The propagator of the attractive delta-Bose gas in one dimension

0.23 MB·English

by  Sylvain Prolhac

The propagator of the attractive delta-Bose gas in
one dimension
2
1 Sylvain Prolhac and Herbert Spohn
‡ §
0
Zentrum Mathematik and Physik Department,
2
Technische Universit¨at Mu¨nchen,
n
D-85747 Garching, Germany
a
J

5 Abstract. We consider the quantum δ-Bose gas on the infinite line. For repulsive

interactions, Tracy and Widom have obtained an exact formula for the quantum
]
h propagator. In our contribution we explicitly perform its analytic continuation
p
to attractive interactions. We also study the connection to the expansion of the
-
h propagator in terms of the Bethe ansatz eigenfunctions. Thereby we provide an
t
a independent proof of their completeness.
m
[

3 PACS numbers: 02.30.Ik 05.30.Jp
v
4
0
4 Keywords: delta-Bose gas, quantum propagator, Bethe ansatz, completeness, analytic
3
. continuation.
9
0
1
1
:
v
i
X
r
a
[email protected]
‡
[email protected]
§
The propagator of the attractive delta-Bose gas in one dimension 2
1. Introduction
Quantum particles on the real line interacting through a δ-potential are governed by
the Hamiltonian
n ∂2 n
H = 2κ δ(x x ) . (1.1)
κ − ∂x2 − j − k
j=1 j j<k
X X
The number of particles, n, is fixed throughout and x = (x ,...,x ) denotes the
1 n
positionsoftheparticles. Wewillrestrictourselves tothebosonicsubspace ofsymmetric
wave functions. Eq. (1.1) is the Hamiltonian of a quantum many-body system which
can be analyzed through the Bethe ansatz. The repulsive interaction, κ < 0, has been
studied in great detail and we refer to [1, 2, 3, 4, 5, 6, 7]. The attractive case, κ > 0, has
received less attention. One reason is that the structure of the Bethe equations is more
complicated. On top, physical applications are not obviously in reach. In the recent
years, there has been renewed interest. We have now available a detailed study of the
eigenfunctions [8, 9, 10, 11] and, as argued by Calabrese and Caux [12], applications
to real materials are in sight. A further motivation comes from the one-dimensional
Kadar-Parisi-Zhang (KPZ) equation [13]. Its replica solution is given in terms of the
propagator of the attractive δ-Bose gas [14, 15] which can be used to obtain exact
solutions for some special initial conditions [16, 17, 18, 11, 19, 20, 21, 22, 23, 24, 25].
In the KPZ context, and also in other cases, one is actually interested in the
quantum propagator x e−tHκ y , t 0. In principle, e−tHκ can be expanded in a sum
h | | i ≥
(integral) over eigenfunctions. But one might hope to have at disposal more concise
expressions for the propagator. In the repulsive case, Tracy and Widom [26] carried
out such a program. The resulting expression we refer to as TW formula, which will
be discussed below, including its relation to the expansion in eigenfunctions. A natural
issue is to extend such a program to the attractive case, which is the topic of our
contribution.
By symmetry the propagator x e−tHκ y can be restricted to the domain Λ =
h | | i
x x ... x Rn. Using the Bose symmetry, H of (1.1) is then defined by
1 n κ
{ | ≤ ≤ } ⊂
n ∂2
H ψ(x ,...,x ) = ψ(x ,...,x ) , x Λ◦ , (1.2)
κ 1 n − ∂x2 1 n ∈
j=1 j
X
with the boundary conditions
∂ ∂
+κ ψ(x ,...,x ) = 0 , (1.3)
(cid:18)∂xj+1 − ∂xj (cid:19) 1 n |xj+1=xj
where the limit x = x is taken from the interior, Λ◦, of Λ. The Hamiltonian H is
j+1 j κ
a self-adjoint operator and x e−tHκ y is continuous in x,y Λ. In particular,
h | | i ∈
n
1
lim x e−tHκ y = δ(x y ) , x,y Λ . (1.4)
j j
t→0h | | i n! − ∈
j=1
Y
Throughout the paper x,y Λ, hence the position of the particles are ordered
∈
increasingly. As will be proved in Appendix A, x e−tHκ y is analytic in κ for otherwise
h | | i
The propagator of the attractive delta-Bose gas in one dimension 3
fixed arguments. Thus our strategy will be to analytically extend the TW formula, valid
for κ 0, to κ > 0. As written, the TW formula becomes singular at κ = 0. Therefore
≤
the main task is to understand the structure of the analytic continuation in κ. As a
result, we will arrive at various formulas for the propagator. One formula will be just
the expansion in Bethe ansatz eigenfunctions, which thus implies their completeness.
