University of Amsterdam
Institute for Theoretical Physics
Bachelor’s project
Exact time evolution in the one-dimensional
attractive Bose gas
Author:
Yuri van Nieuwkerk
Student ID:
6179703
Supervisor:
Prof. Dr. Jean-S´
ebastien Caux
Second corrector:
Dr. Davide Fioretto
April 1, 2014 - July 30, 2014
15 ECTS
Contents
1 Introduction 2
2 Theory 3
2.1 Basics of the Lieb-Liniger model . . . 3 2.2 The attractive Lieb-Liniger model . . . 3 2.3 Density operator: expectation values . . . 5
3 Density form factors 6
3.1 N-state . . . 6 3.2 N-1:1-state . . . 7
4 Exact time evolution 10
4.1 N-state . . . 10 4.2 N-1:1-state . . . 17
5 Conclusions and perspective 22
Appendix A Generic density form factor 23
A.1 Definitions . . . 23 A.2 Inner-string blocks . . . 24 A.3 Complete U-matrix . . . 27
Appendix B N-state determinant 27
Appendix C N-1:1-state determinant 29
Abstract
Numerical work by Kaminishi et al. on time evolution in the one dimensional repulsive Bose gas is extended to the attractive scenario, using analytical techniques. Time evolution of the density operator’s expectation values is described, starting from a Bethe Ansatz. Analytical results are obtained for the density operator’s matrix elements, enabling the calculation of its time-dependent expectation values. For so-called N -strings in a localized initial state, these calculations are performed, leading to formulas describing the decay of such quasi-particles. Techniques to treat more complicated initial states in quasi-ionized form are investigated, offering an outlook on future theoretical work.
1 INTRODUCTION
1
Introduction
In a very recent article [6], Kaminishi, Sato and Deguchi have described the equilibration of localized density distributions in the repulsive one-dimensional Bose gas. Starting from a narrow spatial notch in the expectation value of the density operator, the authors have used numerical methods to describe the ensuing time-evolution of the system, producing a number of snapshots of its evolving density profile. The present paper extends this work to the attractive one-dimensional Bose gas, focusing on density peaks, instead of notches. Analytical results, rather than numerical ones, have been obtained for so-called N -strings, describing the density operator’s expectation values at all times and positions, leading to similar snapshots that show the evolution of a density profile. To cover more complicated strings, several techniques have been explored, possibly leading to further extensions to the work of Kaminishi, Sato and Deguchi in the near future.
One-dimensional bosonic systems have received considerable attention in the past few years, due to their combined theoretical importance and experimental realizability. The advancements of both cold-atom exper-iments and optical waveguides have paved the way for almost ideal realizations of the one-dimensional Bose gas. Harmonic confinement techniques such as described in [12] and [2] can create “cigar-shaped” traps, to which the techniques explained in this paper may be applied with only little theoretical adaptations, as shown in [13], for example. Other instances include “doughnut-shaped”, optically induced potentials in which a quasi-one-dimensional r´egime can be obtained [18]. Alongside with these experimental developments, the theoretical description of the attractive one-dimensional Bose gas has been explored exhaustively, notably by Caux and Cal-abrese, whose paper [3] is one of the primary sources for this project. The attractive one-dimensional bose gas is thus an exceptionally valuable test for the techniques of integrability, since it is both realizable experimentally and solvable analytically.
This analytical approach is reviewed in Section 2, describing the relevant aspects of the Bethe Ansatz technique to find the system’s eigenvalues and eigenstates. Characteristics of the attractive bose gas, such as the formation of bound states, are discussed. The density operator, whose expectation values will be calculated, is introduced alongside with its matrix elements at x=0, t=0, often referred to as form factors. These form factors are investigated in Section 3, leading both to generic but complicated expressions in terms of momentum-eigenvalues and to specific closed-form descriptions for important classes of eigenstates, so-called N -states and N -1:1-states. Section 4 uses these formulas to exactly describe the time evolution of a number of localized density distributions, forming the main result of this Bachelor’s project. Next, techniques are described to treat more complicated states and form factors, offering a tantalizing outlook on possible future theoretical work.
2 THEORY
2
Theory
2.1
Basics of the Lieb-Liniger model
The one-dimensional Bose gas was first described by Elliott Lieb and Werner Liniger [10] as a system of N bosons, interacting only at the point of contact. The corresponding Hamiltonian, setting ~ = 2m = 1, describes the kinetic energy of N free particles in its first term:
H = −~ 2 2m N X j=1 ∂2 ∂x2 j + 2cX hj,ki δ (xj− xk) . (2.1)
Only a zero range delta function in the second term makes the particles interact. It is therefore no coincidence that the so-called Bethe Ansatz [1] provides a solution to the accompanying Schr¨odinger equation: an important part of the Bethe Ansatz wavefunction consists of plane waves, which solve the free particle Hamiltonian. The Bethe Ansatz wavefunction reads
Ψ (x1. . . xN|λ1. . . λN) = X P ∈SN
A(P )eiPNj=1λP (j)xj, (2.2)
where P ranges over all possible permutations of the N indistinguishable particles and λi are the system’s pseudo-momenta, often called rapidities. Only in the coefficients A(P ) does the interaction show itself: as particles tunnel accross each other, the delta function’s interaction parameter c leads to a mutual scattering phase. This phase can be described by integrating the Schr¨odinger equation over an -environment around the interaction point [7], eventually leading to the following coefficients:
A(P ) = sgn(P ) Y 1≤j<k≤N
sgn(xk− xj)sgn(λk− λj)e
i
2sgn(xk−xj)φ(λP (k)−λP (j)), (2.3)
containing the scattering phase
φ(λ) = 2 arctanλ
c. (2.4)
Completeness and uniqueness of 2.2 have been proven in, among others, [4] and [9]. By applying periodic boundary conditions, an implicit relation between all rapidities in the system is obtained in the form of the following Bethe equations:
eiλjL= (−1)N −1e−iPkφ(λj−λk)=Y
k6=j
λj− λk+ ic λj− λk− ic
. (2.5)
Putting these equations into logarithmic form reveals their explicitly quantized nature:
λjL + X
k
φ(λj− λk) = 2πIj, (2.6)
since these equations map an evenly spaced set of quantum numbers {Ij} to a discrete, but not necessarily evenly spaced, set of rapidties {λj}. The quantum numbers Ijare integers if N is odd and half-odd integers if N is even. The Bethe equations show that the essentially local delta-interaction between mere nearest-neighbours causes all rapidities in the system to be in a strong causal contact, even if the associatd particles are highly delocalized.
The set of rapdities obtained from 2.6 can be proven to be real in the case of repulsive interactions, where c > 0. The case of attractive interactions, however, does allow for complex rapidities. This leads to a number of interesting effects.
2.2
The attractive Lieb-Liniger model
In the attractive case, the interaction paramter becomes negative, and a new, positive parameter is introduced, via −c = ¯c > 0. The Bethe equations then become
eiλαL= Y
β6=α
λα− λβ− i¯c λα− λβ+ i¯c
2.2 The attractive Lieb-Liniger model 2 THEORY
Following [3], the effects of introducing a complex rapidity λα = λ + iη, (λ, η ∈ R) can be investigated. Inserting this rapidity into the Bethe equations and taking the limit as L → ∞
lim L→∞e iλαL= lim L→∞e iλL−ηL= lim L→∞ Y β6=α λα− λβ− i¯c λα− λβ+ i¯c = ( 0, η > 0 ∞, η < 0. (2.8)
For these equations to be satisfied, the case of η > 0 requires the numerator in the third expression to approach zero like O e−ηL. It follows that there must exist some rapidity λβ = λα− i¯c + O e−ηL. See (a) in Figure 1. Similarly, the case η < 0 requires the denominator to vanish like O e−|η|L, prompting the existence of some rapidity λβ0 = λα+ i¯c + O e−|η|L, as depicted at (b) in Figure 1. The proof by Vladimirov [19] of the invariance of solutions under complex conjugation shows that we are dealing with so-called strings, lying symmetrically about the real axis.
Re Im
(a)
(b)
Figure 1: Symmetric string solutions in the complex plane. Visible are a 4-, 5- and 2-string, drawn in red, blue and green respectively. This 11-particle state will be denoted as a 5:4:2-state.
The strings described above can be parametrized in the following way:
λj,aα = λjα+ i¯c
2 (j + 1 − 2a) + iδ j,a
α , a = 1, . . . j, (2.9)
where j denotes the string length, a is an inner-string index and α is a unique string-label. The δαj,a are exponentially small deviations from the string structure, which will be treated to leading order in Appendix A. Recent work by Sykes et al. [17] has described these string deviations in more detail.
Whilst the Bethe Equations relate all rapidities in the system to one another, a reduced form of these equations has been derived by Caux and Calabrese [3], relating the string centers of the above string solutions:
Reduced Bethe Equations. For quantum numbers Iαj, which are integers if N is odd and half-odd integers if N is even, the string centers λjα are given by the implicit relation
jλjαL − X (k,β) Φjk λjα− λ k β = 2πI j α., (2.10) where Φjk(λ) = (1 − δjk) φ|j−k|(λ) + 2φ|j−k|+2(λ) + . . . + 2φj+k−2(λ) + φj+k(λ) (2.11) φj(λ) = 2 arctan 2λ j¯c . (2.12)
Armed with these string center solutions, the energy and momentum of string states can be calculated, up to the exponentially small corrections δαj,a. For a string’s total impulse, its symmetry in the complex plane leads to the simple expression
P(j,α)= jλjα. (2.13)
The string’s total energy has been calculated by McGuire [11], in as early as 1964:
E(j,α)= j λjα 2 − ¯c 2 12j(j 2− 1). (2.14)
Since this energy diverges negatively like ∼ j3, the usual thermodynamic limit leads to infinitely negative energies. The limit to consider is therefore the one where L → ∞, while N remains fixed.
