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Density fluctuations in the 1D Bose gas
Panfil, M.K.
Publication date
2013
Link to publication
Citation for published version (APA):
Panfil, M. K. (2013). Density fluctuations in the 1D Bose gas.
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The Algebraic and the
Thermodynamic Bethe Ansatz
In this chapter we will study in depth the 1D Bose gas and as the title suggests we focus on two parallel descriptions: the algebraic and the thermodynamic. The algebraic Bethe Ansatz allows for a description of the Lieb-Liniger Hamiltonian in the language of operators and thus not referring directly to the coordinate representation of wave functions. This description, however more abstract, led to many new and striking results that are of a great practical value. The main aim of the first part of the chapter is to obtain an expression for the form factor of the density operator, that is
|h{µ}N
j=1|ˆρ(0)|{λ}Nj=1i|2
k{λ}N
j=1kk{µ}Nj=1k
, (3.1)
where|{λ}Nj=1i, |{µ}Nj=1i are eigenvectors of the Lieb-Liniger Hamiltonian parametrized by a set of rapidities1, and
k{λ}Nj=1k =
q h{λ}N
j=1|{λ}Nj=1i (3.2)
is the norm of the eigenfunction|{λ}Nj=1i.
The approach presented here follows the work of the Leningrad school and was developed in the late 70’s and throughout the 80’s. The summary of this approach is presented in [18], we also refer there for references to original papers.
In the second part of the chapter we will develop a description of the Lieb-Liniger model that is practical when the number of particlesN → ∞ while keeping the density n = N/L constant. This is usually called a thermodynamic limit. Considering the thermodynamic
1
Wavefunctions presented in Chapter2we would write as Ψ({xj}Nj=1; {λj}j=1N ) = h{xj}Nj=1|{λj}Nj=1i
limit is important for a number of reasons. First of all by sending L→ ∞ we diminish the role of the periodic boundary conditions and therefore we focus on the bulk proper-ties of the gas. In experimental realizations one usually does not have periodic boundary conditions and therefore the thermodynamic limit gives us a connection between the ex-periment and the theory. Moreover the thermodynamic limit (however in some situations quite cumbersome, see Chapter 4) in general leads to a more tractable expressions and simplifies the studies. Finally, in the thermodynamic limit we can describe the thermal equilibrium in a compelling way.
But let us start with the algebraic method.
3.1
The Algebraic Bethe Ansatz
The Lieb-Liniger model possesses a large number of integrals of motion. ForN particles one can in principle construct N linearly independent operators { ˆQj}Nj=1 that are in
involution with the Hamiltonian, [H, ˆQj] = 0 for all j = 1, . . . , N .
The proof of the existence of the set of conserved charges relies on the observation that the set of rapidities {λ}Nj=1 is invariant under the time evolution up to a permutation of its elements. Therefore the value of any symmetric function of {λ}Nj=1 is a conserved quantity. We can then construct N linearly independent symmetric functions of N variables. These functions can be thought of as eigenvalues of operators{ ˆQj}Nj=1 acting
on the eigenstate given by a set of rapidities {λ}Nj=1. Thus naturally these operators would commute. This shows that a set { ˆQj}Nj=1 of operators being in involution must
exist.
Actually an explicit construction of conserved charges from the Lieb-Liniger Hamiltonian is quite cumbersome. Expressions for only a first few conserved charges in terms of the physical creation/annihilation operators are known [48]. Instead we should consider a generating function of conserved charges, a transfer matrix. As we should see the ABA provides a natural representation for the transfer matrix.
To this end let us define the transfer matrix in the following way. Introducing an arbitrary complex numberλ we can write2
τ (λ) =
∞
X
i=0
λiQˆi. (3.3)
2There are different ways of defining the generating function. These different definitions lead to
different relations between the operators ˆQ and the transfer matrix τ (λ). Since, anyway, we are not interested in an explicit formula for ˆQ’s we do not care about these relations and choose the simplest definition.
Now, from the very definition of τ , its values commute, that is
[τ (λ), τ (µ)] = 0, (3.4)
From the definition it also follows that τ (λ) has common eigenvectors with the Lieb-Liniger Hamiltonian. Therefore we set our attention on the eigenvalue problem of op-erator τ (λ) for any value of λ. The whole idea behind the ABA is to find a way to generateτ (λ). Following this spirit we can ask the following question: Is there a differ-ent way of definingτ (λ) (different from eq. (3.3)) such that eq. (3.4) would be naturally fulfilled? The answer is yes, we can defineτ (λ) to be a trace of an n-dimensional matrix of operators
τ (λ) = Tr T (λ), T (λ) is a n× n matrix. (3.5) The elements of the matrix T (λ) are again operators acting in the same Hilbert space as the Lieb-Liniger Hamiltonian. Substituting eq. (3.5) into the commutator (3.4) we obtain
0 = [τ (λ), τ (µ)] = [Tr T (λ), Tr T (µ)] = Tr T (λ) Tr T (µ)− Tr T (µ) Tr T (λ) (3.6) = Tr (T (λ)⊗ T (µ)) − Tr (T (µ) ⊗ T (λ)) , (3.7) where we used that product of traces is a trace of tensor product. We see also that in order to secure the commutation relation (3.4) the tensor products T (λ)⊗ T (µ) and T (µ)⊗ T (λ) need to be connected through a similarity transformation. Therefore let us assume that there is a(2n)× (2n) matrix of complex numbers such that
R(µ, λ)T (µ)⊗ T (λ)R−1(µ, λ) = T (λ)⊗ T (µ). (3.8) If the matrix elements of T (λ) were just complex numbers not operators than R(µ, λ) would simply be a permutation matrix Π. Thus any deviation from the permutation matrix in the R matrix signals that the matrix elements of T (λ) are non-commuting operators.
Following from the definition (3.8) theR-matrix is an intertwining operator, acting on the tensor product of transfer matrices it interchanges theirs arguments. Can any matrix do this? We have seen that deformations of the permutation matrix are good candidates but can we draw more concrete conclusions? Consider a triple product of transfer matrices.
By intertwining the 3 matrices in two equivalent ways we arrive at the following equality T (λ)⊗ T (µ) ⊗ T (ν) = R−1(λ, µ)⊗ I I⊗ R−1(λ, ν) R−1(µ, ν)⊗ I × (T (ν) ⊗ T (µ) ⊗ T (λ)) × (R(µ, ν) ⊗ I) (I ⊗ R(λ, ν)) (R(λ, µ) ⊗ I) = I⊗ R−1(µ, λ) R−1(λ, ν)⊗ I I⊗ R−1(λ, µ) × (T (ν) ⊗ T (µ) ⊗ T (λ)) × (I ⊗ R(λ, µ)) (R(λ, ν) ⊗ I) (I ⊗ R(µ, λ)) . (3.9) Thus theR-matrix fulfills the Yang-Baxter equation (c.f 2.1.1)
(R(µ, ν)⊗ I) (I ⊗ R(λ, ν)) (R(λ, µ) ⊗ I) =
= (I⊗ R(λ, µ)) (R(λ, ν) ⊗ I) (I ⊗ R(µ, λ)) , (3.10) with a freedom of multiplying the R-matrix by an arbitrary function of µ and ν. Therefore we have managed to describe the system possessing infinite number of integrals of motion in terms of the n× n dimensional matrix T (λ) of operators and a 2n × 2n dimensional matrixR(λ) of complex number that defines the intertwining operation (3.8). All we have to do now is it to invert the logic. Let us assume that we have a proper R-matrix, fulfilling the Yang-Baxter equation (3.10). For the Lieb-Liniger model the rightR-matrix is the following
R(λ− µ) = f (λ− µ) 0 0 0 0 g(µ− λ) 1 0 0 1 g(µ− λ) 0 0 0 0 f (λ− µ) , (3.11) with g(λ− µ) = ic λ− µ, (3.12a) f (λ− µ) = 1 + ic λ− µ = λ− µ + ic λ− µ . (3.12b)
Equivalently we can writeR(λ− µ) = ˆΠ + g(λ, µ) ˆI with ˆΠ the permutation matrix and ˆ
Having the properR-matrix we can write out the matrix elements of the transfer matrix
T (λ) = A(λ) B(λ) C(λ) D(λ)
!