The issue of completeness for the attractive δ-Bose gas on the line has been
studied before. In his thesis, Stephen Oxford [27] proves completeness of the
generalized eigenfunctions defined as bounded Bethe ansatz eigenfunctions. He uses
functionalanalyticmethodstoconstruct theHilbertspaceisometryfromthegeneralized
eigenfunctions and thereby the spectral representation of H . A similar strategy is used
κ
by Babbitt and Thomas [28] for the ground state representation of the ferromagnetic
Heisenberg model on Z. Heckman and Opdam [29] exploit the fact that the δ-Bose gas
turns up in the representation theory of graded Hecke algebras. (We are grateful to
Bal´azs Pozsgay for pointing out this reference.) They have results for the case when the
interaction strength is allowed to be pair dependent. But only for H their expression
κ
simplifies and they arrive at a Plancherel formula, which is the completeness relation.
For the system on the line, studied here, the set of admissible wave numbers
is known explicitly. For a bounded system, in particular with periodic boundary
conditions, the discrete set of wave numbers are the solutions to the Bethe equations,
a coupled system of n transcendental equations. Completeness becomes more difficult
to establish and to our knowledge only for the repulsive case a completeness proof is
available [30].
The article is organized as follows. In Section 2, we recall the Tracy and Widom
formula for the propagator in the repulsive case, and rewrite it in terms of Bethe
eigenstates. In Section 3, we summarize our main results on the propagator with
attractive interactions. These results are proved in Section 4, by performing explicitly
the analytic continuation to κ > 0. A further rewriting represents the propagator in
terms of the known Bethe eigenstates. The special case of the propagator with all
particles starting and ending at 0 is handled in Section 5. In Appendix A, we prove
that the (imaginary time) propagator is an analytic function of the coupling.
2. δ-Bose gas with repulsive interaction (κ < 0)
Let S betheset of alln! permutations of theintegers between 1andn. Inthe following,
n
we use the notations
n
. (2.1)
≡
j<k 1≤j<k≤n
Y Y
For κ < 0 the TW formula states (in the notation of [26] the strength of the potential
is called c = κ)
−
1
x e−tHκ y = dq ... dq
h | | i n!(2π)n 1 n
Rn
Z
The propagator of the attractive delta-Bose gas in one dimension 4
n n
q q iκ
σ(j) − σ(k) − eiqσ(j)(xj−yσ(j))e−tqj2 . (2.2)
q q +iκ
σ(j) σ(k)
σX∈Sn jY<k − Yj=1(cid:16) (cid:17)
σ(j)>σ(k)
The connection to the eigenfunction expansion can be seen by symmetrizing over all
permutations of the q . We introduce a new permutation τ S and replace q by q .