2.3 Density operator: expectation values 2 THEORY
2.3
Density operator: expectation values
The following chapters focus on obtaining exact expressions for the expectation value of the following density operator:
ρ(x, t) =X j
δ (x − xj(t)) , (2.15)
which returns a value of 1 at all particle positions xj, whilst yielding zero everywhere else. The string states on which this operator will be left to work have inital states |Ψ0i, which will be decomposed into momentum eigenfunctions. These eigenfunctions are characterized by a set {λ} of N rapidties, as dictated by the Bethe Equations:
|Ψ0i = X
{λ}B.E.
k({λ}) |{λ}i . (2.16)
Using the definition of the quantum expectation value, the initial state can be propagated in time and shifted in space to reduce the problem to the expectation value of ρ(0, 0):
hρ(x, t)i = h ˜Ψ0| ρ(x, t) | ˜Ψ0i k ˜Ψ0k2 = h ˜Ψ0| U ∗(t)T∗(x)ρ(0, 0)T (x)U (t) | ˜Ψ 0i k ˜Ψ0k2 , (2.17) where U (t) = e−i ˆHt; (2.18) T (x) = e−i ˆpx. (2.19)
Since the basisvectors |{λ}i are eigenvectors of both U (t) and T (x), we let these operators work on their eigenstates and fill in the corresponding eigenvalues, using the identities 2.13 and 2.14. Denoting, for example, by E({µ}) the energy eigenvalue of the state characterized by the set of rapidities {µ}, we can then express the density expectation value as
hρ(x, t)i = 1 k ˜Ψ0k2 X {µ}B.E. X {λ}B.E.
k∗({µ})k({λ})ei[E({µ})−E({λ})]tei[P ({µ})−P ({λ})]xh{µ}| ρ(0, 0) |{λ}i . (2.20)
The complete evaluation of such sums will be carried out in Section 4, for N -strings and N − 1-strings. But before this can be done, an explicit expression must be found for h{µ}| ρ(0, 0) |{λ}i, the matrix elements of the density operator at x = 0, t = 0. These matrix elements are referred to as the system’s density form factors and will be denoted by F ({λ}, {µ}), for simplicity. They will be calculated for N -states and N -1:1-states in Section 3. But before this is done, it is interesting to point out a general property of these form factors, which has been formulated by Izergin et al. [5] in the following theorem:
Form Factor Reduction Theorem. If two rapidities µ1λ1get close to each other, the form factor can be reduced as follows: h{λj}Nj=1|ρ(x)| {µj}Nj=1i λN→µN −−−−−→ −i¯c µN − λN N −1 Y j=1 µN− µj− i¯c µN − µj λj− λN − i¯c λj− λN − N −1 Y j=1 µN − µj+ i¯c µN − µj λj− λN+ i¯c λj− λN λN=µN × h{λj}N −1j=1 |ρ(x)| {µj}N −1j=1 i + Regular expressions. (2.21) Here, “Regular epressions” designates all terms in the form factor that do not diverge as λN → µN. The above theorem will prove useful in Section 4.
3 DENSITY FORM FACTORS
3
Density form factors
In order to compute expressions for the various form factors, which will be denoted by F ({µ}, {λ}) for conciseness, the work of Nikita Slavnov is an important source. From [15], it can be concluded that the form factor between two states, characterized by generic sets of rapidities {µ} and {λ} respectively, can be expressed as
F ({µ}, {λ}) = (Pµ− Pλ) N Y j=1 Vj+− Vj− N Y j,k=1 λj− λk+ ic µj− λk detNU˜jk(λp) Vp+− Vp− , (3.1) where Vj±= N Y m=1 µm− λj± ic λm− λj± ic , (3.2) ˜ Ujk(λp) = δjk+ i µj− λj Vj+− Vj− N Y m6=j µj− λj λm− λj (K(λj− λk) − K(λp− λk)) . (3.3)
and λpis a free parameter. In Appendix A this expression is used to calculate specific form factors for an N -state and an N -1:1--state. A set of notes by Jacopo de Nardis was taken as a starting point for these calculations. It has been adapted to an accessible form and a number of corrections have been made. The results of these appendices will be transformed into closed-form expressions in the following sections, before they can be used to compute expectation values in Section 4.
3.1
N-state
From Appendix A, it can be concluded that
FN({µ}, {λ}) = iN −1¯c C[λ, µ]det N U ({µ}, {λ}), (3.4) with C[λ, µ] = QN (j,α,a),(k,β,b), (k,β,b)6=(j,α,a+1) λj,aα − λk,bα − i¯c QN (j,α,a),(k,β,b) µj,aα − λk,bβ Y (j,α) ij−1. (3.5)
Using B.15 from Appendix B, the determinant can be filled in:
det Uij= M −1Y b=2 Vb1 ¯ c hXM a=1 Vai 2 . (3.6)
This expression is simplified one step further by observing the fact that
N X a=1 Va = N X a=1 QN m=1(µm− λa) QN m6=a(λm− λa) = (µ − λ) N X a=1 N Y m6=a µ − λ + i¯c(a − m) i¯c(a − m) = (µ − λ) N = Pµ− Pλ (3.7) Inserting the determinant thus obtained into 3.4 allows an explicit computation of the form factor for an N-string: FN(µ, λ) = (−1)N −1¯c QN (j,α,a),(k,β,b), (k,β,b)6=(j,α,a+1) λj,a α − λk,bα − i¯c QN (j,α,a),(k,β,b) µj,aα − λk,bβ det N U ({µ}, {λ}) (3.8) = (−1)N −1¯c QN a,b,b6=a+1(λa− λb− i¯c) QN a,b(µa− λb) N −1Y b=2 Vb1 ¯ c h Pµ− Pλ i2 (3.9) = (−1)N −1 QN a=1 h Qa b=1(λb− λa+ i¯c)Q N b=a+2(λb− λa+ i¯c) i QN b=1Qb "N −1 Y b=2 Qb Qb−1 l=1(λl− λb) QN l=b+1(λl− λb) # h Pµ− Pλ i2 , (3.10)
3.2 N-1:1-state 3 DENSITY FORM FACTORS
in which the expression Qb =Q N
a=1(µa− λb) was used. Next, cancel the Qb’s, use λb = λb−1− i¯c and set β = b − 1: FN(µ, λ) = (−1)N −1 Q1QN N Y a=1 a Y b=1 (λb− λa+ i¯c) N Y b=a+2 (λb− λa+ i¯c) "N −1 Y b=2 1 Qb−1 l=1(λl− λb−1+ i¯c)Q N l=b+1(λl− λβ+ i¯c) # h Pµ− Pλ i2 (3.11) =(−1) N −1 Q1QN (i¯c)2N −1 N −1 Y b=1 (N − b) N Y b=1 (N + 1 − b)hPµ− Pλ i2 (3.12) =i(¯c) 2N −1 Q1QN N Γ2(N )hPµ− Pλ i2 . (3.13)
Also, investigate the denominator of 3.13.
Q1QN = N Y m=1 (µm− λ1) N Y m=1 (µm− λN) (3.14) = N Y m=1 µ − λ −i¯c 2(2m − 2) N Y m=1 µ − λ +i¯c 2(2N − 2m) (3.15) = ¯c2N N Y m=1 µ − λ ¯ c 2 + (N − m)2 ! . (3.16)
Reversing the index m and inserting this expression into 3.13, compactly expresses the form factor as
FN(µ, λ) = i ¯ c N Γ2(N ) [Pµ− Pλ] 2 QN −1 m=0 µ−λ ¯ c 2 + m2 (3.17) ≈ icN [P¯ µ− Pλ] 2 (µ − λ)2 πµ−λ¯c sinhhπµ−λc¯ i . (3.18) (3.19) It can be concluded that the expression for the N-string density form factor reads
FN(µ, λ) = i
πN3(µ − λ) sinhhπµ−λ¯c i .
(3.20)
The factor of i is unexpected: ˆρ is a hermitian operator, and since its off-diagonal elements are clearly invariant under exchange of µ and λ, they are expected to be real in order to satisfy the hermitian property.
3.2
N-1:1-state
In order to compute the form factor between two N -1:1-states with rapidites {µ}, M and {λ}, Λ respectively, expression A.33 can be used again:
FN −1:1({µ}, M, {λ}, Λ) = iN −1¯c C[λ, µ]det
N U ({µ}, M, {λ}, Λ). (3.21)
Inserting C.21 and A.34 gives
FN −1:1({µ}, M, {λ}, Λ) = i(−1)N QN (j,α,a),(k,β,b), (k,β,b)6=(j,α,a+1) λj,a α − λk,bα − i¯c QN (j,α,a),(k,β,b) µj,aα − λk,bβ | {z } (a) 1 ¯ c(Pλ− Pµ) 2 ( M −2 Y b=2 Vb) | {z } (b)
[2VΛ− ic(VΛ−ic− VΛ+ic)] .
