Excitations of the Gapped
XXZ Heisenberg Spin-1/2 Chain
In Partial Fulfillment
of the Requirements for the Degree
MSc Physics and Astronomy
Theoretical Physics
At the Institute for Theoretical Physics, UvA September 2017 - August 2018
60 ECTs
Author:
Supervisor/Examiner:
Rebekka Koch Prof. Dr. Jean-S´ebastien Caux
Student number:
Second Examiner:
Acknowledgements iii
Abstract v
1 Introduction 1
1.1 The Power of the Bethe Ansatz . . . 1
Notion of Integrability . . . 1
Role of Integrable Systems in Many-Body Physics . . . 2
Heisenberg Spin Chains . . . 2
Correlation Functions and Form Factors . . . 3
Numerical Methods - ABACUS . . . 4
Classification of States . . . 4
1.2 Goal and Outline of this Thesis . . . 5
2 The XXZ Heisenberg Spin Chain 7 2.1 The One-dimensional Heisenberg Model . . . 7
2.2 Symmetries of the XXZ Spin Chain and the Dimensionality of its Hilbert Space . . . 9
2.2.1 Ground State and Excitations . . . 11
3 The Coordinate Bethe Ansatz 13 3.1 Coordinate Bethe Ansatz for XXZ spin chains . . . 13
A Single Flipped Spin . . . 13
Two Flipped Spins . . . 14
Arbitrary Number of Flipped Spin . . . 15
3.1.1 Bethe Equations in Terms of Rapidities . . . 17
Bethe Equations of the gapped XXZ Heisenberg Spin Chain . . . . 17
Logarithmic Bethe Equations . . . 18
Energy and Total Momentum Parametrized with Rapidities . . . 19
3.2 Complex Rapidities and String States . . . 20
3.3 Bethe-Takahashi Equations . . . 21
String Hypothesis and Bethe Equations . . . 21
String Hypothesis and Logarithmic Bethe Equations . . . 23
Energy and Total Momentum for the String Hypothesis . . . 24
4 Classification of States 25 4.1 Possible Quantum Number Sets . . . 25
4.2 Possible Symmetry Transformations . . . 27
Definition of the Symmetry Transformation Sj,α . . . 29
Properties of the Symmetry Transformation Sj,α . . . 30
4.3 Restrictions on the Quantum Number Sets . . . 31
Restrictions for Case (a) . . . 32
Restrictions for Case (b) . . . 33
4.4 Uniqueness and Completeness . . . 36
Uniqueness . . . 36
Completeness . . . 39
4.5 Counting Sets of Quantum Numbers . . . 40
Rectangular Young Diagrams . . . 40
Young Diagrams with only Distinct Rows . . . 41
A. Two Particles . . . 41
B. Three Particles . . . 43
C. Arbitrary Number of Particles . . . 47
Arbitrarily Shaped Young Diagram . . . 49
4.6 Cardinality and Completeness of Sets . . . 51
4.7 The Phase Space of the Anti-ferromagnet . . . 55
5 String Deviations 59 5.1 Completeness of Heisenberg spin chains . . . 59
String Deviations in the Gapped XXZ Case . . . 60
Narrow and Wide Pairs . . . 61
String Deviations for 2-Strings . . . 62
6 Conclusion 67 Summary and Conclusion . . . 67
Outlook . . . 68
A Bethe Equations for the XXX and the Gapless XXZ Case 69 The Isotropic Case . . . 69
The Gapped Case . . . 70 B Supplemental Equations to Determine String Deviations 73
First of all, I want to thank my supervisor Jean-S´ebastien Caux. With his energy and contagious enthusiasm he reminded me once more, that doing physics is actually fun. I enjoyed this year of collaboration and I am especially grateful for his patience when an-swering my questions - even the trivial ones -, his support in finding a PhD position, and above all, making my attendance at the summer school in Les Houches possible.
I want to thank the whole group - or better say gang - of J.-S. for supporting me and helping me out whenever they could. For instance, not a single member missed my test talk for the interview in Dresden and them giving me extremely useful feedback helped to make it a success. Moreover, it was a great atmosphere within the gang and I really enjoyed working alongside so many great young physicists.
Furthermore, I want to thank all my family! Specifically, my grandparents Helga, Ludwig and Hans, my parents Franz and Brigitte and my wonderful siblings Teresa, Sophia and Johannes. Everyone of you supported my stay in Amsterdam in one way or other and enabled me a joyful and successful two years. It is unfortunately not given to have a lov-ing and supportlov-ing family, but I know that I have one and I am extremely grateful for that.
I want to say thank you to all my friends, all these inspiring and fun people, especially the ones that I got to know during the last two years. You made my time in Amsterdam such a great experience and you are the reason, that my farewell from Amsterdam will be so immensely difficult.
Last but not least, I want to thank my boyfriend Julian for all his professional and mental support. Without you, I would have never graduated. Thanks for all the days and nights you spent helping me out whenever I almost drowned in my self-made chaos or whenever my confidence as a physicist threatened to leave me for good. You never let me down and I am so happy about all the time that we have spent and will spend together.
The magnetic properties of many materials are accurately captured by the Heisenberg model. If the lattice structure possesses a much stronger interaction in one spatial di-mension it behaves as an effective one-didi-mensional system described by a Heisenberg spin chain. Also in experiments with cold atoms Heisenberg spin chains have proven useful as they provide the theoretical backbone for the spin-spin interactions of the cold atoms.
The exact eigenstates of Heisenberg spin chains are provided by the famous Bethe ansatz. The eigenstate basis is one of the key ingredients to calculate form factors, correlation functions and other physical observables that are then measured in experiments. In order to obtain all eigenstates in the framework of the Bethe ansatz, a complete classification of states is needed.
In the specific case of the gapped XXZ Heisenberg spin-1/2 chain the still missing classi-fication of states in terms of quantum numbers is presented in this thesis. Only by using the periodicity of the momenta describing the wave function of an eigenstate, the full classification is obtained. Counting the eigenstates then gives rise to the completeness of the eigenstate basis. However, deviations of bound states have to be taken into account, that hinder a rigorous proof of completeness by solely counting states. These deviations are partially discussed as well.
Such a classification of states is especially useful for algorithms like ABACUS (by J.-S. Caux) that are designed to effectively calculate e. g. spin-spin correlation functions. For the gapped XXZ Heisenberg spin-1/2 chain, ABACUS is missing such a classification which leads to an overcounting of states when calculating certain types of correlation func-tions. The main motivation of this thesis is to fix the overcounting problem of ABACUS.
Introduction
”You can recognize truth by its beauty and simplicity.
When you get it right, it is obvious that it is right – at least if you have any experience – because usually what happens is that more comes out than goes in.“
Richard Feynman [1]
1.1
The Power of the Bethe Ansatz
It was Enrico Fermi’s simplicity in approaching problems [2, pp. 193-195] that inspired Hans Bethe to formulate his famous ansatz [3], nowadays known as coordinate Bethe ansatz. It was published almost a century ago, in 1931, during Bethe’s research visit in Rome, which he primarily undertook to meet with Fermi. The Bethe ansatz reveals the exact eigenstates and eigenfunctions of a certain class of one-dimensional models. In [3], Bethe explicitly found, thanks to his great intuition, the solutions to one particular system - the isotropic Heisenberg spin-1/2 chain. Heisenberg spin chains are the one-dimensional versions of the Heisenberg model, which is an effective model describing spin interactions in atomic lattices, introduced around three years earlier by Heisenberg [4] (and almost simultaneously by Dirac) in order to explain magnetism.
Bethe wanted to generalize this idea to two or three dimensional lattices (at least he states so in his paper [3]) but he either never tried or, more likely, tried and failed. Fact is, his method is not transferable to arbitrary models especially not higher dimensional models and the reason why is integrability.
Notion of Integrability
The notion of integrability comes from classical mechanics. A well-known example, that is often used to illustrate the difference between integrability and non-integrability, is the Kepler problem and its extension to a three body problem. For the former, a general solution to the differential equations for arbitrary initial conditions is found by integration and is therefore coined an integrable model. For the latter, there is no general closed-form solution [5]: The three body problem is in general non-integrable. The reason lies in the number of conservation laws, if one follows the Liouville notion of integrability [6]. A classical system with Hamiltonian H is integrable if it has a 2n-dimensional phase space and n first integrals in involution, e.g. n linearly independent functions f that are Poisson-commuting among each other and with the Hamiltonian H.
