The Central Spin Problem and the Richardson Equations

Master’s Project

The Central Spin Problemand the Richardson Equations

Author:Evert Bosdriesz

Supervisor:Dr. Jean-Sebastien Caux

Universiteit van Amsterdam

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Instituut voor Theoretische Fysica

December 16, 2009

Abstract

In this work, we present a numerical method to solve the Richardson equa-tions for an arbitrary system. These equations are the Bethe Ansatz Equa-tions of the central spin model, which describes the hyperfine coupling ofan electron spin to a spin bath. In order to study the decoherence of theelectron spin, all eigenstates, and thus all the solutions the Richardson Equa-tions, need to be obtained. The occurrence of divergences in the equationsfor certain critical values of the coupling strength greatly complicates anynumerical approach. In contrast to previous efforts, in order to deal withthis we do not perform a change of variables but use an as simple as possiblemethod to allow for maximal flexibility.

Acknowledgements

First of all, I would like to thank my supervisor, Jean-Sebastien Caux;your enthusiasm as much as your patience and guidance made this project avery positive experience for me. Very useful were the discussions I had withAlexander Faribault.

There are many more people at the ITFA I would like to thank; Myfellow students for all the interesting discussions over a cup of coffee or two,and of course Bianca and Jonneke.

Not all in life is physics, so I would also like to thank all my friends -you know who you are - for keeping my mind of it every once in a while.

Last but definitely not least, I would like to thank my girlfriend Claartjeand my family, for always being there for me and for all their support.

Thank you all!

Contents

1 Introduction 2

2 The Central Spin Problem 42.1 Quantum Bits and Quantum Dots . . . . . . . . . . . . . . . 42.2 Hyperfine Coupling and Decoherence . . . . . . . . . . . . . . 62.3 The Central Spin Model . . . . . . . . . . . . . . . . . . . . . 72.4 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Semi-Classical Approximation . . . . . . . . . . . . . . . . . . 12

3 The Algebraic Bethe Ansatz 153.1 The Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Yang-Baxter Equation . . . . . . . . . . . . . . . . . . . 173.3 The Bethe Ansatz Equations . . . . . . . . . . . . . . . . . . 183.4 Construction of the Hamiltonian . . . . . . . . . . . . . . . . 203.5 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . 24

4 Numerical Procedure 274.1 General Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Results 415.1 Some Individual States . . . . . . . . . . . . . . . . . . . . . . 415.2 The Full Hilbert Space . . . . . . . . . . . . . . . . . . . . . . 425.3 Large g Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusion and Prospects 48

1

1Introduction

One of the holy grails of modern physics, theoretically as well as experimen-tally, is the realization of a quantum computer. However, while the potentialof quantum computation is enormous, so are the challenges. For instance, itis very difficult to make quantum bits (q-bits), the elementary information-processing units that form the building blocks of a quantum computer. Apromising candidate for a scalable realization of these q-bits are electronstrapped in a quantum dot (q-dot). One of the main limiting factors is thedecoherence of the electron spin state due to interactions with its environ-ment, most notably the hyperfine coupling to nuclear spin states. This isdescribed by the central spin model. This will therefore be the subject ofthis thesis.

The ultimate goal is to obtain an exact solution of the central spin prob-lem. To treat this problem exactly, we need to know all the eigenstates ofthe combined electron and nuclear spin system. And, in order to obtainthese eigenstates, we need to solve the so called Richardson equations, alarge set of coupled, non-linear equations. These generally cannot be solvedanalytically, we must therefore content ourselves with numerical solutions.

The main aim of this project is to write a computer program that isable to solve the Richardson equations numerically. Other attempts at thishave been made; however, these all made (unphysical) assumptions aboutthe interaction strength between the electron and nuclear spins. It is ourambition to construct an implementation that is as general as possible andwhere no such assumptions need to be made. This thesis is not a finished,stand alone project. Rather, it hopefully is a first step towards an exactsolution of the central spin problem.

In chapter 2 we will begin by discussing in some detail the central-spin

2

3

problem, and its relevance to quantum dots. The decoherence of the electronspin state is described by the decoherence function, which involves a doublesum over all eigenstates of the system. Fortunately, there is a mappingbetween the central spin Hamiltonian and the well known BCS Hamiltoniandescribing type I superconductivity. Because this model is known to beintegrable, we can use the Algebraic Bethe Ansatz (ABA) to construct allits eigenstates. This will be the subject of chapter 3. In order to actuallycalculate something using the ABA, we have to solve the Bethe AnsatzEquations; in this case better know as the Richardson equations. Thesecannot be solved analytically, which brings us to the most important part ofthis thesis. In chapter 4 we will discuss in detail the numerical procedure wedeveloped to solve the Richardson equations. In chapter 5 we will presentthe first results that we obtained. This can be seen in part as a test run ofthe program, and part as some first steps towards a better understandingof the general structures of the solutions to the Richardson equations. Inchapter 6 we will conclude with some remarks on the way to go from here.

