Spin chains with integrability-preserving impurities

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Spin chains with

integrability-preserving impurities

Jorran de Wit

Theoretical Physics

University of Amsterdam

A thesis submitted for the degree of: Master of Science Submission date: 17 July 2018

Supervisor: Prof. dr. Jean-Sébastien Caux Second examiner: Prof. dr. Kareljan Schoutens

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Acknowledgements

In particular, I would like to thank my supervisor Jean-Sébastien Caux, J-S. During the last year, you have shown me what a person’s will to achieve really is, without losing eye for detail. Your immense drive to do; not talk is definitely inspiring, if not contagious. Thank you for supervising my second year project, and thanks for the many enjoyable morning coffee discussions about anything and everything. Cheers!

Kareljan Schoutens, I would like to thank you for, without any hesitation, taking on the examiner’s role in my project. Thank you for your honest look on this thesis and the related presentation.

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Abstract

We will analyse the Yang-Baxter integrable spin-1₂ XXX model. For this model, the form of the monodromy matrix that returns the XXX model in the algebraic Bethe ansatz is known. Now we will evaluate the matrix away from the homogeneous limit that is used to generate the XXX model’s Hamil-tonian. The resulting Hamiltonian mimics the XXX model with additional spin interactions terms including next-nearest neighbour interactions. We extend domain of the impurities from the real axis to the complex plane by imposing some restrictions, with substantiated arguments that would guar-antee physical relevance, on how to move away from the homogeneous limit. We analyse the two-particle sector and we show how the impurities lead to the disappearing of complex solutions and validate these predictions with numerical analysis of the newly obtained inhomogeneous XXX Bethe equations.

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List of Figures

7.1 The numerical solution of the critical system size for which a complex two-string solution collapses to the real axis for the first state (n = 1) in a system with two real-valued impurities: α = 0 (blue line), α = 2 (orange line) andα = 4 (green line). . . 57 7.2 Solutions to equation (7.16) under interpretation of N_c(n=1) in a system

with two real-valued impurities: α = 0 (blue line), α = 2 (orange line) andα = 4 (green line). . . 58 7.3 Solutions to equation (7.16) under interpretation of N_c(n)in a system with

two real-valued impurities. The blue line is for n= 1, the orange line for

n= L − 1. . . 59

7.4 The numerical solution of the critical system size for which a complex two-string solution collapses to the real axis for the first state (n= 1) in a system with two complex-valued impurities: β = 0 (blue line), β = 0.4 (orange line),β = 0.6 (green line) and β = 2.0 (red line). . . 60 7.5 Solutions to equation (7.16) under interpretation of N_c(n=1) in a system

with two complex-valued impurities: β = 0.0 (blue line), β = 0.4 (or-ange line),β = 0.6 (green line) and β = 2.0 (red line). . . 61 7.6 Solutions to equation (7.16) under interpretation of N_c(n)in a system with

two complex-valued impurities. . . 62 7.7 Solutions to equation (7.16) under interpretation of N_c(n)in a system with

two complex-valued impurities. . . 62 8.1 Complex solutions to Bethe equations for N = 24, M = 2. The blue

dashed line only indicates where solutions are expected in the perfect string solution. . . 64 8.2 Complex solutions to Bethe equations for N = 24, M = 2 with two

im-purities with value x = 1.7. . . 65 8.3 Complex solutions to Bethe equations for N = 24, M = 2 with one

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8.4 Complex solutions to Bethe equations for N = 24, M = 2 with one con-jugate complex pair of impurities. . . 66

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Chapter 1 Introduction

In 1930, Felix Bloch introduced a method of solving the eigenfunctions of ferromagnets that was based on the idea of using combinations of plane waves[2]. In this method however, Bloch obtained too many eigenvalues thus his ‘theory’ was not correct due to overcounting. One year later, Hans Bethe introduced a method to calculate eigen-functions and eigenvalues for ‘one-dimensional metals’ that solved the over counting of eigenvalues[3]. After realisation that spin waves with only real solutions for mo-mentum do not yield enough solutions, he proposed the idea of complex (conjugate) solutions to complete the counting of total number of solutions. In the years that fol-lowed, the ‘Bethe ansatz’ was used to for solve for properties of, among others, the Heisenberg model.

Years later, starting in the late 70s, an algebraic generalization of the Bethe ansatz was developed, now commonly known as the algebraic Bethe ansatz[12, 16]. Up to today, the Bethe ansatz is a very popular method for solving quantum integrable systems. A very popular teaching example and research object is the isotropic version of the spin chain as also discussed by Bethe himself, the XXX model:

H_{X X X} = N X j=1 Sj· Sj+1− 1 4. (1.1)

The algebraic Bethe ansatz procedures analyse the properties of R-matrices which makes it possible to generate operators for eigenstates known as Bethe states. These R-matrices are matrices describing scattering processes that satisfy the Yang-Baxter equations, first appearing in the article of Rodney J. Baxter in 1971 to solve the eight-vertex model[8]. Many cases of homogeneous lattice models, or: a combination of equally structured

R-matrices, have been studied ever since including of course the study of Heisenberg models[17, 18, 20].

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This thesis is based on work from Andrei[13] and Frahm [27]. In these articles, the Heisenberg models have been rederived with an impurity in the R-matrices. We will follow the procedures of the algebraic Bethe ansatz to generate operators including the spin-1₂ XXX Hamiltonian. Instead of taking the homogeneous limit, we see how the addition of inhomogeneities, or impurities, has an effect on the operators that are gen-erated from the R-matrices. We see how these impurities develop extra spin interactions supplementary to the original nearest-neighbour couplings.

The thesis is structured as follows: chapter 2 and 3 explain the mathematical underlying principles that will later be used in the algebraic Bethe ansatz. In chapter 4 the algebraic Bethe ansatz is explained and this chapter will be concluded with the explicit example of the spin-1₂ XXX model. In chapter 5, we will move away from the homogeneous limit: the impurities are introduced. In this chapter the consequences of adding an impurity will be shown as well as the restrictions of what impurities may be to be physically relevant. In chapter 6, the notion of complex solutions, also known as string states, are more extensively described, such that in chapter 7 certain behaviour of string states under influence of impurities could be predicted by looking at ‘collapsing strings’. In chapter 8, the strategy of numerically solving the Bethe equations is briefly described and concluded with some actual calculations.

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Chapter 2 Classical integrability

The idea of the algebraic Bethe ansatz is naively interpreted as the quantum mechanical version of a mathematical concept that is used to solve nonlinear partial differential equations, called the inverse scattering method. For pedagogical reasons, I will start by explaining the classical version of what may be considered a generalisation of the Fourier transform. Subsequently, the algebraic Bethe ansatz will follow more naturally in the upcoming chapter.