The power of the Bethe ansatz - coordinate and algebraic - is that it solves most quantum integrable models. In contrast to the classical case however, it is less clear what a good notion of integrability is [7]. It is clearly not enough to change the Poisson brackets into commutation relations, since then every finite dimensional quantum systems would be integrable. In fact, quantum integrable systems have infinitely many conservation laws in contrast to non-integrable ones. A more accurate statement is thus, that systems are integrable if they satisfy the Yang-Baxter equation (after the independent work of Yang [8]
and Baxter [9]). In one dimensional integrable systems, the eigenstates are characterized by a set of quantum numbers leading to quasi particles that represent the fundamental excitations of the system. Two particles scatter with each other by either preserving (identity) or exchanging (permutation) their momenta and thereby neither get created nor destroyed in a scattering event. Additionally, many-body scattering factorizes into two-body scattering events in which the order does not matter. This is in words roughly what the Yang-Baxter equation states.
The Young-Baxter equation is one of the key ingredients for the algebraic Bethe ansatz developed in the 1980s by the Leningrad (St. Petersburg) school under the leadership of Faddeev. Faddeev also developed the inverse scattering method for quantum integrable systems [10, 11] which together with the algebraic Bethe ansatz allows for calculating form factors and correlation functions that can eventually be measured in experiments.
Role of Integrable Systems in Many-Body Physics
Due to their exact eigenstates and eigenenergies, integrable models are ideal systems to play with. Nevertheless, there are just a small number of integrable systems and one could wonder why they might be of any interest?
The fact that there are exactly solvable models in the past have led to developing better numerical and analytical methods. Integrable models serve as reference models, able to cross-check algorithms and approximate methods, since they provide exact analytical an-swers as well. They not only play this role in quantum many body physics but they have also proven very useful in high energy physics for similar reasons.
Additionally, due to many breakthroughs in experiments with cold atoms in the last two decades integrable models have had a new drive, as it is now possible to realize almost any interaction strength and geometry of quantum many-body systems in the lab. This has lead to new physical questions being raised. Especially in the strong interacting regime, for which methods like perturbation theory fail, tools like the Bethe ansatz are crucially needed to find theoretical answers. For instance, the exact time evolution, thermodynam-ics or correlation functions can be derived for integrable Hamiltonians and have shown to give precise predictions for many experiments [12].
One of the most famous examples is the Quantum Newton Cradle [13] and just recently, by using advanced integrable techniques, it has been shed some light on its dynamics [14]. Out-of-equilibrium physics has gained more and more attention in the field of integrabil-ity, not least because of good experimental realizations: By switching lasers or magnetic fields on and off the state of cold atoms during an experiment suddenly changes (quantum quench). Alternatively, a system can also be driven by applying periodic signals (Floquet dynamics). Both lead to interesting effects in integrable systems for which appropriate the-oretical tools have been developed (e.g. for quantum quenches [15] and Floquet dynamics [16]) and research is still ongoing.
Heisenberg Spin Chains
One of the most fruitful integrable quantum systems solved by the Bethe ansatz is the Heisenberg spin chain, in which only nearest neighbor spin couplings are taken into ac-count. Special cases are for instance the isotropic spin chain (XXX model) which is exactly the spin chain examined in Bethe’s original paper [3], or the gapless XXZ spin chain for which the coupling between spins in one spatial direction is reduced. Introducing
an anisotropy favoring the (anti)-alignment of spins in one spatial direction gives another example - the so-called gapped XXZ model, which has shown to effectively describe many materials such as Sr2CuO3, SrCuO2 [17], CsCoCl3 [18] and CsCoBr3 [19], [20].
The fundamental excitations of spin chains are delocalized spin waves that can be imag-ined as flipped spins over a ground state. These flipped spins can also form bound states, so-called string states. The ground state, ‘unbound’ states and string states then span the Hilbert space of the Hamiltonian.
String states are complicated to verify in experiments and are often only very indirectly measured, for instance by using quantum quenches [21]. Just recently string states for the gapped XXZ spin chain have been directly measured for the first time in SrCo2V2O8
using high-resolution tetrahertz spectroscopy [22].
Correlation Functions and Form Factors
It was around the beginning of this millennium that great progress was made in extracting quantitative results from integrable models, especially from Heisenberg spin chains. That is, it was found by Kitanine et al. that form factors [23] and correlation functions [24] for finite size spin chains at zero temperature can be derived by using the algebraic Bethe ansatz.
Of course, pioneering work beforehand has paved the way: We have already mentioned the algebraic Bethe ansatz and the quantum inverse scattering method [25]. Moreover, the important norm of Bethe wave functions was already known since the late sixties [26], [27], and another crucial step was taken by Slavnov in 1989, when he calculated the overlap (inner product) of Bethe states [28].
We will give some more explanations in the following, highlighting the importance of structure factors and correlation functions. The Fourier transformation of the two-point correlation function hSja(t)Sj¯a0(0)i at zero temperature is the dynamical structure factor
(DSF) Sa¯a(k, ω) Sa¯a(k, ω) = 1 N N X j,j0=1 exp(−ik(j − j0)) Z ∞ −∞ dt exp(iωt)hSja(t)Sj¯a0(0)i (1.1)
where Sja are the spin-1/2 operators of the jth site, N is the system size and α = z, +, −. A more detailed description of the spin operators is given later on. The notation Sj¯astands for the hermitian conjugate (Sja)†. The dynamical structure factor is experimental acces-sible by measuring the cross section in neutron scattering experiments and is therefore of great interest.
Reshaping the dynamical structure factor makes it more suitable for numerical calcula-tions. Hence, another Fourier transform of the operators Ska = P
jexp(−ikj)Sja is used
and after summation over the lattice sites and integration over time we get the so-called Lehmann series representation
Sa¯a(k, ω) = 2π N
X
µ
|hλ0|Ska|µi|2δ(ω − Eµ+ E0). (1.2)
Here, |λ0i is the ground state with ground state energy E0, the states |µi with energy Eµ
are called intermediate eigenstates and the matrix element is the form factor hλ0|Ska|µi. It
exponentially growing in system size. The convenient property however is, that not a lot of states are significantly contributing to the DSF. Therefore, applying certain sum rules that determine the important summands give the DSF up to 99, 9% even though only a few states were considered.
Numerical Methods - ABACUS
The calculation of the DSF including form factors or correlation functions remains a very difficult task and advanced numerical methods have to be applied. Designed for these calculations is for instance ABACUS (Algebraic Bethe Ansatz-based Computation of Uni-versal Structure factors) [29], a state-of-the-art algorithm developed by J.-S. Caux to calculate correlation functions for the prototypical Bethe ansatz solvable models including spin-1/2 Heisenberg spin chains. A significant advantage of ABACUS is that it can deal with a large number of sites N thanks to the efficient implementation of the sum rules. Finding the eigenenergies by exactly diagonalizing the Hamiltonian is in principle possible, but becomes in most cases unsuitable when larger systems are considered: As the Hilbert space increases exponentially in system size one runs into the curse of dimensionality. Nevertheless, this method can serve as comparison for very small system sizes.
ABACUS has shown to give excellent results for various systems: The first calculations done involved the dynamical structure factor for the gapless XXZ spin chain in a mag-netic field [30, 31]. Various calculations of correlation functions followed for XXX and XXZ spin chains [32], [33] as well as for other integrable models like the Lieb-Liniger model [34]. Among others, ABACUS has also been applied to study quench dynamics in the Lieb-Liniger model [35] and it was additionally involved for analyzing the dynamics of the Quantum Newton Cradle [14].
Furthermore, the spectrum of low energy excitations in the anti-ferromagnetic XXZ spin chain, so called spinons, and the corresponding correlators have been analyzed with the support of ABACUS [36], [37]. They had been measured in experiments [19], [20] before-hand. In the gapped regime however, only ’unbound’ states have been involved in this analysis.
To yield quantifiable theoretical results for experiments involving string states (for instance [22]) the corresponding form factors have to be included in the DSF. Unfortunately, ABA-CUS runs into a problem with the representation of eigenstates in terms of quantum numbers in the case of the gapped XXZ-Heisenberg spin chain. This is to be specified in the following section.
Classification of States
One of the key ingredients needed to calculate the DSF, and in fact also almost any other physical quantity, is the eigenstate basis spanned by the mentioned excitations of the spin chain. Therefore, we have to find a unique description of quantum numbers that help to construct all the eigenstates. Broadly speaking, for a spin chain with N sites there are N possible quantum numbers that combined in all possible ways (there might be some restrictions) describe the different states. They only have to obey the generalized Pauli principle, i.e. no two quantum numbers belonging to the same (the state defining) set can coincide. In the case of the gapped spin chain however, there is too much of a choice for quantum number sets. It turns out that two different combinations still may describe the same eigenstate and hence the description in terms of quantum numbers is not unique.
