Correlation functions of the open XXZ chain II

LPENSL-TH-03/08

arXiv:0803.3305v1 [hep-th] 23 Mar 2008

Correlation functions of the open XXZ chain II

N. Kitanine1 , K. K. Kozlowski2 , J. M. Maillet3 , G. Niccoli4 , N. A. Slavnov5 , V. Terras6

Abstract We derive compact multiple integral formulas for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulas follow from several effective re-summations of the multiple integral representation for the elementary blocks obtained in our previous article (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic field and decay algebraically in terms of the distance to the boundary.

1

Introduction

The Hamiltonian of the Heisenberg XXZ spin-1/2 finite chain [1] with diagonal boundary conditions (namely with longitudinal boundary magnetic fields) is defined as [2, 3] H= 1

M −1 X m=1




x x y y z z z σm σm+1 + σm σm+1 + ∆ σm σm+1 − 1 + h− σ1z + h+ σM .

(1.1)

LPTM, Universit´e de Cergy-Pontoise et CNRS, France, [email protected] Laboratoire de Physique, Universit´e de Lyon, ENS Lyon et CNRS, France, [email protected] 3 Laboratoire de Physique, Universit´e de Lyon, ENS Lyon et CNRS, France, [email protected] 4 DESY Theory group, Hamburg, Germany, [email protected] 5 Steklov Mathematical Institute, Moscow, Russia, [email protected] 6 Laboratoire de Physique, Universit´e de Lyon, ENS Lyon et CNRS, France, [email protected], on leave of absence from LPTA, Universit´e Montpellier II et CNRS, France 2

1

M

This is a linear operator acting in the quantum space H = ⊗ Hm , Hm ≃ C2 , of m=1

± , σ z denote local spin operators dimension 2M of the chain. In this expression, σm m (acting as Pauli matrices) at site m, ∆ is the anisotropy parameter and h± are the boundary (longitudinal) magnetic fields. We have recently developed a method to compute the so-called elementary blocks of correlation functions for this model (see [4], that we refer to as Paper I in the following) in the framework of the (algebraic) Bethe ansatz [5–18] for boundary integrable systems [2, 3, 19–31]. The results essentially agree with previous expressions derived from the vertex operator approach [32, 33]. The purpose of the present paper is to obtain the physical spin correlation functions for this model, in particular, the one point functions for the local spin operators at distance m from the boundary as well as several two point functions (like boundary-bulk correlation functions). There are numerous physical interests in such quantities that can be measured in actual experiments [34–46]. In much the same way as in the bulk case [47–52], the computation of the physical correlation functions amounts to obtain effective re-summations of the multiple integral representations derived for the elementary blocks. For example, the one point z i, functions at distance m from the boundary, such as the local magnetization hσm m can be written as a sum of 2 elementary blocks. We will show how to obtain compact expressions for such objects, typically involving the sum of only m terms, each containing multiple integrals whose integrants have a structure similar to the one of the elementary blocks. In the free fermion point we are able to compute these multiple integrals (and hence the corresponding correlations functions) almost completely by reducing them to single integrals. For instance the local magnetization and the density of energy profiles (a quantity of interest in the study and the understanding of the interplay between quantum entanglement and quantum criticality [53–61]) are expressed as single integrals. Hence, their asymptotic behavior at long distance m from the boundary can be explicitly evaluated. In addition to the bulk constant value they exhibit Friedel type oscillations [44–46, 60, 61], algebraically decaying with the distance m, their amplitudes being rational functions of the boundary magnetic field, in agreement with field theory predictions [38–42, 62–70]. We start this paper with a short technical introduction concerning the algebraic Bethe ansatz approach to the open XXZ spin-1/2 chain subject to diagonal boundary magnetic fields. This preliminary section is followed in Section 3 by a reminder of the method proposed in [4] to compute correlation functions of open integrable models in the framework of algebraic Bethe ansatz. In Section 4 we obtain formulae for the action of local operators on arbitrary boundary states in a form suitable for taking later on the thermodynamic limit. Using these results, we derive a series representation for the generating function hQm (κ)i of bulk-boundary σ z correlation functions in Section 5. This formula is the boundary analogue of the original series [49] in the bulk case. In Section 6, we obtain a formula for hQm (κ)i alternative to the one + inferred in Section 5. We also give multiple integral representations for hσm+1 σ11 i and for the local density of energy. These formulae are obtained by a direct resummation

2

of the corresponding elementary blocks. It is worth stressing that we actually have two representations for their integrand. The first one is in the spirit of the bulk case [52] and involves the Izergin determinant representation [71] for the partition function of the six vertex model with domain wall boundary conditions. The second one involves the Tsuchiya [72] determinant representation for the partition function of the six vertex model with reflecting ends. The next section is devoted to the free fermion point. For that case are able to reduce the multiple integrals to one dimensional ones. This allows us to write the leading asymptotics of the local magnetization and of + the density of energy profiles as well as of the hσm+1 σ1− i correlation function. Our conclusions are presented in the last section.

2

The open XXZ spin-1/2 chain

The spectrum of H can be obtained by algebraic Bethe ansatz (ABA) [3]. The central tool of this method is the boundary monodromy matrix, which will be defined after we introduce some necessary notations. Here and in the following we adopt the standard parameterizations ∆ = cosh η and h± = sinh η coth ξ± . Let R : C → End(V ⊗ V ), V ≃ C2 , be the R-matrix of the six-vertex model, obtained as the trigonometric solution of the Yang-Baxter equation:   1 0 0 0 0 b(u) c(u) 0 b b  R(u) = sinh (u + η) R(u), with R(u) = (2.1) 0 c(u) b(u) 0 , 0 0 0 1

and

b(u) =

sinh u , sinh(u + η)

c(u) =

sinh η . sinh(u + η)

(2.2)

The bulk monodromy matrix T (λ) ∈ End(V0 ⊗ H), V0 ≃ C2 , is defined as an ordered product of R matrices:   A (λ) B (λ) T0 (λ) = R0M (λ − ξM ) . . . R01 (λ − ξ1 ) = . (2.3) C (λ) D (λ) [0] The subscript 0 labels here the two-dimensional auxiliary space V0 , whereas subscripts m running from 1 to M refer to the quantum spaces Hm of the chain. Besides, we attach an inhomogeneity parameter ξm to each site m of the chain. We recall that T (λ) satisfies the Yang-Baxter algebra, on V0 ⊗ V0′ ⊗ H: R00′ (λ − µ) T0 (λ) T0′ (µ) = T0 (λ) T0′ (µ) R00′ (λ − µ) .

(2.4)

Let us also introduce the two boundary matrices, K± (λ) = K (λ ± η/2; ξ± ), where K (λ; ξ) is the 2 × 2 matrix acting on the auxiliary space:   sinh (λ + ξ) 0 . (2.5) K (λ; ξ) = 0 sinh (ξ − λ) [0]

3

The boundary monodromy matrix U (λ) [3] is built out of a product of T (λ) and K+ (λ), namely1  t0 A (λ) B (λ) t0 t0 t0 t0 b U0 = T0 (λ) K+ (λ) T0 (λ) = , (2.6) C (λ) D (λ) [0]

where

Tb0 (λ) = R10 (λ + ξ1 − η) . . . RM 0 (λ + ξM − η) = (−1)M

M Y

[sinh (λ + ξj ) sinh (λ + ξj − η)] T0−1 (−λ + η) .