, (3.13)
and use the intertwining relation (3.8) to establish commutation relations between op-erators A(λ), B(λ), C(λ) and D(λ). It is merely a technical task that requires a bit of patience and careful comparison of left and right hand sides of eq. (3.8). For this reason we give here only the final result for a few of the relations.
A(µ)B(λ) = f (µ, λ)B(λ)A(µ) + g(λ, µ)B(µ)A(λ), (3.14a) D(µ)B(λ) = f (λ, µ)B(λ)D(µ) + g(µ, λ)B(µ)D(λ), (3.14b) [C(λ), B(µ)] = g(λ, µ) (A(λ)D(µ)− A(µ)D(λ)) . (3.14c) Additionally, all the operators commute with themselves, for example [A(µ), A(λ)] = 0. The complete list of the commutation relations can be found in [18].
Finally to complete the description recall the vacuum state |0i that we have introduced in the beginning of Chapter 2. Since the vacuum is an eigenstate of the Lieb-Liniger Hamiltonian (with a zero eigenvalue) it also must be an eigenstate ofτ (λ) = A(λ)+D(λ). What are the vacuum eigenvalues? For the Lieb-Liniger Hamiltonian they should be the following
a(λ) = e−iλL, d(λ) = eiλL. (3.15)
This completes the construction of the algebraic description of the Lieb-Liniger model. We are now standing in front of an empty canvas with a new palette of colors at hand. What we can draw with it shall be seen in the next sections of this chapter. Before that let us summarize what have been done so far.
Starting from the existence of the infinite set of conserved charges we have shown that it implies the existence of an R-matrix, that defines commutation relations between matrix elements of a matrix T (λ). In turn, the trace of T (λ) is a generating function of the conserved charges. The choice of R-matrix (3.11) and the vacuum eigenvalues (3.15) can only be justified by showing that the operatorτ (λ) indeed contains the Lieb-Liniger Hamiltonian. This is usually done by first constructing a lattice version of the model. Based on it we could explicitly construct the matrix T (λ) and then derive the R-matrix as well as obtain the vacuum eigenvalues. Doing this would certainly enrich our understanding of the Algebraic Bethe Ansatz and allowed for a consistent description. However we are not going to do this. Instead let us be more pragmatic and assume that
the R-matrix (3.11) and the vacuum eigenvalues (3.15) are indeed proper ones for the Lieb-Liniger model.
3.1.1 Construction of Eigenstates
Let us get back to the empty canvas and see how we can construct eigenstates of the Lieb-Liniger Hamiltonian. We are going to use the algebra of the operators building the transfer matrix. Eigenstates of the Hamiltonian are simultaneously eigenstates of A(λ) + D(λ). Thus to actually construct the eigenstates we can use one of the other two operators, B(λ) or C(λ). The usual choice is to treat B(λ) as an creation operator and C(λ) as an annihilation operator. Therefore we write
C(λ)|0i = 0, ∀ λ ∈ C, (3.16)
and we can use the B(λ) operator to create particles, thus we want the following state
|{λ}Nj=1i = N
Y
j=1
B(λj)|0i, (3.17)
to be an eigenstate of τ (λ). We note first that the product on the r.h.s is well-defined because the B operators commute with each other. Consider now an action of A(µ) on |{λ}Nj=1i. Using commutation relation (3.14a) betweenA(µ) and B(λ) we can commute
theA(µ) through all the B’s operators. The result is of the following structure
A(µ)|{λ}Nj=1i = a(µ) N
Y
j=1
f (µ, λj)|{λ}Nj=1i + additional terms, (3.18)
where the first term comes from the first part of the commutation relation (3.14a). The additional terms are picked up each time we commute these two operators. Using that theB’s commute with each other we can easily capture the general structure
A(µ)|{λ}Nj=1i = a(µ) N Y j=1 f (µ, λj)|{λ}Nj=1i + N X i=1 ΛiB(µ) N Y j=1, j6=i B(λj)|0i. (3.19)
The coefficient Λi is also easy to capture. As it has the same form for each i we can
consideri = 1. Then Λ1 comes from picking the functiong(λ1, µ) once and N− 1 times
of A(λ1) on the vacuum. Generalizing to arbitrary i we obtain Λi= a(λi)g(λi, µ) N Y j=1, j6=i f (λi, λj). (3.20)
Λi is a function of µ and is different from zero. Thus |{λ}Nj=1i is not an eigenstate of
A(µ). But it still can be an eigenstate of A(µ) + D(µ). Therefore consider an action of the operator D(µ). Comparing the commutation relation (3.14a) between A and B with the commutation relation (3.14b) between D and B we immediately arrive at the following equation D(µ)|{λ}Nj=1i = d(µ) N Y j=1 f (λj, µ)|{λ}Nj=1i + N X i=1 ˜ ΛiB(µ) N Y j=1, j6=i B(λj)|0i, (3.21) with ˜ Λi= d(λi)g(µ, λi) N Y j=1, j6=i f (λj, λi). (3.22)
Thus we conclude that
τ (µ)|{λ}Nj=1i = a(µ) N Y j=1 f (µ, λj) + d(µ) N Y j=1 f (λj, µ) |{λ}N j=1i, (3.23)
if and only if the following set of equations is fulfilled
a(λi)g(λi, µ) N Y j=1, j6=i f (λi, λj) =−d(λi)g(µ, λi) N Y j=1, j6=i f (λj, λi), ∀i = 1, . . . , N. (3.24)
Using now eq. (3.12) and (3.15) we obtain the explicit expression for the Bethe equations
exp (iλjL) =− N Y k=1 λj − λk+ ic λj − λk− ic . (3.25)
The equivalence between equations (3.25) and (2.19) introduced in the previous chapter (see2.1.1) follows from the identity2i arctan x = ln(1 + ix)− ln(1 − ix) and can be easily established. Thus the logarithmic form of (3.25) reads
λj = 2π LIj− 1 L N X k=1 θ (λj− λk) , j = 1, . . . , N, (3.26)
In this way we have constructed the eigenstates of the Lieb-Liniger Hamiltonian. This construction parallels the results of the Coordinate Bethe Ansatz. We will now spend some time studying mathematical properties of the Bethe equations. Their mathematical structure depends on the sign ofc (we remind that for c > 0 the interaction is repulsive while for c < 0 it is attractive). Therefore we will consider both cases separately. For the repulsive interaction we will show the following:
• For each set of quantum numbers {Ij} there is an unique solution to the Bethe
equations.