ABACUS is also still missing a correct classification of all states for the gapped XXZ spin chain and as a consequence, the same eigenstates are either counted multiple times or too few eigenstates are considered distorting the DSF. We therefore seek for stricter conditions on the set of quantum numbers. By doing so, we will classify the eigenstates in terms of quantum numbers. Such a classification of states has already been done for the unbounded excitations in the XXZ-Heisenberg spin chain [36]. We will complete the classification of states in this thesis by extending it to the string states.
1.2
Goal and Outline of this Thesis
This thesis aims at completing ABACUS by providing a classification for all quantum number sets and hence, for all eigenstates of the gapped XXZ-Heisenberg spin chain. The full classification of states is of course not solely important for ABACUS but for every application the complete eigenstate basis is needed for. However, ABACUS is the prompt profiteer and will be ready - once the correct choice of quantum numbers is implemented - to compute the DSF straight away.
During the classification the question of completeness of states is inevitable. Thus, it is shown by counting admissible quantum number sets that their total number matches the expected number of states and hence gives rise to the completeness of the Bethe equations for the XXZ-model. However, completeness of states has to be discussed hand in hand with string deviations - deformations of string states - as has been pointed out by Hagemans et al [38]. Besides, in small system sizes (small number of sites) string deviations become more and more important and they have to be considered in order to calculate the exact eigenstate basis.
The structure of the thesis is as follows. In chapter 2, the model in our focus - the gapped XXZ Heisenberg spin-1/2 chain - is introduced and in chapter 3 the coordinate Bethe ansatz is presented, giving the analytical tools to calculate eigenstates. The classification of quantum number sets and a proof of completeness of the sets is done in chapter 4. Chapter 5 deals with some aspects of string deviations before the thesis ends with a conclusion and an outlook in chapter 6.
The XXZ Heisenberg Spin Chain
2.1
The One-dimensional Heisenberg Model
This chapter deals with some properties of the Heisenberg model in one dimension (1d) and the next chapter with the Coordinate Bethe Ansatz in order to find its exact Eigen-states. All of it is well known physics and covered by several papers, books, and lecture notes. These two parts of the thesis were written based on mainly two lecture notes [39] and [40] and the book [41] that are to be read for more details and - more importantly - are not repeatedly cited. All other sources used are referenced at the appropriate text passages.
In one dimension, the Heisenberg model describes spins with nearest neighbor interaction on a chain or on a ring, if periodic boundary conditions (PBC) are imposed. Its most general Hamiltonian (with PBC) reads
HXY Z = N X j=1 JxSjxSj+1x + JySjyS y j+1+ JzSjzSj+1z − hz N X j=1 Sjz, (2.1)
where N is the number of sites and Sα
j with α ∈ x, y, z are the spin projection operators
in α direction. The spin operators act on the spin sitting on site j and due to the PBC, SN +j = Sj. Jx, Jy and Jz are the coupling constants of the different spacial dimensions
x, y and z, respectively. Finally, the last sum encodes the effect of an external magnetic field along the z-axis on the spin chain with field strength hz.
If the three coupling constants Jx, Jy and Jztake distinct values, we call this model XY Z
Heisenberg model, which was introduced by Sutherland [42] in 1970 by relating it to the eight-vertex model and exactly solved by Baxter [9] one year later. If only Jz differs from
Jx = Jy, it is called the XXZ Heisenberg model and likewise, if all coupling constants are
equal Jx= Jy = Jz, it is called the XXX Heisenberg model which represents the isotropic
version of the Heisenberg spin chain. The latter equals the model Bethe initially solved [3]. If Jz = 0, one arrives at either the XY model, solved by Lieb, Schultz and Mattis in 1961
[43], or the XX model, depending on whether Jx 6= Jy or Jx= Jy, respectively. The latter
corresponds to non interacting spinless fermions on a lattice which can by obtained by a Jordan-Wigner transformation [44].
In this thesis we consider only the XXZ Heisenberg model in zero external field in which case the Hamiltonian reads
HXXZ = J N X j=1 SjxSj+1x + SjySj+1y + ∆ SjzSj+1z −1 4 , (2.2)
where ∆ · J = Jz and we denote ∆ the anisotropy. Also, a non physical constant of
−1/4 · Jz is added that will come in handy later on.
We have not specified the multiplicity of the spin system yet. In general, Heisenberg spin chains can be solved for arbitrary spin [45, 46], but throughout this thesis, only spin-1/2
systems are considered. In this case, the spins can either have the magnetic quantum number 1/2 or −1/2 referred to as
”spin up“ or”spin down“, respectively. The spin-1/2 operators follow the SU (2) symmetry relations
[Sjα, Skβ] = i~δjkαβγSjγ with α = x, y, z, (2.3)
where δjk is the Kronecker delta and αβγ is the totally antisymmetric Levi-Civita symbol.
They furthermore can be rewritten in terms of annihilation and creation operators S±j = Sjx± iSjy (2.4) with commutation relations
[Sjz, Sj±] = ±~Sj±, [S + j , S − j ] = 2~S z j, (2.5)
that act on a state as
Sjz|±ij = ±~ 2|±ij, S ± j |∓ij = ~|±ij, S ± j |±ij = 0 (2.6)
where |+ij denotes the spin up and |−ij the spin down state of a spin on the jth site.
They can be represented by the well known Pauli matrices σα j as
Sjα = ~ 2σ
α
j (2.7)
with the Pauli matrices defined as
σxj = 0 1 1 0 ! , σjy = 0 −i i 0 ! , σjz= 1 0 0 −1 ! . (2.8)
From now on we will set ~ = 1.
In terms of the raising and lowering operators, the Hamiltonian reads
HXXZ = J N X j=1 1 2 Sj+Sj+1− + Sj−Sj+1+ + ∆ SjzSzj+1−1 4 . (2.9)
Again, one can distinguish several cases in which the parameters J and ∆ take different regions in parameter space. The sign of the coupling J determines whether the spin chain is ferromagnetic or anti-ferromagnetic. In the former case, J is negative and the spins have to align to minimize the system’s energy. The added constant −1/4 sets the ferromagnetic ground state energy to zero. In the latter case, J is positive and the spins tend to anti-align. Here, the −1/4 ensures that only anti-aligned spins contribute to the overall sum. The Eigenstates of the ferromagnet and the anti-ferromagnet are the same but changing the sign of J flips the spectrum and thus, their ground state energy and the low-energy excitations differ greatly from one each other.
Obviously, if the anisotropy |∆| = 1, one recovers the isotropic XXX Heisenberg spin chain. The sign of the anisotropy ∆ also determines the magnetic properties. If J > 0 but ∆ ≤ −1, the system changes again from a ferromagnet to an anti-ferromagnet and vice versa (up to a rotation of 180◦ around the z-axis). It effectively boils down to changing
the sign in front of the planar term SjxSj+1x + SjySj+1y and Yang and Yang [47] have shown that this does not change the spectrum of the Hamiltonian.
If we take J > 0 for now, one can distinguish three phases that are crucially depended on the value of the anisotropy ∆: the ferromagnetic (∆ ≤ −1), the anti-ferromagnetic quantum critical (−1 < ∆ < 1) and the anti-ferromagnetic phase (∆ ≥ 1). In the last phase, the spectrum of the anti-ferromagnet is gapped between the ground state and a quasi degenerate ground state [48]. If ∆ → ∞, we arrive at the classical Ising model in 1d and the system is no longer gapped but the ground state energy is exactly degenerate (see next section). If |∆| < 1, the system has a continuous spectrum [48] and is referred to as ‘gapless’. It is then in its quantum-critical phase as the competing planar term (Sx
jSj+1x + S y jS
y
j+1) becomes more important than the term in z-direction. However, the
ground state is still anti-ferromagnetic.
In this thesis, the main focus lies on the XXZ Heisenberg model without external field with its Hamiltonian displayed in eq. (2.9) that is
• anti-ferromagnetic (J > 0) • gapped (∆ > 1)
• a spin-1/2 system.
2.2
Symmetries of the XXZ Spin Chain and the
Dimensionality of its Hilbert Space
Lets define the α projection of the total spin operator as Sαtot=X
j
Sjα with α ∈ x, y, z. (2.10)
The XXX Hamiltonian (∆ = 1) commutes with all of these three projection operators [HXXX, Stotα ] = 0 ∀ α ∈ x, y, z, (2.11)
which is easily obtained by using the commutation relations (2.3) and the XXX Hamil-tonian (2.2) in terms of the spin projection operators Sjα. In particular, one uses
[SjαSj+1α + SjβSj+1β + SjγSj+1γ , Sjγ] = [Sjα, Sjγ]Sj+1α + [Sjβ, Sjγ]Sj+1β
= i~αβγ(SjαSβj+1− SjβSj+1α ) (2.12) with α 6= β 6= γ and α, β, γ ∈ x, y, z. This vanishes because it is summed over all sites j and PBC were assumed.