j n j τ(j)
∈
Replacing then σ by τ−1 σ, one finds
◦
1
x e−tHκ y = dq ... dq
h | | i n!2(2π)n 1 n
Rn
σ,Xτ∈SnZ
n n
q q +iκ
j − k eiqj(xσ−1(j)−yτ−1(j))e−tqj2 . (2.3)
q q iκ
j k
τ−1(jj,)Yk<=τ1−1(k) − − Yj=1(cid:16) (cid:17)
σ−1(j)>σ−1(k)
One notes the factorization
n n n
q q +iκ q q +iκ q q iκ
j k j k j k
− = − − −
q q iκ q q iκ q q +iκ
j k j k j k
j,k=1 − − j<k − − j<k −
τ−1(j)Y<τ−1(k) τ−1(j)Y<τ−1(k) τ−1(j)Y>τ−1(k)
σ−1(j)>σ−1(k) σ−1(j)>σ−1(k) σ−1(j)<σ−1(k)
n n
q q +iκ q q iκ
j k j k
= − − − . (2.4)
q q iκ q q +iκ
j k j k
j<k − − j<k −
σ−1(j)Y>σ−1(k) τ−1(j)Y>τ−1(k)
We introduce the Bethe eigenstates of the Hamiltonian (1.1) with repulsive interaction
n n
1 q q +iκ
ψ(x;q) = j − k eiqjxσ−1(j) , (2.5)
n! q q iκ
j k
σX∈Snσ−1(jj)Y><σk−1(k) − − Yj=1
with momenta q = (q ,...,q ) Rn. The eigenstate ψ(x;q) has energy
1 n
∈
n
E(q) = q2 . (2.6)
j
j=1
X
In terms of Bethe eigenstates, Eq. (2.3), (2.4) rewrite as
1
x e−tHκ y = dq ... dq ψ(x;q)ψ(y;q) e−tE(q) , (2.7)
h | | i (2π)n 1 n
Rn
Z
where (...) denotes complex conjugation. At t = 0, (2.7) reduces to the completeness
relation for the Bethe eigenstates,
1
11 = dq ... dq ψ(q) ψ(q) . (2.8)
(2π)n 1 n| ih |
Rn
Z
A proof of the orthonormality of the Bethe eigenstates can be found e.g. in [11],
Appendix A.
The propagator of the attractive delta-Bose gas in one dimension 5
3. δ-Bose gas with attractive interaction (κ > 0)
As shown in Appendix A, the propagator is an analytic function of the coupling κ for
t > 0. By analytic continuation of (2.2) from κ < 0 to κ > 0, we will derive in Section
3 an exact expression for the propagator in the attractive case κ > 0.
Before stating the two main theorems, a few definitions are needed. We call
D the set of the M-tuples ~n = (n ,...,n ) such that n 1, j = 1,...,M and
n,M 1 M j
≥
n +...+n = n. For ~n D , the clusters Ω (~n) Ω , j = 1,...,M are defined by
1 M n,M j j
∈ ≡
Ω = n +...+n +1,...,n +...+n . (3.1)
j 1 j−1 1 j
{ }
From the Bethe ansatz point of view, the clusters will correspond to bound states of the
particles. The function r r, acting on 1,...,n , is defined by
~n
≡ { }
r(a) = s for a = n +...+n +s Ω . (3.2)
1 j−1 j
∈
More visually, one has
a 1 ... n n +1 ... n +n ... n n +1 ... n
1 1 1 2 M
−
Cluster Ω Ω ... Ω . (3.3)
1 2 M
r(a) 1 ... n 1 ... n ... 1 ... n
1 2 M
Finally, we call S′(~n) (respectively S′′(~n)) the subset of S containing only the
n n n
permutations σ (resp. τ) such that for all j = 1,...,M and a,b Ω with a < b
j
∈
one has σ−1(a) < σ−1(b) (resp. τ−1(a) > τ−1(b)).
In the attractive case, the following expression for the propagator is proved in
Section 4.
Theorem 1. For fixed ~n, let µ , j = 1,...,M, be arbitrary real numbers satisfying the
j
constraint
n < µ 0 . (3.4)
j j
− ≤
For κ > 0 and x,y Λ, one has
∈
n κn−M M
x e−tHκ y = (n !(n 1)!) dq ... dq
h | | i n!M!(2π)M j j − 1 M
RM
MX=1 ~n∈XDn,M Yj=1 Z
M
ei(qj+iκ(µj+r(a)−1))(xσ−1(a)−yτ−1(a))e−t(qj+iκ(µj+r(a)−1))2
σ∈XSn′(~n)τ∈XSn′′(~n)Yj=1aY∈Ωj(cid:16) (cid:17)
M
(q +iκ(µ +r(a))) (q +iκ(µ +r(b)))+iκ
j j k k
− . (3.5)
× (q +iκ(µ +r(a))) (q +iκ(µ +r(b))) iκ
j,Yk=1 aY∈Ωj (cid:18) j j − k k − (cid:19)
j6=kσ−1(ab)∈>Ωσk−1(b)
τ−1(a)<τ−1(b)
All the apparent poles in the integrand cancel except for simple poles at q + iκµ =
j j
q + iκ(µ + n ) and q + iκ(µ + n ) = q + iκµ , j < k. The integrand vanishes at
k k k j j j k k
q +iκµ = q +iκµ and q +iκ(µ +n ) = q +iκ(µ +n ).