Liouville-Arnold integrable systems The classical version of the notion of integra-bility is a well-known concept. Suppose having a system with Hamiltonian H, living in a finite-dimensional phase space M , with n degrees of freedom: dim M = 2n. For any function f on M , the equations of motion are ˙f = {H, f }. The system is said to

be Liouville-Arnold integrable if it has n equations of motions such that their Poisson brackets{, } vanish [15]:

1. There are n integrals of motion: {H, Fi} = 0, i= 1, . . . , n;

2. All are in involution: {Fi, Fj} = 0, ∀i, j.

2.1 Lax pairs

The concept of Lax pairs is used to describe non-linear systems in a linear fashion. The notion of Lax pairs can be described as the pair of two linear, either scalar or matrix, operators L and M . The Lax pairs are defined such that the Hamiltonian evolution equa-tions are related with linear operators, as proposed by Peter Lax in 1968[7, 21]. In a classical integrable system, one can associate a self-adjoint operator L, the Lax operator, with an unitary operator u, such that under evolution of time t, the Lax operator would

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change with t, but remain unitarily equivalent. Hence, there will be a one-parameter family of unitary operators u(t), such that:

L(t) = u(t)L(0)u−1(t). (2.1)

Therefore, any function of the Lax operator I(L) that is invariant under L → uLu−1_{is a}

constant of motion. Now, taking the time derivative:

d L(t) d t = ˙L(t) = du d tL(0)u −1_{(t) + u(t)L(0)}du−1(t) d t , (2.2)

and defining M = ˙uu−1_{. Observe that the time derivative of u}−1_{(t) is}du−1(t)

d t = −u−1(t) du(t)

d t u−1(t),

so that the Lax equation naturally appears:

˙L= [M, L]. (2.3)

The operator M is required to be chosen such that the Lax equation, or operator ˙L, and

Mdo not contain differential operator terms. Thus, L and M may be chosen to be either scalar of matrix operators.

Importantly, the eigenvaluesλ of the linear operator L are independent of time, ∂tλ = 0.

Hence, for any integrable system, if you have its Lax pairs, you should be able to find sufficiently many integrals of motion that are in involution, to show that the system is an integrable system.

Example: Korteweg-de Vries equation The modern concept of Lax pairs is used in solving integrable systems and was first used in 1967 to solve the Korteweg-de Vries (KdV) equation. Similarly, the idea of the Lax pairs [5, 10] is demonstrated in the following. Suppose we have the Lax pair[36]:

L= ∂ 2 ∂ x2 − u M = α ∂3 ∂ x3 − B(x, t) ∂ ∂ x − C(x, t) ∂ u ∂ x, (2.4)

with u= u(x, t). Now clearly, the Lax equation reads:

∂ L ∂ t = − ∂ u ∂ t = α ∂3 ∂ x3 − B ∂ ∂ x − C ∂ u ∂ x, ∂2 ∂ x2 − u . (2.5)

Expanding the commutator to retrieve the explicit form for ˙L: [M, L] = 2∂ B ∂ x − 3α ∂ u ∂ x ∂2 ∂ x2 + ∂2_B ∂ x2 + 2 ∂ C ∂ x − 3α ∂ u2 ∂ x2 ∂ ∂ x +∂ 2_C ∂ x2 + B ∂ u ∂ x − α ∂3_u ∂ x3. (2.6)

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In this expression, the ˙L still has differential operator terms. The operators B and C thus have to satisfy the following to solve for this problem:

3α∂ u ∂ x = 2 ∂ B ∂ x ∧ 3α ∂2_u ∂ x2 = ∂2_B ∂ x2 + 2 ∂ C ∂ x. (2.7)

These equations may be solved by integration. Plugging the expressions for B and C back into (2.5) yields:

∂ u ∂ t − 3 4αu ∂ u ∂ x + 1 4α ∂3_u ∂ x3 = 0, (2.8)

which, forα = 4, is the well-known KdV equation.

2.2 Zero curvature condition

In the previous section we have seen the Lax pair concept for linear operators L and

M. However, it is possible to find a Lax pair of matrices as well. Again consider the operator L, with its eigenfunction f : L f = λf . Take the derivative of this eigenvalue problem:

∂

∂ tL f = ˙L f + L ˙f, (2.9)

which can be evaluated using the Lax equation:

(L − λ)( ˙f + M f ) = (L − λ)C(t)f = ˙λf . (2.10)

As mentioned in the previous section, the eigenvalues of this problem do not depend on t: ˙λ = 0. Hence if f is an eigenfunction of L, then so is ˙f + M f . We now have an overdetermined system. By requiring the Lax equation to hold, we see how integrability is a compatibility condition for overdetermined systems to obtain a solvable system of matrix differential equations:

∂ f

∂ t + M f = 0. (2.11)

More generally, defining a vector F = (f0, f1, ..., fn−1)T, where fk = ∂xkf, and the Lax

pair: L= ∂ n ∂ xn + un−1(x, t) ∂n−1 ∂ xn−1+ ... + u1(x, t) ∂ ∂ x + u0(x, t), (2.12) M = ∂ n ∂ xn + vn−1(x, t) ∂n−1 ∂ xn−1 + ... + v1(x, t) ∂ ∂ x + v0(x, t). (2.13)

The equation L f = λf is now equivalent to the first order matrix equation:

∂ F

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where U_L=       0 1 0 . . . 0 0 0 1 . . . 0 .. . ... ... ... ... 0 0 0 . . . 1

λ − u0 −u1 −u2 . . . −un−1

      . (2.15)

Also similar to the scalar differential equation case, we consider∂tf + M f = 0, and

assume the Lax equations hold. The matrix differential equation above can be used to solve the position derivative of the latter:

∂ ∂ x ∂ ∂ tf + M f = 0, (2.16)

and express every higher order derivative∂n

x f in terms ofλ and lower order derivatives.

This results in another equation:

∂ F

∂ t = VMF. (2.17)

The two equations combined now give:

∂2 ∂ x∂ tF = 0 − ∂ ∂ x(VMF) = ∂ ∂ t(ULF), (2.18)

which leads us to the zero curvature representation of the Lax pair[37]: 0=∂ UL

∂ t − ∂ VM

∂ x + [UL, VM]. (2.19)

Example: Korteweg-de Vries equation Define two matrices U_Land V_M as such[36]:

U_L= ₀ ₁ λ −α 6u 0 , (2.20) and V_M = α 6 ∂ u ∂ x −4λ −α3u −4λ2+λα3 u+ α2 18u 2₊α 6 α 6 ∂2_u ∂ x2 −α₆∂ u_{∂ x} . (2.21)

Plugging these two matrices into the zero curvature representation of the Lax equation (2.19), we obtain: 0 0 −α6 ∂ u ∂ t + αu∂ u∂ x +∂ u 3 ∂ x3 0 = 02×2. (2.22)

The compatibility condition thus reads: −α 6 ∂ u ∂ t + αu ∂ u ∂ x + ∂ u3 ∂ x3 = 0, (2.23)

which is the famous Korteweg-de Vries (KdV) equation, atα = 6. Hence, the integrals of motion that come from the matrix L (2.20), describe, among others, the KdV system. Note how we now have arrived at the KdV equations without the need of having to solve higher order problems as for example the problem (2.7).

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2.3 Monodromy matrix

Consider the zero curvature condition as a compatibility condition of a new defined system of differential equationsΦ(x, t|λ) which depend on a spectral parameter λ ∈ C. Hence[30]: ∂ ∂ x + U(x|λ) Φ(x, t|λ) = 0 ∂ ∂ t + V (x|λ) Φ(x, t|λ) = 0. (2.24)

A special solution to these equations is the one of the transition matrix T(x, y|λ): ∂

∂ x + U(x|λ)

T(x, y|λ) = 0, T(x, x|λ) = 1, x ≥ y. (2.25)

can find another identity of the transition matrix:

∂yT(x, y|λ) − T(x, y|λ)U(y|λ) = 0, (2.26)

This identity, combined with (2.25), leads us to the important property of the transition matrix:

T(x, z|λ)T(z, y|λ) = T(x, y|λ), y ≤ z ≤ x. (2.27)

The trivial identity det T = exp(Tr ln T) and the latter lead us an expression for the determinant of the transition matrix in terms of the function of the system V(x):

det T(x, y|λ) = exp− Z x

y

Tr U(z)dz. (2.28)

The monodromy matrix T(λ) is an important case of the transition matrix, such that the transition matrix describes the transition of x over the complete domain of the system:

T(λ) = T(x = L, x = 0|λ). (2.29)