(2.7)

j=1

This boundary monodromy matrix satisfies the reflection algebra first introduced in [20]: t

R00′ (−λ + µ) U0t0 (λ) R00′ (−λ − µ − η) U0′0′ (µ) t

= U0′0 (µ) R00′ (−λ − µ − η) U0t0 (λ) R00′ (−λ + µ) . (2.8) ′

The commuting charges of the XXZ spin-1/2 chain with diagonal boundary conditions are realized by the one-parameter family of transfer matrices: T (λ) = tr0 [ U0 (λ) K− (λ)] , and the Hamiltonian (1.1) is obtained in terms of the derivative

(2.9) dT (λ) |λ=η/2 in the dλ

homogeneous case ξi = η/2, i = 1, . . . , M . Common eigenstates of all transfer matrices (and thus of the Hamiltonian (1.1) in the homogeneous case) can be constructed by successive actions of B (λ) operators on the reference state |0 i which is the ferromagnetic state with all the spins up. More precisely, the state 2 | {λ}n1 ib ≡ B (λ1 ) . . . B (λn ) |0 i (2.10) is a common eigenstate of the transfer matrices if the set of spectral parameters {λ}n1 ≡ {λj }1≤j≤n is a solution of the Bethe equations yj (λj ; {λ}n1 ) = yj (−λj ; {λ}n1 ) ,

j = 1, . . . , n,

(2.11)

where yj (x; {λ}n1 ) =

yˆ (x; {λ}n1 ) , s (λj , x − η)

yˆ (x; {λ}n1 ) = −a (x) d (−x) sinh (x + ξ+ − η/2) sinh (x + ξ− − η/2) n Y × s (x − η, λl ) .

(2.12)

l=1

1 Note that it corresponds to the matrix U+ of our previous article (I). Since we consider only the ‘+’ case in the present article, we do not specify it in the notations. 2 In order to lighten the formulae we have slightly changed the notation with respect to the one in Paper I. Namely, the vector | {λ}n 1 ib corresponds to |ψ+ ({λ}) i in [4]. Such a boundary state should in particular be distinguished from the corresponding bulk state that we merely denote | {λ}n 1 i.

4

Here and in the following, s (λ, µ) denotes the function s (λ, µ) = sinh (λ + µ) sinh (λ − µ) ,

(2.13)

and the functions a (λ) and d (λ) stand respectively for the eigenvalues of the bulk operators A(λ) and D(λ) on the pseudo-vacuum |0 i: a (λ) =

M Y i=1

sinh (λ − ξi + η) ,

d (λ) =

M Y i=1

sinh (λ − ξi ) .

(2.14)

Of course it is also possible to implement the Bethe ansatz starting from the dual state h0 | and acting on it with C (λ) operators: n b h{λ}1

| ≡ h0 |C (λ1 ) . . . C (λn ) .

(2.15)

The description of the ground state of H in the half-infinite chain depends on the regime. One should distinguish the two domains −1 < ∆ ≤ 1 (massless regime) and ∆ > 1 (massive regime): αj = λj ,

ζ = iη > 0,

αj = iλj , ζ = −η > 0,

π π with − < ξ˜− ≤ , 2 2 π ˜ ˜ ξ− = −ξ− + iδ , with ξ− ∈ R, 2 ξ− = −iξ˜− ,

for − 1 < ∆ ≤ 1, for ∆ > 1,

where δ = 1 for |h− | < sinh ζ and δ = 0 otherwise. Thus, to a given set of roots {λj } corresponds a set of variables {αj } given by the previous change of variables. Note that the nature of the ground state rapidities depends on the value of the boundary field h− . Indeed, when ξ˜− < 0 or ξ˜− > ζ/2, the ground state of the Hamiltonian (1.1) is given in both regimes by the maximum number N of roots λj corresponding to real (positive) αj such that cos p(λj ) < ∆. In the thermodynamic limit M → ∞, these roots λj form a dense distribution on an interval [0, Λ] of the real or imaginary axis. Their density ρ(λj ) = lim [M (λj+1 − λj )]−1 (2.16) M →∞

satisfies the integral equation 2πρ (λ) +

ZΛ

−Λ

2i sinh η i sinh(2η) ρ (λ) dλ = , s (λ − µ, η) s (λ, η/2)

(2.17)

with Λ = +∞ in the massless regime, and Λ = −iπ/2 in the massive one. The density can be expressed in terms of usual functions:  1   −1 < ∆ < 1;  ζ cosh (πλ/ζ) ,   (2.18) ρ (λ) = i Q sinh nζ 2 θ3 (iλ; −ζ)   , 1 < ∆.  π n≥1 cosh nζ θ4 (iλ; −ζ) 5

ˇ (corresponding However, when 0 < ξ˜− < ζ/2, the ground state also admits a root λ to a complex α ˇ ) which tends to η/2 − ξ− with exponentially small corrections in the large M limit. In that case, the density of real roots is still given by the solution of (2.17).

3

The ABA approach to correlation functions

A zero temperature correlation function is the normalized expectation value, in the ground state of the Hamiltonian (1.1), of some local 3 operator Om , hOm i =

N b h{λ}1

|Om | {λ}N 1 ib

N b h{λ}1

| {λ}N 1 ib

,

(3.1)

where the parameters λ are the solutions of the ground state Bethe equations. In order to compute such a correlation function, one should first derive the action of the corresponding local operator on the boundary state | {λ}N 1 ib , and then evaluate the resulting scalar products. We have constructed in [4] a method to solve this problem. This method is based on a revisited version of the quantum inverse problem, first introduced in [47, 73] for the XXZ spin chain with periodic boundary conditions. Once a local operator is reconstructed in terms of the entries of the bulk monodromy matrix, its action on boundary states can then be computed thanks to the decomposition of boundary states in terms of bulk states and to the Yang-Baxter commutation relations. We shall now recall the main points of our method.

3.1

The bulk inverse problem revisited

ij Proposition 3.1 (Solution of the bulk inverse problem) [47,73] Let Em be an th elementary matrix acting non-trivially only on the m site of the chain, then

ij Em =

m−1 Y k=1

m   Y (A + D) (ξk ) tr T0 (ξm ) E0ij (A + D)−1 (ξk ) .

(3.2)

k=1

Note that, thanks to the crossing symmetry of the R-matrix, one can recast the inverse of the bulk transfer matrix at inhomogeneity parameter (A + D)−1 (ξk ) in terms of the transfer matrix at shifted parameter (A + D) (ξk − η), namely (A + D)−1 (ξk ) =

(A + D) (ξk − η) . a (ξk ) d (ξk − η)

(3.3)

It is worth pointing out that the products of elementary matrices on the first m sites of the chain define a basis in the space of local operators Om , so that (3.2) allows one to define a reconstruction for all such operators. However, this reconstruction is 3

m

i.e. acting non-trivially only in ⊗ H k . k=1

6

especially convenient when one wants to obtain the action on a bulk Bethe state; indeed, in such a case, the product of bulk transfer matrices merely produces a numerical factor. This is no longer the case when one acts on a boundary Bethe state. The theorem below allows one to reconstruct local operators in a way adapted to an action on boundary states. Theorem 3.1 [4] For any set of inhomogeneity parameters {ξi1 , . . . , ξin }, the product of bulk operators Tǫin ǫ′i (ξin ) . . . Tǫi1 ǫ′i (ξi1 ) Tǫ¯i1 ǫ¯′i (ξi1 − η) . . . Tǫ¯in ǫ¯′i (ξin − η) n

1

n

1

(3.4)

vanishes if, for some k ∈ {i1 , . . . , in }, ǫk = ǫ¯k . Thus we have : Corollary 3.1 A product of elementary matrices acting on the first m sites of the chain can be expressed as a single monomial in the entries of the bulk monodromy matrix: ǫ ǫ′1

E11

′

ǫm ǫm . . . Em =

m Y  i=1

−1 a(ξi ) d(ξi − η)

× Tǫ′1 ǫ1 (ξ1 ) . . . Tǫ′m ǫm (ξm ) Tǫ¯m ǫ¯m (ξm − η) . . . Tǫ¯1 ǫ¯1 (ξ1 − η) (3.5)

with ¯ǫi = ǫ′i + 1 (mod 2). This result represents a strong simplification. Indeed, it means that, over the monomials appearing in the reconstruction of a local operator (3.2), only one is non-zero. We shall now explain how to compute the action of this non-vanishing monomial on an arbitrary (bulk or boundary) state. 2m