j j k k j j j k k k
The propagator of the attractive delta-Bose gas in one dimension 6
ComparedtotheTWformula(2.2),ourresult(3.5)ismorecomplicated. Physically,
the complication can be traced to the presence of bound states for the attractive case:
thenparticlesarearrangedinM clustersofsizen ,..., n ,hencetheextrasummations
1 M
over M and ~n. Furthermore, Eq. (3.5) contains a summation over two permutations
σ and τ instead of only one for the TW formula (2.2). In the special case x = y = 0
discussed in Section 5, both summations over σ and τ can be eliminated.
For the attractive case, the propagator can also be written in terms of a summation
over the eigenstates of the Hamiltonian (1.1). The Bethe eigenfunctions for attractive
interaction are (see [11], Eq. (B.26) and (B.48); in [11], r(a) is equal to r(σ(a)) with
our notations, and Ω to σ−1(Ω ))
j j
n−M M M
ψ(x;M,~n,q) = κ 2 nj!(nj 1)! ei(qj+iκ(r(a)−n2j−12))xσ−1(a)
√n! −
Yj=1q σ∈XSn′(~n)Yj=1aY∈Ωj(cid:16) (cid:17)
M (q +iκ(r(a) nj)) (q +iκ(r(b) nk))+iκ
j − 2 − k − 2 , (3.6)
× (q +iκ(r(a) nj)) (q +iκ(r(b) nk)) iκ
Yj<k aY∈Ωj (cid:18) j − 2 − k − 2 − (cid:19)
σ−1(ab)∈>Ωσk−1(b)
with M = 1,...,n, ~n D and q RM. Eq. (3.6) is an eigenfunction of the
n,M
∈ ∈
Hamiltonian (1.1) with eigenvalue
M κ2
E(M,~n,q) = n q2 (n3 n ) . (3.7)
j j − 12 j − j
j=1 (cid:18) (cid:19)
X
The relation of the propagator with the Bethe eigenfunctions is stated as next theorem
in terms of (3.6) and (3.7).
Theorem 2. For κ > 0 and x,y Λ, one has
∈
n
1
x e−tHκ y = dq ... dq
h | | i M!(2π)M 1 M
RM
MX=1 ~n∈XDn,M Z
ψ(x;M,~n,q) ψ(y;M,~n,q) e−tE(M,~n,q) . (3.8)
As in the case of repulsive interaction discussed in Section 2, taking t = 0 yields the
completeness relation for the Bethe eigenstates (3.6). Their orthonormality is proved in
[11], Appendix B.
4. Analytic continuation from κ < 0 to κ > 0
In this section, the TW formula (2.2) for the propagator is extended by analytic
continuation to the attractive case κ > 0. Theorem 1 and Theorem 2 are proved.
4.1. Contribution of the residues
The contours of integration in (2.2) can be moved freely as long as the denominators
q q iκ keep a strictly positive imaginary part. In particular, if the integration is
j k
− −
The propagator of the attractive delta-Bose gas in one dimension 7
shifted to q R+iλ(n j), j = 1,...,n, with λ > 0, we obtain a formula valid for all
j
∈ −
κ such that (q q iκ) = (k j)λ κ > 0 for j < k, i.e. for all κ < λ. One obtains
j k
ℑ − − − −
n
1
x e−tHκ y = dq
h | | i n!(2π)n a
a=1(cid:18)ZR+iλ(n−a) (cid:19)
Y
n n
q q +iκ
a − b eiqa(xσ−1(a)−ya)e−tqa2 . (4.1)
q q iκ
a b
σX∈Snσ−1(aaY)><σb−1(b) − − aY=1(cid:16) (cid:17)
In the following, we want to further move the contours of integration, but this time
the contours will have to cross poles of the integrand, which will add several new terms
resulting from the residues at these poles, symbolically,
✲
✛✘
= (4.2)
× ×
✛
✚✙
✲
If one denotes by j k the action of taking the residue at q = q + iκr with r Z,
j k
→ ∈
the terms, obtained after moving the contours of integration, correspond to collections
of j k such that, for each ℓ = 1,...,n, ℓ ... appears only once in the collection
→ →
(since after taking the residue at q = q + iκr, the integrand no longer contains q ).