Later, we shall see how the trace of the transition matrix can be used to generate con-stants of motion. This, of course, is of great interest in the notion of integrability. Its trace we will from here on call the transfer matrix:

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Example: Nonlinear Schrödinger equation In the example of the classical nonlin-ear Schrödinger (NS) equation, it becomes clnonlin-ear how the structure of the monodromy matrix generates constants of motions, among which: a Hamiltonian of the system. Consider the monodromy matrix T(λ) that is a solution to (2.25) with elements that are functionals of the fields ψ(x) which have periodic boundary conditions: ψ(x) =

ψ(x + L):

T(λ) = A(λ) B(λ) C(λ) D(λ)

. (2.31)

The trace of the monodromy matrix is thusτ(λ) = A(λ) + D(λ). We will now derive an explicit expression for the trace of the monodromy matrix. Consider the matrix

U_L= U(x|λ) and an off-diagonal matrix Ω(x): U(x|λ) = iλ 2σ3+ Ω(x), ∗ _{Ω(x) =} 0 ipcψ∗(x) −ipcψ(x) 0 . (2.32)

Suppose the monodromy matrix T(x, y|λ) is written as a transformation of the diagonal matrix D(x, y|λ):

T(x, y|λ) = G(x|λ)D(x, y|λ)G−1(y|λ), (2.33)

where G is constructed as an asymptotical expansion inλ of off diagonal matrices G_n, which may be interpreted as gauge fields related to the transformation of the latter:

G(x|λ) = 1 + A(x|λ), A(x|λ) = ∞ X n=1 λ−n_G n(x). (2.34)

The equation (2.25), multiplied with G(x|λ) from the left, now becomes:

∂x+ W(x|λ)

D(x, y|λ) = 0, (2.35)

such that the new matrix W(x|λ) is given by:

W(x|λ) = iλ

2 G

−1_(x|λ)σ

3G(x|λ) + G−1(x|λ)Ω(x)G(x|λ) + G−1(x|λ)∂xG(x|λ), (2.36)

which we can require to be diagonal. This requirement implicitly defines the matrix G. The latter may be multiplied by G(x|λ):

W(x|λ) + A(x|λ)W(x|λ) = iλ 2σ3+ iλ 2σ3A(x|λ) + Ω(x) + Ω(x)A(x|λ) + ∂xA(x|λ), (2.37) ∗_σ

3represents the 2× 2 Pauli matrix

1 0 0 −1

.

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This equation can easily be solved recursively for G_n(x), by inspecting the coefficients

λ−n_{in the expansion of A}_(x|λ): G₁(x) = iσ₃Ω(x) G₂(x) = iσ₃(∂_xiσ₃Ω(x)) G₃(x) = iσ₃∂_xG₂(x) − iσ₃G₁(x)Ω(x)G₁(x) .. . G_n(x) (2.41)

Because we demanded W(x|λ) to be a diagonal matrix, the equation (2.35) has a so-lution for D:

D(x, y|λ) = exp

Z y

x

W(z|λ)dz. (2.42)

Recall at this point that we imposed periodic boundary conditions on the fieldsψ, hence the trace of the monodromy matrix is now equal to that of the trace of D for the tran-sition over the imposed domain:

τ(λ) = Tr D(L, 0|λ) = Tr T(L, 0|λ). (2.43)

At this point, we can see how τ(λ) is a generator of constants of motion and we can explicitly write them out recursively:

τ(λ) = Tr exp Z 0 L W(z|λ)dz = Tr exp Z 0 L  iλ 2 σ3+ ∞ X n=1 λ−n_Ω(x)G n(x) d x = Tr exp −iλL 2 σ3+ Z 0 L ∞ X n=1 λ−n_Ω(x)G n(x)d x | {z } iσ3cP_∞ n=1λ−nIn . (2.44)

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We are done now, the integrals of motions are essentially found now. In the case of this example withΩ(x) as per (2.32) we find:

I1= Q = Z L 0 ψ∗_{(x)ψ(x)d x} _(2.45) I2= P = Z L 0 ψ∗_(x)∂ xψ(x)d x (2.46) I3= H = Z L 0 ∂xψ∗∂xψ + cψ∗ψ∗ψψ d x. (2.47)

2.4 Classical r-matrix

In the previous section, we have found a way of finding integrals of motion I_n, only now we are still left with the other requirement of integrability, namely: {In, Im} = 0 ∀n, m.

To start with, in the notion of the Lax equation, we will obtain more knowledge about the Poisson structure of the L-matrices: {Li j, Lkl}. Say, we have the matrix Lax pair

M, L of two N× N matrices, and L is diagonalized L = UΛU−1, with elements λi. The

canonical basis of the Lax pair matrices is(e_{i j})_kl = δ_ikδ_jl. Hence, the Lax matrix L may be written as:

L=X

i j

L_{i j}e_{i j}. (2.48)

In a more general sense, let L be a tensor product of multiple matrices, and let Lidenote

the i-th L matrix embedded in the tensor product such that L1 = L ⊗ 1 ⊗ ... ⊗ 1, etc. etc.

To confirm the involution property of the eigenvalues of L, we will compute the diago-nalised L matrix’s Poisson bracket and assume its eigenvalues commute: {λi,λj} = 0:

{Li, Lj} = {UiΛiUi−1, UjΛjU−1j } = UiΛi{Ui−1, Uj}ΛjU−1j + Ui{Λi, Uj}Ui−1ΛjU−1j + {Ui, Uj}ΛiUi−1ΛjUj−1+ UjΛjUiΛi{Ui−1, U −1 j } + UjΛjUi{Λi, U−1j }U −1 i + UjΛj{Ui, Uj−1}ΛiUi−1 + UjUiΛi{U_i−1,Λj}U−1_j + UjUi{Λi,Λj}U_i−1U−1_j + Uj{Ui,Λj}ΛiUi−1U−1j = UiUj{Λi,Λj}Ui−1U −1 j + [ri j, Li] − [rji, Lj], (2.49) where r_{i j} = q_{i j}+ 1₂[k_{i j}, L_j], q_{i j} = U_j{Ui,Λj}Ui−1U −1 j and ki j = {Ui, Uj}Ui−1U −1 j . We

as-sume the eigenvalues commute, and using this to evaluate the Jacobi identity†of Poisson brackets:

[Li,[ri j, rik] + [ri j, rjk] + [rk j, rik] + {Lj, rik} − {Lk, ri j}] + permutations = 0. (2.50) †_{{ f , {g, h}} + {g, {h, f }} + {h, { f , g}} = 0}

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Any function that is constant over the phase space commutes with a function that de-pends on that phase space. If the r-matrices happen to be independent of the phase space of L_i, this identity is heavily reduced to the illustrious classical Yang-Baxter equa-tion:

[ri j, rik] + [ri j, rjk] + [rik, rjk] = 0. (2.51)

Now that we have found the classical Yang-Baxter equation, we could use it to find out more information about the Poisson structure of the system of equations that we have. In particular, we want to get to know the Poisson structure of the transition matrix, such that it could tell us more on the requirements of integrability for the methods described in the previous section. So, ideally we obtain the result for:

T(λ), T(µ) , (2.52)

or better, from the last section:

{In, Im}. (2.53)

Recall the transition matrix T(x, y|λ) that satisfies equations (2.26) and (2.25). The matrix T is a functional of fieldsψ and φ. For any functional A and B of fields ψ and

φ, the poisson bracket is written using the variation principle [15]:

{A, B} = −i Z dz δA δψ(z) δB δφ(z)− δA δφ(z) δB δψ(z) , (2.54)

where the variation of the functionals:

δA = Z dz δA δψ(z)δψ(z) + δA δφ(z)δφ(z) . (2.55)

Using these expressions the matrix elements of T may be written as:

T_i1i2(x, y|λ), T_k1k2(x, y|µ) =

We need an expression for the variationsδT and δU. Using equation (2.25), one could find the variationδT under a changing potential U → U + δU:

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Until now, the reasoning is general for any system satisfying the equation (2.25). Now suppose we are familiar with the scattering information about the system, by virtue of the Yang-Baxter equation. The r-matrices now each relate to two vector spaces, with each vector space related to its spectral parameter, eg. r12= r12(λ, µ) in V ⊗ V ⊗ 1 and

r₁₃= r₁₃(λ, ν) in V ⊗1⊗V . If the system satisfies the Yang-Baxter equation, the Poisson

brackets of two matrices U of the system may be written as:

U(x|λ), U(y|µ) = δ(x − y)r(λ, µ), U(x|λ) ⊗ 1 + 1 ⊗ U(y|µ). (2.60) This is the last step to show that:

Again, the trace of the matrix T is τ(λ) = Tr T(λ) = P_iT_ii(λ). Hence for the tensor

product of the two T matrices holds: Tr(A ⊗ B) = P_iP_jA_iiB_{j j} = Tr ATr B. Using the

cyclic property of the trace, this shows easily how the existence of a r-matrix ensures that:

{τ(λ), τ(µ)} = 0, ∀ λ, µ. (2.62)

What was left from the previous section, where it was shown how the zero curvature condition generates the monodromy matrix T(λ), was to show that the resulting inte-grals of motion I_n also satisfied the requirements for integrability. It is an easy task to plug equation (2.44) into the last equation and see how the integrals of motion satisfy: {In, Im} = 0.

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Chapter 3 Quantum integrability

It follows naturally that one wants to understand the notion of quantum integrability, in a similar fashion as for the classical integrable systems. As in the classical case, it would be favourable if, for quantum mechanical systems, we were able to differentiate the er-godic systems, the non-integrable systems, and the non-erer-godic systems, the integrable systems. However, the definition of quantum integrability is much more ambiguous as will be shown later. A naive approach would be to use the classical definition of inte-grability and transform it into a quantum version.

The naive definition of quantum integrability is an appropriate definition, and it is the definition we use as a basis for the Bethe Ansatz in the following chapter[35]. Intu-itively, it is the use of quantum commutators i[, ] with quantum operators instead of poisson brackets in the classical version of integrability. More concretely, we say a sys-tem is quantum integrable if it posses a complete set of commuting quantum operators

Q_j such that:

1. [H,Qi] = 0 for i = 1, . . . , dim(H),

2. [Q_i, Q_j] = 0 ∀i, j.

3.1 Naive integrability

The step from classical to quantum integrability is not as trivial as it seems to be. The requirement of having a complete set of quantum operators is thus very ambiguous. This is easily demonstrated by taking the classically integrable harmonic oscillator in its quantum mechanical form[24]:

ˆ H = ω ˆ n_i+1 2 with nˆ_i = c†_ic_i; [ci, c † i] = δi, j. (3.1)

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For the Hamiltonian, there exists an orthonormal basis{|n〉} of energy eigenstates: ˆ

n|n〉 = n |n〉 , n∈ Z≥. (3.2)

Its projection operator is hereby defined as: ˆ

P(m) = |m〉〈m| , ˆP(n), ˆP(m) = 0. (3.3)

We can however relabel the basis into a basis of two labels|n〉 = |n(n1, n2)〉 → |n1, n2〉.

The transformation of n into the ‘new basis’ may now, not uniquely, be written such that the eigenvalue n is a function of n1 and n2: n= n1+

1

2(n1+ n2)(n1+ n2+1). This makes

sure the eigenvalues of the Hamiltonian are still non-degenerate in n: ˆ H_{|n〉 = ˆ}H_|n₁, n2〉 = ω n1+ 1 2(n1+ n2)(n1+ n2+ 1) + 1 2 |n1, n2〉 , n1, n2 ∈ Z≥. (3.4)

By inspection of the behaviour of the sequence of states|n〉 for the labels n1 and n2, it

is possible to recognise the pattern of|n〉 for changing n1 and n2. Observe the pattern

for fixed n1= 0, with incrementing n2:

|0〉 , |1〉 , |3〉 , |6〉 , |10〉 , . . . , (3.5)

for n1= 1 and incrementing n2:

|2〉 , |4〉 , |7〉 , |11〉 , |21〉 , . . . , (3.6)

etc. etc.

With the complete pattern revealed, operators ˆn₁ and ˆn₂ are:

ˆ n₁= ∞ X k=0 ∞ X l=0 k ˆP l+1 2(l + k)(l + k + 1), (3.7) ˆ n₂= ∞ X k=0 ∞ X l=0 l ˆP k+1 2(k + l)(k + l + 1). (3.8)

Because the two operators are just linear combinations of ˆP, they themselves commute [ˆn1, ˆn2] = 0. Recall the, not unique, definition we used of the new labels n1 and n2.

Because of this, the Hamiltonian is to be written: ˆ H = ω ˆn1+ 1 2(ˆn1+ ˆn2)(ˆn1+ ˆn2+ 1) + 1 2, (3.9) and: [ˆn1, ˆH] = 0, [ˆn2, ˆH] = 0. (3.10)

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Now, define a classical non-integrable system of n= 2 degrees of freedom: ˆ H_c= ∞ X n=0 E_nPˆ_n. (3.11)

The operatorP_n is, again, the operator projecting a vector onto energy eigenstates. We do not know anything about the structure of the eigenvalues, but we are able to define a function that sets the structure of the eigenvalues such that their structure matches with the quantum system above.

εn= g−1(En), En= n +

1

2, n= 0, 1, 2, . . . . (3.12)

The ‘new Hamiltonian’ now reads: ˆ H_c= g−1( ˆH) ˆH= ∞ X n=0 g−1(E_n)E_nPˆ_n= ∞ X n=0 εnPˆn. (3.13)

In the description above, we have seen how a quantum-equivalent system with equidis-tant eigenvalues, we can find two operators the commute with the Hamiltonian similar to (3.7) and (3.8). Hence, it seems possible for the quantum-variant of our classical non-integrable system to be quantum integrable since it now is possible to have enough commuting operators.

The steps followed could also be done in reversed order. In that case, it is possible to show that a set of commuting quantum operators may be reduced into one quantum operator commuting with the Hamiltonian. Therefore, the simple transformation of the classical Liouville integrability definition into a naive quantum variant does not give us a rigorous basis that could separate quantum systems into integrable and non-integrable categories.

3.2 Transfer matrix

With the scheme for classical systems in mind, we will quantize these to arrive at the quantum mechanical version of the inverse scattering method. Consider a one-dimensional lattice model of N sites. In the classical case where the system of differen-tial equations Φ(x, t|λ) depends on the continuous spatial coordinate, these will now be quantized to the sites m∈ Z: Φ(x, t|λ) → Φ(m, t|λ). Hence, in this case the L matrix is now a L operator shifting the functionΦ from one site to its neighbour [25]:

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The transition matrix T(n, m|λ) in the quantum case describes the transition of site m to site n+ 1:

T(n, m|λ) = L(n|λ)L(n − 1|λ) . . . L(m + 1|λ)L(m|λ), (3.15)

and the monodromy matrix T(λ) = T(N, 1|λ) being a special case of the transition ma-trix that runs over the entire lattice of the system. The elements Ti j of the monodromy

matrix T(λ) are now quantum operators, acting in the Hilbert space of the Hamilto-nian H∈H. As a result, the monodromy matrix itself acts in the auxiliary spaceAand quantum spaceH: A_⊗H.