3.2 N-1:1-state 3 DENSITY FORM FACTORS
First, rewrite (a):
QN (j,α,a),(k,β,b), (k,β,b)6=(j,α,a+1) λj,aα − λk,bα − i¯c QN (j,α,a),(k,β,b) µj,aα − λk,bβ = N −1 Y a=1 QN −1 b6=a+1λ a− λb− i¯c QN −1 b=1 µa− λb | {z } a6=N 6=b N −1 Y a=1 λa− Λ − i¯c µa− Λ | {z } b=N,a6=N N −1 Y b=1 Λ − λb− i¯c M − λb | {z } a=N,b6=N Λ − Λ − i¯c M − Λ | {z } a=b=N . (3.23)
Combine the a 6= N 6= b-factor with (b) in 3.22:
N −1 Y a=1 QN −1 b6=a+1λ a− λb− i¯c QN −1 b=1 µa− λb 1 ¯ c(Pλ− Pµ) 2( N −2 Y b=2 Vb) (3.24) =1 ¯ c 1 ˜ Q1Q˜N −1 (i¯c)2N −3(N − 1) Γ2(N − 1) [Pµ− Pλ] 2 N −2 Y b=2 M − λb Λ − λb (3.25)
which has been calculated completely analogously to 3.9, with uppermost factors omitted in every product. Next, redefine ˜Qb=QN −1a=1 µa− λb, and calculate
˜ Q1Q˜N −1= ¯c2N −2[(N − 2)!] 2 µ − λ ¯ c 2 N −2 Y a=1 " µ − λ ¯ ca 2 + 1 # . (3.26) Inserting gives 1 ¯ c (i¯c)2N −3(N − 1) Γ2(N − 1) [Pµ− Pλ] 2 ¯ c2N −2[(N − 2)!]2µ−λ ¯ c 2 QN −2 a=1 µ−λ ¯ ca 2 + 1 N −2 Y b=2 M − λb Λ − λb (3.27) =i 2N −3 ¯ c (N − 1) [Pµ− Pλ] 2 (µ − λ) π sinhhµ−λc¯ πi N −2 Y b=2 M − λb Λ − λb (3.28)
Combining with 3.22 and 3.23 leads to
FN −1:1({µ}, M, {λ}, Λ) = (−1)Ni2N −3 | {z } =i (N − 1) [Pµ− Pλ]2 (µ − λ) π sinhhµ−λ¯c πi [2V
Λ− ic(VΛ−ic− VΛ+ic)] N −2 Y b=2 M − λb Λ − λb N −1 Y a=1 λa− Λ − i¯c µa− Λ N −1 Y b=1 Λ − λb− i¯c M − λb −i¯c M − Λ (3.29)
Finally, insert the following functions:
VΛ= QN k=1(µk− Λ) QN −1 k=1(λk− Λ) (3.30) VΛ−i¯c= QN k=1(µk− Λ + i¯c) QN −1 k=1(λk− Λ + i¯c) 1 i¯c (3.31) VΛ+i¯c= QN k=1(µk− Λ − i¯c) QN −1 k=1(λk− Λ − i¯c) 1 −i¯c. (3.32)
After a number of algebraic steps, this leads to the following expression for the form factor of an N − 1 : 1 string:
3.2 N-1:1-state 3 DENSITY FORM FACTORS FN −1:1({µ}, M, {λ}, Λ) =iπ (Pµ− Pλ)2 µ − λ (N − 1) sinhhπµ−λc¯ i Λ − λ −i¯cN2 M − λ +i¯cN 2 − i¯c Λ − λ +i¯cN2 M − λ −i¯cN 2 + i¯c " Λ − λ + i¯cN 2 − i¯c Λ − λ − i¯cN2 Λ − µ −i¯cN 2 Λ − µ +i¯cN2 − i¯c 1 + i¯c M − Λ +Λ − λ − i¯cN 2 + i¯c Λ − µ −i¯cN2 + i¯c Λ − µ +i¯cN2 Λ − λ + i¯cN2 1 − i¯c M − Λ − 2 # . (3.33)
where the squared difference in total momenta is given by (Pµ−Pλ)2= (N −1)2(µ − λ) 2
+(N −1) (µ − λ) (M − Λ)+ (M − Λ)2.
To check this result, it can be verified that as Λ and M get close, the part of FN −1:1({µ}, M, {λ}, Λ) that scales like 1 M −Λ is equal to −i¯c M − Λ " M − µ +i¯cN2 M − µ −i¯cN 2 + i¯c Λ − λ −i¯cN2 Λ − λ +i¯cN 2 − i¯c − Λ − λ − i¯cN 2 Λ − λ +i¯cN 2 − i¯c M − µ −i¯cN2 M − µ +i¯cN 2 − i¯c # iπ(N − 1) 2(µ − λ) sinhhπµ−λ¯c i | {z } FN −1(µ,λ) , (3.34) which, after some manipulations, turns out to be the exact requirement posed by the Form Factor Reduction Theorem.
4 EXACT TIME EVOLUTION
4
Exact time evolution
4.1
N-state
In equation 3.20, it has been concluded that for the state containing a single N -string,
FN(µ, λ) = πN3(µ − λ) sinhhπ(µ−λ)c¯ i . (4.1)
We can now turn to equation 2.20, and fill in this important unknown factor. For an initial state containing only single N-strings, written generically as
|Ψ0i = X λB.E. k(λ) |λi , (4.2) this means hρ(x, t)i = 1 k ˜Ψ0k2 X µB.E. X λB.E.
k∗(µ)k(λ)ei[E(µ)−E(λ)]tei[P (µ)−P (λ)]xhµ| ρ(0, 0) |λi (4.3)
= 1 k ˜Ψ0k2 X µB.E. X λB.E. k∗(µ)k(λ)eiN[µ2−λ2]teiN [µ−λ]x πN 3(µ − λ) sinhhπ(µ−λ)c¯ i , (4.4)
Where both the form factor and the eigenvalues, as stated in equations 2.14 and 2.13, have been inserted. For the energy-eigenvalues this means, explicitly:
E(µ) − E(λ) = N µ2− ¯c 2 12N (N 2− 1) − N λ2+c¯2 12N (N 2− 1) = N µ2− λ2 . (4.5) Now before the time evolution for a specific initial state is computed, it is interesting to give a theoretical prediction of its behavior at x = 0.
Theoretical interlude: density evolution at x = 0.
For the case of a single N -string, the reduced Bethe equations 2.10 simplify to
N Lλ = 2πI, (4.6)
The scattering term is zero, as there is only one quasi-particle in the system, leaving no room to scatter with other particles. Since the quantum numbers I are spaced at integer intervals, this means that ∆λ =
2πIn+1
LN −
2πIn
LN =
2π
LN. It is clear that ∆λ becomes infinitesimal as L → ∞. If the integrand contains no divergences between the lattice points defined by the Bethe equations, we can write
X λ =X λ ∆λ ∆λ = LN 2π X λ ∆λ−−−−→L→∞ LN 2π Z dλ. (4.7)
Since the form factor 4.1 is finite for all λ and µ, we can apply 4.7 to the generic N -string result and describe the time evolution of the density operator’s expectation value at x = 0 as follows:
hρ(0, t)i = 1 k ˜Ψ0k2 L2N5 4π ∞ Z Z −∞ dµdλk∗(µ)k(λ)eiN[µ2−λ2]t (µ − λ) sinhhπ(µ−λ)c¯ i . (4.8)
To this integral, it is suitable to apply the following method: Stationary Phase Approximation. Consider an integral of the form
I = Z
4.1 N-state 4 EXACT TIME EVOLUTION
Then in all regions where dxdf (x) 6= 0, the function eiN f (x) is highly oscillatory and destructive interference causes the integral to be zero. It is only in a small environment around so-called stationary points p, where
d
dxf (x)|x=p= 0, that the integral returns a nonzero value. Over these regions, g(x) can be treated as a constant and f (x) can be Taylor-expanded around p to second order, where the first-order term is zero by definition:
f (x) = f (p) +f 00(p)
2 (x − p)
2+ . . . . (4.10)
Summing over all stationary points p gives the stationary phase approximation:
I ≈X p g(p)eiN f (p) Z dxeiNf 00 (p)2 (x−p) 2 dx, (4.11)
where the remaining integral is, depending on the boundaries, of an easily evaluated Gaussian form.
x
fHxL
Figure 2: A rapidly oscillating function (purple) and a more slowly varying one (blue), the product of which is conditional to the Stationary Phase Approximation.
Applying the Stationary Phase Approximation to 4.8, we see that both for the µ- and the λ-integral, the function eiN[µ2−λ2]t is rapidly oscillating for all N 1t. Conveniently, it has a phase which is already equal to its second-order Taylor expansion around the only stationary point, 0. We can therefore treat all other functions as constants and use the limit
lim µ→λ (µ − λ) sinhhπ(µ−λ)¯c i = ¯ c π (4.12) to approximate 4.8 as hρ(0, t)i ≈ 1 k ˜Ψ0k2 L2N5 4π ¯ c πk ∗(0)k(0) ∞ Z −∞ dµeiN tµ2 Z ∞ −∞ dλe−iN tλ2 (4.13) = k ∗(0)k(0) k ˜Ψ0k2 ¯ cL2N5 4π2 r π iN t r π −iN t= k∗(0)k(0) k ˜Ψ0k2 ¯ cL2N4 4πt ∼ 1 t ∀N 1 t. (4.14)
The initial density peak is thus expected to decay like 1t, with a factor of porportionality determined by the exact initial state.
Constructing initial state |Ψ0i
Defining an inital state is not without ambiguity. In order to construct a perfectly localized N -string, all N -string momentum-eigenfunctions have to be included:
|Ψ0i = 1 L X λ eiN λx0|λi . (4.15)
Here, |λi denotes an N-string with string centre rapidity λ and thus with total momentum N λ. The sum runs over all values of λ as allowed by 4.6, the reduced Bethe equations for a single N string. 4.15 can therefore be seen as a discrete Fourier transform from momentum space to position space at position x0. Furthermore,
4.1 N-state 4 EXACT TIME EVOLUTION
Caux and Calabrese [3] state that the norm of a single N string is kλk2= ¯cLN2. The norm of 4.15 can thus be computed: kΨ0k2= Ψ∗0Ψ0= 1 L2 X µ X λ ei(λ−µ)x0hµ|λi = 1 L2 X µ X λ ei(λ−µ)x0kλk kµk δ λ,µ (4.16) = 1 L2 X λ kλk2= 1 L2 X λ ¯ cLN2. (4.17)
Using 4.7, this means that
kΨ0k 2 = ¯cN 2 L X λ → lim Λ→∞ ¯ cN2 L LN 2π Z Λ −Λ dλ = lim Λ→∞ ¯ cN3Λ π , (4.18)
which is clearly infinite.