3.2

Action on bulk and boundary states

Before stating the lemma which explains how to derive the action of the former monomial on a bulk state, we recall that the action of A(µ) or D(µ) on a bulk state QN N | {λ}1 i ≡ j=1 B(λj )|0 i produces two kinds of terms: the direct term, where all rapidities remain unchanged, and indirect terms where one λj is replaced by µ. Lemma 1 (Action on a bulk state) [4] The action on a bulk state | {λ}N 1 i of a string of operators ǫ′i ,...,ǫ′i

Oǫi11 ,...,ǫinn = Tǫ′i ǫin (ξin ) . . . Tǫ′i ǫi1 (ξi1 ) Tǫ¯i1 ¯ǫi1 (ξi1 − η) . . . Tǫ¯in ¯ǫin (ξin − η) 1 {z } {z }| | n (2)

(1)

with ¯ǫl = ǫ′l + 1 (mod 2), satisfies the restrictions:

• The only non-zero contributions of the tail operators (2) come from 7

(3.6)

(i) the indirect action of all A(ξl − η) operators;

(ii) the direct action of all D(ξl − η) operators. • In what concerns the head operators (1),

(iii) if ǫ′l = 1, the action of the operator Tǫ′l ǫl (ξl ) (i.e. A(ξl ) or B(ξl )) does not result in any substitution of a parameter ξi − η;

(iv) if ǫ′l = 2, the action of the operator Tǫ′l ǫl (ξl ) (i.e. D(ξl ) or C(ξl )) substitutes ξl − η with ξl ; moreover, if there were others parameters ξj − η, j 6= l, in the initial state, they are still present in the resulting state.

This lemma enables us to compute the action of local operators on any (arbitrary) bulk state. In order to compute the action on a boundary state, we use the fact that the latter can be decomposed in terms of bulk states: Proposition 3.2 (Boundary-bulk decomposition) [4], [74] Let | {λ}n1 ib be an arbitrary boundary state, then it can be expressed in terms of bulk states as X B H{σ ({λ}n1 ) | {λσ }n1 i, | {λ}n1 ib = (3.7) i} σi =± i=1,...,n

with B H{σ ({λ}n1 ) i}

=

n Y

HσBj (λj )

j=1

Y sinh(λσ − η) rs . · σ sinh(λrs ) 1≤rb Q Sσ ({λ}α+ , {ξ}γ+ ; {λ}α− ) = Q s (ξb , λa ) s (ξb , ξa ) a∈α −

a,b∈γ+ a>b

×

b∈γ+

Y yˆ(ξb ; {λ}α+ ∪α− ) sinh(2ξb + η) Y sinh(2λσa − η) sinh(2λa ) , (3.16) sinh(2ξb ) yˆ(λσa ; {λ}α+ ∪α− ) sinh(2λa + η) a∈α

b∈γ+

+

for any value of σa ∈ {+, −}, a ∈ α+ , where yˆ is the function defined in (2.12). When M is large, the matrix elements of Se reduce to ( ˇ, 2iπM sinh (λa − ξ− + η/2) Ψ (λa , ξb ) if λa = λ (3.17) Seab ∼ ˇ, M →∞ ρ−1 (λa ) Ψ (λa , ξb ) if λa 6= λ

the corrections being of order O (1/M ), and Ψ (λ, ξ) =

ρ (λ − ξ) − ρ (λ − η + ξ) . 2 sinh (2ξ − η) 9

(3.18)

The determinant structure of the scalar product as well as peculiarities of the coefficients Cαm enable us to write: hOm i =

1 Mm

X

m {ν}I ⊂{λ}N 1 ∪{ξ}1

Hm ({ν}I , {ξ}m 1 ) (1 + O (1/M )) ,

(3.19)

|I|=m

in which the coefficient Hm ({ν}I , {ξ}m 1 ) can be computed generically. Taking the thermodynamic limit M → +∞ we recast the sums over replaced rapidities λ into integrals: Z N 1 X X σ σi f (λi ) −→ dλ ρ (λ) f (λ) , M →+∞ M σ =± i=1

i

∀f ∈ C 0 (C ) .

(3.20)

C

In the boundary model the contour of integration C depends on the anisotropy parameter ∆ and on the boundary field h− .

4

Action of local operators on boundary states

Using Corollary 3.1, Lemma 1 and the boundary-bulk decomposition of Proposition 3.2, it is easy to compute the action of a product of elementary matrices of the form (3.5) on an arbitrary boundary state. This computation was explicitely performed in [4], and enabled us there to obtain some expressions for the elementary building blocks of correlation functions. The aim of the present article is to obtain such expressions for physical correlation functions, and in particular for one-point functions. Therefore, if we want to use the method recalled in Section 3, the main problem is to obtain some resummed formulas directly for the action of the local spin operators we consider. This is the purpose of the present section. In the first part of this section, we derive the action of the operator Qm (κ) ≡

m Y i=1

m m Y
Y (A + D)−1 (ξi ) (A + κD) (ξi ) Ei11 + κEi22 = i=1

(4.1)

i=1

on arbitrary boundary states. hQm (κ)i can be interpreted in the boundary model as the generating function of the magnetization at a distance m from the boundary: h

z 1 − σm i = Dm ∂κ hQm (κ)i|κ=1 , 2

(4.2)

where Dm is the lattice derivative : Dm um ≡ um+1 − um . Then, in the second part of this section, we give the formulas for the action of the z 1−σm 22 12 = σ + and E 21 = σ − on arbitrary boundary local spin operators Em = 2 , Em m m m 11 = 1 − E 22 . 11 states. Note that the action of Em follows from the fact that Em m

10

4.1

Action of Qm (κ)

We start by computing the action of Qm (κ) on an arbitrary bulk state, and then infer from this formula its action on arbitrary boundary states. Proposition 4.1 The action of Qm (κ) on an arbitrary bulk state | {λ}N 1 i can be expressed as Qm (κ) | {λ}N 1 i=

m X X

n=0 Pλ ; Pξ

Rnκ (Pλ , Pξ ) | {ξ}γ+∪ {λ}α− i .

(4.3)

In the above formula, we sum over all possible partitions Pλ and Pξ of the sets {λ}N 1 and {ξ}m 1 into subsets {λ}α+ ∪ {λ}α− and {ξ}γ+ ∪ {ξ}γ− respectively, satisfying the constraint on the cardinality | α+ |=| γ+ |= n: Pλ : {λ}N 1 = {λ}α+ ∪ {λ}α− , Pξ :

{ξ}m 1

| α+ |= n ,

= {ξ}γ+ ∪ {ξ}γ− ,

(4.4)

| γ+ |= n .

(4.5)

The coefficient Rnκ (Pλ , Pξ ) splits into two parts,


Rnκ (Pλ , Pξ ) = R (Pλ , Pξ ) Snκ {ξ}γ+ , {λ}α+ ,

the first one having a product structure,   Q Q f (λb , λa ) a (λa ) a∈α+

R (Pλ , Pξ ) =

Q

a∈γ+

 a (ξa )

b∈α−

Q

f (λb , ξa )

b∈α− ∪α+



Y

a∈γ−

Q

f (ξb , ξa )

b∈γ+

Q

(4.6)

f (λb , ξa )

,

(4.7)

b∈α+

and the second one, which depends here only on the subsets {λ}α+ and {ξ}γ+ , being given as a ratio of two determinants,   
 1 . (4.8) Snκ ({ν}n1 , {µ}n1 ) = detn Mκ {µ}n1 , {ν}n1 det−1 n sinh (νk − µj + η) The entries of the matrix Mκ read 

Mκ {µ}n1 , {ν}n1




jk

= t (νk , µj ) − κ t (µj , νk )

n n Y f (µa , µj ) Y f (µj , νa ) , f (µj , µa ) f (νa , µj )

a=1 a6=j

(4.9)

a=1

and the functions f and t stand for t (λ, µ) =

sinh η , sinh (λ − µ) sinh (λ − µ + η)

11

f (λ, µ) =

sinh (λ − µ + η) . sinh (λ − µ)

(4.10)