ℓ m ℓ
Each term thus corresponds to a forest (a set of trees), for example
1
❄
1 2,2 7,3 5,4 7 2 4 3 (4.3)
{ → → → → } ⇔
❆ ✁
❆❯ ✁☛ ❄
7 5 6
The particular trees obtained in this fashion depend on the order in which the contours
are moved. Here, we choose to move first the contour for q in such a way that it
n−1
crosses only the pole at q = q +iκ. Then, we move the contour for q in such a
n−1 n n−2
way that it crosses only the poles at q = q +iκ and q = q +iκ (in which case
n−2 n n−2 n−1
we still have an integration over both q and q ), or only the pole at q = q +2iκ
n−1 n n−2 n
(in which case the residue at q = q +iκ has been taken). We continue in this fashion
n−1 n
until in the final step the contour for q is moved.
1
Inprinciple, aftermoving thecontours, thepropagator x e−tHκ y will beexpressed
h | | i
as a sum over forests. In fact, it turns out that during this procedure there are many
cancellations which remove all the forests which contain trees with “branches”: in other
words, only the forests with merely “branchless” trees (like a, a b, a b c,
→ → →
a b c d, ...) remain after these cancellations. Instead of a sum over
→ → →
forests, we end up with a sum over partitions of 1,...,n (each element of the partition
{ }
corresponding to one of the branchless trees of the forest).
In the context of the distribution of the leftmost particle in the asymmetric
simple exclusion process, the procedure described here bears some similarity with the
The propagator of the attractive delta-Bose gas in one dimension 8
transformation from Theorem 3.1 to Theorem 3.2 in [31], where contours of integration
are moved from small to large circles. A complication in our context is that we need a
summation over all partitions of 1,...,n and not just over subsets of 1,...,n . We
{ } { }
expect that in the case of the full transition probability for the asymmetric exclusion
process an expression with an integration over large circles would require summing over
all partitions of 1,...,n .
{ }
A proof of the previous statements is based on induction w.r.t. an integer ℓ such
that all the contours for q , ..., q have already been moved.
ℓ+1 n−1
We introduce a few notations. For a boolean condition c, 11 is defined to be
{c}
equal to 1 if c is true and 0 otherwise. For ~n D , the set P (~n) contains all the
n,M n
∈
partitions A~ = A ,...,A of 1,...,n with A = n , j = 1,...,M. The partition
1 M j j
{ } { } | |
~
A verifies A ... A = 1,...,n and for j = k A A = . The partitions are not
1 M j k
∪ ∪ { } 6 ∩ ∅
ordered, i.e. the partition B~ = A ,...,A is considered to be the same element
R(1) R(M)
{ }
of P (~n) as A~ for all R S . Each A is called a cluster, and will correspond to a
n M j
∈
bound state of particles in the Bethe ansatz point of view. For a partition A~, we define
d (a), a = 1,...,n, (abbreviated as d(a) to lighten the notation) to be the rank of a in
A~
its cluster A , starting with rank 0 for the largest element of the cluster, rank 1 for the
j
second largest, ..., and rank A 1 for the smallest element of A .
j j
| |−
With these notation, the following lemma can be stated.