• All solutions are real.
• There is no solution with any two rapidities coinciding. Equivalently the rapidities and thus the quantum numbers obey the Pauli principle.
In the case of attractive interaction the situation is more involved. First of all we will demonstrate that there is a special class of solutions called strings solutions. For the string solutions we will show similar properties as for the repulsive case. Moreover we should assume that in fact string solutions are all solutions to the Bethe equations with c < 0. This, not yet proven, theorem is usually called the string hypothesis. In what follows we will assume that it holds.
3.1.2 Repulsive Interactions
Let us start with the repulsive case. The Bethe equations (3.26) can be obtained as an extremum condition of the following action (Yang-Yang action [49])
S = 1 2L N X j=1 λ2 j − 2π N X j=1 λjIj+ 1 2 N X j,k Z λj−λk 0 θ(µ)dµ. (3.27)
By considering ∂S/∂λj we easily arrive at (3.26). This simple result is actually of great
value because it relates the existence of a solution of the Bethe equations to the existence of an extremum of the Yang-Yang action. The later can be easily investigated by con-sidering a matrix of second derivatives, ∂2S/∂λ
j∂λk. If this matrix has all eigenvalues
smaller/bigger than zero the action has an extremum and there is a solution to the Bethe equations. To this end consider the following expression
N X j,k vj ∂2S ∂λjλk vk= L N X j=1 v2j + N X j>k K(λj, λk) (vj− vl)2, (3.28)
where we defined a new function
K(λ) = ∂φ(λ)
∂λ =
2c
λ2+ c2. (3.29)
Thus we immediately obtain that for c > 0 the matrix of second derivatives is positive definite and there is always a unique solution to the Bethe equations.
Now let us prove that the solutions are real. This can be simply done using the product form (3.25) of the Bethe equations. Let λmax and λmin be rapidities with respectively
the largest and the smallest imaginary parts, we then have to two inequalities
|exp(iλmaxL)| = N Y k=1 λmax− λk+ ic λmax− λk− ic ≥ 1 → Im λmax≤ 0, (3.30) |exp(iλminL)| = N Y k=1 λmin− λk+ ic λmin− λk− ic ≤ 1 → Im λmin≥ 0. (3.31) The only way that these two inequalities can be fulfilled is if Imλj = 0, j = 1, . . . , N .
Thus all the solutions to the Bethe equations forc > 0 are real.
Finally we will prove the Pauli principle. We will show that there is no solution with λj = λk for j 6= k. Since the Bethe equations are symmetric functions of rapidites
without losing generality we can set j = 1 and k = 2. Let us start with the expression for the action ofτ (µ) on a general state (3.23)
ˆ τ (µ)|{λ}Nj=1i = τ(µ)|{λ}Nj=1i + N X i=1 Λi+ ˜Λi B(µ) N Y j=1, j6=i B(λj)|0i, (3.32) with Λi= a(λi)g(λi, µ) N Y j=1, j6=i f (λi, λj), (3.33) ˜ Λi= d(λi)g(µ, λi) N Y j=1, j6=i f (λj, λi). (3.34)
Bethe equations follow from a requirement thatΛi+ ˜Λi = 0 for i = 1, . . . , N . Let us take
carefully the limit λ1 → λ2. For i = 3, . . . N . the limit can be easily taken and yields
equations similar to the usual Bethe equations (3.25)
exp (iλjL) =− λj− λ1+ ic λj− λ1− ic 2 NY k=3 λj− λk+ ic λj− λk− ic , j = 3, . . . , N, (3.35)
The cases i = 1, 2 has to be treated with care since f (λ, µ) = 1 + ic/(λ− µ) diverges when arguments coincide. Taking the limit we obtain
0 = lim λ1→λ2 h Λ1+ ˜Λ1 B(λ2) + Λ2+ ˜Λ2 B(λ1) i = 2g(λ2, µ) a(λ2) N Y j=3 f (λ2, λj)− d(λ2) N Y j=3 f (λj, λ2) B(λ2) − ic ∂ ∂λ2 g(λ2, µ) a(λ2) N Y j=3 f (λ2, λj) + d(λ2) N Y j=3 f (λj, λ2) B(λ2) − icg(λ2, µ) a(λ2) N Y j=3 f (λ2, λj) + d(λ2) N Y j=3 f (λj, λ2) B0 (λ2). (3.36)
This equality can be split in two parts, one dependent on B(λ2) and the other one
dependent on B0(λ2). For the equality to hold these two parts must vanish separately:
we obtain two equations
0 = a(λ2) N Y j=3 f (λ2, λj) + d(λ2) N Y j=3 f (λj, λ2), (3.37) and 0 = 2 a(λ2) N Y j=3 f (λ2, λj)− d(λ2) N Y j=3 f (λj, λ2) − ic ∂ ∂λ2 a(λ2) N Y j=3 f (λ2, λj) + d(λ2) N Y j=3 f (λj, λ2) . (3.38)
Plugging in all the functions and simplifying for the first equation we obtain
eiλ2L=− N Y j=3 λj− λ2+ ic λj− λ2− ic . (3.39)
These equation combined with equations (3.35) for j = 3, . . . has only real solutions. This follows from the same argument as before. The second equation readily reduces to logarithmic derivative 0 = 4 c − i ∂ ∂λ2 log a(λ2) N Y j=3 f (λ2, λj)− d(λ2) N Y j=3 f (λj, λ2) . (3.40)
Now it is enough to show that this equation does not have solution for real rapidities. To this end we perform the derivative and obtain
0 = 4 c + L + N X j=3 2c c2+ (λ j− λ2)2 . (3.41)
Forc > 0 this equation cannot be solved for real rapidities. This finish the proof of the Pauli principle for the repulsive gas.