There are many different ways in which the initial state can be regularized. A first approach is to fix the value of the maximum allowed momentum, Λ. This restriction of the initial state’s spectrum of momenta is often called a UV-cutoff. In short,
|Ψ0i = 1 L Λ X λ=−Λ eiN λx0|λi (4.19) kΨ0k 2 =cN¯ 3Λ π . (4.20)
Setting the point of localization in 4.15 to x0 = 0 leads to coefficients k = k∗ = L1 in 4.14. The Stationary Phase Approximation of the density evolution at x = 0 then gives
hρ(0, t)i ≈ k ∗(0)k(0) k ˜Ψ0k2 ¯ cL2N4 4πt = N 4Λt ∀N 1 t. (4.21)
Another option is to give the initial state a Gaussain momentum distribution:
|Ψ0i = 1 L X λ e−αλ2eiN λx0|λi . (4.22)
The momentum-space standard deviation in terms of α is given by
σ = r
1
2α. (4.23)
The norm of initial state 4.22 is finite:
kΨ0k 2 = Ψ∗0Ψ0= 1 L2 X µ X λ e−α(µ2+λ2)eiN (λ−µ)x0hµ|λi (4.24) = 1 L2 X µ X λ e−α(µ2+λ2)eiN (λ−µ)x0cLN¯ 2δ λ,µ (4.25) = ¯cN 2 L X λ e−2αλ2 →cN¯ 2 L LN 2π Z dλe−2αλ2 (4.26) = ¯cN 3 2π r π 2α = ¯ cN3 2√2απ. (4.27)
It can be observed that in the limit as α → 0, the norm indeed diverges, in accordance with 4.18. To calculate the predicted density at x = 0, we observe that for localization point x0= 0 we again have k∗(0) = k(0) = L1, so filling in 4.14 leads to hρ(0, t)i ≈ k ∗(0)k(0) k ˜Ψ0k2 ¯ cL2N4 4πt = r α 2π N t ∀N 1 t. (4.28)
4.1 N-state 4 EXACT TIME EVOLUTION
Time evolution for a Gaussian distribution Consider the initial state
|Ψ0i = 1 L X λ e−αλ2eiN λx0|λi (4.29) with norm kΨ0k 2 = ¯cN 3 2√2απ. (4.30)
Setting x0= 0 leads to coefficients k(λ) = e
−αλ2
L . Inserting this into 4.4, the generic formula for the density operator’s expectation value, leads to
hρ(x, t)i = 1 L2 1 kΨ0k2 X µ X λ eiN (µ−λ)xeiN (µ−λ)(µ+λ)te−α(µ2+λ2) πN 3(µ − λ) sinhhπ(µ−λ)c¯ i (4.31)
Since the integrand contains no singularities, we can convert the sums to integrals according to 4.7. Next, change coordinates: µ =u + v 2 u = µ + λ λ = u − v 2 v = µ − λ =⇒ dµdλ = ∂(µ, λ) ∂(u, v) dudv = 1 2 1 2 1 2 − 1 2 dudv = 1 2dudv (4.32) λ2+ µ2=1 2u 2+ v2 , (4.33) to obtain hρ(x, t)i = 2 √ 2απ ¯ cN3L2 L2N2 4π2 Z ∞ −∞ Z ∞ −∞ dudv 2 e iN vxeiN vute−α 2[u 2+v2] vπN3 sinhπv ¯ c (4.34) = r 2α π N2 4¯c Z ∞ −∞ dveiN vxe−α2v 2 v sinhπv ¯ c Z ∞ −∞ du cos [N vut] e−α2u 2 (4.35) = r 2α π N2 4¯c Z ∞ −∞ dveiN vxe−α2v 2 v sinhπv ¯ c e −N 2 t2 v2 2α r π α √ 2. (4.36)
In the first step, the imaginary part of the integrand vanishes because it is an odd function integrated over a symmetric interval. Now use ˜v = π¯cv:
hρ(x, t)i = ¯cN 2 2π2 Z ∞ −∞ d˜veiN ¯πcx˜ve− α¯c2 2π2v˜ 2 v˜ sinh [˜v]e −N 2 t2 ¯c2 ˜v2 2απ2 (4.37) = ¯cN 2 2π2 Z ∞ −∞ d˜v cos N ¯c π x˜v exp −˜v2¯c 2 π2 α 2 + N2t2 2α | {z } (?) ˜ v sinh [˜v], (4.38)
where again, the imaginary part integrates to zero for symmetry reasons. We now note that for N α, the factor (?) in the integrand is extremely sharply peaked around the origin. The other factor, sinh[˜v˜v] can thus be assumed to be constant over the relevant region, using lim˜v→0sinh[˜v˜v] = 1. This leads to
hρ(x, t)i = ¯cN 2 2π2 Z ∞ −∞ d˜v cos N ¯c π x˜v exp −˜v2¯c 2 π2 α 2 + N2t2 2α (4.39) = N 2 2√π 1 q α 2 + N2t2 2α exp " − N 2x2 4α2 +N2α2t2 # (4.40)
4.1 N-state 4 EXACT TIME EVOLUTION
For N α, this becomes
hρ(x, t)i = N t r α 2πexp −α 2 x2 t2 . (4.41)
This function has been plotted in Figure 4.1 for various values of t. We see that at x = 0,
hρ(0, t)i = N t
r α
2π, (4.42)
which is exactly equal to the result 4.28, calculated with the Stationary Phase Approximation. As a check, calculate the average density in the limit L → ∞:
1 L Z ∞ −∞ dx hρ(x, t)i = √1 2 N L r α π 1 t Z ∞ −∞ dx exp −α 2 x2 t2 (4.43) = √1 2 N L r α π 1 t r 2πt2 α = N L, (4.44) as it should be.
Directly integrating 4.40 gives exactly the same result.
-4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N -4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N -4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N -4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N -4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N -4 -2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N
4.1 N-state 4 EXACT TIME EVOLUTION
Time evolution for a UV-cutoff distribution
We now turn to an initial state, which is constructed using a UV-cutoff and which is localized at x0, for simplicity.
|Ψ0i = 1 L Λ X λ=−Λ eiN λx0|λi , x 0= 0 (4.45) kΨ0k 2 =¯cN 3Λ π . (4.46)
Inserting this state into 4.4 and again converting sums to integrals in the absence of singularities,
hρ(x, t)i = 1 k ˜Ψ0k2 Λ X −Λ Λ X −Λ 1 L2e iN[µ2−λ2]t eiN [µ−λ]x πN 3(µ − λ) sinhhπ(µ−λ)¯c i , (4.47) = π ¯ cN3Λ 1 L2 L2N2 4π2 Z Λ −Λ dµ Z Λ −Λ dλeiN (µ−λ)xeiN (µ−λ)(µ+λ)tπN 3(µ − λ) sinhhπµ−λc¯ i (4.48) = N 2 4¯cΛ Z Λ −Λ dµ Z Λ −Λ dλeiN (µ−λ)xeiN (µ−λ)(µ+λ)t (µ − λ) sinhhπµ−λc¯ i . (4.49)
Once more, change coordinates according to 4.32,:
hρ(x, t)i = N 2 8¯cΛ Z 2Λ −2Λ du Z 2Λ −2Λ dveiN vxeiN vut v sinhπv ¯ c (4.50) = N 2 8¯cΛ Z 2Λ −2Λ du Z 2Λ −2Λ
dv [cos (N v(x + ut)) + i sin (N v(x + ut))] v sinhπv
¯ c
. (4.51)
The imaginary part of the integrand vanishes since it leads to an odd function evaluated over a symmetric interval in v. Applying ˜v =πvc¯ then gives
hρ(x, t)i = N 2 8¯cΛ ¯ c2 π2 Z 2Λ −2Λ du Z 2¯πcΛ −2¯c πΛ d˜v cos N ¯c π ˜v(x + ut) ˜v sinh ˜v (4.52)
Now, observe that for large values of v, the integrand is highly suppressed by the hyperbolic sine, as is illustrated in Figure 4. - 10 - 5 5 10 v - 1.0 - 0.5 0.5 1.0 Integrand@ vD
Figure 4: A Mathematica-sketch of the behavior of the integrand in 4.52, which is clearly suppressed very strongly outside of a certain range.
Since 2¯πcΛ is already a fairly large number, the range of integration can be expanded to the entire real line without a large error in the integral, leading to
4.1 N-state 4 EXACT TIME EVOLUTION hρ(x, t)i = N 2 8¯cΛ ¯ c2 π2 Z 2Λ −2Λ du Z ∞ −∞ d˜v cos N ¯c π ˜v(x + ut) v˜ sinh ˜v (4.53) = ¯cN 2 8π2Λ Z 2Λ −2Λ du π 2 1 + cosh N ¯πcπ(x + ut) (4.54) =¯cN 2 8Λ " tanhN ¯c(x+ut)2 N ¯ct #u=2Λ u=−2Λ , (4.55)
so the density-time evolution of a localized N -string with a Gaussian cutoff can be described as
hρ(x, t)i = N 8Λt tanhN ¯c(x + 2Λt) 2 − tanh N ¯c(x − 2Λt) 2 . (4.56)
To check the validity of this result, we compute
1 L Z ∞ −∞ dxρ(x, t) = 1 L Z ∞ −∞ dx N 8Λt tanhN ¯c(x + 2Λt) 2 − tanh N ¯c(x − 2Λt) 2 = N L8Λt8Λt = N L, (4.57) which is the expected result. It is also interesting to compare te behavior of 4.56 at x = 0 to the expectation obtained with the Stationary Phase Approximation. Here,
hρ(0, t)i = N 8Λt[tanh(N ¯cΛt) − tanh(−N ¯cΛt)] = N 4Λttanh(N ¯cΛt) ≈ N 4Λt ∀N 1 t. (4.58) This is in exact accordance with 4.21, the prediction for the density distribution at x = 0 of an initial state with a UV-cutoff. - 4 - 2 0 2 4 x 0.5 1.0 1.5 2.0 2.5 3.0 Units of N
Figure 5: Time evolution of hρ(x, t)i from t ≈ 0 to t = 1, using a Gaussian Cutoff of Λ = 10, and setting ¯c = 10. In Figure 5 and 6, the behavior of 4.56 can be observed, which leads to a number of general conclusions: • Although the Gaussian distribution and the distribution with a UV-cutoff differ considerably in their exact
shapes, the overall behavior is remarkably similar: both decay like ∼ 1t for t N1, with a comparable overall form.