The above theorem appears as a non-trivial generalization of the action of Qm (κ) on bulk Bethe eigenvectors [49]. Indeed, when | {λ} i is not an eigenstate of the bulk Q transfer matrix, then m (A + D) (ξi ) does not act by multiplication any more. Of i=1 course our result reproduces the previous case when we send the parameters λ to a solution of the bulk Bethe equations. Proof — The proof goes by induction on m. Property (4.3) is obvious for m = 1. Assume that it holds for some m. To prove its validity for m + 1 we have to compute Qm+1 (κ) | {λ}N 1 i=

(A + κD) (ξm+1 ) Qm (κ) (A + D) (ξm+1 − η) | {λ}N 1 i. a(ξm+1 )d(ξm+1 − η)

(4.11)

Let us first reproduce the coefficient Rnκ (Pλ , Pξ ) in the case when the partition Pξ is such that ξm+1 6∈ {ξ}γ+ . The corresponding state | {λ}α− ∪ {ξ}γ+ i can only be obtained by the direct action of (A + κD) (ξm+1 ). In order to reproduce the claimed form of the coefficient Rnκ it is enough to prove that (A + κD) (ξm+1 − η) acts directly. Suppose that this is not the case. Then Qm (κ) acts on a state containing ξm+1 − η. In virtue of Lemma 1, the action of Qm (κ) on these states cannot replace ξm+1 − η. Thus (A + D) (ξm+1 ) exchanges ξm+1 − η with ξm+1 , which leads to a contradiction. We still have to reproduce the coefficient Rnκ (Pλ , Pξ ) corresponding to states | {λ}α− ∪ {ξ}γ+ i such that ξm+1 ∈ {ξ}γ+ . Theorem 3.1 yields the decomposition: Qm+1 (κ) =

κD (ξm+1 ) Qm (κ) A (ξm+1 ) Qm (κ) D (ξm+1 − η) + A (ξ − η) , {z } a(ξm+1 )d(ξm+1 − η) | m+1 {z } a(ξm+1 )d(ξm+1 − η) | (1)

(2)

whereas Lemma 1 ensures that

• (1) only acts directly; indeed A (ξm+1 ) cannot replace ξm+1 − η by ξm+1 ; • (2) acts indirectly and thus D (ξm+1 ) only acts by substitution. The formula for Rnκ (4.6) follows after computing the resulting actions and rearranging the sums thanks to the re-summation formula provided by the contour integral: 0=

I

R∪R+iπ

r Y
dz f (z, xa ) κ Sn+1 {ξ}n1 ∪ {z} , {λ}n+1 . 1 sinh (z − ξn+1 ) a=1 f (ξn+1 , xa )

(4.12)

Note that the parameters xa appearing in the contour integral (4.12) are generic.



Using the boundary-bulk decomposition of Proposition 3.2, one can now deduce from Proposition 4.1 the action of Qm (κ) on arbitrary boundary states. Corollary 4.1 The action of Qm (κ) on an arbitrary boundary state | {λ}N 1 ib reads: Qm (κ) | {λ}N 1 ib

=

m X X

n=0 Pλ ; Pξ

Rκn (Pλ , Pξ ) | {ξ}γ+∪ {λ}α− ib .

12

(4.13)

The sum over partitions is defined as in Theorem 4.1, and the coefficient Rκn can be expressed as X Rκn (Pλ , Pξ ) = Rσ (Pλ , Pξ ) Snκ ({ξ}γ+, {λσ }α+ ) , (4.14) σi =± i∈α+

where Snκ ({ν}n1 , {µ}n1 ) is the bulk function defined in (4.8), while Rσ (Pλ , Pξ ) is the boundary dressing of (4.7):    Q Q  σ σ σ a(λa ) f (λb , λa ) f (−λb , λa ) Rσ (Pλ , Pξ ) =

a∈α+

b∈α

−    Q  Q Q σ f (λb , ξa ) f (λb , ξa )f (−λb , ξa ) a(ξa )

a∈γ+

b∈α−

b∈α+

×

Y

a∈γ−

Q

f (ξb , ξa )

b∈γ+

Q

b∈α+

B ({λ}α+ ) H{σ} α +

f (λσb , ξa )

H B ({ξ}γ+ )

. (4.15)

B ({λ}α+ ) and H B ({ξ}γ+ ) stand for the boundary-bulk coefficients (3.8) Here H{σ} α +

associated respectively to {λ}α+ , {σ}α+ , and to {ξ}γ+ , {σ}γ+ = {1, . . . , 1}. Proof — The proof is a straightforward consequence of the boundary-bulk decomposition (3.7) applied to Proposition 4.1. More precisely, expressing the boundary state σ N |{λ}N 1 ib in terms of the bulk states |{λ }1 i, and using (4.3), we get Qm (κ) | {λ}N 1 ib

=

m X X

X

n=0 Pλ ; Pξ σi =± 1≤i≤N

B κ σ H{σ} ({λ}N 1 ) Rn (Pλσ , Pξ ) | {ξ}γ+∪ {λ }α− i.

We now use the fact that Q f (−λσb , λσa ) H B Y a∈α+ {σ}α+ ({λ}α+ ) B B Q H{σ} ({λ}N H1,{σ} ({ξ}γ+ ∪ {λ}α− ), 1 )= σ α− H B ({ξ}γ+ ) f (−λb , ξa ) b∈α− a∈γ +

B ({ξ}γ+ ∪ {λ}α− ) is the boundary-bulk coefficient of |{ξ}γ+ ∪ {λ}α− ib where H1,{σ} α −

in terms of |{ξ}γ+ ∪ {λσ }α− i. Note that the first factor of this product combines with the products over b ∈ α− in the expression (4.7) of R(Pλσ , Pξ ), and that the resulting factor,  Q  f (λσb , λσa ) f (−λσb , λσa ) Y a∈α+ , Q  σ , ξ ) f (−λσ , ξ ) f (λ a a b b b∈α −

a∈γ+

is actually independant of the value of σi for i ∈ α− . It enables us to reconstruct the boundary state |{ξ}γ+ ∪ {λ}α− ib , with a coefficient which reduces to (4.14). 

13

4.2

Action of local spin operators

− , σ + and E 22 on bulk and We list here the action of the local spin operators σm m m boundary states. We omit the proofs since, although a little more technical, they parallel the one concerning the action of Qm (κ). − , E 22 and σ + on an arbitrary bulk state |{λ}N i Proposition 4.2 The action of σm m m 1 can be expressed as

− σm

22 Em

|{λ}N 1

i=

|{λ}N 1

m−1 X

n=0 P − ,Pξ λ

i=

+ |{λ}N σm 1 i=

X

m−1 N X X

Rn− (Pλ− , Pξ ) |{ξ}γ+ ∪ {λ}α− i, X

n=0 c1 =1 P 22 ,Pξ

lim

λN+1 →ξm

Rn22 (Pλ22 , Pξ ) |{ξ}γ+ ∪ {λ}α− i,

λ

m−1 N N +1 X X X

X

n=0 c1 =1 c2 =1 P + ,Pξ c2 6=c1 λ

Rn+ (Pλ+ , Pξ ) | {ξ}γ+ ∪ {λ}N 1 \ {λ}α e+ i,

in which the sums run over the following partitions Pξ : {ξ}m 1 = {ξ}γ+ ∪ {ξ}γ− ,

Pλ− : {λ}N 1 = {λ}α+ ∪ {λ}α− ,

with

| γ+ |= n + 1,

(4.16)

with

| α+ |= n,

(4.17)

Pλ22 : {λk }1≤k≤N = {λ}α+ ∪ {λ}α− ,

with

| α+ |= n,

(4.18)

Pλ+ : {λk }1≤k≤N = {λ}α+ ∪ {λ}α− ,

with

| α+ |= n.

(4.19)

k6=c1

k6=c1 ,c2

We also define the following partitions, associated respectively to (4.18) and to (4.19), Peλ22 : {λ}N 1 = {λ}α e+ ∪ {λ}α− ,

with

+1 Peλ+ : {λ}N = {λ}αe+ ∪ {λ}αe− , 1

with

α e+ = α+ ∪ {c1 },

α e+ = α+ ∪ {c1 , c2 }.