3.1.3 Attractive Interactions
For c < 0 the situation is different. In order to simplify the notation let us define ¯
c =−c > 0. In this subsection we follow [50]. Denoting again byλmaxandλminrapidities
with the largest and the smallest imaginary parts we have the following inequalities due to the Bethe equations (3.25)
|exp(iλmaxL)| = N Y k=1 λmax− λk− i¯c λmax− λk+ i¯c ≤ 1 → Im λmax≥ 0, (3.42) |exp(iλminL)| = N Y k=1 λmin− λk− i¯c λmin− λk+ i¯c ≥ 1 → Im λmin≤ 0. (3.43) Thus we see that the imaginary parts of rapidities are not necessary equal to zero. Let us assume that there is a complex rapidityλj = λ + iη. We have
η > 0 : N Y k=1 λj− λk− i¯c λj− λk+ i¯c = exp (iλL− ηL) → 0, (3.44) η < 0 : N Y k=1 λj− λk− i¯c λj− λk+ i¯c = exp (iλL− ηL) → ∞, (3.45)
In order for the left hand sides of the Bethe equations to scale to zero or∞ there must be a second rapidity with the imaginary part different by −i¯c and with exponentially vanishing correction that is ifλj = λ + iη then there exists l = 1, . . . , N and l6= j such
that λl = λ− iη − i¯c + O (exp(−ηL)). Thus we conclude that in the case of attractive
interactions there are complex solutions of the form of complex conjugated pairs. This allows for a general parametrisation of the rapidities. Whenever there are rapidities with the same real part and non-zero imaginary part they can be written in the following way
λj,aα = λjα+ i¯c
2(j + 1− 2a) + iδ
j,a
where λjα are real numbers and δ are the exponentially small corrections in the system
size L. We can interpret these solutions as strings in the complex plane. Index j sets a length of a string, a is an intra-string index and α indexes strings of the same length j. Thus states containing strings can be classified by a string content, that is a numberNj
of strings of lengthj that fulfills obvious relations (Ns is number of strings)
N = jXmax j=1 jNj, Ns= jXmax j=1 Nj. (3.47)
As we shall see strings are much like bound states, they represent new forms of particles. Hints of this can be seen from the Bethe equations. To this end we need to rewrite the Bethe equations (it is convenient to start with the product form (3.25)) in terms of the real part of the rapidities (the string-centersλjα’s). This can be easily done by first
separating the product into inter- and intra-string parts
exp iλj,aα L= Y
(k,β)6=(j,α) k Y b=1 λjα− λkβ+ ic2¯(j− k − 2(a − b + 1)) λjα− λkβ+ ic2¯(j− k − 2(a − b − 1)) Y b6=a a− b + 1 a− b − a, (3.48) where we have neglected the exponentially small corrections because they do not con-tribute to the leading order. Second step is to take product over all rapidities (a = 1, . . . , j) within the string (α, j), we obtain
exp ijλjαL= Y (k,β)6=(j,α) j Y a=1 k Y b=1 λjα− λkβ+ i¯c2(j− k − 2(a − b + 1)) λjα− λkβ+ i¯c2(j− k − 2(a − b − 1)) . (3.49)
The right-hand side can be brought to the form resembling the original Bethe equations by defining a new function
Ejk(λ) = e|j−k|(λ)e2|j−k|+2(λ) . . . e2j+k−2(λ)ej+k(λ), (3.50) ej(λ) = λ− ij¯2c λ + ij¯2c, (3.51) and then exp ijλjαL= Y (k,β)6=(j,α) Ejk λjα− λkβ, j = 1, . . . , jmax, α = 1, . . . , Nj (3.52)
Thus the Bethe equations forN particles reduce to Ns equations for string centers. The
in a logarithmic form jλjα = 2πI j α L + X (k,β) Φjk λjα− λkβ , (3.53) where Φjk(λ) = (1− δjk)φ|j−k|(λ) + 2φ|j−k|+2(λ) +· · · + 2φj+k−2(λ) + φj+k(λ), (3.54) φj(λ) = 2 arctan 2λ j¯c . (3.55)
From the very definition the string centers are all real numbers. The question of existence and uniqueness of the solution remains open. Moreover also the Pauli principle is difficult to prove. We should gather all these problems under the string hypothesis assumptions. Existence and uniqueness of the solution is rather a mathematical problem that we can overcome by just explicit construction of the solutions. This will be our strategy in Chapter6. The Pauli principle is more cumbersome. To see this consider the condition (3.41) that ultimately led to the Pauli principle in the repulsive gas. Considering state of the attractive gas with only 1-strings the condition reads
0 = L−4 ¯ c − N X j=3 2¯c ¯ c2+ (λ 2− λj)2 . (3.56)
The right hand side (c.f. eq. (3.86)) is equal to Lρt(λ2) with 1 ≥ ρt(λ) ≥ 0. Therefore
the Pauli principle holds as long as ρt(λ2) 6= 0. This is a sensible assumption because
ρt(λ2) = 0 means that there are no rapidities equal to λ2 (see section 3.2) and therefore
contradicts the assumption that λ1 = λ2. This is not a complete proof of the Pauli
principle but at least it hints that states not obeying the Pauli principle are not important in the the thermodynamic limit. Thus in what follows we assume that the Pauli principle holds also for the attractive gas.
3.1.4 Scalar Products and Form Factors
So far we have fulfilled the first step of our plan, we have described the Hilbert space of the Lieb-Liniger Hamiltonian. Now it is time to complete the second part: find the action of physical operators in this space. We are interested in expression like this: let us choose an operator, for example the density operator. We want to calculate its expectation value between any two Bethe states, and for physical applications what we
actually need is also a normalization of this expression FFρ {λi}Ni=1,{µi}Ni=1 = h{λi} N i=1|ˆρ(0)|{µi}Ni=1i q h{λi}Ni=1|{λi}Ni=1i q h{µi}Ni=1|{µi}Ni=1i . (3.57)
We should start however with an even more general problem, namely with the calculation of scalar products. Usually one defines the scalar product as a mapping from the vectors in the Hilbert space to complex numbers. It turns out such a general expression is not yet known. What is known however is a scalar product between a generic vector in the Hilbert space and one of the Bethe states. This formula was derived by N. Slavnov in [51] and it will serve our purposes as it is enough to compute both the numerator (once the action of the operator ρ(0) is known) and the denominator. Since the Hamiltonian commutesˆ with the particle number operator scalar products between states with different particle number are all identically zero.
The derivation of the expression for the scalar product relies on a recursive relation connecting the scalar product withN− 1 particles with the residue of the scalar product withN particles when two of the rapidities approach each other. The expression for the scalar product follows from solving the recursion relation. The existence of the residue in the scalar product simply follows from the commutation relations between operators B(λ) and C(µ) (3.14). The full derivation of the formula for the scalar product is rather lengthy and mostly technical and therefore we present here only the final answer [51]
* 0 N Y j=1 C(λj) d(λj) N Y j=1 B(µj) d(µj) 0 + = (ic)−N QN j,k=1(λj− µk+ ic) QN j>k=1(λj − λk) (µj− µk) det N Mjk, (3.58) Mjk = c2 µj− λk 1 (λk− µj + ic) + exp (−iLµl) (µj− λk+ ic) N Y m=1 µj− λm+ ic λm− µj+ ic ! . (3.59)
In this expression the set {λ}Nj=1 fulfills the Bethe equations (3.26). The set {µj}Nj=1 is an arbitrary set of complex numbers.
Considering limitµj → λj for allj = 1, . . . , N of the scalar product (3.58) we obtain the
following expression for the norm of the Bethe state [51]
h{λj}Nj=1|{λj}Nj=1i = (Lc)N N Y j6=k λj− λk+ ic λj− λk det N ,Gjk {λj} N j=1 , (3.60) Gjk {λj}Nj=1 = δjk " 1 + 1 L N X m=1 K(λj− λm) # − 1 LK(λj− λk), (3.61) with G known as the Gaudin matrix. The fact that the norm of the Bethe state is proportional to the matrix of the form (3.61) was conjectured by M. Gaudin shortly after
the original paper by E.H. Lieb and W. Liniger using the Coordinate Bethe Ansatz [52]. The complete proof was first obtained by V. Korepin [53].