• As can be seen in Equations 4.58 and 4.42, the speed of decay at x = 0 is linearly proportional to Λ, the point of UV-cutoff, and to the standard deviation σ in the case of a Gaussian distribution. The latter conclusion is obtained by comparing 4.42 to 4.23.
• The predictions of these decay speeds, made with the Stationary Phase Approximation, converge very quickly. For both density distributions, they are exact for N 1t.
• In the UV-cutoff distribution, the shape of the decaying wavepacket is strongly determined by the interaction parameter ¯c. For small ¯c, the wavepacket is smooth, whereas for large values of ¯c its edges fall off almost instantly.
4.2 N-1:1-state 4 EXACT TIME EVOLUTION - 4 - 2 0 2 4 x 0.1 0.2 0.3 0.4 0.5 Units of N
Figure 6: Comparison between UV-cutoff density profiles at t = 1
10, for ¯c = 2 (blue) and ¯c = 100 (purple). It can be seen that for a large interaction parameter ¯c, the profile becomes cut-off very sharply.
Finally, it is interesting to note the effect of extreme localization on density time evolution. In both distri-butions just described, starting with a more localized distribution considerably speeds up the decay process. As can be seen in equations 4.56, 4.41, 4.42 and 4.58, respectively sending α to zero or Λ to infinity causes the speed of decay to increase, until, at perfect localization, it becomes inifinite, instantaneously “exploding” the particle. This infinitely quick decay process is not well described by the above equations and it seems that at this point, a perfectly localized wavepacket can not be time-evolved in a well-defined manner.
4.2
N-1:1-state
To compute the time evolution of an N -1:1-state, the more complicated form factor 3.33 must be used. Although this form factor can be shown to be completely regular using the Form Factor Reduction Theorem, this is only true for λ, µ, Λ and M satisfying the Bethe equations. In between these points, in the so-called off-shell regions, the form factor might show divergences. This means that sums over rapidities that are dictated by the Bethe equations cannot simply be converted to integrals in the way it has been done in Section 4.1. A more careful route must be taken, to secure a regular integrand. One possible approach is to rewrite the original sum as a sum over contour integrals in the complex plane. Deforming these contours, whilst subtracting contributions from unwanted off-shell poles, can lead to a calculable integral whilst avoiding the divergences lurking behind the naive transformation from sums to real integrals, as applied in Section 4.1.
This method is now explored in a general sense. Inserting concrete functions and coefficients is an interesting starting point for further research, but has not been done in this project. Write the initial state in its generic form |Ψ0i = X λ,Λ k(x, t, λ, Λ) |λ, Λi (4.59) and define k(λ, Λ) = T (x, λ, Λ)U (t, λ, Λ)˜q k(λ, Λ) kΨ0k2 , (4.60)
where T and U are the usual translation and time-evolution operators, as introduced in 2.19.
The expectation value of the density operator can be separated into an off-diagonal and a diagonal part:
hΨ0| ρ(x, t) |Ψ0i = X λ,Λ X (µ,M )6=(λ,Λ) k∗(λ, Λ)k(µ, M )F (λ, Λ, µ, M ) +X λ X Λ |k(λ, Λ)|2F diag.(λ, Λ). (4.61)
The diagonal part will lead to a constant contribution and can be added later. The off-diagonal part is less trivial, however. As mentioned above, it can be expressed as a sum over contour integrals:
hΨ0| ρ(x, t) |Ψ0iO.D.= X λ,Λ k∗(λ, Λ) X (µ,M )6=(λ,Λ) I I Cµ,M dz1 2π dz2 2πk(z1, z2) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1). (4.62)
4.2 N-1:1-state 4 EXACT TIME EVOLUTION where we use Q1(z1, z2) = L(N − 1)z1− Φ(z1, z2) (4.63) Q2(z1, z2) = Lz2+ Φ(z1, z2) (4.64) σ(z1, z2) = ∂Q1 ∂z1 ∂Q1 ∂z2 ∂Q2 ∂z1 ∂Q2 ∂z2. (4.65)
Here, the expressions 4.63 and 4.64 take on values of 2π times an integer on all points satisfying the Bethe equations, thus leading to a simple pole in the denominator of 4.62. Combined with the expression for σ, obtained from the multi-dimensional residue formula in [14], this serves to exactly pick out those values of the integrand that were originally being summed over.
Re Im Cλ/Λ Γ± Bethe singularity non−Bethe singularity
Figure 7: Deformation of the original contours Cλ/Λaround solutions to the Bethe equations (blue) to an overall contour Γ±, at Im(z) = ± (red). Singularities that do not satisfy the Bethe equations have to be subtracted, by integrating around them in the opposite direction.
As described in Fig 7, a next step is to deform the little contours encompassing single solutions to the Bethe equations to an overall contour Γ containing all solutions, passing from ∞ to −∞ at Im(z) = and back at Im(z) = −. Poles on the real axis that do not satisfy the Bethe equations will have to be subtracted by small clockwise contours around them. Inspecting 3.33, we see that these poles occur when either M → Λ or µ → λ, but not both at the same time, since this last case is already covered by the diagonal contribution. As M → Λ, µ has to satisfy the Bethe equations, in order to hava a nonzero residue. The same is true for µ → λ, where M has to satisfy the Bethe equations. As an example, one of these subtracted poles of the latter type looks like
I Cµ=λ I CM dz1 2π dz2 2πk(z1, z2) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) (4.66) = I Cµ=λ dz1 2π 2πi 2πk(λ, M ) [F (λ, Λ, z1, M )] σ(z1, M ) (eiQ1(λ,M )− 1)i∂Q2 ∂z2|z2=M (4.67) =2πi 2πk(λ, M ) Res z1=λ [F (λ, Λ, z1, M )] σ(λ, M ) (eiQ1(λ,M )− 1)∂Q2 ∂z2|z1=λ,z2=M . (4.68) We thus obtain X µ X M I I Cµ,M dz1 2π dz2 2πk(µ, M ) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) = I Γ dz1 2π I Γ dz2 2πk(z1, z2) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) | {z } (a) −X µ ik(µ, Λ) Res z2=Λ [F (λ, Λ, µ, z2)] σ(µ, Λ) (eiQ2(µ,Λ)− 1)∂Q1 ∂z1|z1=µ,z2=Λ −X M ik(λ, M ) Res z1=λ [F (λ, Λ, z1, M )] σ(λ, M ) (eiQ1(λ,M )− 1)∂Q2 ∂z2|z1=λ,z2=M | {z } (b) . (4.69) Subtracted poles, (b)
4.2 N-1:1-state 4 EXACT TIME EVOLUTION Q1(λ, M ) = L(N − 1)λ − Φ(λ, M ) (4.70) = L(N − 1)λ − Φ(λ, Λ) | {z } 2πn +Φ(λ, Λ) − Φ(λ, M ) (4.71) =⇒ eiQ1(λ,M )= ei(Φ(λ,Λ)−Φ(λ,M )) (4.72) Q2(µ, Λ) = LΛ + Φ(µ, Λ) (4.73) = LΛ + Φ(λ, Λ) | {z } 2πm −Φ(λ, Λ) + Φ(µ, Λ) (4.74) =⇒ eiQ2(µ,Λ)= ei(−Φ(λ,Λ)+Φ(µ,Λ)), (4.75)
where the Reduced Bethe Equations 2.10 were used to simplify the first two terms in both cases. Since summants in the last two terms of 4.69 are completely regular, taking the limit as L → ∞ and converting sums to integrals is now well-defined:
X µ = Z ∞ −∞ dµ 2π(N − 1)L (4.76) X M = Z ∞ −∞ dM 2π L. (4.77)
This means that the the subtracted terms in 4.69become
(b) = −i Z ∞ −∞ dµ 2π k(µ, Λ)σ(µ, Λ) ei(−Φ(λ,Λ)+Φ(µ,Λ))− 1 Res z2=Λ [F (λ, Λ, µ, z2)] (N − 1)L ∂Q1 ∂z1|z1=µ,z2=Λ −i Z ∞ −∞ dM 2π k(λ, M )σ(λ, M ) ei(Φ(λ,Λ)−Φ(λ,M ))− 1 Res z1=λ [F (λ, Λ, z1, M )] L ∂Q2 ∂z2|z1=λ,z2=M , (4.78) where ∂Q1 ∂z1 z 1=µ,z2=Λ = (N − 1)L (4.79) ∂Q2 ∂z2 z 1=λ,z2=M = L, (4.80) to first order in L.