Lemma 1. Let ℓ be an integer between 0 and n 1. For fixed M = 1,...,n, let ǫ ,
j
−
j = ℓ+1,...,M be distinct numbers with 0 ǫ < 1. Then, for 0 < κ < λ one has
j
≤
n κn−M M ℓ
x e−tHκ y = (n !(n 1)!) dq
h | | i n!(2π)M j j − j
MX=1 ~n∈XDn,M Yj=1 Yj=1(cid:18)ZR+iλ(n−j) (cid:19)
M ℓ M
dq 11 11
× j {Aj={j}} {σ−1(a)>σ−1(b)}
j=Yℓ+1 ZR−iκǫj !A~∈XPn(~n)σX∈Sn Yj=1 Yj=1 a,Yb∈Aj
a<b
M
ei(qj+iκd(a))(xσ−1(a)−ya)e−t(qj+iκd(a))2
×
Yj=1aY∈Aj(cid:16) (cid:17)
M
(q +iκd(a)) (q +iκd(b))+iκ
j k
− . (4.4)
× (q +iκd(a)) (q +iκd(b)) iκ
j,Yk=1 aY∈Aj (cid:18) j − k − (cid:19)
j6=k b∈Ak
a<b
σ−1(a)>σ−1(b)
Proof. The constraint on the ǫ , j = ℓ+1,M, implies that (4.4) is well defined since all
j
the poles are at q = q +iδ with (q ) = (q )+δ.
j k j k
ℑ 6 ℑ
For ℓ = n 1, the identity between the expressions (4.1) and (4.4) of x e−tHκ y
− h | | i
is immediate. All the clusters must have size 1, and only M = n contributes. Since the
poles for q are at q = q iκ, j = 1,...,n 1, the contour for q can be moved freely
n n j n
− −
from R to R iκǫ , provided 0 ǫ < 1 and κ > 0.
n n
− ≤
We now proceed to prove the general identity by induction in ℓ: we assume that
The propagator of the attractive delta-Bose gas in one dimension 9
the expression (4.4) for x e−tHκ y is valid for ℓ 1 and will establish the expression
h | | i ≥
with ℓ replaced by ℓ 1.
−
For given ~n, we want to move the contour of integration for q from R+iλ(n ℓ) to
ℓ
−
R iκǫ with 0 ǫ < 1 and ǫ different from all the other ǫ , k = ℓ+1,...,M. In order
ℓ ℓ ℓ k
− ≤
to accomplish this, one needs to take into account the residues of the poles at q = z for
ℓ
κǫ < (z) < λ(n ℓ). The only poles for q are at z = q iκ, j = 1,...,ℓ 1, and
ℓ ℓ j
− ℑ − − −
at z = q +iκ(d(c)+1), m = ℓ+1,...,M, c A . In the first case, using κ < λ and
m m
∈
j ℓ 1, one finds (z) > λ(n ℓ), which implies that these poles do not contribute
≤ − ℑ −
when moving the contour for q . In the second case, using 0 < κ < λ, 0 ǫ < 1 and
ℓ m
≤
ℓ + d(c) + 1 n, one has κǫ 0 < (z) < λ(n ℓ), which implies that all these
ℓ
≤ − ≤ ℑ −
poles contribute a residue (with a factor 2iπ corresponding to a clockwise contour
−
integration).
Moving the contour for q produces several terms: one term corresponding to the
ℓ
integration over q R iκǫ , for which the integrand still depends on q , and one term
ℓ ℓ ℓ
∈ −
for each c A , m = ℓ + 1,...,M, for which the residue at q = q + iκ(d(c) + 1)
m ℓ m
∈
has been taken. The latter term corresponds to merging the cluster A = ℓ and the
ℓ
{ }
cluster A . Assuming σ−1(ℓ) > σ−1(c) (otherwise, the pole vanishes), this term is equal
m
to
κn−M M ℓ−1 M
( 2iπ) (n !(n 1)!) dq dq
− n!(2π)M j j − j j
j=1 j=1(cid:18)ZR+iλ(n−j) (cid:19)j=ℓ+1 ZR−iκǫj !
