.100) into Eq. (2.72) we get for the transfer matrix (2.91)

(i sin2)L(0) = Tr0 (L0L10 2010). (2.101)The trace on the right hand side of this expression is taken over the auxiliary space. An

alternative representation for the right hand side of Eq. (2.101) is

U (i sin2)L(0) = LL1L1L2 3221. (2.102)Bearing in mind Eq. (2.99), we recognize in Eq. (2.102) the shift operator:

Uaj U1 = aj1, a = x, y, z, j = 1, 2, . . . , L , (2.103)

where 0 L. In the continuum models the momentum operator, P, generates translations:eiaP(x)eiaP = (x + a). In the case of the lattice model, Eq. (2.103) indicates that theoperator P defined by

U = eiP (2.104)

is a generator of the lattice translations, that is, Bloch quasi-momenta.

For the first derivative of () at = 0 the calculations similar to those used to getEq. (2.103) give

(i sin2)L+1 ()|=0 =L

m=1

LL1 m+1mLm1(m|0)m1m2 21. (2.105)

Here the notation Lm1(m|0) is used for Lm1(m|)|=0; the L-operator is given byEq. (2.71) and the subscript m1 indicates that the operator acts in the tensor product of

37


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the quantumspaces. Combining Eqs. (2.102) and (2.105) and taking into account Eq. (2.99)we arrive at Eq. (2.96). Higher Im can be obtained similarly. The locality of higher Im followsfrom the structure of the derivatives of the transfer matrix and can be seen rather easy. Notethat I1 is exceptionally non-local since it is proportional to the Bloch quasimomentum,

I1 = iL ln[sinh(2i)] P, (2.106)and the latter is the logarithm of the shift operator, acting in all lattice sites, Eqs. (2.102)and (2.104).

2.3.2 Eigenfunctions and spectrum of the integrals of motion

We are ready to describe the eigenfunctions of the transfer matrix (2.91). We start withintroducing the pseudovacuum

|0 =L

j=1| j, (2.107)

where | j is the spin-up state in the j-th lattice site. The down-left entry of the L-operatorL(n|) vanishes when acting onto the | n. This implies for the monodromy matrix (2.72)

C()|0 = 0, A()|0 = a()|0, D()|0 = d()|0, (2.108)where

a() = [i cosh( i)]L, d() = [i cosh( + i)]L. (2.109)One can check that the commutation relations between the z-component of the total spinand the entries of the monodromy matrix are

[Sz, B()] =

B(), [Sz, C()] = C(), [Sz, A()] = [Sz, D()] = 0. (2.110)

Thus, the operators B and C resemble the creation and annihilation operators of the har-monic oscillator, with L/2 Sz counting the number of excitations. A difference with theharmonic oscillator is that B and C commute non-canonically, Eq. (2.81). Nonetheless, wetake the states

|N({j}) =N

j=1

B(j)|0 (2.111)

as a candidate for the eigenfunctions of the transfer matrix. To move A() and D() pastB() one can use Eqs. (2.83) and (2.88), respectively. Such an operation generates un-wanted terms g(, )B()A() and g(, )B()D(). It can be shown (see, for example,

section VII.1 of the book [5]) that all unwanted terms can be canceled out by the properchoice of the parameters j in Eq. (2.111). These parameters (called rapidities) should satisfythe Bethe equations:

a(n)

d(n)

Nj=1j=n

f(n, j)

f(j , n)= 1, n = 1, . . . , N , (2.112)

where a and d are given by Eq. (2.109). They are often written in the logarithmic form:

Lp0(j) +N

k=1

(j

k) = 2 nj

N + 1

2 , j = 1, . . . , N , (2.113)

38


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where

p0() = i lncosh( i)cosh( + i)

, p0() = p0(), p0(+) = 2 (2.114)

and

() = i ln sinh( + 2i)sinh( + 2i) , () = (), (+) = 4. (2.115)

The parameters nj take arbitrary integer values. Equation (2.113) is the Bethe equa-tion (2.19) obtained by the Coordinate Bethe Ansatz.

We can now exploit the full power of the Algebraic Bethe Ansatz. For the transfermatrix (2.91) acting onto the state (2.111) with j satisfying Eq. (2.112) we get

()|N({j}) = (, {j})|N({j}), (2.116)

where(, {j}) = a()

Nj=1

f(, j) + d()N

j=1

f(j, ). (2.117)

This expression, being combined with Eq. (2.95) gives the spectrum of Im :

Im+1({j}) =N

j=1

vm(j) + Lm, m = 0, 1, . . . , L 1, (2.118)

where

vm(j) = mp0(j )

m

=0

, m = i m

mln[sinh( 2i)]

=0

, (2.119)

and p0 is defined by Eq. (2.114). In getting Eq. (2.118) we used thatm

md()

=0

= 0 form = 0, 1, . . . , L 1.

The spectrum of the Hamiltonian (2.6) follows from Eqs. (2.96) and (2.118). It followsfrom Eqs. (2.107), (2.110), and (2.111) that

Sz|N({j}) =

L

2 N

|N({j}). (2.120)

Therefore, the spectrum of the Hamiltonian (2.5) is

Eh =N

j=1

0(j) hL, (2.121)

where

0() = 2h 2sin2 2

cosh( + i)cosh( i) . (2.122)

We thus recovered results for the spectrum obtained by the Coordinate Bethe Ansatz.

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3 Discussion and open horizons

3.1 Normalization of wave functions

We know how to calculate norms of the wave functions written in the form (2.111) [7].

No solution available through the Coordinate Bethe Ansatz wave functions, Eqs. (1.47)and (2.12).