The coefficients Rn− (Pλ− , Pξ ), Rn22 (Pλ22 , Pξ ) and Rn+ (Pλ+ , Pξ ) are given as Q sinh(ξa − ξ) a∈γ+ − − − Rn (Pλ , Pξ ) = R(Pλ , Pξ ) lim Q Sbn ({λ}α+ , {ξ}γ+ ; ξ, ∅), ξ→ξm sinh(λa − ξ)

(4.20) (4.21)

(4.22)

a∈α+

e22 , Pξ ) sinh η Rn22 (Pλ22 , Pξ ) = R(P λ × lim

ξ→ξm

Q

f (λa , λc1 )

a∈α+

a∈γ+

sinh(ξa − ξ)

a∈e α+

sinh(λa − ξ)

Q

Y

Sbn ({λ}α+ , {ξ}γ+ ; ξ, {λc1 }),

14

(4.23)

Rn+

Pλ+

, Pξ




 2  Y Y
+ e f (λa , λci ) sinh η = R Pλ , Pξ f (λc2 , λc1 )

a∈γ+

sinh(λN +1 − ξa + η)

a∈e α+

sinh(λN +1 − λa + η)

× Q

a∈α+

i=1

Q


Sbn {ξ}γ+ , {λ}α+ ; λN +1 , {λc1 , λc2 } . (4.24)

Here R(Pλ , Pξ ) is given by (4.7), and the structure of the factor Sbn ({ξ}n+1 , {λ}n1 ; ξ, {µ}p1 ) 1 is similar to (4.8): n n+1 Q Q

sinh(ξb − λa + η) Q sinh(ξa − ξb ) sinh(λb − λa ) a>b a>b h i c({λ}n1 , {ξ}n+1 ; ξ, {µ}p ) , (4.25) × detn+1 M 1 1

Sbn ({ξ}n+1 , {λ}n1 ; ξ, {µ}p1 ) = Q 1

a=1 b=1

c are obtained as where the matrix elements of M
  
 n+1 c {λ}n1 , {ξ}n+1 ; λn+1 , {µ}p M , {ξ}n+1 , 1 1 jk = Mκ(λj ,{µ}) {λ}1 1 jk with

(4.26)

′

κ(λj , {µ}) = (1 − δj,n+1 )

n Y f (µi , λj ) i=1

f (λj , µi )

,

in which δij denotes the Kronecker symbol and Mκ is defined as in (4.9). Using again the boundary-bulk decomposition, we are now in position to list the action of local spin operators on boundary states. − , E 22 Corollary 4.2 With the same notations as in Proposition 4.2, the action of σm m + on an arbitrary boundary state |{λ}N i takes the form and σm 1 b − σm |{λ}N 1 ib =

22 Em

|{λ}N 1 ib

m−1 X

n=0 P − ,Pξ λ

=

+ σm |{λ}N 1 ib =

X

m−1 N X X

− R− n (Pλ , Pξ ) |{ξ}γ+ ∪ {λ}α− ib ,

X

n=0 c1 =1 P 22 ,Pξ

lim

λN+1 →ξm

22 R22 n (Pλ , Pξ ) |{ξ}γ+ ∪ {λ}α− ib ,

λ

m−1 N N +1 X X X

X

n=0 c1 =1 c2 =1 P + ,Pξ c2 6=c1 λ

N + R+ n (Pλ , Pξ ) | {ξ}γ+ ∪ {λ}1 \ {λ}α e+ ib .

The boundary coefficients R− , R22 and R+ have a structure similar to their corresponding bulk counterparts: Q sinh(ξa − ξ) X a∈γ+ − − − Rn (Pλ , Pξ ) = Sbn ({λ}α+ , {ξ}γ+ ; ξ, ∅), Rσ (Pλ , Pξ ) lim Q ξ→ξm sinh(λa − ξ) σ =± i

i∈α+

a∈α+

15

X

22 R22 n (Pλ , Pξ ) =

σi =± i∈e α+

Rσ (Peλ22 , Pξ ) sinh η Q

ξ→ξm

R+ n

Pλ+ , Pξ Q 




=

X

σi =± i∈e α+

f (λa , λc1 )

a∈α+

a∈γ+

sinh(ξa − ξ)

a∈e α+

sinh(λa − ξ)

Q

× lim

Y

Sbn ({λ}α+ , {ξ}γ+ ; ξ, {λc1 }),

 2  Y

Y
+ σ σ σ σ e f λa , λci Rσ Pλ , Pξ f λc2 , λc1 sinh η a∈α+

i=1

 f (−λN +1 , ξa ) sinh(λN +1 − ξa + η)
a∈γ+  Sbn {ξ}γ+ , {λσ }α+ ; λN +1 , {λσci } , × Q  f (−λN +1 , λσa ) sinh(λN +1 − λσa + η) a∈e α+

where Rσ is defined as in (4.15) and Sbn is the bulk quantity (4.25).

5

Correlation functions in the half-infinite chain

We apply the results of the previous section to derive the expectation values of z i, and of hσ + σ − i in the ground state of the the generating function hQm (κ)i of hσm 1 m+1 half-infinite chain. These are the boundary analogues of the results published in [49].

5.1

The generating function hQm (κ)i

Proposition 5.1 The generating function hQm (κ)i is obtained, in the thermodynamic limit M → +∞, as the homogeneous limit of the quantity m X

1 hQm (κ)i = 2 n=0 (n!)

I

dn z (2iπ)n

Z

CD

Γ+ ({ξ}m 1 )

m Y n Y f (zb , ξa ) d λ W− ({λ}n1 , {z}n1 ) f (λb , ξa ) a=1 n

b=1

× detn [Mκ ({λ} , {z})] detn [Ψ (λj , zk )] ,

(5.1)

in which Mκ is given by (4.9), and W− is the boundary dressing,
W− {λ}n1 1 , {z}n1 2

=

n2 Q

j=1 n1 Q

j=1

×

sinh (zj + ξ− − η/2) sinh (λj + ξ− − η/2) n2 n1 Q Q

a=1 b=1 n2 Q

ab a>b Q Q Sσ ({λ}α+ , {ξ}γ+ ; {λ}α− )−1 , (5.5) × sinh(ξb − λa + η) a∈α+ b∈γ+

in which Sσ ({λ}α+ , {ξ}γ+ ; {λ}α− ) is the function defined in (3.16). Then, using the reduced scalar product formula (3.15) and absorbing the sums over partitions Pξ into auxiliary z integrals5 , we obtain the former
representation. ˇ for large positive boundary field since we Note that the contour contains Γ+ λ ˇ have to absorb the contribution coming from the replacement of the complex root λ as explained in [4].  5

We refer the reader to [49] for technical details.

17

5.2

− The ground state expectation value hσ1+ σm+1 i

Using the same method as for the generating function hQm (κ)i, we can also com− pute the ground state expectation value hσ1+ σm+1 i. It gives Z I n+1 m−1 Y Y dzk Z n+1 X sinh(ξ1 + ξ− − η/2) + − dλn+2 dλk hσ1 σm+1 i = n!(n + 1)! 2iπ n=0 Γ+ ({ξ}m+1 ) 1

×

n+1 Q

m+1 Y

b=1 n Q

a=2

b=1

×

n+1 Q

×

b=1 n+2 Q b=1 n+1 Q b=1

CA

f (zb , ξa ) Y n n+1 sinh(λb − ξ1 ) Y sinh(zb − ξm+1 ) sinh(λb − ξm+1 ) sinh(zb − ξ1 ) b=1 f (λb , ξa ) b=1

sinh(λb − λn+1 + η)

b=1 n+1 Q 

CD k=1

k=1

n+2 Q b=1

sinh(λn+2 − λb + η)

 sinh(zb − λn+1 + η) sinh(λn+2 − zb + η)

sinh(ξ1 + λb − η) sinh(ξ1 + zb − η)

W− {λ}n+2 , {z}n+1 1 1




h
i c {λ}n , {z}n+1 ; ξm+1 detn+1 M 1 1

× detn+2 [Ψ(λj , ξ1 ), Ψ(λj , z1 ), . . . , Ψ(λj , zn+1 )] .