In the case of attractive interactions and presence of strings in Bethe state, the expression for the norm simplifies and involves only the string centers [50]
h{λj}Nj=1|{λj}Nj=1i = ¯cNs Y j jNj Y (k,β)>(j,α) Fjk λjα− λkβ, (3.62) where Fjk(λ) = λ2+ c¯ 2(j + k) 2 λ2+ c¯ 2(j− k) 2. (3.63)
3.1.5 The Density Operator
Finally, the last piece that we are missing is the expression for the form factor of the density operator in the Bethe eigenstates basis. Again skipping all the technical details let us just sketch briefly how one obtains such an expression. To this end we divide the system in two parts, from 0 to x and from x to L. We can the define the transfer matrices TL,R(λ) in both parts with the same R-matrix and which product gives the
transfer matrix of the whole system (Fig. 3.1). The matrix elements between the left and right matrices commute. Similarly the vacuum state is given by the tensor product of the left and right vacuum. The eigenstates of the left(right) subsystem easily follows. This whole construction is useful because now we can write down the operator of a particle number in the left subsystem
ˆ NL=
Z x 0
dx Ψ†(x)Ψ(x). (3.64)
and we know its action on the left state
ˆ NL NL Y j=1 B(λj)|0iL= NL NL Y j=1 B(λj)|0iL. (3.65)
Therefore we can compute the expectation value of ˆNL between any two N -particle
Bethe eigenstates of the whole system by considering summation over all partitions of the N particles into left and right subsystems. In principle such a summation is not trivial, because the creation operator for the whole system is not just a sum of creation operators for the subsystems but rather
Figure 3.1: Depiction of the two site model.
what follows directly from the definitionT (λ) = TL(λ)TR(λ). Finally having the
expres-sion for the form factor of ˆNL the density operator form factor follows from taking the
derivative with respect tox3. Referring to [18,54] for the details we give here the final
result hµ|ρ(0)|λi = (Pµ− Pλ) N Y j=1 Vj+− Vj− YN j,k=1 λj− λk+ ic µj− λk detN(δjk+ Ujk) Vp+− Vp− , (3.67) with Vj± = N Y m=1 µm− λj± ic λm− λj± ic , (3.68) Ujk = µj− λj Vj+− Vj− K (λj− λk)− K (λj− λp). (3.69)
Here an integerp and a complex number λpare in fact arbitrary numbers not necessarily
belonging to the set{λj}Nj=1[54]. One usually takes advantage of this freedom and either
setsλp to be one of the rapidities and therefore reduces the size of the matrix4, or sends
λp to infinity. The later yields a representation of the form factor that we will use in
Chapter4.
With this we conclude the first part of this Chapter. We sketched here how the Algebraic Bethe Ansatz solves the problem of diagonalization of the Lieb-Liniger Hamiltonian and how within it the form factors of the density operator can be computed.
3.2
Thermodynamic Bethe Ansatz
In this section we study the thermodynamic limit of the Lieb-Liniger model by sending the number of particles to infinity while keeping the density of the gas constant. We focus
3
In fact one usually considers form factor of exp( ˆNL) but it does not change the logic.
4This is especially useful for numerical evaluation of the determinant as it slightly reduces the
com-plexity of the problem. For example, using the LU decomposition of the matrix the comcom-plexity of the determinant evaluation is O(N3).
here on the repulsive interactions. Many of the results of this section easily generalize to the attractive interactions. For the details we refer to Chapter6.
As number of rapidites increases it is not practical to handle all of them at the same time. Thus we have to find a more appropriate way of dealing with the Bethe states in the thermodynamic limit. This is our first task in this section. Once it is done we will be able to rewrite many of the expressions from the previous sections in the new way, but also we will be able to discuss new issues. The most important is a description of the gas at the finite temperature. For example we will be able to compute the partition function.
The thermodynamic of the Lieb-Liniger model was solved completely by Yang and Yang [49]. However first results in the thermodynamic limit, such as density of ground state rapidities appeared already in the work of Lieb and Liniger [21]. Exhaustive description of the thermodynamic limit is also contained in [18].
3.2.1 Density of Quantum Numbers
Let us start with the Bethe equations (3.26). Consider a set of quantum numbers I = {Ij}Nj=1 and corresponding to them rapidities {λj}Nj=1 solving the Bethe equations. The
setI is a subset of A where
A = (
{2m : m ∈ Z} : N is odd
{2m + 1/2 : m ∈ Z} : N is even (3.70)
and A is isomorphic with Z. Considering set I as a possible choice of quantum numbers from the A there is also a set of not used quantum numbers ˜I = A/I. Let us define the following density functions
ρ(x) = 1 L X a∈I δx− a L , ρh(x) = 1 L X a∈ ˜I δx− a L , (3.71) ρt(x) = ρ(x) + ρh(x) = 1 L X a∈A δx− a L . (3.72)
In the thermodynamic limit the total density functionρt(x) becomes constant and equal
to 1. The ρ(x) specifies completely the set of quantum numbers (it is equivalent to a characteristic function of set I). The notion of the entropy now naturally arises when we observe that slightly different sets of I yield in the thermodynamic limit the same ρ(x). Lρ(x)dx is equal to a number of particles with quantum numbers within a window (Lx, L(x + dx)) while Lρh(x)dx is a number of empty slots (holes) in the same interval.
configurations is equal to
# of configurations= [(Lρ(x)dx)!] [(Lρh(x)dx)!] [(Lρt(x)dx)!]
. (3.73)
The entropy dS(x) associated with the interval dx is equal to5
dS(x) = L (ρ(x) log ρ(x) + ρh(x) log ρh(x)) dx +O(1/L). (3.74)
The total entropy is then S = L
Z ∞ −∞
dx (ρ(x) log ρ(x) + ρh(x) log ρh(x)) . (3.75)
Using the density functionρ(x) we can also express physical quantities that are explicit functions of quantum numbers, the example being the number of particles (the number of quantum numbers) and the momentum (sum of the quantum numbers)
N = L Z ∞ −∞ dx ρ(x), (3.76a) P = L Z ∞ −∞ dx xρ(x). (3.76b)
In order to get further we need to find a way of describing rapidities in the thermodynamic limit.
3.2.2 Density of Rapidities
The idea is simple, in the same way like quantum numbers are related to the rapidities through the Bethe equations the density of quantum numbers can be related to the density of the rapidities. To this end let us first show that it makes sense at all to talk about the density of rapidities.