3.2 How to go from Coordinate to Algebraic BA: a solution of theInverse Scattering Problem

The state (2.111) resembles the state of the harmonic oscillator. The Hamiltonian of theXXZ chain, Eq. (2.6), written through the operators A, B, C, and D is a non-local functionof A and D operators taken at some point

H

ln(A + D). (3.1)

The problem of representation of the local fields through the operators A, B, C, and D, oftenreferred to as the Quantum Inverse Scattering Problem, is solved for the XXZ Heisenbergchain [11, 12, 13, 21]:

()j =

j1i=1

(A + D)(j)B(i)L

i=j+1

(A + D)(j). (3.2)

The discussion of the solution to Quantum Inverse Scattering Problem for other models lies

outside the scope of the present lecture.

3.3 Correlation functions

When the spectrum of a model is the same as of free fermions (which is the case for theTonks-Girardeau gas and for the XXZ chain at = 0) one can write the correlation functionsas determinants of infinite-dimensional matrices with relatively simple entries. Numerics andanalytical methods works quite well for such determinants, as one can see for the Tonks-Girardeau gas in section 1.5.

Away from the points of free-fermion spectra, the analytic expression for the correlationfunction are tremendously complex, if exist. The book [5] provides a very decent accountof approaches and results on the subject until year 1993. The solution of the QuantumInverse Scattering Problem boosted the efforts in the direction of getting closed-form analyticexpressions for the correlation functions. However, their analysis still requires extensivenumerics at some stage - so extensive that the required computational time is compatible withthe computational time required for purely numerical analysis. One can check Refs. [22, 23]for further details.

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3.4 Limitations of Algebraic Bethe Ansatz

The ABA scheme does not work if there is no pseudovacuum |0 above which creation andannihilation can be defined. This is the case for the totally anisotropic (XYZ) Heisenbergspin chain:

H = M

m=1

Jx

(x)m

(x)m+1 + Jy

(y)m

(y)m+1 + Jz

(z)m

(z)m+1

. (3.3)

This model is exactly solvable [24] but Functional Bethe Ansatz should be used.

References

[1] H. A. Bethe. Zur theorie der metalle. i. eigenwerte und eigenfunktionen der linearenatomkette. Z. Phys., 71:205, 1931.

[2] E. H. Lieb and W. Liniger. Exact analysis of an interacting bose gas. i. the generalsolution and the ground state. Phys. Rev., 130:1605, 1963.

[3] L. Tonks. The complete equation of state of one, two and three-dimensional gases ofhard elastic spheres. Phys. Rev., 50:955, 1936.

[4] M. Girardeau. Relationship between systems of impenetrable bosons and fermions inone dimension. J. Math. Phys., 1:516, 1960.

[5] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin. Quantum Inverse ScatteringMethod and Correlation Functions. Cambridge University Press, Cambridge, 1993.

[6] M. Gaudin. La fonction donde de Bethe. Masson, Paris, 1983.

[7] V. E. Korepin. Calculation of norms of bethe wave-functions. 86:391, 1982.

[8] T. Giamarchi. Quantum Physics in One Dimension. Oxford University Press, Oxford,2004.

[9] F. D. M. Haldane. Demonstration of the luttinger liquid character of bethe-ansatz-soluble models of 1-d quantum fluids. Phys. Lett. A, 81:153, 1981.

[10] E. H. Lieb. Phys. Rev., 130:1616, 1963.

[11] N. Kitanine, J.M. Maillet, and V. Terras. Form factors of the xxz heisenberg spin-1/2finite chain. Nucl. Phys. B, 554:647, 1999.

[12] F. Gohmann and V. E. Korepin. Solution of the quantum inverse problem. J. Phys. A,33:1199, 2000.

[13] N. Kitanine, J.M. Maillet, and V. Terras. Correlation functions of the xxz heisenbergspin-1/2 chain in a magnetic field. Nucl. Phys. B, 567:554, 2000.

[14] A. Lenard. Momentum distribution in the ground state of the one-dimensional systemof impenetrable bosons. J. Math. Phys., 5:930, 1964.

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[15] V. I. Smirnov. A course of higher mathematics. Pergamon, Oxford, 1964.

[16] K. B. Efetov and A. I. Larkin. Correlation functions in one-dimensional systems witha strong interaction. Sov. Phys. JETP, 42:390, 1976.

[17] H. G. Vaidya and C. A. Tracy. One-particle reduced density matrix of impenetrable

bosons in one dimension at zero temperature. Phys. Rev. Lett., 42:3, 1979.[18] M. Jimbo, T. Miwa, Y. Mori, and M. Sato. Density matrix of an impenetrable bose gas

and the fth painleve transcendent. Physica D, 1:80, 1980.

[19] M. Olshanii and V. Dunjko. Short-distance correlation properties of the lieb-linigersystem and momentum distributions of trapped one-dimensional atomic gases. Phys.Rev. Lett., 91:090401, 2003.

[20] E. Lieb, T. D. Schultz, and D. C. Mattis. Ann. Phys. (N. Y.), 16:407, 1961.

[21] J. M. Maillet and V. Terras. On the quantum inverse scattering problem. Nucl. Phys.

B, 575:627, 2000.

[22] J.-S. Caux and J. M. Maillet. Computation of dynamical correlation functions of heisen-berg chains in a magnetic field. Phys. Rev. Lett., 95:077201, 2005.

[23] J.-S. Caux, R. Hagemans, and J. M. Maillet. Computation of dynamical correlationfunctions of heisenberg chains: the gapless anisotropic regime. J. Stat. Mech., 2005,2005.

[24] R. J. Baxter. Ann. Phys. (N. Y.), 70:323, 1972.

notes on bethe ansatz

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