The trace over the monodromy matrix, the transfer matrixτ(λ), is of significant impor-tance for the algebraic Bethe ansatz in the following chapter. As per its definition, the transfer matrix acts in a Hilbert space and similarly to the classical case, it is a generator of constants of motion: τ(λ) ≡ TrAT(λ) = exp ∞ X n=0 αn n!Qn(λ − ξ) n_:_H_→_H_. _(3.16) Or: Q_n = α−1_n d n dλnlnτ(λ)|λ=ξ. (3.17)

3.3 Yang-Baxter equation

Consider the transition matrix T(n, m|λ) describing the transition between two sites

n, m within the lattice. This matrix is constructed from the L-matrices, which in the quantum case has matrix elements of quantum operators. Hence, we assume the ele-ments Li j of the L-matrices commute like a generic quantum operator:

[Li j(α|λ), Lkl(β|µ)] = 0, if α 6= β. (3.18)

This is known as ultra-locality, because it essentially says that the matrix elements of a L(α|λ) operator only act nontrivially in their own local quantum space at the site

α. With the assumption of ultra-locality, one can now easily derive the commutation

relations of the quantum monodromy matrix. To start with, take two transition matrices from siteα to β with different spectral parameters commute:

˜ R(λ, µ) T(α, β|λ) ⊗ T(α, β|µ) = ˜ R(λ, µ) L(α|λ) ⊗ L(α|µ) . . . L(β|λ) ⊗ L(β|µ) . (3.19)

The ˜Rmatrix exists such that it commutes all L matrices acting on the same site ˜R(λ, µ)(L(α|λ)⊗ L(α|µ)) = (L(α|µ) ⊗ L(α|λ))˜R(λ, µ). With the existence of such a ˜R matrix, one can

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easily see how two transition matrices, both from siteα to β, also commute with the ˜R matrix, by simply applying the commutation relation step by step onto equation (3.19):

˜ R(λ, µ) T(α, β|λ) ⊗ T(α, β|µ) =T(α, β|µ) ⊗ T(α, β|λ) ˜ R(λ, µ). (3.20)

As a result of the definition of the transfer matrixτ(λ), and the invariance of the trace under cyclic permutations, it is trivial to see that any two transfer matrices, although not equal per seτ(λ) 6= τ(µ), may always commute:

[τ(λ), τ(µ)] = 0, ∀λ, µ. (3.21)

Now do the same steps of deriving a commutation relation again for a triple tensor product of monodromy matrices, and use the commutation relation (3.20) to solve it. Since the above commutator describes the relation for two transfer matrices, or monodromy matrices to say, it is possible to arrive at two different expressions of our triple product: T1T2T3= ˜R−112T2T1R˜12T3‡ = ˜R−1 12R˜−113T2T3T1R˜13R˜12 = ˜R−1 12R˜−113˜R−123T3T2T1R˜23˜R13R˜12, (3.22) and: T1T2T3= ˜R−123T1T3T2R˜23 = ˜R−1 23R˜−113T3T1T2R˜13R˜23 = ˜R−1 23R˜−113˜R−112T3T2T1R˜12˜R13R˜23. (3.23)

Merging the two expressions: ˜

R−1₂₃R˜−1₁₃R˜−1₁₂T₃T₂T₁R˜₁₂R˜₁₃˜R₂₃= ˜R−1₁₂R˜−1₁₃R˜−1₂₃T₃T₂T₁R˜₂₃R˜₁₃˜R₁₂

(. . . )

1 = ˜R12R˜13R˜23R˜−112R˜−113R˜−123, (3.24)

results in the famous Yang-Baxter equation: ˜

R₁₂(λ, µ)˜R₁₃(λ, ν)˜R₂₃(µ, ν) = ˜R₂₃(µ, ν)˜R₁₃(λ, ν)˜R₁₂(λ, µ). (3.25) This independently derived in 1968 and 1971 by C.N. Yang and R. J. Baxter respectively, relation is all we need to generate an integrable model such as for example the sine-Gordon, Bose gas or Heisenberg model.

‡_{For convenience, the following notation is used here: T}

i= 1⊗· · ·⊗ ith space

z }| {

T(λi) ⊗ · · ·⊗1. Similarly, the ˜Ri j

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Chapter 4 Algebraic Bethe ansatz

The algebraic Bethe ansatz (ABA) is a technique to reveal physical properties of an inte-grable system by analysing its algebraic structures. It builds on the previously obtained results of the Yang-Baxter equation that described how two monodromy matrices T(λ) and T(µ) are related. The following shows the analysis such that we could eventually identify the XXX model in the algebraic structures.

4.1 Scattering profile

The starting point of the Algebraic Bethe Ansatz is the knowledge that for our system a R-matrix exists. As we could have seen before, this describes the relation with the monodromy matrix:

R12(λ, µ)T1(λ)T2(µ) = T2(µ)T1(λ)R12(λ, µ). (4.1)

At this point, we will start losing generality, and get to a more specific example, ob-viously more towards the XXX model. The first start would be to retrieve a specific structure of the R matrix. We do not want it to be a constant matrix. This could for example be the unitary matrix or permutation matrix. One simple, and once again: not unique, form for the matrix could thus be:

R(λ, µ) =    a(λ, µ) 0 0 0 0 b(λ, µ) c(λ, µ) 0 0 c(λ, µ) b(λ, µ) 0 0 0 0 a(λ, µ)   . (4.2)

A specific choice of functions a(λ, µ), b(λ, µ) and c(λ, µ) can be associated with a cer-tain collection of models. In appendix C, multiple examples are shown of R matrices and the model they have as a result, for the sake of motivation for the reader.

Now that we have a more restricted form for the R matrix, we can also say more about the monodromy matrix T(λ) that is still completely unknown at this point. Given the

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form of our R matrix, the T matrix should be a 2× 2 block with matrix elements in a Hilbert space: T(λ) = A(λ) B(λ) C(λ) D(λ) . (4.3)

With the Yang-Baxter relation (4.1), the R-matrix (4.2) and the general form of T(λ) as per (4.3), we essentially retrieve 16 commutation relations. The commutation relations are explicitly worked out for this case of the R-matrix (4.2) in appendix C.

These commutation relations help us solving the eigenstates of the generator τ(λ). To start with, we can interpret the monodromy matrix as a matrix with creation and annihilation operators: B(λ) and C(λ). We assume the existence of some vacuum state, which in the case of a Heisenberg chain does represent the state of all spins pointing up. This vacuum state we want to be an eigenstate of the transfer matrix with eigenvalues

a(λ) and d(λ). Looking at the structure of the T matrix (4.3), one could also recognise

the C(λ) element as an annihilation operator:

A(λ) |0〉 = a(λ) |0〉 , D(λ) |0〉 = d(λ) |0〉 , C(λ) |0〉 = 0. (4.4) Note how any eigenstate of τ(λ) can never depend on the parameter λ as a direct consequence of the commutation relation of transfer matrix (3.21). Of course, this does not hold for the eigenvalues itself which may depend onλ.

We also assume there exists a dual of the vacuum state for which B(λ) is the annihilator of the vacuum dual:

〈0| A(λ) = a(λ) 〈0| , 〈0| D(λ) = d(λ) 〈0| , 〈0| B(λ) = 0, (4.5)

and, of course, the eigenstates are orthonormal:

〈0|0〉 = 1. (4.6)

It seems natural to define the state that is the product of the creation operator working on the vacuum: B(λ) |0〉 = |λ〉. However, since we have no explicit expressions for the elements of the monodromy matrix, this might not be as trivial as it seems. We should make a more specific choice here of the monodromy matrix. In lattice models, such as our XXX model, usually one defines the monodromy matrix as a product of local

L-operators:

T(λ) = L_N(λ)L_N₋₁(λ) . . . L1(λ). (4.7)

All of these L_j-matrices act in their own local space and some auxiliary space: A_⊗H_j. We now ‘redefine’ our vacuum state as the product of local vacuum states: |0〉 = ⊗N

j=1|0〉j.