Let us consider a difference between the two consecutive quantum numbers (Ij+1> Ij)
2π L (Ij+1− Ij) = λj+1− λj+ 1 L N X k=1 (θ(λj+1− λk)− θ(λj− λk)) . (3.77)
The difference of sums can be bounded both from below and from above. First of all since Ij+1 > Ij then λj+1 > λj and since θ(λ) is a monotonously increasing function of
5
λ we obtain 1 L N X k=1 (θ(λj+1− λk)− θ(λj− λk))≥ 0. (3.78)
On the other hand θ(λ) is continuous and its derivative is bounded so it is Lipshits continuous
θ(λj+1− λk)− θ(λj− λk)≤
2
c(λj+1− λj) , (3.79)
By combining both bounds we obtain 2π L (Ij+1− Ij)≥ λj+1− λj ≥ 2π L Ij+1− Ij 1 + 2n/c. (3.80)
This shows that in the thermodynamic limit, as long as the difference between two consecutive quantum numbers stays finite (is not of orderN ), the difference between the consecutive rapidities goes to zero but rapidities do not accumulate at one point. Thus we can define a rapidities density function together with a mapping from the density of quantum numbers to the density of rapidities. This mapping is provided through the Bethe equations. Let us define by analogy to (3.71) the density as the following function
ρ(λ) = 1 L N X i=1 δ (λ− λi) , (3.81)
and let us define a functionx(λ) (x∈ R)
2πx(λ) = λ + 1 L N X i=1 θ (λ− λi) . (3.82)
From the definition it follows that x(λj) = Ij/L. Moreover from the properties of the
Dirac delta function we have the following relation
ρ(λ) = ρ(x) dx dλ x=x(λ) . (3.83)
In an analogous way we write the density of holes and the total density as functions of rapidity ρh(λ) = ρh(x) dx dλ x=x(λ) , (3.84) ρt(λ) = dx dλ x=x(λ) , (3.85)
where in the second line we used that ρt(x) = 1. The r.h.s. of the second equality can
be computed from the definition ofx(λ) (eq. (3.82))
ρt(λ) = 1 2π 1 + 1 L N X i=1 K(λ− λi) ! . (3.86)
Finally rewriting the summation as an integration with the help of ρ(λ) (eq. (3.81)) we obtain an integral equation for ρ(λ) and ρh(λ)
ρ(λ) + ρh(λ) = 1 2π+ 1 2π Z ∞ ∞ K(λ− µ)ρ(µ)dµ. (3.87)
Once we know the relation betweenρ(λ) and ρh(λ) this equation can be solved and yields
the distribution of rapidities.
At first it might be surprising that in order to specify the distribution of rapidities we also need to know the distribution of holes. In the case of the distribution of quantum numbers, knowledge of ρ(x) was sufficient to describe the thermodynamic limit of the Bethe state. Here, in the space of rapidities, it is not enough because the total density is not a constant function. As the mapping between the quantum numbers space and rapidity space is non-linear the original equidistant lattice on which quantum numbers live (the set A) gets distorted resulting in the λ dependent total density. Thus in or-der to completely specify the distribution of rapidities we need both ρ(λ) and ρt(λ) or
equivalentlyρ(λ) and a ratio of ρ(λ) to ρt(λ).
The ratio of ρ(λ) to ρt(λ) will occur quite often in the forthcoming considerations. Its
interpretation is simple, it is a filling function, it gives a ratio of a number of particles in a interval(x, x + dx) as compared with the maximal number of particles in the same interval. We will denote it byϑ(λ)
ϑ(λ) = ρ(λ) ρt(λ)
, 0≤ ϑ(λ) ≤ 1. (3.88)
For the number of the particles, momentum, energy and entropy we have simple expres-sions N [ρ] = L Z ∞ −∞ ρ(λ) dµ, (3.89a) P [ρ] = L Z ∞ −∞ λρ(λ) dµ, (3.89b) E[ρ] = L Z ∞ −∞ λ2ρ(λ) dλ, (3.89c) S[ρ, ρh] = L Z ∞ −∞
Despite the above quantities we are also able now to take the thermodynamic limit of more complicated expressions. Recall the formula for the norm of the Bethe state (3.60)
h{λj}Nj=1|{λj}Nj=1i = (Lc)N N Y j6=k λj− λk+ ic λj − λk det N G, (3.90)
with Gaudin matrix (eq. (3.61))
Gjk = δjk " 1 + 1 L N X m=1 K(λj− λm) # − 1 LK(λj− λk) ! . (3.91)
Upon comparing with eq. (3.86) the determinant of the Gaudin matrix can be written as det N G = (2π) N N Y j=1 ρt(λj) det N δjk− 1 2πLρt(λj) K(λj− λk) . (3.92)
The thermodynamic limit of the prefactor is not yet well-defined (the divergences ap-pearing in the thermodynamic limit should cancel once the norm is combined into a form factor) but the determinant already has a proper thermodynamic limit. To see this let us expand it in powers of the kernelK(λ, µ)
det N δjk− 1 2πLρt(λj) K(λj− λk) = 1− N X j=1 K(λj, λj) 2πLρt(λj) +· · · + (3.93) + N X j1,...,jN=1 N Y a=1 (2πLρt(λja)) −1 det N K(λj1, λj1) . . . K(λj1, λjN) . . . . K(λjN, λj1) . . . K(λjN, λjN) . (3.94)
Each term in the expansion has a proper thermodynamic limit, for example the first order term reads
N X j=1 K(λj, λj) 2πLρ(λj) = 1 2π Z ∞ −∞ K(µ, µ)ϑ(µ)dµ +O(1/L). (3.95)
All higher order terms have a proper thermodynamic limit as well. Thus the determinant of the Gaudin matrix becomes, in the thermodynamic limit, the Fredholm determinant
det N Gj,k Th. Lim. −−−−−→ det R2,ϑ 1− Kˆ 2π ! , (3.96)
where the Fredholm determinant is defined on R2 and the filling function ϑ(λ) plays a role of a weight function. ˆK is now an operator acting on R2. Recall that the Fredholm
determinant is well-defined only for a trace-class operator. That is we should check that Z ∞ −∞ K(µ, µ)ϑ(µ) dµ = 2 c Z ∞ −∞ ϑ(µ) dµ <∞, (3.97)
This condition, the integrability of ϑ(µ), is naturally fulfilled, the whole construction relies on it.
Altogether the norm of the Bethe state after a partial thermodynamic limit reads
h{λj}Nj=1|{λj}Nj=1i = (2πLc)N N Y j=1 ρ(λj) N Y j6=k λj − λk+ ic λj− λk det R2,ϑ 1− Kˆ 2π ! . (3.98)
3.2.3 Excitations and the Back-flow Function
So far we are able to describe states that follow continuous distribution of the rapidities. That is we have assumed that the difference between the neighboring quantum numbers is of order ∼ 1. This not always is the case. The simple counter-example is an excited state above the ground state. The way to accommodate for such states is described below.
Consider two Bethe states |λi = |{Ijλ}Nj=1i and |µi = |{Ijµ}Nj=1i which differ by n quan-tum numbers. The subset of quanquan-tum numbers present only in the first state we call {Ij−}nj=1and present only in the second state{Ij+}nj=1. The numbern specifies number of
excitations in the state|µi with respect to the state |λi and we assume that n/L vanishes in the thermodynamic limit. Thus both states differ only by a handful of excitations. We would like to find now a way to specify the state|µi in the thermodynamic limit as an excited state above the state|λi. To this end it is convenient to distinguish the subset of rapidities corresponding to the quantum numbers {Ij±}nj=1 by writing {µ+j }nj=1 and {λ−j}nj=1respectively. All the other rapidities in the excited state are shifted with respect
to the original rapidities. We expect this shift to be of order1/L and we will show this in a moment. To this end it is convenient to define an extra set of artificial rapidities {µ−j }nj=1 that correspond to {λ
−
j }nj=1 but are shifted just like all the other rapidities in
the excited state (see Fig. 3.2). Furthermore let us adopt the following notation. A sum accompanied with a single prime,P0, means we are leaving out terms corresponding to {µ+j }nj=1 in the sum. A double prime,
P00
k ω
a) b)
Figure 3.2: In an interacting theory creation of a particle-hole excitation influences all the particles. a) the ground state configuration. b) a single particle-hole excited states. Rapidities of all the particles are shifted with respect to the ground state values (dashed lines) proportionally to the back-flow functionF (λ|µ+, µ−) (3.102). Note that for the repulsive interactions a hole attracts the rapidities.