Therefore, the XXX Heisenberg spin chain has a global SU (2) symmetry. However, the XXZ Heisenberg spin chain’s symmetry is broken by the anisotropy ∆ and thus, the XXZ Hamiltonian only commutes with the z-projection of the total spin operator
[HXXZ, Stotz ] = 0. (2.13)
It therefore possesses a global U (1) symmetry.
In addition, the XXX and XXZ spin chains also possess a discrete translational sym-metry, i. e. the spins can be shifted by multiples of the lattice spacing from one site to
another.
It is worthwhile mentioning that for certain values of the anisotropy ∆, the spin chain has furthermore the sl2 loop algebra as symmetry [49]. This only can happen in the quantum
critical phase, when the anisotropy is parametrized as
∆ = cos(η) = 1 2(q + q
−1) with 0 ≤ η ≤ π, (2.14)
at roots of unity
q2n= 1 with n ∈ N+, and n 6= 1. (2.15)
This will be briefly revisited in subsequent discussions about the completeness of the Bethe ansatz.
Another important property of the spin chain is its magnetization, determined by the z-projection of the total spin operator Stotz . Its eigenvalue gives the magnetization that is also defined as
Stotz = N
2 − M (2.16)
with M denoting the number of flipped spins. Because of (2.13), the Hilbert space H of the whole spin chain separates into subspaces HM of fixed magnetization, e. g. with a
fixed number of flipped spins.
To analyze the dimensionality of the total Hilbert space H lets write down an immediate basis - the basis spanned by the basis states |bi that are obtained by taking tensor products of the individual states (spin up or spin down) of all the N sites
|bi =
N
O
j=1
|±ij. (2.17)
Here, |±ij means either |+ij or |−ij depending on the state |bi and every |±ij is
inde-pendent of the state |±ik of another site k such that there are 2N different combinations
to build the product in (2.17). Hence, there are 2N linearly independent basis states |bi, indicating the dimensionality of the Hilbert space dim(H) = 2N.
Each subspace HM with fixed M has then the dimensionality dim(HM) = MN, since
the binomial coefficient calculates the number of all the different possibilities to place M flipped spins on N sites. Summing the dimensionality of the subspaces and using the binomial theorem, one recovers the dimensionality of the whole Hilbert space
N X M =0 dim(HM) = N X M =0 N M = 2N = dim(H). (2.18)
It is important to note that the Hamiltonian HXXZ is not diagonal in this basis.
Diago-nalizing the Hamiltonian HXXZ is exactly what is achieved by the Bethe Ansatz which is
2.2.1
Ground State and Excitations
It is very intuitive, that the ferromagnet is in its ground state if all the spins are aligned and indeed, such a state is an eigenstate of the Hamiltonian. The ground state in zero external field is degenerate and is given by the two states
|+, +, +, . . . , +, +, +i and |−, −, −, . . . , −, −, −i, where, again, ‘+’ stands for spin-up and ‘−’ for spin-down.
The disturbances or excitations of the ferromagnetic order are spin waves, a concept which was firstly introduced by F. Bloch [50] in 1930 and almost 30 years later, in 1957, detected for the first time by Brockhouse [51] in a neutron scattering experiment. These low energy excitations are quantized and coined magnons. They are delocalized all over the spin chain and, as they change the magnetization Stotz of the spin chain by one, are spin-1 particles (bosons) [52, p. 98]. More remarkably, the magnon states are the exact eigenstates of the Hamiltonian and the magnon-magnon-scattering generalizes in such a way to higher order magnon-scattering that the whole eigenbasis of the Hamiltonian is found by studying the magnon-magnon scattering within the Bethe ansatz framework, discussed in the next chapter. Instead of scattering off, magnons also can form a bound state or so called string state that travels as one entity around the spin chain.
Naively one would think, that the ground state of the anti-ferromagnet is given when all spins are anti-aligned. A state in such order is called N´eel-ordered state, going back to Louis N´eel who was the first to describe this type of ordering [53]. However, N´ eel-ordered states are not even eigenstates of the Hamiltonian. Only in the classical Ising limit (∆ → ∞), the degenerate ground state is given by the two N´eel-ordered states:
|+, −, +, . . . , −, +, −i and |−, +, −, . . . , +, −, +i
The true ground state of the anti-ferromagnet cannot be visualized in a classical picture but it has still magnetization Stotz = 0. One rather has to picture the anti-ferromagnetic ground state as the N/2-magnon state, when N is even. If N is odd, the situation is more complicated as there are two degenerate ground states with Stotz = ±1/2. We will stick to the easy case and assume from now on, that N is always even. The magnons in the anti-ferromagnetic ground state are all single excitations (no bound states).
The exact ground state and its energy are rigorously obtained by the Bethe ansatz. This has been proven by Yang and Yang [54] and they furthermore showed that for the ground state the solution to the Bethe equations is also unique. As mentioned before, the ground state is non degenerate, but possesses the energy gap between the ground state and the quasi degenerate ground state which scales with system size as exp(−const(∆)N ) [55, p. 7].
In the anti-ferromagnetic case, the low-energy excitations are best understood when going back to the Ising limit. Flipping a spin, which is essentially removing a magnon, followed by subsequent spin flips creates N´eel-ordered domains on which boundaries two spins are pointed in the same direction:
| . . . , −, +, −, +, −, +, +, +, −, +, . . .i → | . . . , −, +, +, −, +, −, +, −, +, +, . . .i We see, that removing one quasi particle has led to two domain walls. We can assign quasi particles - so-called spinons - to these domain walls, also occurring for finite ∆. They are
fractional emergent particles with spin 1/2 (fermions), first described by Faaddev and Takhtadzhyan [56] in 1981.
The spinon spectrum is as well obtained by the Bethe ansatz. Therefore, let us finally turn to unfold this very powerful ansatz.
The Coordinate Bethe Ansatz
Hans Bethe presented in his famous paper
”Zur Theorie der Metalle“ (English: ”About the Theory of Metals“) [3] in 1931 a method that is nowadays known as
”Coordinate Bethe Ansatz“. With this method he was able to find the exact eigenvalues and -functions of the isotropic XXX Heisenberg spin chain. With Orbachs parametrization of the anisotropy ∆ this exact ansatz was successfully generalized to the XXZ spin chain in 1958 [57]. The coordinate Bethe ansatz was later extended to a variety of other models, for instance to δ−interacting bosons in 1d (Lieb-Liniger model) by Lieb and Liniger [58] in 1963 (see [39] and [41]).
In the following, we only present the coordinate Bethe ansatz for spin chains with main focus on the gapped XXZ case.
3.1
Coordinate Bethe Ansatz for XXZ spin chains
We seek the exact Eigenstates of the XXZ Hamiltonian (2.9), i.e. we want to diagonalize it. In order to do so, lets start with a reference state: the state |0i with only spin up states with magnetization N/2 (number of flipped spins M = 0)
|0i =
N
O
j=1
|+ij. (3.1)
This state is trivially an eigenstate of the Hamiltonian and the only state of the subspace HM =0 as N0 = 1.
A Single Flipped Spin
Lets continue with the next subspace HM =1that has exactly one flipped spin. To account
for the translational invariance of the spin chain, we make an ansatz for the eigenstate |ψ1i and write it as a superposition of all N states with one flipped spin
|ψ1i = N X j=1 Ψ1(j) · Sj−|0i = N X j=1 Ψ1(j)|ji. (3.2)
Here, we have labeled the state |ji = Sj−|0i as the state with one flipped spin on site j and Ψ1(j) is the corresponding amplitude.
Projecting the Schr¨odinger equation HXXZ|ψ1i = E1|ψ1i onto the bra state hj| = h0|Sj+
reveals E1hj|ψ1i = hj|HXXZ|ψ1i E1Ψ1(j) = hj|J N X k=1 1 2 S + kS − k+1+ S − kS + k+1 + ∆ SkzSk+1z −1 4 |ψ1i E1Ψ1(j) = J 2 (Ψ1(j − 1) + Ψ1(j + 1)) − J ∆Ψ1(j), for 1 < j < N, (3.3)
Thus, |ψ1i is an Eigenstate if its amplitude Ψ1 satisfies equation (3.3) as well as the
periodic boundary condition Ψ(1) = Ψ(N + 1). The latter is revealed by doing the same derivation in (3.3) explicitly for site 1 and site N .