We can assume the form of L-matrices looks similar as to (4.3), however acting now in one of the local vacua:

L_j(λ) =Aj(λ) Bj(λ) C_j(λ) D_j(λ)

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For these local operators acting in local vacua, the same rules apply as to (4.4). Using the definition (4.7), we can trivially find the relation between the ‘global’ and ‘local’ eigenvalues: a(λ) = M Y j=1 a_j(λ), d(λ) = M Y j=1 d_j(λ). (4.9)

Also at this point, is clear that B(λ) is a combination of eigenvalues aj(λ), d(λ) and

annihilation operator C(λ).

Now, we are able to interpret the operator B(λ) as being a creation operator such that it creates the ‘Bethe state’, such that:

M Y j=1 B(λ_j) |0〉 =λ_j M . (4.10)

As can be seen in the commutation relations in appendix D, the ordering of the multiple creation operators is immaterial.

4.2 Bethe equations

Following up on the vacuum eigenstate for the transfer matrix in the previous section, we will now construct other eigenstates of the transfer matrix. We can essentially re-quire the state (4.10) to be an eigenstate of the transfer matrix. This means we are looking for: τ(λ) λ_j M = τ(λ) M Y j=1 B(λ_j) |0〉 = τ(λ|{λ_j_}_M)λ_j M . (4.11)

Again, we can use the commutation relations to see if it is possible to generate such a state. The transfer matrixτ(λ) = A(λ) + D(λ) that acts on the state may be evaluated in its two separate parts. The first one being:

A(λ) M Y j=1 B(λ_j) |0〉 = Λ M Y j=1 B(λ_j) |0〉 + M X l=1 ΛlB(λ) M Y j=1 B(λ_j) |0〉 , (4.12)

withΛ and Λ_l being a function of spectral parameters:

Λ = a(λ) M Y j=1 b−1(λ_j,λ), Λ_l= −a(λ_l)c(λl,λ) b(λ_l,λ) M Y j6=l b−1(λ_j,λ_l). (4.13)

The second part is the D(λ) operator working on the state as:

D(λ) M Y j=1 B(λ_j) |0〉 = ¯Λ M Y j=1 B(λ_j) |0〉 + M X l=1 ¯ ΛlB(λ) M Y j=1 B(λ_j) |0〉 , (4.14)

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with ¯Λ and ¯Λ_l being a function of eigenvalues: ¯ Λ = d(λ) M Y j=1 b−1(λ_j,λ), Λ¯_l= −d(λ_l)c(λ, λl) b(λ, λ_l) M Y j6=l b−1(λ_l,λ_j). (4.15)

Hence, for the Bethe state to be an eigenvector of theτ operator, it should be clear that we should demand thatΛ_l+ ¯Λ_l = 0. This requirement gives us M Bethe equations, for any general form of the monodromy matrix:

a(λ_j) d(λ_j) M Y j6=l b(λ_j,λ_l) b(λ_l,λj) = 1, j= 1, . . . , M. (4.16)

4.3 Bethe eigenstates

We have constructed the Bethe eigenstates in the previous section proceeding from the vacuum state|0〉. We would of course also want to define the dual of the Bethe state. In a similar way, we let the C operator work on the dual of the vacuum:

λj M = 〈0| M Y j=1 C(λ_j). (4.17)

Also, the ordering of C operators is immaterial as can be seen in appendix D. For the dual to be an eigenstate of the transfer matrix the same restrictions hold: the Bethe equations as per (4.16).

The next step is to evaluate the scalar product of the Bethe states for two different sets of solutions to the Bethe equations{λj}M and{µk}N:

λj M µ_j N = 0 M Y j=1 C(λ_j) N Y k=1 B(µ_k)0. (4.18)

In case M 6= N , obviously the two sets of solutions are not equal. In this case the corre-sponding states are orthogonal if the solutions do satisfy the Bethe equations. As before, we use the commutation relations in appendix D and work out the matrix elements of the transfer matrix:

λj M τ(λ) µ_j M = τ(λ|{λj}M)λj M µ_j M. (4.19)

Interestingly, this has as a result that the eigenstates are orthogonal, only ifτ(λ|{λ_j}M) 6=

τ(λ|{µk}M), which means that the set of solutions must differ for the Bethe states to be

orthogonal:

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Now that we know what the orthogonality restrictions are, we are left with the question of what the scalar product, or norm, of the Bethe states would be:

N({λ_j_}_M) =λ_j M λ_j M. (4.21)

Hence, the norm of the Bethe states can only be found in a more explicit form if we make a choice for the elements of the monodromy matrix. In our case of the XXX model, the norm is[29]: N({λ_j_}_M) = ηM M Y j6=k λj− λk+ η λj− λk detΦ({λj}M), (4.22)

whereΦ is a M × M matrix with elements given by:

Φjk= − ∂ ∂ λk ln a(λ j) d(λ_j) M Y j6=k b(λ_j,λl) b(λ_k,λ_j) . (4.23)

4.4 L

-operators

At this point, we have our R-matrices depending each on their own combination of complex parametersλ, µ and ν. The Yang-Baxter equation still holds if we do a change of variablesλ → λ0(λ). In most cases, it should also be possible to apply a change of variables in such a way that for example:

R(λ, ν) = R(λ − ν). (4.24)

Furthermore, the Yang-Baxter equations are defined such that it is possible to pick an unique parameter for each (sub)space it acts on. Hence, from here on, we introduce ultra-local fixed complex parameters for the L-operators, which we will call

inhomo-geneity parametersξ_j, j= 1, . . . , N. The L-operators as per (3.15), we can now rewrite such that each site has its own parameterξ_j, and thus the monodromy matrix is influ-enced by the set of these inhomogeneity parameters:

T(λ|{ξ_j}N) = T(λ) = LN(λ − ξN)LN−1(λ − ξN−1) . . . L1(λ − ξ1). (4.25)

4.5 Trace identities for the XXX model

Now that we have obtained all requirements for diagonalization into the Bethe basis, we want to find the constants of motion of the system. Fixing the function in the R-matrix (4.2) will define the model, which in our case of the XXX model, we will do as follows:

a(λ) = 1, b(λ) = λ

λ + η, c(λ) = η

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with some constantη ∈ C. As a result, this gives: d(λ) = N Y j=1 b(λ − ξ_j). (4.27)

Now, given our monodromy matrix being a product of L-operators with site-dependent parameters{ξj}N as per (4.25), this leads to the following:

T(λ) =b(λ − ξ_N)1 + c(λ − ξ_N)P_aNb(λ − ξ_N₋₁)1 + c(λ − ξ_N₋₁)P_aN

−1

× . . .b(λ − ξ₁)1 + c(λ − ξ₁)P_a₁. (4.28)

In the simplest case, we will evaluate the transfer matrix operator τ(λ) to the point where ξj = ξ and η = 2ξ = i. This should get the transfer matrix into its evaluated

form into a product of solely permutation matrices:

τ(ξ) = TrAPa1

P1NP1N−1. . . P12. (4.29)

Now it is essentially a matter of finding all conserved charges using (3.17):

Q0 = ln τ(ξ) = iP (4.30) Q₁ =η 2 d dλlnτ(λ)|λ=ξ= H (4.31) ...