Consider the difference betweenλjandµj forµj not in the set{µ+j }nj=1. Using the Bethe
equations (3.26) we obtain L (λj− µj) =− N X k=1 θ(λj − λk)− θ(µj− µk) =− N X k=1 0 θ(λj − λk)− θ(µj− µk) + n X k=1 θ(µj− µ+k) =− N X k=1 00 θ(λj− λk)− θ(µj − µk) + n X k=1 θ(µj− µ+k)− θ(µj − µ−k).
The large summation can be simplified by expanding the second term aroundλj− λk N X k=1 00 θ(λ j− λk)− θ(µj− µk)= N X k=1 00K(λ j− λk) (λj− µj − λk+ µk) +O(1/L), which yields L (λj − µj) 1 + 1 L N X k=1 K (λj− λk) ! = N X k=1 00 K(λj− λk) (λk− µk) + n X k=1 θ(µj− µ+k)− θ(µj− µ−k). (3.99)
Using eq. (3.86) forρt(λ) and defining a back-flow function Fλ|{µ+ j }nj=1,{µ−j }nj=1 = Lρt(λ) (λj− µj) (3.100) we obtain 2πF (λ) = 1 L N X k=1 K(λ− λk) F (λk) ρt(λk)− n X k=1 θ(µj− µ+k)− θ(µj− µ−k). (3.101)
The first summation has a straightforward thermodynamic limit and therefore F (λ) fulfills the following integral equation
2πF (λ) = n X k=1 θ(λ− µ+k)− θ(λ − µ−k)+ Z ∞ −∞ K(λ− µ)F (µ)ϑ(µ) dµ. (3.102)
The function F (λ) describes a flow of rapidities caused by creating particle-hole exci-tations specified by {µ+j}nj=1 and {µ−j }nj=1 over a state described by the filling function ϑ(λ). The integral equation (3.102) is a linear equation and therefore
Fλ|{µ+ j }nj=1,{µ−j }nj=1 = n X i=1 Fλ|µ+ j, µ − j . (3.103)
In what follows we will consider only a single particle-hole excitation, generalization to multiple particle-hole excitations follows from the linearity of the back-flow function. The back-flow function allows us to express the momentum and the energy of an excited state. For the momentum we have
k = N X j=1 (µj − λj) = µ+− µ−+ N X j=1 00 (µj− λj) = µ+− µ−+ 1 L N X j=1 F (λj) ρt(λj) = µ+− µ−+ Z ∞ −∞ F (λ)ϑ(λ) dλ. (3.104)
Similarly for the energy we obtain
ω = N X j=1 µ2j − λ2 j = µ+2− µ−2+ N X j=1 00 µ2j− λ2 j = µ+2− µ−2+ 2 Z ∞ −∞ λF (λ)ϑ(λ) dλ +O(1/L). (3.105)
It is interesting to note that due to the properties of the back-flow function both mo-mentum and energy of the excited state can be rewritten in a form not involving the back-flow function. To this end we define the resolvent L(λ, µ) of the kernel K(λ, µ)
through the following relation Z ∞ −∞ (δ(λ− µ) + L(λ, µ)) δ(µ− ν) − 1 2πK(µ, ν)ϑ(µ) dµ = δ(λ− ν), (3.106) From eq. (3.106) two equivalent expressions arise
1 2π Z ∞ −∞ (δ(λ− µ) + L(λ, µ)) K(µ, ν)ϑ(µ) dµ = L(λ, ν), (3.107a) Z ∞ −∞ L(λ, µ) δ(µ− ν) − 1 2πK(µ, ν)ϑ(µ) dµ = K(λ, ν)ϑ(λ). (3.107b) Furthermore sinceK(λ, µ) is a symmetric function, the resolvent L(λ, µ) has the following property [49]
L(λ, µ)ϑ(λ) = L(µ, λ)ϑ(µ). (3.108)
We can use the resolvent to invert the integral equations involving the kernel K(λ, µ). For the back-flow function the expression is really simple
F (λ|µ+, µ−) =
− Z µ+
µ−
L(λ, ν)ϑ−1(ν) dν. (3.109)
Also the total density can be written as
ρt(λ) = 1 2π+ 1 2π Z ∞ −∞ L(λ, µ) dµ. (3.110)
For the momentum of the excited state we have then
k = µ+− µ−+ Z ∞
−∞
ρ(µ) θ(µ+− µ) − θ(µ−− µ)dµ. (3.111) The energy of the excited state can be written with the help of a function that naturally arises while studying a thermal equilibrium. Therefore we give an alternative expression for the energy only after introducing the thermal equilibrium.
3.2.4 Thermal Equilibrium
Let us write down the partition function for theN particles using quantum numbers to span the Hilbert space
ZN = 1 N ! X I1,...,IN exp −E {Ij}Nj=1 T . (3.112)
In the thermodynamic limit the sum over quantum numbers can be replaced by a func-tional integral in the space of density functions of rapidities and holes. Since we take the thermodynamic limit at fixed density of the particles, the functional integral should be restricted to the densities of rapidities yielding the same density n. Thus we write
Z = Z D (ρ(λ), ρh(λ)) δ L Z ∞ −∞ ρ(λ)dλ− Ln exp [S[ρ, ρh]− E[ρ]/T ] , (3.113)
with S[ρ, ρh] and E[ρ] given by (3.89d) and (3.89c) respectively. The entropy and the
energy are both extensive quantities, they scale linearly withL. Thus the integral can be evaluated using the saddle-point method. To this end we first rewrite the delta function in the integral representation
δ L Z ∞ −∞ ρ(λ) dλ− Ln = 1 2πi Z i∞ −i∞ exp hL T Z ∞ −∞ ρ(λ) dλ− n dh. (3.114) which yields for the partition function
Z = C Z D (ρ(λ), ρh(λ)) exp S[ρ, ρh]− E[ρ]/T + Lh T Z ∞ −∞ ρ(λ)dλ− n . (3.115) Variation of the exponent with respect toρ(λ) and ρh(λ) leads to the following equation
for the saddle-point configuration [49]
−h + λ2+ T log ρ(λ) ρh(λ) − T 2π Z ∞ −∞ K(λ− µ) log 1 + ρ(µ) ρh(µ) dµ = 0. (3.116) Let us define a new function, closely connected to the filling functionϑ(λ),
(λ) =−T log ρ(λ) ρh(λ) . (3.117)
The saddle-point condition translates into the following integral equation for (λ) [49]
(λ) = λ2− h − T
2π Z ∞
−∞
K(λ− µ) log (1 + exp (−(µ)/T )) dµ. (3.118)
Therefore the partition function in the saddle-point approximation equals
Z = exp(−F [ρ, ρh])× (1 + O(1/L)) , (3.119a)
F [ρ, ρh] = L Z ∞ −∞ ρ(λ) λ2− (λ)− T (ρ(λ) + ρh(λ)) log (1 + exp (−(λ)/T )) dλ, (3.119b) where the densities ρ(λ) and ρh(λ) fulfill the integral equations (3.87) and (3.118) with
strength c and chemical potential h (or particle density n). In thermal equilibrium we need to specify two of them, a value of the third one follows consistently.