A plane wave ansatz Ψ1(j) = exp(ikj) solves (3.3) and leads to the following expression
for the energy E1
E1exp(ikj) + J ∆ exp(ikj) = J 2 exp(ik(j − 1)) + J 2 exp(ik(j + 1)) ⇔ E1 = J (cos(k) − ∆). (3.4)
Furthermore, by imposing the boundary condition we get exp(ikN ) = 1
⇒ kI˜=
2π ˜I
N , with ˜I = 0, 1, . . . , N − 1. (3.5) From that requirement it is seen, that only quantized momenta are allowed. Additionally, there are N distinct momenta kI˜, each of which leads to a linearly independent Eigenstate
|Ψ1i =PNj=1exp(ikI˜j)|ji. This is in agreement with the dimensionality of the one magnon
subspace dim(HM =1) = N1 = N .
Two Flipped Spins
Lets move on to the next sector with two flipped spins - the two magnon sector. Even though the same steps are followed as in the last sector, this is more complicated but luckily, with the two magnon sector in hand, we are able to generalize the wave functions to the arbitrary M ≤ N/2 sector.
We start again with a superposition of all states with two flipped spins |ψ2i = X j1<j2 Ψ2(j1, j2)|j1, j2i = X j1<j2 Ψ2(j1, j2) · S−j1S − j2|0i. (3.6)
To avoid double counting states in the superposition, j1 is chosen to be always smaller
than j2. When projecting the Schr¨odinger equation HXXZ|ψ2i = E2|ψ2i onto the bra
hj1, j2| = h0|Sj+1Sj+2, one has to distinguish two cases: Either the flipped spins occupy sites
next to each other or not. The former case gives (E2+ J ∆)Ψ2(j1, j2) =
J
2 (Ψ2(j1− 1, j2) + Ψ2(j1, j2+ 1)) , for 2 < j1+ 1 = j2< N, (3.7) and the latter
J
2 (Ψ2(j1− 1, j2) + Ψ2(j1, j2− 1) + Ψ2(j1+ 1, j2) + Ψ2(j1, j2+ 1))
= (E2+ 2J ∆)Ψ2(j1, j2), for 2 < j1+ 1 < j2 < N. (3.8)
Following Bethe’s logic [3], this is solved by a wave function generalized from the former sector that is not only a simple superposition of two plane waves but has also an interacting term (momentum exchanging term)
Ψ2(j1, j2) = A12exp(ik1j1+ ik2j2) + A21exp(ik2j1+ ik1j2),
Substituting this wave function into (3.8) gives the expression for the energy
(E2+ 2J ∆)Ψ2(j1, j2) =
J Ψ2(j1, j2)
2 (exp(ik1) + exp(−ik1) + exp(ik2) + exp(−ik2)) ⇔ E2 = J (cos(k1) + cos(k2) − 2∆). (3.10)
Inserting it into (3.7), using this energy expression and recalling that j1+ 1 = j2 gives a
relation for its amplitudes A12 and A21
A12
A21
= −1 + exp(ik1+ ik2) − 2∆ exp(ik1) 1 + exp(ik1+ ik2) − 2∆ exp(ik2)
≡ − exp(−iΦ(k1, k2)), (3.11)
where Φ(k1, k2) is the scattering phase shift function defined as
Φ(k1, k2) = 2 arctan ∆ sink1−k2 2 cosk1+k2 2 − ∆ cos k1−k2 2 ! . (3.12)
By setting A12 = exp(−2iΦ) and A21 = − exp(2iΦ) the wave function (3.11) is rewritten
as Ψ2(j1, j2) = exp(ik1j1+ ik2j2− i 2Φ(k1, k2)) − exp(ik2j1+ ik1j2+ i 2Φ(k1, k2)). (3.13) We have not yet imposed the periodic boundary conditions, given by the four equations that one obtains when deriving equation (3.7) and (3.8) for site 1 and N at a time. These equations boil down to the conditions
Ψ2(j2, j1+ N ) = Ψ2(j1, j2), Ψ2(j2− N, j1) = Ψ2(j1, j2). (3.14)
In analogy to the M = 1 sector, the PBC restrict the momenta, since
exp(ik1N ) = − exp(−iΦ(k1, k2)), exp(ik2N ) = − exp(iΦ(k1, k2)) (3.15)
and in logarithmic form
N k1+ Φ(k1, k2) = 2π ˜I1, N k2+ Φ(k1, k2) = 2π ˜I2, (3.16) with I˜1, ˜I2 = 1 2, . . . , N − 1 2 and ˜I1 6= ˜I2.
Note, that this time the quantum numbers are given by half odd integers.
Arbitrary Number of Flipped Spin
At this point, one can already generalize the ansatz to arbitrary M . However, we will restrict ourselves to M ≤ N/2. The reason lies in the choice of the reference state: We could have started with the state of only downwards pointed spins as reference state and as well recover all the magnon states by flipping spins up. Because of this mirror symmetry it is enough to only consider M ≤ N/2. The wave function for arbitrary M then reads
ΨM(j1, . . . , jM) = Y M ≥a>b≥1 sgn(ja− jb)× (3.17) X PM (−1)[P ]exp i M X a=1 kPaja+ i 2 X M ≥a>b≥1 sgn(ja− jb)Φ(kPa, kPb) (3.18)
with the eigenstate |ΨMi = X j1<j2···<jM ΨM(j1, j2, . . . , jM)|j1, j2, . . . , jMi. (3.19) and the PBC ΨM(j1, j2, . . . , jM) = ΨM(j2, j3. . . , jM, j1+ N ) (3.20) in every index.
From the PBC it follows that
exp(ikaN ) = (−1)M −1exp(−i
X
b6=a
Φ(ka, kb)), with a = 1, . . . , M. (3.21)
That are the so called Bethe equations. They are M such equations guaranteeing consis-tency. Every set of M momenta ka, that satisfy these equations, gives an eigenstate of the
system. The momenta have fermionic properties and thus, the momenta in a set have to be distinct.
These Bethe equations are visualized as follows:
k1→
(a) 1-Magnon Sector
k1→ k2 k3 kM −1 kM . . . (b) M-Magnon Sector
The one magnon sector (depicted on the left) consist of one wave package with momentum k1. Moving it exactly once around the closed chain (depicted as circle) does not change
the wave. In the M -magnon sector however (shown on the right), the wave package with momentum k1 will scatter with the remaining M − 1 waves while moving once around the
chain. This explains the M − 1 factors on the right hand side of equation (3.21) decrypting the phase shifts the moved wave picks up from each scattering event.
In logarithmic form the Bethe equations are N ka+
X
b6=a
Φ(ka, kb) = 2π ˜Ia, with a = 1, . . . , M, (3.22)
connecting M quantum numbers to the momenta. The eigenenergy is given by
EM = J M
X
a=1
(cos(ka) − ∆) (3.23)
and we can define the total momentum as sum of the individual momenta P =X
a
Before we take the next step and write the Bethe equations in terms of rapidities (quasi momenta) - the probably better known formulation - a few things should be noticed. First, the momenta ka have a 2π periodicity, implying that the wave function does not
change when a momentum ka is shifted by ±π. Since we are describing quasi particles on
a lattice and the Fourier transformation of a δ-function δ(x) is given by exp(ikx), this is not surprising. Fixing the quantum numbers ˜I to the range 0, . . . , N − 1 implies that we only consider the first Brillouin zone.
Second, the many-body-wave function is only dependent on a scattering shift function of two particles. That means in particular, the many-body-interaction factorizes into two-body interactions, a remarkable observation and the very manifestation of integrability. This is no longer true for the Heisenberg model on a 2d and 3d lattice, that are as a matter of fact non-integrable.
Third, up to now the anisotropy ∆ and the interaction strength J have not been specified yet, though they crucially determine the physics of the spin chain. Therefore, the Bethe equation introduced here are valid for all XXX and XXZ models. A distinction of the different models will happen in the next step, when a parametrization in terms of rapidities is introduced. In the next section, the Coordinate Bethe ansatz is further derived for the gapped XXZ spin chain to illustrate the method, but for the sake of completeness the solutions to the XXX case and the gapless XXZ case are indicated as well.
3.1.1
Bethe Equations in Terms of Rapidities
It is handy to find a parametrization of the momenta in such a way, that the scattering shift function is no longer dependent on two momenta (two variables) but only dependent on their difference (one variable). This has also the advantage of bringing the scattering-phase-shift function in a translational invariant form.