Of course, Q1 = H is of great interest. As a result of the commuting transfer matrices,

the derivative of the logarithm may be used as _dλd lnτ(λ) = τ−1(λ)_dλd τ(λ). To evaluate this, we need the following identities:

L_j(ξ, ξ_j) = P_{a j} (4.32) L_j(ξ, ξ_j)−1= P_{a j} (4.33) d dλ Lj(λ, ξj) λ=ξ= 1 η[1 − P]a j. (4.34)

Using these identities, one can easily evaluate Q1, where for convenience we define

Πk...l a = Pak...Pal: Q₁= η 2τ −1_(ξ) d dλτ(λ)|λ=ξ = 1 2TrA Π1...N a ₁ η 1 − P_aNΠNa−1...1+ PaN 1 − P_aN₋₁ΠNa−2...1 + · · · + ΠN...2 a 1 η 1 − P_a₁ = 1 2 N X j=1 P − 1_{j j}₊₁. (4.35)

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To arrive at the last expression, we have repeatedly applied the identity Pa bPbc= PacPa b.

The last step in the identification of our Hamiltonian would be to use the relation be-tween Pauli matrices and permutation matrices: P_ασα_i ⊗ σα_j =2P − 1_{i j}. Thus, with the specific choice of the R matrices as per (4.26), we have generated Q1, which is the

Hamiltonian for the XXX model:

Q1= HX X X = N X j=1 S_j· Sj+1− 1 4. (4.36)

Bethe equations for XXX

Now that we have chosen the form of the R-matrix, we also can find a more specific form for the Bethe equations (4.16). Plugging in the choices for a− d, we find that the Bethe equations for the XXX model are:

λ j+ i/2 λj− i/1 N = M Y k6= j λj− λk+ i λj− λk− i , j= 1, . . . , M. (4.37)

In logarithmic form these read as:

θ1(λj) − 1 N M X k6= j θ2(λj− λk) = 2π J_j N. (4.38)

Here, we use the function θ_n(λ) = 2 arctan2_nλ and will continue to do so throughout the rest of the text. The quantum numbers Jj are half-integers for N+ M being even,

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Chapter 5 Spin impurities

Now that we have seen how specific choices of the R-matrices can lead us to a Hamil-tonian. However, the resulting Hamiltonians are not per se obvious with the choices for the R-matrices and its elements. Given that we have the XXX model solutions from the previous chapter, we will see how ultra-local impurities in the L-operators affect the results of the Bethe ansatz. Our starting point will be our defined monodromy ma-trix with a set of inhomogeneity parameters (4.25). Where in the previous chapter, to obtain the XXX model, we took the homogeneous limitξ_j = ξ, ∀j, we now look at the algebraic structure of the monodromy matrix and the Bethe equations, away from this limit. In the following, we use the notation:

ξj = ξ + xj. (5.1)

5.1 Bethe equations

We can still take over the construction of eigenstates as done in chapter 4. The require-ments of the Bethe state to be an eigenstate of the transfer matrix, we have already derived for an arbitrary set of inhomogeneity parameters{xj}N (4.16). Given the value

of arbitrary impurities{xj} and the same form of R-matrices we used in (4.26), we see

the Bethe equations read in their most general form for the XXX model: 1 QN k=1b(λj− i/2 − xk) Y l6= j b(λ_j− λl) b(λ_l− λj) = 1, (5.2) and, with b(λ) = λ λ+i explicit: M Y l6= j λj− λl+ i λj− λl− i = N Y k=1 λj− xk+ i/2 λj− xk− i/2 . (5.3)

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Also, the actual eigenvalues of the transfer matrix with impurities read: τ(λ|{λj}M) = M Y j=1 λj− λ + i λj− λ + N Y j=1 λ − xj− i/2 λ − xj+ i/2 M Y j=1 λj− λ + i λj− λ . (5.4)

It can be shown that the eigenvalues of the Hamiltonian do not depend on the impuri-ties: E({λ_j_}_M) = η 2 d dλlnτ(λ|{λj}M)|λ=ξ (5.5) = i 2τ −1_(λ|{λ j}M) M X j=1  −(λj− λ) + (λj− λ + i) (λj− λ)2 M Y k6= j λl− λ + i λj− λ + N X j=1 (λ + i/2 − xj) − (λ − i/2 − xj) (λ + i/2 − xj)2 M Y k6= j λ − i/2 − xk λ + i/2 − xk M Y l=1 λl− λ + i λl− λ + M X j=1 −(λj− λ) + (λj− λ + i) (λj− λ)2 N Y k=1 λ − i/2 − xk λ + i/2 − xk M Y l6= j λl− λ + i λl− λ (5.6) = i 2τ −1_(λ|{λ j}M) M X j=1 i (λj− i/2)(λj+ i/2) τ(λ|{λj}M) + N X j=1 i (λ + i/2 − xj)(λ − i/2 − xj) N Y l=1 λ − i/2 − xj λ + i/2 − xl M Y k=1 λk− λ + i λk− λ (5.7) = M X j=1 −2 4λ2 j + 1 . (5.8)

Parametrising the impurities as xj = αj+ iβj, one can write down the Bethe equations.

As will be discussed in further detail in §5.4, if we want to consider impurities to be complex valued, they are added in conjugate pairs. In this case, one is able to write down the Bethe equations as follows:

M Y l6= j λj− λl+ i λj− λl− i = N Y k=1 λj− αk+ i 2(1 − 2βk) λj− αk− i 2(1 + 2βk) , (5.9) M Y l6= j (−1)1+ i(λj− λl) 1− i(λj− λl) = (−1)λj− i 2 λj+ i 2 δN%2,1bN /2c_Y k=1 (−1)21+ 2i λj−αk 1−2βk 1− 2iλj−αk 1−2βk 1+ 2iλj−αk 1+2βk 1− 2iλj−αk 1+2βk , (5.10)

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which, after taking the logarithm, leads to: δ1,N %2θ1(λj) + N/2 X k=1 θ1−2βk(λj− αk) + θ1+2βk(λj− αk) = M X l6= j

θ2(λj− λl) + πi(M − 1 + N) mod 2πi, (5.11)

δ1,N %2θ1(λj) + N/2 X k=1 θ1−2βk(λj− αk) + θ1+2βk(λj− αk) − M X l6= j θ2(λj− λl) = 2πJj. (5.12)

Recall the arctangent written asθ_n(λ) = 2 arctan2_nλ. The Bethe quantum number J_jthus is either an integer or half-integer for respectively N+ M is odd or N + M is even. This observation will be decisive when solving the complex solutions of the Bethe equations.