To determine the residues, we can use the Form Factor Reduction Theorem 2.21, since in both cases, two rapidities get close to each other, leading to a residue, whilst all non-singular parts have zero residue. In general:
FN({λj}, {µj}) = h{λj}Nj=1|ρ(x)| {µj}Nj=1i = (4.81) −i¯c µN − λN N −1 Y j=1 µN − µj− i¯c µN − µj λj− λN− i¯c λj− λN − N −1 Y j=1 µN − µj+ i¯c µN − µj λj− λN + i¯c λj− λN λN=µN × h{λj}N −1j=1 |ρ(x)| {µj}N −1j=1i + Regular expressions (4.82) = −i¯c µN − λN N −1 Y j=1 µN − µj+ i¯c µN − µj λj− λN+ i¯c λj− λN N −1 Y j=1 µN − µj− i¯c µN − µj+ i¯c λj− λN− i¯c λj− λN+ i¯c − 1 λN=µN × FN −1({λj}, {µj}) + Reg. (4.83) = −i¯c µN − λN N −1 Y j=1 µN − µj+ i¯c µN − µj λj− λN+ i¯c λj− λN N −1 Y j=1 eiφ(µN−µj)e−iφ(λN−λj)− 1 λN=µN × FN −1({λj}, {µj}) + Reg. (4.84)
To these form factors. the well-known residue formula for a simple pole [8] can be applied:
Res z=z0
f (z) = lim z→z0
4.2 N-1:1-state 4 EXACT TIME EVOLUTION
At the same time, we know from [3] that
N −1 Y j=1
eiφ(µN−µj)e−iφ(λN−λj)= ei[Φ(M −µ)−Φ(Λ−λ)]= ei[−Φ(µ,M )+Φ(λ,Λ)]. (4.86)
In the particular case where M → Λ, this means that the residue of 4.84 becomes Res z2=Λ [FN(λ, Λ, µ, z2)] = (4.87) = − i¯c Λ − µ + i¯c 2N Λ − µ −i¯2c(N − 2) λ − Λ +i¯2cN λ − Λ −i¯2c(N − 2) h ei(Φ(λ,Λ)−Φ(µ,Λ))− 1i λN=µN × FN −1(λ, µ). (4.88) Inserting this result into the first term of 4.78, it turns out that apart from a difference in sign, the exponentials cancel: i Z ∞ −∞ dµ 2π k(µ, Λ)σ(µ, Λ) ei(−Φ(λ,Λ)+Φ(µ,Λ))− 1 Res z2=Λ [F (λ, Λ, µ, z2)] (4.89) = ¯c Z ∞ −∞ dµ 2πk(µ, Λ)σ(µ, Λ) " Λ − µ +i¯2cN Λ − µ −i¯2c(N − 2) λ − Λ + i¯2cN λ − Λ −i¯2c(N − 2)FN −1(λ, µ) # (4.90) = ¯c Z ∞ −∞ dµ 2πk(µ, Λ)σ(µ, Λ)ζ(λ, Λ, µ, M )FN −1(λ, µ). (4.91)
In the case where λ → µ, similar operations can be performed, leading to the following generic expression for the subtracted terms:
(b) = ¯c Z ∞ −∞ dµ 2πk(µ, Λ)σ(µ, Λ)ζ(λ, Λ, µ, M )FN −1(λ, µ) + ¯c Z ∞ −∞ dM 2π k(µ, Λ)σ(µ, Λ) ˜ζ(λ, Λ, µ, M )F1(λ, µ). (4.92) Complete contour, (a)
Recall part (a) of equation 4.78, containing all poles within the expanded contour Γ:
I Γ dz1 2π I Γ dz2 2πk(µ, M ) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) (4.93) The integral along path Γ is a sum of integrals of the forms R
R+i and
R
R−i. The exponentials in the denominator of the integrand, however, simplify greatly in the limit L → ∞. For on the paths at Im(z) = ±i, the denominator behaves like
(eiQ1(z1±i,z2±i)− 1)(eiQ2(z1±i,z2±i)− 1)
=(ei(L(N −1)z1−Φ(z1±i,z2±i))∓L(N −1)− 1)(ei(Lz2+Φ(z1±i,z2±i))∓L− 1)
L→∞ −−−−→
(
∞, for paths R − i
1, for paths R + i, for L → ∞.
(4.94)
This means that the only path with a nonzero integral as L → ∞ is the path where both inner- and outer integrals run at Im(z) = +i. Their signs cancel, and we can rewrite (a) in 4.69 as
lim L→∞ I Γ dz1 2π I Γ dz2 2πk(µ, M ) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) = ∞ Z Z −∞ dµ 2π dM 2π k(µ, M )F (λ, Λ, µ, M )σ(µ, M ). (4.95) In this last expession, F (λ, Λ, z1, z2) contains residues at z1= µ = λ and z2= M = Λ. We therefore seperate both integrals into their Cauchy principal value and the residues of these poles, leading to
4.2 N-1:1-state 4 EXACT TIME EVOLUTION lim L→∞ I Γ dz1 2π I Γ dz2 2πk(µ, M ) F (λ, Λ, z1, z2)σ(z1, z2) (eiQ1(z1,z2)− 1)(eiQ2(z1,z2)− 1) = P V ∞ Z −∞ dµ 2πP V ∞ Z −∞ dM 2π k(µ, M )F (λ, Λ, µ, M )σ(µ, M ) −P V ∞ Z −∞ dµ 2π iπ 2πM =ΛRes[F (λ, Λ, µ, M )] k(µ, M ) −iπ 2πResµ=λ " P V ∞ Z −∞ dM 2π F (λ, Λ, µ, M )k(µ, M ) # +iπ 2πResµ=λ iπ 2πM =ΛRes[F (λ, Λ, µ, M )] k(µ, M ). (4.96)
The above residues are exactly equal to those calculated in 4.88. Now that all singularities have been treated explicitly, the outer sums in 4.61, running over λ and Λ, can be converted to integrals without further problems, leading to the following expression for the off-diagonal contributions to the density operator’s expectation value:
hΨ0| ρ(x, t) |Ψ0iO.D.= (N − 1)L2) 4π2 ∞ Z Z −∞ dλdΛk∗(λ, Λ) " ∞ Z Z −∞ dµ 2π dM 2π k(µ, M )F (λ, Λ, µ, M )σ(µ, M ) ¯ c Z ∞ −∞ dµ 2πk(µ, Λ)σ(µ, Λ)ζ(λ, Λ, µ, M )FN −1(λ, µ) + ¯c Z ∞ −∞ dµ 2πk(µ, Λ)σ(µ, Λ) ˜ζ(λ, Λ, µ, M )F1(λ, µ) # , (4.97)
where care must be taken to treat the first inner integral in the way that was described in equation 4.96. Inserting all functions k, k∗, σ, ζ and the appropriate form factors leads to the desired off-diagonal contribution. The diagonal contribution is a simple constant. Calculating the resulting integrals is an interesting starting point for further research into composite string states.
5 CONCLUSIONS AND PERSPECTIVE
5
Conclusions and perspective
Taking work by Caux, Calabrese [3] and Slavnov [15] as a reference, this project has extended the recent article [6] by Kaminishi, Sato and Deguchi to the realm of the attractive one-dimensional Bose-gas. Whereas Kaminishi et al. have studied the equilibration of localized density notches in the repulsive scenario using numerical techniques, the present paper gives an analytical description of density peaks, rather than notches, for the attractive case.
To do so, the analytical treatment of the model has been investigated, using Bethe Ansatz techniques to find eigenstates and eigenvalues. Next, the formation of string states was explored, in which groups of eigenvalues share the same real value. Generic expressions were found to describe the density operator’s matrix elements, so-called form factors. These expressions were then specialized to specific formulas in the case of N -states and N -1:1-states. Using these form factors, the evolution of localized density peaks could be calculated for the case of an N -string.
This has led to a number of conclusions. First of all, a state of perfect localization cannot be properly time-evolved using the explored techniques, leading to an infinitely high speed of decay. More realistic initial localizations were proposed in the form of a Gaussian momentum distribution and a UV-cutoff. Although different in appearance, these initial states show remarkable parallels in the time-evolution of their density profiles, both decaying like 1
t. The corresponding factor of proportionality was found to be linear in Λ, which is the point of UV-cutoff, and to the distribution’s standard deviation for the case of a Gaussian initial state. The precise formulas for this decay speed have turned out to be equal to predictions from a stationary phase approximation in the r´egime where t N1. Using these precise formulas, a number of movies and snapshots has been produced, showing the complete density profile of a localized N -string distribution for all points in time. In order to treat more general string states, a first theoretical hurdle was taken, by exploring the conversion of a sum over Bethe states to an integral on the real line. This problem has been solved by a method of contour integration around Bethe eigenvalues, opening the door towards a more general description of density time evolution in the one-dimensional Bose gas.
The recommendation to use these techniques in further research is far more than a formally required statement. Indeed, the machinery to calculate matrix elements, which has been described for specific cases in the appendices, can be expanded to treat instances of arbitrary N -M :M -strings as well, which might then be time evolved by the contour integration techniques described above. It would be interesting to see how the speed of decay of such composite string states depends on the number of particles in each string. This particle number might serve as an effective string mass, slowing the decay process of larger strings compared to smaller ones. Besides from such fundamental questions, even the N -string results are a valuable starting point for more application-centered research. Among many, a recent proposal is to use them in the context of optical waveguides, where states of bound photons might occur, possibly behaving much like the strings in this project. Clearly, both theoretical generalizations and experimental applications make the realm of attractive one-dimensional Bose gases into a fertile soil for future research, where the enthusiasm ignited by the present project will hopefully find its further use.
A GENERIC DENSITY FORM FACTOR
These appendices have been based on notes by Jacopo de Nardis, which he was so kind to share. A number of corrections have been made and the text has been adapted, to make it accessible for a reader at bachelor’s level.
A
Generic density form factor
Starting at the expression from [15],
F ({µ}, {λ}) = (Pµ− Pλ) N Y j=1 Vj+− Vj− N Y j,k=1 λj− λk+ ic µj− λk detNU˜jk(λp) Vp+− Vp− , (A.1) where Vj±= N Y m=1 µm− λj± ic λm− λj± ic , (A.2) ˜ Ujk(λp) = δjk+ i µj− λj Vj+− Vj− N Y m6=j µj− λj λm− λj (K(λj− λk) − K(λp− λk)) . (A.3)
the free parameter λpcan be taken to infinity, after which straightforward manipulations lead to the starting point of the coming calculations:
F ({µ}, {λ}) = iN −1¯c N Y j,k λj− λk− i¯c µj− λk det N U ({µ}, {λ}) , (A.4)
using the following functions:
Ujk({µ}, {λ}) = δjk Vj+− Vj− i + QN a=1(µa− λj) QN a6=j(λa− λj) K(λj− λk) + 1 ¯ c (A.5) Vj± = N Y a=1 µa− λj∓ i¯c λa− λj∓ i¯c , (A.6) K(λ) = − 2¯c λ2+ ¯c2. (A.7)
A.1
Definitions
Referring to [3] and equation 2.9, the string rapidities can be written as:
λj,aα = λjα+i¯c
2 (j + 1 − 2a) + iδ j,a
α , a = 1, . . . j (A.8)
where α is the string label, a the inner string index and j is the string length. The factor QN
j,kλj− λk− i¯c in A.4 now contains a number of problematic factors: the instances where j and k are part of the same string and satisfy k = j + 1 lead to a situation where the δ’s in A.8 enter the product at leading order. By calling these inner string-indices a and b, we can explicitly calculate all occurrences of such factors which have order of magnitude δ:
(λa− λb− i¯c)|b=a+1= i¯c (b − a − 1) + iδαj,a− iδ j,b α |b=a+1= i δj,aα − δ j,a+1 α (A.9) Y (j,α) j Y a j Y b (λa− λb− i¯c) = Y (j,α) ij−1 j−1 Y a=1 δαj,a− δj,a+1 α j Y a j Y b6=a+1 λa− λb− i¯c . (A.10)
It will turn out in A.29 that each factor δj,a
α − δαj,a+1 in A.10 can be absorbed into the determinant of Ujk, thus canceling exactly.