The parametrization is dependent on the anisotropy ∆, therefore, the three cases (a) |∆| = 1, (b) |∆| > 1, and (c) |∆| < 1 are distinguished. We treat the first case in the following, and the others are displayed in the appendix A.
The different parametrization of these three cases goes back to Orbach [57] and the anisotropy ∆ for the anti-ferromagnet reads
(a) ∆ = 1, (b) ∆ = cosh(η), (c) ∆ = cos(η) (3.25)
It has the following important advantage: We don’t have to care about the phase transition in ∆ of the spin chain as it is hidden inside the parametrization for real η.
Bethe Equations of the gapped XXZ Heisenberg Spin Chain
We start by setting
exp(ik) = sin(λ + i
η 2)
sin(λ − iη2). (3.26)
in which we denote λ as rapidity and we have set η = arcosh ∆. This choice becomes more obvious when substituting the above parametrization (3.26) into equation (3.11), which
had indirectly defined the scattering phase shift function exp(−iΦ(k1, k2)) = − 1 +sin(λ1+i η 2) sin(λ2+i η 2) sin(λ1−iη2) sin(λ2−iη2) − 2∆
sin(λ1+iη2) sin(λ1−iη2) 1 +sin(λ1+i η 2) sin(λ2+i η 2) sin(λ1−iη2) sin(λ2−iη2) − 2∆
sin(λ2+iη2) sin(λ2−iη2)
= −2 cos(λ1− λ2) − 2∆ cos(λ1− λ2+ iη) + 2(∆ − cosh(η)) cos(λ1+ λ2) 2 cos(λ1− λ2) + 2∆ cos(λ1− λ2− iη) + 2(∆ − cosh(η)) cos(λ1+ λ2)
= −sin(λ1− λ2+ iη) sin(λ1− λ2− iη)
≡ exp(−iΦ(λ1− λ2)). (3.27)
After inserting (3.27) and (3.26) into the Bethe equations (3.21), they become
Bethe Equations sin(λj + iη/2) sin(λj − iη/2) N = M Y k6=j sin(λj− λk+ iη) sin(λj− λk− iη) with j = 1, . . . , M. (3.28)
Logarithmic Bethe Equations
We can work on equation (3.26) to gather an explicit expression for the parametrization of the momenta k(λ) k(λ) = −i ln sin(λ + i η 2) sin(λ − iη2)
= −i ln sin(λ) cosh(
η
2) + i cos(λ) sinh( η 2)
sin(λ) cosh(η2) − i cos(λ) sinh(η2) = π − 2 arctan tan(λ) tanh(η/2) − 2π λ π + 1 2 mod (2π) ≡ π − Θ1(λ) mod (2π), (3.29)
with Θj(λ) the below defined kernel. The floor-function b. . . c is added to take care of the
branch cuts of the complex logarithm. It furthermore ensures that the information about the different branches of the tangent is not lost after taking the arctangent.
A very similar calculation to (3.29) can be performed for the scattering phase shift function (see last line of (3.27))
Φ(λ1− λ2) = 2 arctan tan(λ1− λ2) tanh(η) + 2π λ1− λ2 π + 1 2 ≡ Θ2(λ). (3.30)
Inserting both into the logarithmic Bethe equations (3.22) we arrive at the logarithmic Bethe functions in terms of rapidities
Bethe Equations in Logarithmic Form 2π NIj = Θ1(λj) − 1 N M X k=1 Θ2(λj− λk), with j = 1, . . . , M (3.31) and Θj(λ) = 2 arctan tan(λ) tanh(j·η2 ) ! + 2π λ π + 1 2 and with Ij ∈ −N 2 + 1 2, . . . , N 2 − 1 2 if M even −N 2 + 1, . . . , N 2 if M odd.
Note, that the Bethe equations in their
”normal“ form (3.28) as well as in their logarithmic form (3.31) are transcendental equations that need to be solved numerically. This is in particular hard in the first case. In the latter case however, we can generate solutions by taking an arbitrary set of M quantum numbers out of the N possible ones, inserting them into the M Bethe equations and solve for it numerically. The kernel Θj(λ) is a monotonic
increasing function which ensures to find a unique solution.
Recall that the momenta have to differ to render the eigenstates independent and so do the rapidities and thus the quantum numbers. One can build MN such sets, which is exactly the expected dimensionality of the Hilbert subspace with a given magnetization dim(HM) = MN. In turns out, that some of the eigenstates are equal due to the 2π
symmetry of the momenta reflected in the quantum number sets. How to avoid the redundant sets is intensively discussed in chapter 4.
Other independent solutions to the Bethe equations, that endow us with the missing eigenstates, are provided by complex rapidities bound together to so called
”string states“.
Energy and Total Momentum Parametrized with Rapidities
Before turning to string states, lets quickly express the eigenenergy and the total mo-mentum in terms rapidities. Using the parametrization for the momo-mentum k(λ) (3.29) to rewrite the energy EM (3.23) we find
EM = J M X a=1 (cos(ka) − ∆) = −J M X a=1 (cos 2 arctan tan(λ) tanh(η/2) + ∆) = −J M X a=1 tanh2(η/2) − tan2(λ) tanh2(η/2) + tan2(λ) + ∆ = −J M X a=1 cosh(η) cos(2λ) − 1 cosh(η) − cos(2λ) + ∆ = −J M X a=1
cosh(η) cos(2λ) − 1 + cosh2(η) − cosh(η) cos(2λ)
cosh(η) − cos(2λ) = −J M X a=1 sinh2(η) cosh(η) − cos(2λ) . (3.32)
Similarly, we find for the total momentum P (3.24) P = M X a=1 ka= M X a=1 π − Θ1(λa) mod (2π) = M π − 2π N M X a=1 Ia mod (2π) (3.33)
in which we have used the logarithmic Bethe equations and the fact that the skew sym-metric kernel Θ2 was summed over symmetrically.
3.2
Complex Rapidities and String States
Complex rapidities can occur, too, which was already realized by Bethe himself [3]. It was shown by Vladimirov [59], that the solutions to the Bethe equations in the XXX as well as in the XXZ case are invariant under complex conjugation. Consequently, all complex rapidities come in complex conjugate pairs and the imaginary parts cancel when for instance the eigenenergy is calculated. Several pairs of complex conjugates may form a so called string state, a bound state, that is considered as one effective quasi particle. The so called string hypothesis assumes that the strings occur in very regular patterns. This is qualitatively understood by going back to the Bethe equations (3.28). In (3.28), the left hand side might either become exponentially large or small for N 1, depending on weather its norm is larger or smaller than 1, respectively. Therefore, the right hand side has also to increase or decrease exponentially. For fixed M , this is possible if for every complex rapidity, there exists another rapidity with imaginary part differing by approximately ηi.
Hence, in case of the gapped XXZ Heisenberg spin chain, a string complex consisting of j bound particles takes the form
λnj,α= λj,α+
iη
2(j + 1 − 2n) + d
n
j,α (3.34)
in which n = 1, . . . , j counts through the j particles in the complex. We refer to j as the length of a string. The number of j-strings (strings with length j) is denoted as Mj,
and α = 1, . . . , Mj labels the different j-strings with the same length j. We call λj,α
the string center and it gives the real part of the complex if one neglects the deviation dnj,α = nj,α+ iδnj,α. For a pair of complex conjugate rapidities, the deviations must be complex conjugates as well because of Vladimirovs statement.
Usually, the string deviations are exponentially small in system size N but there are some exceptions for large but finite sizes [38], [60]. For the purpose of calculating physical quantities like correlation functions, the excessive deformed strings might be neglected as there are only a small fraction of those [38], [60]. Nevertheless, it is of great importance to be aware of the string deviations especially for small system sizes. String deviations are further discussed in chapter 5.
For now, we use the string hypothesis and assume all eigenstates of the spin chain are described by (undeformed) strings. We thus neglect all deviations for now. Because of the above mentioned exceptions, it is doubted whether the whole Hilbert space is exhausted only with the string hypothesis in hand. However, in the thermodynamic limit N → ∞,
it has provided accurate results: The exact thermodynamics for the XXX chain where given by Takahashi [61], for the gapped XXZ chain by Gaudin [62] and for the gapless XXZ chain by Takahashi and Suzuki [63].
As a start, we will use the string hypothesis for the gapped XXZ spin chain to derive the Bethe-Takahashi equations in the next section. In the subsequent chapter, we reason by a simple counting argument that the number of Bethe-Takahashi solutions coincides with the dimensionality of the Hilbert space.