(5.6)


c {λ}n , {z}n+1 ; ξm+1 In this expression, W− denotes the boundary quantity (5.2), M 1
1 c {λ}n , {z}n+1 ; ξm+1 , ∅ defined in (4.26), is a simplified notation for the matrix M 1 1 CD is the contour (5.4), and CA denotes the following contour (A-type contour): 
ˇ , if − ζ/2 < ξ˜− < 0, ] −Λ + η ; Λ + η [ ∪ Γ− λ CA = (5.7) ] −Λ + η ; Λ + η [ , otherwise.

In the homogeneous limit, this results simplifies into − hσ1+ σm+1 i=

×

×

m−1 X n=0

n+1 Y a=1

n+1 Q

sinh ξ− n!(n + 1)!

I

Γ+ (η/2) k=1

sinh(za + η/2) sinh(za − η/2)

dzk 2iπ

m Y n 

sinh(λb − λn+1 + η)

b=1 n+1 Q 

n+1 Y

a=1

n+2 Q b=1

Z n+1 Y

dλk

CD k=1

sinh(λa − η/2) sinh(λa + η/2)

Z

dλn+2

CA

m

n+2 Q

b=1 n+1 Q b=1

sinh(λn+2 − λb + η)

sinh(λb − η/2) sinh(zb − η/2)

, {z}n+1 W− {λ}n+2 1 1




 sinh(zb − λn+1 + η) sinh(λn+2 − zb + η) b=1 h i c({λ}n , {z}n+1 ; η/2) detn+2 [Ψ(λj , η/2), Ψ(λj , z1 ), . . . , Ψ(λj , zn+1 )] , × detn+1 M 1 1 18

6

An alternative resummation

6.1

Bulk type resumations

We have obtained in the previous sections a series representation for the generating function. It happens, just as in the bulk case [52], that it is also possible to derive a totally different representation for hQm (κ)i. The latter is based on a re-summation of its expansion with respect to elementary blocks : m m Y X
11 22 κs Fs , hQm (κ)i = h Ei + κEi i =

(6.1)

s=0

i=1

where X 1 ǫ ǫ ǫ ǫ hE1π(1) π(1) . . . Emπ(m) π(m) i, Fs = s! (m − s)! π∈Σm

ǫi =



2, i = 1...s, 1, i = s + 1...m,

(6.2)

and Σm is the group of permutations of m elements. These elementary blocks were computed in [4]. They can be written as multiple integrals in the half-infinite size limit: Z Y Z Y m s detm [Ψ (λi , ξj )] ǫ1 ǫ′1 m−s ǫm ǫ′m dλi m  hE1 . . . Em i = (−1) dλi
 Q sinh ξij sinh ξ ij − η CA i=s+1 CD i=1 ij

m Q i,j

sinh (λi + ξj − η)

sinh (λij − η) sinh λij − η




s Y

p=1

 

ip −1

Y

j=1

sinh (ξj − λp )

m Y

j=ip +1



sinh (ξj − λp − η)

 ip −1 m m m Y Y sinh (ξi + ξ− − η/2) Y  Y × sinh (ξj − λp + η). (6.3) sinh (ξj − λp ) sinh (λi + ξ− − η/2) i=1

p=s+1



j=ip +1

j=1

The indices ip are defined by  {i : 1 ≤ i ≤ m, ǫ′i = 2} = {i1 < · · · < is } , {i : 1 ≤ i ≤ m, ǫi = 1} = {is+1 > · · · > im } .

(6.4)

For simplicity, we consider from now on the massless regime (although all what follows can be performed in the massive regime as well). In that case, η = −iζ and the contours of integration CD and CA depend on the boundary magnetic field h− as follows: range of ξ− D − contour A − contour ˜ ζ/2 ξ− > 0 CD = R Γ + λ CA = R − iζ
S ˇ −ζ/2 < ξ˜− < 0 CD = R CA = {R − iζ} Γ− λ ˇ = η/2 − ξ− , and that Γ± (z) is a small loop around z of index ±1. We recall that λ 19

We now perform a change of variables in the A-type contours: λ′A = λA − iζ. Moreover, we shift the inhomogeneities around zero δi = ξi + iζ/2, and define  1/2 (ǫi = 1) for A-type, ai = 3/2 − ǫi = (6.6) −1/2 (ǫi = 2) for D-type.

This gives ǫ

hE1π(1)

ǫπ(1)

ǫ

. . . Emπ(m)

ǫπ(m)

i = (−1)[π]

Z

ds λ

CD

Z

CeA

h i e (λi , δj ) detm Ψ Q dm−s λ s (δi , δj ) ik Y

×

Here

m Y

j,k=1

sinh (λj + δk − iaj ζ)

m Y

j=1




sinh (ξ− + δj ) . sinh (λj + ξ− − iaj ζ)

e (λ, δ) = Ψ (λ, δ − iζ/2) = ρ (λ − δ) − ρ (λ + δ) , Ψ 2 sinh 2δ 
ˇ − η if − ζ/2 < ξ˜− < 0, ] −Λ ; Λ [ ∪ Γ− λ CeA = ] −Λ ; Λ [ otherwise.

(6.7)

(6.8) (6.9)

and (−1)[π] is the signature of the permutation. One can compute the sum over permutations (6.2) just as in the bulk case [52]. It leads to the following integral representation for Fs : Proposition 6.1 (Bulk-type resummation) The generating function of the spin correlation function hQm (κ)i can be expressed as hQm (κ)i = with Fs =

1 s! (m − s)!

Z

CD

ds λ

Z

CeA

m X

κs Fs

(6.10)

s=0

h i e (λi , δj ) Y m detm Ψ sinh (ξ− + δj ) Q dm−s λ sinh (λj + ξ− − iaj ζ) s (δi , δj ) j=1

il

where ϕ (z) =

sinh 2z κ−i(z+iζ/2)/ζ . s (z, iζ/2)

(6.30)

This boundary-type resummation yields an integrand not only symmetric in {λ} but also invariant under a reversal of any integration variable λ. These properties allow to compute completely the so called emptiness formation probability at ∆ = 1/2 and for vanishing boundary magnetic fields. It also allows to obtain the leading asymptotics of this quantity at the free fermion point [75].

7

The free fermion point

7.1

Local magnetization at distance m

The first (and the most important) application of the re-summation methods given above is the magnetization profile. This one-point function z hσm i = 1 − 2Dm ∂κ hQm (κ)i |κ=1 ,

can be computed at the free fermion point by using the two different types of resummations for the generation function. We give here both derivations. 7.1.1

First method

In the free fermion point, the nth term of the series (5.1) behaves as (κ − 1)n . Thus, after taking the κ derivative and sending κ to 1, only the n = 1 term survives.

24

At ζ = π/2, we have m X

hQm (κ)i =

n=0

(κ − 1)n

(2π)2n (n!)2

Z

I

n

d λ

CD

n

d z

Γ+ ({ξ})

m Y n Y tanh (λb − ξa )

a=1 b=1

tanh (zb − ξa )

 n  Y sinh (zj + ξ− + iπ/4) sinh 2λj sinh (λj + ξ− + iπ/4) j=1     1 1 ×detn detn . sinh (λj − zk ) s (λj , zk )

×

Thus 6 , for m ≥ 2, z hσm i=

(−1)m π

Z

CD

dλ

sinh (λ − ξ− − iπ/4) [tanh (λ + iπ/4)]2(m−1) . sinh (λ + ξ− + iπ/4) cosh2 (λ + iπ/4)

(7.1)

(7.2)

ˇ we get, for m ≥ 2, Computing, if it exists, the residue at λ z hσm i = −2Θ (h− − 1)

(−1)m + π

Z

h2− − 1 h2m − h− + i tanh (λ + iπ/4) [tanh (λ + iπ/4)]2(m−1) dλ , (7.3) 1 + ih− tanh (λ + iπ/4) cosh2 (λ + iπ/4)

R

where Θ (x) is the Heaviside step function. The standard ∆ = 0 change of variables, eip = − tanh (λ − iπ/4) ,

(7.4)

yields z hσm i

h2 − 1 (−1)m = −2Θ (h− − 1) −2m + π h−

Zπ 0

dp e−2i(m−1)p

e−ip + ih− . eip − ih−

(7.5)

z i displays Friedel type oscillations induced by the boundary. Moreover it Thus hσm decays as 1/m when m → +∞:   2 (−1)m h− 1 z , m >> 1. (7.6) hσm i = +O πm h2− + 1 m2

√ Here we recover the results of [69], since we have h− = 2α− in Bilstein’s notations. When | h− |→ ∞ we conclude from (7.5) that the first site is totally decoupled z i from the others as hσm m≥2 goes to its bulk average value 0. Actually in this limit the model is in correspondence with a Kondo model with a spin 1/2 impurity [29]. 6

We do not give hσ1z i as it corresponds to ∂κ hQ1 (κ)i without taking the lattice derivative.