We will show now that (λ) can be naturally understood as a dressed energy. First inverting relation (3.117) we find
ϑ(λ) = 1
1 + exp ((λ)/T ), (3.120)
and in the limit of c → ∞ or c → 0, ϑ(λ) becomes the Fermi-Dirac or Bose-Einstein distribution function respectively. The form of this equation suggests that indeed (λ) is an excitation energy. To provide a further support let us consider the energyω of the excited state (3.105). Simple manipulations of the r.h.s yield
ω = Z µ+ µ− dν Z ∞ −∞ 2λ (δ(ν− λ) + L(ν, λ)) dλ . (3.121)
At the same time0(λ) = d(λ)/dλ fulfills the following integral equation 0(λ) = 2λ + 1
2π Z ∞
−∞
K(λ− µ)ϑ(µ)0(µ) dµ. (3.122)
Resolving this equation shows that a term in square brackets in (3.121) is exactly equal to 0(ν) and we obtain
ω = (µ+)− (µ−). (3.123)
Therefore indeed(λ) specifies the energy of the excitations.
Let us mention a few properties of the dressed energy (λ). One easily notices that for small enough λ, (λ) < 0, while for λ → ∞, → ∞. Moreover 0(λ) > 0 for λ > 0
and thus(λ) is a continuous and monotonic function for positive arguments6. Therefore
there exists aq > 0 such that (q) = 0 and is positive/negative for λ ≷ q.
The Ground State
In the limit T = 0 we obtain that ϑ(λ) = 0 for |λ| > q and ϑ(λ) = 1 for |λ| < q. This yields thatρ(λ) = 0 for |λ| > q and ρh(λ) = 0 for |λ| < q. Therefore the density ρ(λ) is
given by a single integral equation (Lieb’s equation [21])
ρT =0(λ) = 1 2π+ 1 2π Z q −q K(λ− µ)ρT =0(µ) dµ. (3.124) 6
withq the Fermi rapidity7 determined through the condition
n = Z q
−q
ρ(λ) dλ. (3.125)
The dressed energy atT = 0 is given by the following linear integral equation
T =0(λ) = λ2− h +
Z q −q
K(λ− µ)T =0(µ) dµ, (3.126)
where the chemical potential is determined through the condition(q) = 0.
The value of the dressed energy around q can be connected with the sound velocity. To this end consider creating a hole just atµ− = q and creating a particle at µ+ > q. We
have vs = ∂ω(k) ∂k k=0 = ∂T =0(λ) ∂λ λ+→q ∂kT =0(λ) ∂λ λ+→q !−1 . (3.127)
Using eq. (3.111) it is easy to show that (this equality holds also at finite temperature) ∂k(λ) ∂λ λ+→q = 2πρ(q). (3.128)
At zero temperature we have that [18] ∂T =0(λ)
∂λ = 2λ + 2
Z q −q
µ ˙F (µ|λ) dµ, (3.129)
where ˙F (µ|λ) fulfills the following integral equation ˙ F (µ|λ) = 1 2πK(λ|µ) + 1 2π Z q −q K(λ, ν) ˙F (ν, µ) dν. (3.130)
On the other hand the derivative ofρT =0(λ) can also be expressed with a help of ˙F (µ|λ)
∂ρT =0(λ)
∂λ =−ρ(q)
˙
F (µ|q) − ˙F (µ|q). (3.131) Substituting the derivative of ρT =0(λ) (3.131) into (3.129) and using that ∂(q)/∂λ =
−∂(q)/∂λ we obtain ∂T =0(λ) ∂λ = 2q− 2 ρ(q) Z q −q µ∂ρT =0(µ) ∂µ dµ = n ρ(q), (3.132)
7The Fermi rapidity q equals Fermi momentum k
and therefore the sound velocity can be written as
vs= n
2πρ2 T =0(q)
, (3.133)
whereq is the zero of the dressed energy (λ).
Before we end let us extend the physical picture of excitations above the ground state. The low-energy excitations we have described already in Chapter 2 (see 2.1.2). Now empowered with the function(λ) we can broaden the description to include excitations of arbitrary energy. To this end it is convenient to introduce two archetypal processes [55]. One is excitation of the particle from one of the Fermi edges and we will refer to it as a particle-like excitation (or type I). Second is an excitation of a particle from within the Fermi line to one of the edges, this type will be referred to as hole-like excitation (or type II). It is easy to write energy of these excitations. Using the dressed energy (λ) and the equality (3.123) we obtain
ω1(λ) = (λ)− (q) = (λ), λ /∈ [−q, q], (3.134a)
ω2(λ) = (q)− (λ) = −(λ), λ ∈ [−q, q], (3.134b)
The equation (3.126) for (λ) can be easily solved in the c→ 0, ∞ limits. At c = 0 we obtain simply the spectrum of free particles (see Fig. 3.3)
ω1(λ) = λ2, ω2(λ) = 0, c→ 0. (3.135)
At an infinite coupling the spectrum is linear at small momentum but later splits into two lines encompassing the single particle-hole continuum (see Fig. 3.3)
ω1(λ) = λ2+ 2kFλ, ω2(λ) =−λ2+ 2kFλ, c→ ∞. (3.136)
This ends the tour through the Thermodynamic Bethe Ansatz that occupied second half of this chapter. We have developed tools that allow for an effective description of the gas with infinite number of particles through various functions fulfilling integral equations. We described also the thermal equilibrium of the gas and its zero temperature limit. In the next chapters we will use and extend the results presented here in various direc-tions. First we show how one can extract from the Algebraic Bethe Ansatz information that allows for an precise description of the zero temperature correlation functions (Chap-ter4). Secondly we will combine the Algebraic Bethe Ansatz, the Lehmann representa-tion and the Thermodynamic Bethe Ansatz to compute the finite temperature correlarepresenta-tion functions using the ABACUS method (Chapter 5). Finally in the last chapter we leave
0.0 0.5 1.0 1.5 2.0 2.5 k[kF] 0 1 2 3 4 5 ω [k 2 F] ω1(k) ω2(k) c =∞ c = 0
Figure 3.3: The dispersion relations (3.134) for the limiting values of the interaction strength. The light blue region represents the one particle-hole continuum. In the limitc→ 0 the continuum shrinks to the single line. As anticipated before (2.1.2), the sound velocityvs is maximal forc =∞ and monotonously decreases and reaches zero asc→ 0.
the equilibrium and describe correlations of the metastable super Tonks-Girardeau gas (Chapter 6).