J_j=

¨

Z N+ M odd

Z + 12 N+ M even

(5.13)

Bethe quantum numbers

Now that we have obtained the form of the Bethe equations we have, as a result, also obtained the additional Bethe quantum numbers{Jj}M. Without the knowledge of

spe-cific solutions to the Bethe equations, we can already gain some first insights in the solutions that are possible. The phase shift functionsθ_n(λ) in the Bethe equations do not divergence, thus it seems reasonable to verify the limits of the Bethe numbers as follows: lim λj→∞ δ1,N %2θ1(λj) + N/2 X k=1 θ1−2βk(λj− αk) + θ1+2βk(λj− αk) − M X l6= j θ2(λj− λl) = 2πJ∞ j (5.14) =2Nπ 2 − 2(M − 1) π 2 . (5.15)

Any set of (finite) impurities, thus, has no affect on the limits of the Bethe quantum numbers:

Jma x_j ({x_j}_≤N) = N− M − 1

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5.2 Single impurity

As with many things in physics, we start with the simplest case, which in our case is where we have a single impurity in the system. The inhomogeneity parameter living on site m will perturb the homogeneous limitξ_m → ξm(x) = ξ + x. Considering a single

impurity, the XXX model its L-matrices, in its evaluated formsλ → ξ, have changed:

L_j(ξ, ξ_j) = ¨ Pa j, if j6= m 1 η−x[ηP − x1]a j, if j= m , (5.17) d dλ Lj(λ, ξj) λ=ξ= ¨₁ η[1 − P]a j, if j6= m η (η−x)2[1 − P]a j if j= m , (5.18) L_j(ξ, ξ_j)−1= ¨ Pa j, if j6= m 1 η+x[x1 + ηP]a j, if j= m . (5.19)

Now, this means the monodromy matrix T(λ) in its evaluated form T(λ = ξ), has consequently changed on the m-th site:

T(ξ) = 1

η − xPaNPaN−1...[ηP − x1]am...Pa2Pa1. (5.20)

Let’s first start with finding the second conserved charge, the Hamiltonian. Similarly to the homogeneous case, we would need to have the trace over the auxiliary spaceAof the transfer matrixτ(ξ) and its derivative, evaluated as λ → ξ:

τ(ξ) = 1 η − xTrA(Pa1)P1NP1N−1...[ηP − x1]1m...P13P12, (5.21) τ(ξ)−1₌ 1 η + xP12P13...[x1 + ηP]1m...P1N−1P1NTrA(Pa1), (5.22) d dλ τ(λ)|λ=ξ= Tr(Pa1) 1 η[1 − P]1NP1N−1... 1 η − x[ηP − x1]1m...P12 + P1N 1 η[1 − P]1N−1... 1 η − x[ηP − x1]1m...P12+ ... + P1NP1N−1... η (η − x)2[1 − P]1m...P12+ ... + P1NP1N−1... 1 η − x[ηP − x1]1m... 1 η[1 − P]12 + PN N−1... 1 η − x[ηP − x1]N m...PN2 1 η[1 − P]N1Tr(PaN). (5.23)

The evaluation of this part is a lengthier procedure compared to that in the homoge-neous case. This is a direct consequence of the three different sectors that occur as a result of the added impurity. Where in the homogeneous case we have N similarly-looking terms in the trace over T(λ), we now have three different terms: n /∈ {m, m+1},

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n= m and n = m+1.∗ The first term looks similar to the one in the homogeneous case, which we are already familiar with. The second term n= m reads:†

τ(ξ)−1 d dλτ(λ)|λ=ξ m = 1 η + xΠ 2...m−1 1 [x1 + ηP]1mΠm1+1...N × ΠN1...m+1 η (η − x)2[1 − P]1mΠ m−1...2 1 (5.24) = η n2_{− x}2[P − 1]mm−1. (5.25)

This term looks slightly different from the terms in the homogeneous case with some dependence on x. The third term n= m + 1 reads:

τ(ξ)−1 d dλτ(λ)|λ=ξ m+1= 1 η + xΠ 2...m−1 1 [x1 + ηP]1mΠ1m+1...N × ΠN...m+2 1 1 η[1 − P]1m+1 1 η − x[1 − P]1mΠm1−1...2 (5.26) = 1 η 1 η2_{− x}2Π 2...m−1 1 [x1 + ηP]1m[P − 1]1m+1 × [1 − P]1mΠm1−1...2 (5.27) = 1 η 1 η2_{− x}2 x2[P − 1]_m_−1m+1_{− xη[P}_mm₋₁_{, P}_mm₊₁] − η2[P − 1]mm+1 . (5.28)

This term is now a product of permutation matrices acting on the impurity’s site and its neighbouring sites.

Note that this is the point where we are able say more on the generality of adding impu-rities to the monodromy matrix. The extra term in the Hamiltonian that we will obtain following from the identifies we have now, may simply be added for any new impurity

x, with observance of the requirement that the different impurities{xi} in the system,

located on sites mi, are well separated: |mi− mj| > 1, ∀i, j.

The Hamiltonian’s impurity term H_m0(x) may now be evaluated:

H0_m(x) = η 2 τ(ξ)−1 d dλτ(λ)|λ=ξ− τX X X(ξ) −1 d dλτX X X(λ)|λ=ξ (5.29) =1 2 1 η2_{− x}2 x2[P − 1]_mm₋₁+ x2[P − 1]_m_−1m+1_{− xη[P}_mm₋₁_{, P}_mm₊₁] − η2[P − 1]mm+1+ (x 2 − η2)[P − 1]m+1m . (5.30)

∗_n_{being the n-th element in the sum of the derivative on the transfer matrix (5.23).} †_{For convenience, we have defined}_Πk...l

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The last step in the evaluation of the Hamiltonian is choosing a value forξ and η, which we will do as in the XXX case: η = 2ξ = i. In operator form, the Hamiltonian that is generated by the impurity on site m thus reads:

H0_m(x) = x2γ(x) S_m₋₁· Sm+1− Sm· (Sm+1+ Sm−1) − 1 4 − 2i xγ(x) [Sm−1· Sm+1, Sm−1· Sm] , (5.31) γ(x) = 1 1+ x2. (5.32)

The full Hamiltonian for a single impurity is thus H(x) = HX X X+ H_m0(x). This case of a

single impurity is similar to the one mentioned in other articles specialised to a spin-1₂ impurity [13, 27, 39]. It is worth mentioning that for real values of x, the resulting Hamiltonian H(x) is a hermitian Hamiltonian.

Extreme impurities

With the resulting Hamiltonian term H_m0(x), we should check if they behave as expected for x → 0. Obviously H_m0 → 0 if x → 0, and the system is just the homogeneous XXX model with N sites again.

In addition, the limit of infinite valued impurities could possibly be of interest. Hence: lim x→∞H 0 m(x) = lim_x_→∞ d2 d x2x 2 d2 d x2 1+ x2 S_m₋₁· Sm+1− Sm· (Sm+1+ Sm−1) − 1 4 − d d x2i x d d x 1+ x2 S_m₋₁· Sm+1, Sm−1· Sm (5.33) = lim x→∞ ₂ 2 . . . − 2i 2x, (5.34) = Sm−1· Sm+1− Sm· (Sm+1+ Sm−1) − 1 4. (5.35)

The resulting term cancels both original coupling terms from the XXX model with the site of the impurity m, and it adds a coupling between the sites m+1 and m−1. Hence, the infinity limit essentially is equal to removing a site from the chain, and N→ N − 1

if x→ ∞.

5.3 PT

-symmetric Hamiltonians

Traditionally, one is taught that any valid Hamiltonian should have the mathematical condition of being hermitian. The validity of such a condition is point of discussion as it is argued that non-hermitian,PT-symmetric Hamiltonians could still incorporate real

Spin chains with integrability-preserving impurities

Spin chains with

integrability-preserving impurities

Jorran de Wit

Theoretical Physics

University of Amsterdam

Acknowledgements

Abstract

Contents

List of Figures

Chapter 1

Introduction

Chapter 2

Classical integrability

2.1

Lax pairs

2.2

Zero curvature condition

2.3

Monodromy matrix

2.4

Classical r-matrix

Chapter 3

Quantum integrability

3.1

Naive integrability

3.2

Transfer matrix

3.3

Yang-Baxter equation

Chapter 4

Algebraic Bethe ansatz

4.1

Scattering profile

4.2

Bethe equations

4.3

Bethe eigenstates

4.4

L

-operators

4.5

Trace identities for the XXX model

Bethe equations for XXX

Chapter 5

Spin impurities

5.1

Bethe equations

Bethe quantum numbers

5.2

Single impurity

Extreme impurities

5.3

PT

-symmetric Hamiltonians