We will extend the definition of in A.8 to cases where a = 0 or a = j + 1: λj,aα,c= λjα+i¯c
2 (j + 1 − 2a) + iδ j,c
A.2 Inner-string blocks A GENERIC DENSITY FORM FACTOR
enabling us to use the now well-defined identity
λj,aα,c± i¯c = λj,a∓1α,c . (A.12)
Here, the δ’s have been labelled by an independent index, to make them invariant under an addition or subtraction of i¯c to a given rapidity.
From equation (C.8) in [3] we adopt the definition of a function Vj,a
α,c, where we distinghuish between an outer-string and an inner-string product in the denominator:
Vα,cj,a= QN b=1 µb− λ j,a α,c Q (k,β)6=(j,α) Qk b=1 λk,bβ,b− λj,aα,c Qj b6=a λj,bα,b− λj,aα,c . (A.13)
A.2
Inner-string blocks
First, we will describe those blocks of the matrix Uab where a and b are inner-string indices from the generic string (α, j).
Let us first calculate A.6. For all cases except V1+ and Vj−, we find, using A.12:
Va±= N Y b=1 µb− λa∓ i¯c λb− λa∓ i¯c = N Y b=1 µb− λa∓1 λb− λa∓1 = QN b=1 µb− λ j,a∓1 α,c Q (k,β)6=(j,α) Qk b=1 λk,bβ,b− λj,a∓1α,c Qj b6=a∓1 λj,bα,b− λj,a∓1α,c 1
λj,a∓1α,a∓1− λj,a∓1α,c ,
(A.14)
where we recognize Vα,aj,a∓1 in the first part of the expression. The last factor is the instance where b = a ∓ 1, which is missing in Vj,a∓1
α,a . To expand this last factor, we use the fact that c = a, as described below A.12. This means that
Va± = V
j,a∓1 α,a
λj,a∓1α,a∓1− λj,a∓1α,a =
Vj,a∓1 α,a i(δj,a∓1α − δαj,a)
= iV
j,a∓1 α,a (δαj,a− δαj,a∓1)
. (A.15)
In the cases of V1+ and Vj−, the respective factors where b = a − 1 = 0 and b = a + 1 = j + 1 are completely absent in A.14, and the expressions simplify to
V1+= Vα,1j,0, (A.16)
Vj− = Vα,jj,j+1. (A.17)
Now let us calculate A.7. Expanding the rapidities, K(λj,aα,a− λj,bα,b) = −2¯c i¯c(b − a) + iδj,aα − δαj,b 2 + ¯c2 = −2¯c −¯c2(b − a)2− 2¯c(b − a)δj,a α − δj,bα −δj,aα − δj,bα 2 + ¯c2 (A.18) In the case where b = a + 1, this reduces to
K(λj,aα,a− λ j,b α,b) = 2¯c 2¯cδαj,a− δαj,b +δαj,a+ δαj,b 2 = 1 δj,aα − δαj,b − 1 2¯c +δj,aα − δj,bα = 1 δj,aα − δj,bα − 1 2¯c + O(δ), (A.19)
where we have used partial fractions in the second step. Similar manipulations can be applied to the b = a − 1 case, leading to the final structure,
K(λj,aα,a− λ j,b α,b) = O(1) b 6= a ± 1, 1 δαj,a−δj,bα − 1 2¯c + O(δ) b = a + 1, − 1 δj,aα −δαj,b − 1 2¯c+ O(δ) b = a − 1. (A.20)
We are now in a position to explicitly describe the inner-string matrix Uab, using the expressions in A.5. On the diagonal, excluding a = 1 and a = j, we have
A.2 Inner-string blocks A GENERIC DENSITY FORM FACTOR Uaa= Vj,a−1 α,a δαj,a− δj,a−1α − V j,a+1 α,a δj,aα − δj,a+1α + QN b=1(µb− λa) QN b6=a(λb− λa) −2 ¯ c + 1 ¯ c = V j,a−1 α,a δαj,a− δj,a−1α − V j,a+1 α,a δj,aα − δj,a+1α −V j,a α,a ¯ c = V j,a−1 α,a δαj,a− δj,a−1α − V j,a+1 α,a δj,aα − δj,a+1α + O(1), a /∈ {1, j}, (A.21)
while the a = 1 and a = j cases lead to
U11= −iVα,1j,0− Vα,1j,2 δαj,1− δαj,2 −V j,1 α,1 ¯ c = −iVα,1j,0− V j,2 α,1 δαj,1− δαj,2 + O(1) Ujj = iV j,j+1 α,j + Vα,jj,j−1 δj,jα − δj,j−1α −V j,j α,j ¯ c = iVα,jj,j+1+ V j,j−1 α,j δj,jα − δj,j−1α + O(1). (A.22)
For the elements lying one step off the diagonal we can use A.20, to obtain
Ua,a−1= − Vj,a α,a δαj,j− δαj,j−1 +V j,a α,a 2¯c + O(δ) = − V j,a α,a δαj,j− δαj,j−1 + O(1) Ua,a+1= Vj,a α,a δαj,j− δj,j+1α +V j,a α,a 2¯c + O(δ) = V j,a α,a δαj,j− δj,j+1α + O(1). (A.23)
All other elements scale according to O(1) and look like
Uab= Vα,aj,a K(λa− λb) + 1 ¯ c (A.24) Since all this concerns an inner-string block, we can omit the stringlengths and -labels. The inner-string matrix then takes the form
−iV0 1 − V2 1 δ1−δ2 − V1 1 ¯ c V1 1 δ1−δ2 + V1 1 2¯c V 1 1 K(λ1− λ3) +1¯c . . . V11 K(λ1− λj) +1c¯ − V22 δ2−δ1 + V2 2 2¯c V1 2 δ2−δ1 − V3 2 δ2−δ3 − V2 2 ¯ c V2 2 δ2−δ3 + V2 2 2¯c . . . .. . V33 K(λ3− λ1) +1c¯ − V33 δ3−δ2 + V3 3 2¯c . .. . . . ... .. . . . .. V j−1 j−1 δj−1−δj + Vj−1j−1 2¯c Vjj K (λj− λ1) +1c¯ . . . Vjj K (λj− λN −2) +1¯c − V j j δj−δj−1 + Vjj 2¯c iV j+1 j + Vjj−1 δj−δj−1 − Vjj ¯ c (A.25)
This matrix can be further simplified, by an operation which leaves the determinant unchanged: we will add the first row to the second, the second to the third, all the way down to adding row j − 1 to row j. The final row then becomes the sum of all rows. It can be easily checked that after the above manipulations, we can compactly express the matrix elements on the last row as follows:
(Uαj)j,b= −iVαj,0δb,1+ iVαj,M +1δb,M + (1 − δb,1) Vb−1b−1− Vbb−1 (δb−1− δb) − Vb−1b−1 2¯c + (1 − δb,M) −V b+1 b+1 − V b+1 b (δb+1− δb) − Vb+1 2¯c + X a6=b±1 VaaK(λa− λb) + 1 ¯ c j X a=1 Vaa. (A.26)
A.2 Inner-string blocks A GENERIC DENSITY FORM FACTOR
It is important to stress that in A.26, δ’s with lower indices are Kronecker symbols, whereas δ’s with upper indices are string deviations. The terms of the form V
b−1 b−1−V
b−1 b
(δb−1−δb) , can be investigated by Taylor-expanding V
j,a α,b. For a given set of rapidities ({µ}, {λ}), Vα,bj,a is a function of λj,aα,b only. However, λj,aα,b depends on δj,b
α and a number of other paramters, which we will denote by xi. Using this information and the chain rule, we can write down a multi-dimensional Taylor-expansion of Vα,bj,a around the point Vj,a
α,c: Vα,bj,a= Vα,cj,a+ δαj,b− δj,c α " ∂λj,aα,b ∂δj,bα ∂Vα,bj,a ∂λj,aα,b # b=c +X i (xi(b) − xi(c)) " ∂λj,aα,b ∂xi ∂Vα,bj,a ∂λj,aα,b # b=c
+ “higher − order terms00
= Vα,cj,a+ δαj,b− δαj,c i ∂Vj,a
α,c ∂λj,aα,c
+ O(δ2).