3.3
Bethe-Takahashi Equations
Using the string hypothesis, we for now assume only string rapidities λnj,αparametrized as
λnj,α= λj,α+
iη
2(j + 1 − 2n). (3.35) The bounded rapidities in a string form perfect patterns in which their imaginary parts differ by exactly iη.
For a given number of flipped spins M , the strings can form different patterns defined by a set of {Mj} where each Mj indicates the number of j-strings. If e. g. Mk = 0, there are
no strings with length k present. We will refer to the set of {Mj} as base and to a j-string
as string of level j. The base {Mj} has to satisfy ∞
X
j=1
jMj = M. (3.36)
The number of string complexes Ns in a base is simply P Mj = Ns and determines the
number of independent parameters.
String Hypothesis and Bethe Equations
To calculate the complex rapidities, the perfect strings (3.35) are inserted in the Bethe Equations (3.28). The big advantage of using string hypothesis is that we can get rid of the complex part in the Bethe equations by multiplying together the j equations for rapidities in the same string. We thus end up with a system of Ns equations
j Y n=1 sin(λnj,α+ iη/2) sin(λn j,α− iη/2) !N = j Y n=1 Y (k,β,m) 6=(j,α,n) sin(λn j,α− λmk,β+ iη) sin(λn j,α− λmk,β− iη) ! . (3.37)
We will treat both hand sides of this equation separately starting with the left hand side (LHS)
sinNλj,α+ iη2(j − 1) +iη2
· ... · sinNλ
j,α+iη2(−j + 1) +iη2
sinNλj,α+ iη2(j − 1) −iη2
· ... · sinNλ j,α+iη2(−j + 1) −iη2 = sinNλj,α+ jiη2 · ... · sinNλ j,α− jiη2 + iη
sinNλj,α+ jiη2 − iη
· ... · sinNλ j,α− jiη2 = sinλj,α+ jiη2 sinλj,α− jiη2 N . (3.38)
The last step is easy to see as soon as one realizes that the first term in the denominator cancels with the second term in the nominator etc. such that only the last term of the denominator and the first term of the nominator survive.
We move on with the somewhat more tedious right hand side (RHS) and split the product into three parts
j Y n=1 Y k6=j, (β,m) sin(λnj,α− λm k,β+ iη) sin(λnj,α− λm k,β− iη) · j Y n=1 Y γ6=α, m sin(λnj,α− λm j,γ+ iη) sin(λnj,α− λm j,γ− iη) · j Y n=1 Y m6=n sin(λnj,α− λm j,α+ iη) sin(λn j,α− λmj,α− iη) . (3.39)
The last part of this expression reduces to a factor one
j Y n=1 Y m6=n sin(λnj,α− λm j,α+ iη) sin(λn j,α− λmj,α− iη) = Y n,m n6=m sin(iη(m − n + 1)) sin(iη(m − n − 1)) = Y n,m n6=m sin(iη(m − n + 1)) − sin(iη(n − m + 1)) = 1, (3.40) since the last product is taken over all n and m (with n 6= m) and there are thus j(j − 1) = j2− j factors which is an even amount regardless of the parity of j.
The second part of (3.39) is written as
Y γ6=α Y n,m sin(λnj,α− λm j,γ+ iη) sin(λn j,α− λmj,γ− iη) = Y γ6=α Y n,m sin(λj,α− λj,γ+iη2(2m − 2n + 2)) sin(λj,α− λj,γ+iη2(2m − 2n − 2)) . (3.41)
First, the product over m is performed. The term 2m − 2n + 2 in the numerator takes the values 4 − 2n, 6 − 2n, . . . , 2j − 2n, 2j + 2 − 2n and the term 2m − 2n − 2 in the denominator the values −2n, 2 − 2n, . . . , 2j − 4 − 2n, 2j − 2 − 2n. Thus, the last two factors of the nominator and the first two of the denominator don’t cancel and we are left with
Y
γ6=α
Y
n
sin(λj,α− λj,γ+iη2(2j − 2n)) sin(λj,α− λj,γ+iη2(2j + 2 − 2n))
sin(λj,α− λj,γ+iη2(−2n)) sin(λj,α− λj,γ+iη2(2 − 2n))
. (3.42)
When performing the product over n in a similar fashion, only one factor cancels; the factor with term 2j − 2n = 0 for n = j reduces against the factor with term 2 − 2n = 0 for n = 1. With the following new notation
¯ ej(λ) =
sin(λ + ijη2 )
sin(λ − ijη2 ) (3.43) this product is expressed as
Y
γ6=α
¯
The very first part of the expression (3.39) works analogues to the second and becomes with the new notation (3.43)
Y
k6=j
¯
ek−j(λj,α− λk,β) · ¯ek−j+2(λj,α− λk,β)2· ... · ¯ej+k−2(λj,α− λk,β)2· ¯ej+k(λj,α− λk,β) .
(3.45) However, if k < j, this expression can be further reduced. In fact, all factors ¯eb−a with
negative b − a cancel with their positive counterparts ¯ea−b. Keep in mind that ¯ek−j occurs
just once whereas ¯e2j−k is squared and, therefore, this product is in the general case Y
k6=j
¯
e|j−k|(λj,α− λk,β) · ¯e|j−k|+2(λj,α− λk,β)2· ... · ¯ej+k−2(λj,α− λk,β)2· ¯ej+k(λj,α− λk,β) .
(3.46) The Bethe equations (3.28) for complex rapidities with the string hypothesis then become
¯ ej(λj,α)N = Y (k,β)6=(j,α) ¯ Ejk(λj,α− λk,β), ∀j with Mj 6= 0 and α = 1, . . . , Mj (3.47) with E¯jk = ¯e (1−δjk) |j−k| · ¯e 2 |j−k|+2· ... · ¯e2j+k−2· ¯ej+k (3.48)
and ¯ej(λ) defined in (3.43). We see, that by using the string hypothesis the Bethe equations
are now only dependent on the real string centers λj,α.
String Hypothesis and Logarithmic Bethe Equations
After logarithmizing the Bethe equations in a similar fashion as done in section 3.1.1, they are named Bethe-Takahashi equations and displayed below
Bethe-Takahashi Equations 2π NIj,α=Θj(λj,α) − 1 N M X k=1 (k,β) Mk X β=1 6=(j,α) Θjk(λj,α− λk,β), (3.49) ∀ j with Mj 6= 0, and α = 1, . . . , Mj, with Θj(λ) = 2 arctan tan(λ) tanh(j·η2 ) ! + 2π λ π + 1 2 , Θjk = (1 − δjk)Θ|j−k|+ 2Θ|j−k|+2+ · · · + 2Θj+k−2+ Θj+k, and with Ij,α∈ − N 2 + 1 2, . . . , N 2 − 1 2 if Mj even − N 2 + 1, . . . , N 2 if Mj odd.
When searching for complex solutions to the Bethe-Takahashi equations, we draw Ns
quantum numbers for the Ns equations from N possible ones. Note, that the string
length j for each equation decides whether the quantum number Ij,α is an integer or a
numbers {Ij,α} , a configuration. As mentioned before, not every possible configuration
{Ij,α} generates a new eigenstate; some might coincide. This will be the main subject of
the next chapter 4. Before turning to this, we seek the expressions for energy and total momentum in terms of complex rapidities.
Energy and Total Momentum for the String Hypothesis
We use the expression for the energy found for real rapidities (3.32) and replace the real rapidities with their complex form (3.35). Hence, we have to sum over all string lengths j, centers α and rapidities within a string n and we derive
EM = −J X j,α,n sinh2(η) cosh(η) − cos(2λnj,α) = −J X j,α j X n=1 sinh2(η)
cosh(η) − cos(2λj,α+ iη(j + 1 − 2n))
= −JX
j,α
sinh(η) sinh(jη) cosh(jη) − cos(2λj,α)
. (3.50)
The sum over n reduces the expression to a compact form which is shown by induction. Similarly, the total momentum is derived by
P = X j,α,n −i ln sin(λ n j,α+ i η 2) sin(λnj,α− iη2) ! =X j,α Ij,απ − Θj(λj,α) mod (2π) = πNs− 2π N X j,α Ij,α mod (2π). (3.51)
Classification of States
4.1
Possible Quantum Number Sets
Before we turn to the restrictions on the possible quantum number sets, let us first consider how many quantum number configurations we are currently accounting for. When we introduced the Bethe-Takahashi equations (3.49) in section 3.3 we allowed for N different quantum numbers per level j. How many configurations are hence possible? To answer this question we have to first look at the different bases. A base fulfills P
jjMj = M .
This equals the partition of M , visualized by Young or Ferrers diagrams.