Y Y Y
ℓ M
11 11
{Aj={j}} {σ−1(a)>σ−1(b)}
A~∈XPn(~n)σX∈SnYj=1 Yj=1 a,Yb∈Aj
a<b
ei(qm+iκ(d(c)+1))(xσ−1(ℓ)−yℓ) e−t(qm+iκ(d(c)+1))2
×
(cid:16)M (cid:17)
ei(qj+iκd(a))(xσ−1(a)−ya)e−t(qj+iκd(a))2
×
jY=1 aY∈Aj(cid:16) (cid:17)
j6=ℓ
M
(q +iκd(a)) (q +iκd(b))+iκ
j k
−
× (q +iκd(a)) (q +iκd(b)) iκ
j,Yk=1 aY∈Aj (cid:18) j − k − (cid:19)
j6=k b∈Ak
j,k6=ℓ a<b
σ−1(a)>σ−1(b)
M
(q +iκd(a)) (q +iκ(d(c)+1))+iκ
j m
−
× (q +iκd(a)) (q +iκ(d(c)+1)) iκ
j,Yk=1 aY∈Aj (cid:18) j − m − (cid:19)
j6=k a<ℓ
j6=ℓ σ−1(a)>σ−1(ℓ)
M
(q +iκ(d(c)+1)) (q +iκd(b))+iκ
m k
−
× (q +iκ(d(c)+1)) (q +iκd(b)) iκ
j,Yk=1 bY∈Ak (cid:18) m − k − (cid:19)
j6=k ℓ<b
k6=ℓ σ−1(ℓ)>σ−1(b)
The propagator of the attractive delta-Bose gas in one dimension 10
(q +iκ(d(c)+1)) (q +iκd(b))+iκ
m m
−
× (q +iκ(d(c)+1)) (q +iκd(b)) iκ
bY∈Am (cid:18) m − m − (cid:19)
b6=c
σ−1(ℓ)>σ−1(b)
((q +iκ(d(c)+1)) (q +iκd(c))+iκ) . (4.5)
m m
× −
The last line of (4.5) contributes a factor 2iκ and the line before contributes
d(c) d(b)+2
− . (4.6)
d(c) d(b)
bY∈Am (cid:18) − (cid:19)
b6=c
σ−1(ℓ)>σ−1(b)
Let us first assume that c = min(A ). Then, for all b A , b = c one has
m m
∈ 6
σ−1(b) < σ−1(c). Together with σ−1(c) < σ−1(ℓ), it implies that all the elements of
the cluster A (except c) contribute in (4.6). This results in a factor (n + 1)n /2.
m m m
Combined with ( 2iπ) and 2iκ, we obtain a factor 2πκ(n + 1)n . The term with
k k
−
c = min(A ) thus corresponds exactly to the term of (4.4) with ℓ replaced by ℓ 1 and
m
−
~ ~ ~
the partition A replaced by B, obtained from A by merging the cluster ℓ with A
m
{ }
(after a renaming of the q , n , ǫ to q , n , ǫ for ℓ+1 j M).
j j j j−1 j−1 j−1
≤ ≤
It remains to show that for c = min(A ), the residues cancel each other. Since
m
6
the σ−1(b) are ordered in the same way as the d(b) for b A , there exists a unique
m
∈
number f A such that for b A , if b f then σ−1(b) < σ−1(ℓ), and if b < f
m m
∈ ∈ ≥
then σ−1(b) > σ−1(ℓ). Since σ−1(c) < σ−1(ℓ), one has necessarily f c (or equivalently
≤
d(f) d(c)). Then, (4.6) rewrites
≥
d(c) d(b)+2
− . (4.7)
d(c) d(b)
bY∈Am (cid:18) − (cid:19)
b6=c
b≥f
The rest of the argument depends on the relative values of d(c) and d(f). If d(f)
≥
d(c) +2, then, there exists b A such that b f and d(b) = d(c) + 2, thus (4.7) is
m
∈ ≥
equal to zero. Since d(f) d(c), the only cases left are d(f) = d(c)+1 and d(f) = d(c),
≥
for which (4.7) rewrites respectively
d(c) d(f)+2 d(c) d(b)+2 (d(c)+1)(d(c)+2)
− − = , (4.8)
d(c) d(f) d(c) d(b) − 2
− bY∈Am (cid:18) − (cid:19)
b>c
and
d(c) d(b)+2 (d(c)+1)(d(c)+2)
− = . (4.9)
d(c) d(b) 2
bY∈Am (cid:18) − (cid:19)
b>c
Let us call c′ the element of A such that d(c′) = d(c)+1 (c′ is the smallest element of
m
A larger that c). One notes that the two previous cases are exchanged when replacing
m
σ−1 by σ−1 θ , with θ the permutation exchanging ℓ and c′. Thus, summing over
ℓ,c′ ℓ,c′
◦
all permutations σ, the residues at q = q +iκ(d(c)+1) cancel.
ℓ m