Here, all contributions of (xi(b) − xi(c)) are zero, since in going from Vα,cj,a to V j,a
α,b, only δ j,c
α changes. We next neglect the contributions of the δ’s in Vα,bj,a and Vα,cj,a, writing Vαj,a = limδ→0Vα,cj,a and accepting an error at ∼ O(δ). We thus find the identity
Vα,bj,a− Vj,a α,c δj,bα − δj,cα = i ∂Vj,a α ∂λj,aα + O(δ). (A.27)
Using this information, taking the limit as δ → 0 and putting the string labels back into place, A.26 becomes
(Uαj)j,b= −iVαj,0δb,1+ iVαj,j+1δb,j+ (1 − δb,1) i∂V j,b−1 α ∂λj,b−1α −V j,b−1 α 2¯c + (1 − δb,j) − i∂V j,b+1 α ∂λj,b+1α −V j,b+1 α 2¯c + X a6=b±1 Vαj,aK(λj,aα − λj,b α ) + 1 ¯ c j X a=1 Vαj,a. (A.28)
To describe rows 1 through j − 1, recall that the operations of row-addition have left them the following, seemingly chaotic, structure:
−iV0 1 − V2 1 δ1−δ2 − V1 1 ¯ c V1 1 δ1−δ2+ V1 1 2¯c . . . −V22−V 2 1 δ2−δ1 + −iV10+ V22 2¯c − V11 ¯ c − V23 δ2−δ3 + V11−V1 2 δ1−δ2 − V22 ¯ c . . . V3 3 K(λ3− λ1) +1c¯ − V2 2−V12 δ2−δ1 + −iV 0 1 + V2 2 2¯c − V1 1 ¯ c + V3 3 2¯c − V3 3−V23 δ3−δ2 + V1 1−V21 δ1−δ2 . . . .. . ... . .. = − V12 δ1−δ2 + O(1) V11 δ1−δ2 + O(1) . . . −∂V2 ∂λ2 + O(1) − V23 δ2−δ3 + ∂V1 ∂λ1 + O(1) . . . −∂V2 ∂λ2 + O(1) − ∂V3 ∂λ3 + ∂V1 ∂λ1 + O(1) . . . .. . ... . .. = − V12 δ1−δ2 + O(1) V11 δ1−δ2 + O(1) . . . O(1) − V23 δ2−δ3 + O(1) . . . O(1) O(1) . . . .. . ... . .. (A.29)
Clearly, the structure simplifies greatly by only retaining terms at leading order, being ∼ 1 δαj,a−δj,bα
. This sensitive dependence on the string deviations will now be treated explicitly, and, in fact, eliminated. In A.10, we found that for each string (j, α), the prefactor contains j − 1 factors of the form δαj,a− δαj,a+1. Since scalar multipliaction of a row by a constant multiplies the determinant by that same constant, we can multiply each row a excluding the last by the corresponding factor δαj,a− δαj,a+1, thus eliminating these j − 1 factors in the prefactor. We have seen that before this operation, the leading order terms for the matrix entries were those terms scaling like ∼ 1
δj,aα −δαj,b
. The multiplication by the prefactor, however, will cause the terms at order ∼ 1 δαj,a−δj,bα
A.3 Complete U-matrix B N-STATE DETERMINANT
become leading at order O(1) and the terms which previously scaled like O(1) to vanish exponentially like O(δ). It is only the last row that remains unchanged. For all other rows, the entries are quite simply given by
(Uαj)a,b= +δa,bVαj,a+1− δb,a+1Vαj,a, a = 1, . . . , j − 1, b = 1 . . . , j, (A.30) where again, the lower-indexed δ’s denote kronecker symbols, and which is valid at leading order.
A.3
Complete U-matrix
Combining expressions A.30 and A.28, we can obtain an expression for the complete and transformed matrix U , which we now know has blocks of the form Uj
α all along its diagonal. Express the remaining positions by sectors ˜
Uα,βj,k. As an example, a state with two strings would give the following matrix:
U = U j α U˜ j,k α,β ˜ Uβ,αk,j Uβk ! . (A.31)
What do the off diagonal-blocks ˜Uα,βj,k look like? They are in fact very simple. All elements above the last row are given by expressions of the form A.24, scaling like O(1) and therefore vanishing exponentially after multiplication by the prefactor. The last row will not be multiplied by this prefactor and so does not vanish. It contains the sum of all the elements of the form A.24 lying in the above j − 1 positions. Its elements are therefore
˜Uj,k α,β j,b = j X a=1 Vαj,a Kλj,aα − λk,bβ +1 ¯ c . (A.32)
To conclude this section, the form factor can be expressed compactly as F ({µ}, {λ}) = iN −1c C[λ, µ]det¯ N U ({µ}, {λ}), (A.33) where we use C[λ, µ] = QN (j,α,a),(k,β,b), (k,β,b)6=(j,α,a+1) λj,a α − λk,bα − i¯c QN (j,α,a),(k,β,b) µj,aα − λk,bβ Y (j,α) ij−1 (A.34)
and where the matrix U is defined by A.24, A.28 and A.32.
B
N-state determinant
The most simple case for which A.33 can be computed explicitly is the case of an N-string at rapidity λ. Using A.24, A.26 and A.32, the matrix U then takes the following form:
−V2 V1 0 . . . . . . 0 0 −V3 V2 0 . . . 0 .. . ... ... ... ... ... 0 0 . . . 0 −VN VN −1 UN 1 UN 2 . . . UN N , (B.1)
where string indices (j, α) have been omitted. Simple Laplace expansion along the last row gives the following expression for the determinant:
det Uij = N −1Y b=2 Vb N X b=1 VbUN b (B.2)
The functions Vi have well-known and fairly simple expressions. The expression for U
N b, as stated in A.26, is more complicated. We will simplify it by finding a convenient form for the termP
a6=b±1V
aK(λa− λb), using the following complex integral:
Ib= 1 2πi I C dz QN k=1(µk− z) QN k=1(λk− z) 2c (λb−1− z)(λb+1− z) (B.3) By choosing the curve C in such a way that it encloses all poles of the integrand, we can ensure that
B N-STATE DETERMINANT
We will slightly rewrite the integrand, visually seperating first order poles from second order poles. For the cases where b 6= 1, b 6= N , we then compute the integral
I(b) = 1 2πi I C dz QN k=1(µk− z) QN k=1,k6=b±1(λk− z) 2¯c (λb−1− z)2(λb+1− z)2 = N X a=1,a6=b±1 QN k=1(µk− λa) QN k=1,k6=a(λk− λa) −2¯c (λb−1− λa)(λb+1− λa) + ∂ ∂λb−1 h QN k=1(µk− λb−1) QN k=1,k6=b±1(λk− λb−1) 2¯c (λb+1− λb−1)2 i + ∂ ∂λb+1 h QN k=1(µk− λb+1) QN k=1,k6=b±1(λk− λb+1) 2¯c (λb−1− λb+1)2 i = 0, (B.5)
which is obtained by applying the following formula to the aforementioned poles λi:
Resz=λif (z) = 1 (m − 1)!z→λlimi dm−1 dzm−1[(z − λi) mf (z)] , (B.6)
which can be found in any standard textbook, such as [8]. For the cases where b 6= 1, 6= N , we now have
N X a=1,a6=b±1 QN k=1(µk− λa) QN k=1,k6=a(λk− λa) −2¯c (λb−1− λa)(λb+1− λa) = − ∂ ∂λb−1 h QN k=1(µk− λb−1) QN k=1,k6=b−1(λk− λb−1) 2c (λb+1− λb−1) i − ∂ ∂λb+1 h QN k=1(µk− λb+1) QN k=1,k6=b+1(λk− λb+1) 2c (λb−1− λb+1) i , (B.7)
where in both righthandside terms, a factor ±(λb+1− λb−1) has been moved into the product over k’s in the denominator. We now observe the following two identities:
−2¯c (λb−1− λa)(λb+1− λa) = −2¯c (i¯c)2(a − b + 1)(a − b − 1) = −2¯c (i¯c)2(a − b + 1)(a − b − 1) = −2¯c (i¯c(a − b))2+ ¯c2 = −2¯c (λa− λb) 2 + ¯c2 = K (λa− λb) (B.8) and λb+1− λb−1= −2i¯c. Using these identities and applying the product rule to B.7 gives
X a6=±b VaK(λa− λb) = − 2¯c λb+1− λb−1 ∂Vb−1 ∂λb−1 + Vb−1 2¯c (λb+1− λb−1) 2 − 2¯c (λb−1− λb+1) ∂Vb+1 ∂λb+1 + Vb+1 −2¯c (λb−1− λb+1) 2 =− i∂V b−1 ∂λb−1 +V b−1 2¯c +i∂V b+1 ∂λb+1 +V b+1 2¯c ∀b /∈ {1, N }. (B.9) For the cases where b = 1 and b = N we use slightly different integrals. For b = 1 we have
C N-1:1-STATE DETERMINANT I(b) = 1 2πi I C dz QN k=1(µk− z) QN k=1,k6=2(λk− z) 2¯c (λ0− z)(λ2− z)2 = N X a=1,a6=1±1 QN k=1(µk− λa) QN k=1,k6=a(λk− λa) −2¯c (λ0− λa)(λ2− λa) + QN k=1(µk− λ0) QN k=1,k6=1±1(λk− λ0) −2¯c (λ2− λ0)2 + ∂ ∂λ2 h QN k=1(µk− λ2) QN k=1,k6=1±1(λk− λ2) 2¯c (λ0− λ2)2 i = 0, (B.10)
where the second righthandside term is due to the pole at z = λ0.
With similar reasoning as used in B.9, we can conclude from the above results that for b = 1,
X a6=b±1 VaK(λa− λb) = iV0+ i∂Vb+1 ∂λb+1 +V b+1 2¯c , b = 1. (B.11)
Completely analogous operations yield the corresponding identity for b = N ,
X a6=b±1 VaK(λa− λb) = −iVN +1− i∂Vb−1 ∂λb−1 +V b−1 2¯c , b = N. (B.12)
Using Kronecker δ’s, we can now combine B.9, B.11 and B.12 into one suggestive identity,
−iV0δ b,1+ iVNδb,N +1+ (1 − δb,1) i∂V b−1 ∂λb−1 −V b−1 2¯c + (1 − δb,N) − i∂V b+1 ∂λb+1 −V b+1 2¯c + X a6=b±1 VaK(λa− λb) = 0, ∀b ∈ 1, . . . , N . (B.13)
Comparing this to A.28, shows that
UN b= 1 ¯ c j X a=1 Vαj,a ∀b, (B.14)
which immediately leads to the final identity for the N-string determinant,
det Uij = N −1Y b=2 Vb1 ¯ c hXN a=1 Vai 2 [N − string] (B.15)
C
N-1:1-state determinant
In the case of an N -1-string combined with a 1-string, we will denote the set of rapidities as {λ}N −1
i=1 , Λ, (C.1)
where the former are the N − 1-string rapidities and the latter is the rapidity for the single string. The typical matrix now looks like
−V2 V1 0 0 0 0 −V3 V2 0 0 0 0 −V4 V3 0 Us 1 U2s U3s U4s Us,m˜ ˜ U1m,s U˜2m,s U˜2m,s U˜3m,s Um 1 (C.2) where ˜ Us,m= N −1 X a=1 Va(K(λa− Λ) + 1 ¯ c) (C.3)