The partition of M gives all the distinct ways to write the natural number M as a sum of other natural numbers. A Young diagram consists of M boxes placed in rows on top of each other, such that a lower row is never longer that an upper row, thus referring to one way of expressing a natural number as sum of others. All Young diagrams with M boxes give the complete partition. A diagram connects to a base as follows: The number of boxes is the number of flipped spins M , the number of rows is the number of independent parameters Ns. The length of a row j is the length of a string or the level, the number of
rows with same length is Mj. Figure 4.11 depicts all Young diagrams up to 8 boxes and
an example of how to connect a diagram to the configuration.
Example:
(# of boxes) ≡ M
(# of rows) ≡ Ns
(length of row) ≡ j
(# of rows with length j) ≡ Mj
M = 8 Ns= 3
M2 = 2
M4 = 1
Figure 4.1: Young diagrams up to eight boxes and connection to the different string bases.
We have now visualized the possible bases for a given magnetization - each base is
rep-1
By R. A. Nonenmacher., CC BY-SA 4.0, https : //commons.wikimedia.org/w/index.php?curid = 4766072, 15.08.18
resented by a Young diagram. For every sector with M flipped spins all possible bases n
{Mj}|PjjMj = M
o
are depicted by all Young diagrams with M boxes. In each base the set of Bethe-Takahashi equations take a slightly different form in which we insert quantum numbers. How many configurations of quantum numbers exist for a given base? Recall, that two quantum numbers in the same level j cannot coincide as the rapidities have fermionic behavior but strings of different levels j are considered different excitations (’particles’) and can share a quantum number.
On each level j one thus can choose Mj quantum numbers out of N possible quantum
numbers. This is visualized as follows:
The N circles stand for the possible quantum numbers that are given by the interval [Imin, Imax]. If Mj is even, Imin = −N/2 + 1/2 and Imax = N/2 − 1/2, and if Mj is odd
Imin = −N/2 + 1 and Imax = N/2. If a quantum number is chosen the circle is filled,
thus there are Mj filled circles per line, representing a level j. The level j are sorted
such that the shortest j is the upper most row. One configuration is depicted in such a diagram that could also be understood as particles occupying spots. Actually, there is a slightly different particle hole analogy to the quantum numbers to which we will come back in section 4.7. For now, let us refer to strings with level j as j-particles. There are Mj j-particles that can occupy N possible spots and thus there are MNj configurations.
That is why there areQ
j N
Mj quantum number sets per base (per Young diagram) where
j runs over all levels (row lengths). Summing the number of configurations over all bases (Young diagrams) for a certain M we get
X {Mj} P j jMj=M Y j N Mj > N M (4.1)
which is clearly more than the expected dimensionality of the Hilbert subspace dim(HM) = N
M: in the case of only real rapidities with M = M1 (Young diagrams with M boxes in a
column, or with M rows with one box each) we already have MN
1 = N
M configurations
of quantum numbers (see section 3.1.1).
It turns out that two quantum number sets may describe the same wave function. As has been pointed out in section 3.1.1, the reason lies in the 2π-periodicity of the momenta k. The parametrization k(λ) in equation (3.26) shows that a ±2π-shift of the momentum ka
induces a ±π-shift of the rapidity λa. A general consequence of the periodicity is that
there exist infinitely many solutions to the Bethe equations, in most of which the rapidi-ties differ by multiples of π and then have the same wave function. This can of course be
remedied by restricting ourselves to the first Brillouin zone. We have already restricted the quantum numbers to N possible ones to account for that. However, this is seemingly not strict enough − there are still too many sets of quantum numbers.
Interestingly, this kind of over counting does not occur in the isotropic XXX chain. It is easy to see that the kernel Θjk of the logarithmic Bethe-Takahashi equations in the
XXX chain goes to a finite value for λ → ∞. Therefore, it exists a quantum number Ij,∞ corresponding to an infinite rapidity λj. The quantum number Ij,∞ is thus an upper limit
of quantum numbers. This imposes already enough restrictions on the choice of quan-tum numbers. Another aspect is that the solutions of the Bethe-equations in the XXX chain with only real rapidities are highest weight states of the irreducible representations of SU (2). Therefore, one has to act with the global spin lowering operator to create the other states which is obtained by adding infinite rapidities to the state (for more details see [39], [64]).
As firstly the gapped XXZ possesses only a U (1) symmetry and the irreducible represen-tations of U (1) are all one dimensional, and secondly the kernel Θ → ∞ for λ → ∞, we do not have any natural restriction on configurations.
Though what is the curse in the gapped XXZ chain could be its cure: The only chance we have is to study how a ±π shift of a rapidity λ changes the quantum number configu-ration and based on this, formulate new restrictions on the sets of quantum numbers. In other words, we seek a complete classification of quantum number sets, that generates the correct amount of states.
The strategy is as follows: We start with defining and studying the symmetry transfor-mations linked to the ±π-jumps of rapidities in section 4.2. From there, we continue with finding restrictions on the choice of quantum numbers in section 4.3. In the subsequent section 4.4 it is argued why the resulting restricted set B of possible configurations is unique and complete. Furthermore in section 4.5, the possible configurations are counted, e.g. the cardinality of B is found. We can use the cardinality to show that the number of configurations matches the dimensionality of the Hilbert subspace which is done in section 4.6. Finally, the particle-hole excitations as an analogy to the quantum numbers and the phase diagram for some values of M and N in the anti-ferromagnetic regime are given in section 4.7.
4.2
Possible Symmetry Transformations
It has just been mentioned that k(λ ± π) = k(λ) ± 2π and therefore, shifting λ by ±π results in the same eigenstate or wave function, a very consequence of the lattice structure. In general, the Bethe-(Takahashi) equations are coupled and if one rapidity is shifted, all the other rapidities have to shift, too. Otherwise they would not give an Eigenstate. Since the momentum k(λ) is dependent only on one rapidity, shifts by π of single rapidities are justified and render the eigenstate unchanged. In the logarithmic Bethe equations however, shifting a single rapidity affects all quantum numbers in the coupled equations. To see this more explicitly, we study the resulting change of a quantum number set {Ij,α}
after a single rapidity has been changed by ±π. Therefore, λ + π is plugged into the Bethe-Takahashi equations (3.49). Lets first look at the kernels Θ and how they transform under
the π-shift, starting with Θj [65] Θj(λ ± π) = 2 arctan tan(λ ± π) tanh(j·η2 ) ! + 2π λ ± π π + 1 2 = 2 arctan tan(λ) tanh(j·η2 ) ! + 2π λ π ± 1 + 1 2 = Θj(λ) ± 2π (4.2)
The kernel Θjk is defined as a sum of Θj (see (3.49)) with
j + k − |j − k| − δjk = 2 min(j, k) − δjk (4.3)
summands. All terms apart from the edge terms occur twice (are multiplied by a factor 2) which accounts for the fact the indices are counted in steps of two. Thus, Θjk transforms
as
Θjk(λ ± π) = (1 − δjk)(Θ|j−k|(λ) ± 2π) + 2Θ|j−k|+2(λ) ± 4π + . . .
· · · + 2Θj+k−2(λ) ± 4π + Θj+k(λ) ± 2π
= Θjk(λ) ± (2 min(j, k) − δjk)2π. (4.4)
Going back to the Bethe-Takahashi equations, the change of quantum numbers depends on their relation to the shifted rapidity center, decrypted in their indices. The quantum number Ij,α corresponding to the shifted rapidity λj,α changes as
˜ Ij,α= N 2π(Θj(λj,α) ± 2π) − 1 2π M X k=1 (k,β) 6= Mk X β=1 (j,α) (Θjk(λj,α− λk,β) ± (2 min(j, k) − δjk)2π) = Ij,α± N ∓ M X k=1 (k,β) 6= Mk X β=1 (j,α) (2 min(j, k) − δjk) = Ij,α± N − 2X k6=j min(j, k)Mk− X β6=α (2j − 1) = Ij,α± N − 2X k6=j min(j, k)Mk− (Mj − 1)(2j − 1) . (4.5)
If another rapidity λl,γ with (l, γ) 6= (j, α) is shifted, the quantum number Ij,α changes,
too, since the kernel Θjk is summed over all rapidities
˜ Ij,α= N 2πΘj(λj,α) − 1 2π M X k=1 (k,β) 6= Mk X β=1 (j,α) (Θjk(λj,α− λk,β) ∓ δklδβγ(2 min(j, l) − δjl)2π) = Ij,α± M X k=1 (k,β) 6= Mk X β=1 (j,α) δklδβγ(2 min(j, l) − δjl).