25

But in this case the impurity is completely screened, and the overall magnetization in zero. We also recover from (7.5), just as expected from the spin reversal symmetry, that z hσm i = 0 when the boundary field vanishes. Actually this observation holds for all ∆ as inferred from the structure of the monodromy matrix (2.6) on the first site. When one sets ξ1 = η/2 and ξ− = 0 then it acts as a diagonal matrix on the first site, a sign of the claimed decoupling. 7.1.2

Second method

Starting from the re-summation formula (6.11) of Proposition 6.1, we implement the simplification due to ζ = π/2. If we perform the change of variables eip = − tanh (λ − iπ/4)

(7.7)

in (6.11) at ζ = π/2 then we arrive at m(m−1)    m Z  s Z Y dpj eipj + e−ipj Y dpj eipj + e−ipj (2i) 2 Fs = s! (m − s)! 2π eipj − ih− 2π e−ipj − ih−

j=1 C

j=s+1

D

×

×

s Y

m Y

k=1 j=s+1 m Y 

j,k=s+1 j>k




 e−ipk − eipj (sin pj + sin pk )

CA s Y

j,k=1 j>k




 e−ipj − e−ipk (sin pj − sin pk )


 eipj − eipk (sin pk − sin pj ) .

(7.8)

The contours of integration are CA = ] 0 ; π [ ,

h− ≥ −1,

CD = ] 0 ; π [ ,

h− ≤ 1,

CA = ] −π ; 0 [ , h− ≥ −1,

Once we introduce the function


C A = ] 0 ; π [ ∪ Γ− e−ip = −ih− , h− < −1, (7.9)
CA = ] −π ; 0 [ ∪ Γ+ eip = −ih− , h− < −1, (7.10)
CD = ] 0 ; π [ ∪ Γ+ eip = −ih− , h− > 1. (7.11)

θκ (p) =



κ p ∈ CD , 1 p ∈ CA ,

(7.12)

we can re-sum the terms Fs into a single m-fold integral for hQm (κ)i: hQm (κ)i =

(2i)

m(m−1) 2

m! ×

m  Y dpj

Z

CA ∪ CD j=1 m Y −ipj

e

j>k≥1

26

eipj + e−ipj θκ (pj ) ipj 2π e − ih−




− e−ipk (sin pj − sin pk ) .

(7.13)

One can then express the generating function as a single determinant hQm (κ)i = detm [U (κ)] , Z eipk − (−1)k e−ipk 1 dp θκ (p) e−ip(j−1) . Ujk (κ) = 2π eip − ih−

(7.14) (7.15)

CA ∪ CD

To simplify this result we add to each row of U (κ) the next one multiplied by ih− : e (m) (κ), hQm (κ)i = detm U Z   ip(k−j) k −ip(j+k) e (m) (κ) = 1 U , j < m, dp θ (p) e − (−1) e κ jk 2π CA ∪ CD

e (m) (κ) = Umk (κ). U mk

It is easy to see that Qm (1) = 1. Computing the first derivative of the generating function one recovers the result already obtained from the series (7.5):

z hσm i

(−1)m = π

Z

dp e−2ip(m−1)

CD

e−ip + ih− . eip − ih−

h2 − 1 (−1)m = −2 −2m Θ (h− − 1) + π h−

7.2

Zπ

dp e−2ip(m−1)

0

e−ip + ih− . eip − ih−

(7.16)

Local density of energy

The local density of energy is another interesting quantity [61] that one can evaluate for the XX0 chain: x x y y Em = hσm σm+1 + σm σm+1 i .

(7.17)

Starting from (6.16) and using the same technique as in the previous sub-section one easily obtains the following representation for the density of energy: Em = −2i

Z

CD

dp 2π

Z

CA

 dq  i(p+q) e + 1 detm+1 [V (p, q)] . 2π

27

(7.18)

The entries of V (p, q) read Z  ′  1 ′ Vjk (p, q) = dp′ eip (k−j) − (−1)k e−ip (j+k) = δjk , 2π CA ∪ CD

1 Vm−1k (p, q) = 2π

Z

ip′ k ′ −ip′ (m−2) e

dp e

CA ∪ CD

′

− (−1)k e−ip k = eip′ − ih−



j < m − 1, h− i

k−m+1 Θ (k − m + 1) ,

eipk − (−1)k e−ipk , eip − ih− eiqk − (−1)k e−iqk Vm+1k (p, q) = e−iqm . eiq − ih− Vmk (p, q) = e−ipm

(7.19)

Finally, the integrals over CA can be represented as Z Z Z . − = CD

CA ∪ CD

CA

(7.20)

Accordingly, Em reduces to a sum of two 3 × 3 determinants: 1 ih− −h2− Em = −2i F (m − 1, m − 1) F (m − 1, m) F (m − 1, m + 1) 0 1 ih− 1 ih− −h2− −2i F (m, m − 1) F (m, m) F (m, m + 1) 0 0 1

where

1 F (j, k) = 2π

Z

dp e−ipj

CD

eipk − (−1)k e−ipk . eip − ih−

,

(7.21)

(7.22)

The computation of these determinants yields Em = −2i (F (m, m) − F (m − 1, m + 1)) − 2h− (F (m, m − 1) − F (m − 1, m)) , (7.23) or, more explicitly, Em

4 2i = − + (−1)m π π

Z

dp e−ip(2m−1)

CD

e−ip + ih− . eip − ih−

(7.24)

The constant term reproduces the bulk result. The influence of the boundary appears in the oscillating term. In the m → ∞ limit the local density of energy behaves as   2 1 4 2 m 1 − h− Em = − + +O (−1) . (7.25) π πm m2 1 + h2−

28

7.2.1

+ Two-point function hσm+1 σ1− i

The preceding method can be successfully applied to compute other types of twopoint functions (like boundary-bulk two point functions), here we give the example + of hσm+1 σ1− i. In the free fermion point, after the usual change of variables and some straightforward but tedious calculations, one obtains from (6.18) a simple determinant formula for this object. + hσm+1 σ1− i = − i detm+1 V +− , (7.26)  π 2π Z Z   1  +− −  dp eip(k−j−1) − (−1)k e−ip(j+1+k) , j ≤ m − 1, Vjk = 2πi +− Vmk =

Z

1 2π

0

π

dp e−ipm

eipk − (−1)k e−ipk , eip − ih−

dp e−ipm

eipk − (−1)k e−ipk eip − ih−

CD +− Vm+1k

1 = 2π

Z

CA

(7.27)

Computing the integrals in the first m − 1 rows and using the fact that sum of the last two rows is δk,m+1 we reduce this representation to a determinant of a m × m matrix i(−1)m detm V˜ +− , 2π m (j + 1)(1 − (−1)k ) + k(1 + (−1)k ) , =(1 + (−1)j−k ) (j + 1)2 − k2 Z eipk − (−1)k e−ipk = dp e−ipm , eip − ih−

+ hσm+1 σ1− i = +− V˜jk +− V˜mk

(7.28) j ≤ m − 1, (7.29)

CD

This determinant can be computed for any value of m. However the result is quite different for m odd or even. The details of the computation are given in Appendix

29

B, here we give only the final result for the two point function 2  2a−1 2a−1 Y Γ(j) Γ(a − 12 )Γ3 (a + 21 ) 2 + −  hσ2a σ1 i = − i 2  π Γ(a)Γ(a + 1) Γ(j + 12 ) j=1

×

Z

CD + hσ2a+1 σ1− i = −

×

  e−ip(2a−1) dp P2a−1 cos p ip e − ih−

22a π2 Zπ 0



(7.30)

2

2a Y

Γ(j)  Γ(a + 32 )Γ3 (a + 12 ) Γ(a)Γ(a + 1) Γ(j + 12 ) j=1



dq cos q

Z

CD

  e−2ipa , dp P2a cos(q − p) ip e − ih−

(7.31)

where Pm (x) are Legendre polynomials. Asymptotic analysis of these expression yields the following leading behavior of + hσm+1 σ1− i   1 + − m − 34 (7.32) 1 + O( √ ) hσm+1 σ1 i =(−1) A(h− ) m m s   Z ∞  dt −4t 1 2 1 A(h− ) = e − . (7.33) exp 4 0 t π(1 + h2− ) cosh2 t

8

Conclusion

In this article we have obtained different types of physical correlation functions of the open XXZ chain from re-summations of the multiple integrals derived in [4] for the elementary blocks. At the free-fermion point, we were able to use these representations to derive explicit results such as the formula for the density of energy profiles, a quantity arising in the study of quantum entanglement in spin chains [61]. Just as in the bulk case, the question concerning the asymptotic behavior of the correlation functions outside of the free-fermion point naturally arises. The problem is of the same order of difficulty as in the bulk model. Indeed, the multiple integrals differ from their bulk counterparts only by factors due to the Z2 symmetry λ → −λ and the presence of boundary fields. One could also wonder if it would be possible to tell something about the dynamical or temperature correlation functions. It seems that this generalization is highly non-trivial. Finally, we would like to stress that our expressions also simplify at other particular points such as ∆ = 1/2. For instance, when ∆ = 1/2, one can already compute completely the so-called emptiness formation probability when h− = 0 [75].

30

Appendices + Asymptotic of the two-point function hσm+1 σ1− i

A

In the last section we obtained a determinant representation (7.29) for the two+ point function hσm+1 σ1− i. This determinant can be computed for any value of m. However the results are quite different for m odd or even. If m is odd: m = 2a − 1, the result can be written in the following form 

detm V˜ +− =2m+1 π m−3 

2

m Y

Γ(j)  Γ(a − 12 )Γ3 (a + 12 ) Γ(a)Γ(a + 1) Γ(j + 21 ) j=1

a X Γ(a − b + 12 )Γ(a + b − 12 ) × Γ(a − b + 1)Γ(a + b) b=1

×

Zπ

dp e−ip(2a−1)

0

eip(2b−1) + e−ip(2b−1) eip − ih−

(A.1)

If m is even: m = 2a, the result is quite similar but there is a very important difference: 2  m Y Γ(a + 32 )Γ3 (a + 12 ) Γ(j)  detm V˜ +− =2m+1 π m−3  Γ(a)Γ(a + 1) Γ(j + 21 ) j=1 × ×

a X b=1

Zπ 0

Γ(a − b + 12 )Γ(a + b + 21 ) b (b + 21 )(b − 12 ) Γ(a − b + 1)Γ(a + b + 1)

dp e−2ipa

e2ipb − e−2ipb eip − ih−

(A.2)

Asymptotic analysis of the prefactors in (A.1) and (A.2) is rather simple, namely: 

2 Γ(a − 12 )Γ3 (a + 12 ) Γ(j)   Γ(a)Γ(a + 1) Γ(j + 21 ) j=1 √   Z ∞    dt −4t 1 1 2π − 1 1 4 (A.3) e − = m m exp 1 + O( ) 2 4 0 t m cosh2 t  2 2a Y Γ(a + 32 )Γ3 (a + 12 ) Γ(j)   Γ(a)Γ(a + 1) Γ(j + 21 ) j=1 √     Z ∞  1 dt −4t 1 1 2π 3 1 + O( ) (A.4) e − = m+1 m 4 exp 2 4 0 t m cosh2 t 2a−1 Y

31

For m odd the sum in (A.1) can be rewritten as follows: a  X Γ(a − b + 12 )Γ(a + b − 21 )  −2ip(a−b) e + e−2ip(a+b−1) Γ(a − b + 1)Γ(a + b) b=1

=

2a−1 X l=0

Γ(l + 12 )Γ(2a − l − 21 ) −2ipl e , Γ(l + 1)Γ(2a − l)

(A.5)

and can be represented in terms of the Legendre polynomials Pm (cos p) 2a−1 X l=0

Γ(l + 21 )Γ(2a − l − 12 ) −2ipl Γ(2a − 21 )Γ( 12 ) 1 3 e = − 2a; e−2ip ) 2 F1 ( , 1 − 2a; Γ(l + 1)Γ(2a − l) Γ(2a) 2 2 =πe−ipm Pm (cos p)

(A.6)

Using Laplace asymptotic formula,     1    2 1 2 1 π , cos p m + Pm (cos p) = − +O 3 πm sin p 2 4 m2

(A.7)

for the remaining integral we obtain the following leading term Zπ 0

  r π 1 e−ipm 1 + O( √ ) dp Pm (cos p) ip = −i e − ih− m m(1 + h2− )

(A.8)

Assembling all the contributions we obtain the following leading term for the twopoint function (for m odd): s   Z ∞    dt −4t 1 1 2 1 + − m − 34 e − exp hσm+1 σ1 i = (−1) m 1 + O( √ ) 4 0 t m π(1 + h2− ) cosh2 t (A.9) The same result holds for m even, but the derivation is a little bit more tricky. The sum in (A.2) can be once again rewritten in a more simple way Zπ 2ipb − e−2ipb Γ(a − b + 12 )(h− )Γ(a + b + 12 ) b −2ipa e dp e eip − ih− (b + 12 )(b − 12 ) Γ(a − b + 1)Γ(a + b + 1) b=1 0 ! Zπ a Γ(a − b + 12 )Γ(a + b + 12 ) e2ip(b−a) 1 1 X 1 dp = + 2 eip − ih− b + 12 b − 12 Γ(a − b + 1)Γ(a + b + 1)

a X

b=−a

=i

a X

b=−a

=iπ

Zπ 0

0

1 2 )Γ(a

1 2)

Γ(a − b + +b+ Γ(a − b + 1)Γ(a + b + 1)

dq cos q

Zπ 0

Zπ

dq e−2iqb cos q

0

dp Pm (cos(q − p))

0

e−imp eip − ih−

32

Zπ

dp

e2ip(b−a) eip − ih− (A.10)

where we introduced an additional integral to be able to express the result once again in terms of the Legendre polynomials. Asymptotic analysis of these integrals gives iπ

Zπ 0

dq cos q

Zπ 0

 π 3 e−imp 2i 2 q dp Pm (cos(q − p)) ip =− e − ih− m 1 + h2−



 1 1 + O( √ ) , m

(A.11) and it leads once again to the same leading term (A.9) for the two-point function.

Acknowledgments J.M. M., N. S. and V. T. are supported by CNRS. N. K., K.K. K., J.M. M. and V. T. are supported by the ANR programm GIMP ANR-05-BLAN-0029-01. N. K., G. N. and V. T. are supported by the ANR programm MIB-05 JC05-52749. N. S. is supported by the French-Russian Exchange Program, the Program of RAS Mathematical Methods of the Nonlinear Dynamics, RFBR-05-01-00498, Scientific Schools 672.2006.1. N. K., G. N. and N. S. would like to thank the Theoretical Physics group of the Laboratory of Physics at ENS Lyon for hospitality, which makes this collaboration possible.

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