2Quantum Bits, Quantum Dots and the

Central Spin Problem

Recently, a technology called spintronics, that uses the spin degree of free-dom as the carrier of information, rather than the mainstream charge-basedelectronics, has emerged [1]. One of the most promising applications of thistechnology are quantum dots, which are a candidate to become a scalablerealization of the quantum bits (q-bits), necessary for quantum computa-tion. This was first proposed by Loss and DiVincenzo [2]. In this chapterwe will discuss the basic principles and challenges of using quantum dots asq-bits.

2.1 Quantum Bits and Quantum Dots

Generally, a quantum dot is an artificially structured system that can befilled with electrons. These electrons can then be manipulated and mea-sured. The dot is coupled via tunnel barriers to a source and a drain reser-voir, from which electrons can tunnel to and from the dot, respectively.Furthermore, by also coupling the system to gate electrodes, the electro-static potential of the dot with respect to the reservoirs can be tuned. (Seee.g. [3] for an excellent review on few electron quantum dots.) There aretwo effects that dominate the properties of such a dot, and that make itpractical as a q-bit. First of all, the spatial confinement of the dot causesquantum effects to influence the electron dynamics, resulting in a discreteenergy spectrum. Secondly, due to the Coulomb repulsion between electronson the dots there is an energy cost for adding extra electrons to it. Becauseof this charging energy, the tunneling of electrons to and from dot can be

4

2.1. Quantum Bits and Quantum Dots 5

Figure 2.1: Schematic picture of a quantum dot. A 2D electron gas is trappedbetween a GaAs layer and an AlGaAs layer (left panel). In the depleted region thenumber of electrons can be tuned using the gate voltage. This is done by tuningthe electrostatic potential of the dot with respect to the source and drain reservoirs(right panel). Only those levels in the dot for which the electrostatic potential islower than both that of the drain and the source will be filled. If there is a level inthe dot that is lower than the source and higher then the drain, a current will flowthrough the dot.

suppressed at low temperature. This phenomenon, called Coulomb blockade,makes it possible to exactly control the number of electrons on the dot.

In principle, a quantum dot can be filled with a large number of electrons.However, for use as a q-bit, we typically want dots filled with only oneelectron, and we will from here on only consider these. If exactly one energylevel has an electrostatic potential lower than that of the source and thatof the drain, exactly one electron will be trapped in the dot. Due to thediscrete energy spectrum, we are able to confine this electron in its groundstate, provided the level spacing is much larger than thermal energy. We nowhave a quantum-mechanical system that we can, to some degree, manipulateand measure, i.e., we have a q-bit.

One of the main challenges with these kinds of q-bit realizations is thatthe electron on the dot will interact with its environment, leading to un-wanted decoherence of the quantum state of the electron, and thus to lossof the information of the q-bit. This problem was already foreseen by Lossand DiVincenzo in 1998, and while much progress has been made (see e.g.[4, 5]), it certainly is not yet solved. Trying to obtain more insight into theseinteractions is the main motivation for this project.

Usually, quantum dots are fabricated by creating a two-dimensional elec-tron gas (2DEG) at the interface of thin a GaAs and an AlGaAs layer. Theelectron will interact with the environment of the GaAs. We will focus onGaAs quantum dots as these are the most widely used in this context. How-ever, regardless of the details of the quantum dot, it will always be in anenvironment, and the electron will always interact with this environment.

Since the magnetic moment of a single electron spin, µB, is very small,

2.2. Hyperfine Coupling and Decoherence 6

and since electric fields affect spins only indirectly, spin states are onlyweakly influenced by their magnetic and electric environment. In general,for electron spins in semiconducting quantum dots, the most important in-teractions with the environment occur via the spin-orbit coupling and thehyperfine coupling with the nuclear spins of the host material. (Virtualexchange processes with electrons in the reservoirs also effect the electron,but this process can be suppressed by reducing the dot-reservoir tunnelingcoupling and/or creating a large energy gap between the dot and the lead[6], so we will not consider this further.) The observable effects of spin orbitcoupling and hyperfine interactions are threefold: Firstly, the eigenstatesare redefined and the energy splittings renormalized. An important con-sequence hereof is that the pure spin-state of the electron is no longer aneigenstate of the system. Secondly, fluctuations in the environment can leadto phase randomization of the electron spin, by convention characterized bythe transverse relaxation time T2. Finally, also due to fluctuations in theenvironment, the electron spin can be flipped, thereby exchanging energywith degrees of freedom in the environment. This process is characterizedby the longitudinal relaxation time T1. Spin-Orbit coupling is beyond thescope of this project, and from here on we will concentrate on the hyperfineinteraction.

2.2 Hyperfine Coupling and Decoherence

Just like the spin of an electron can interact with the spin of “its” atomicnucleus through the hyperfine interaction, an electron spin in a quantumdots interacts with the nuclear spins in its environment. The difference isthat the spin in a dot interacts with many nuclear spin (typically in theorder of 106). This considerably complicates things.

The Hamiltonian for the hyperfine interaction is given by

Hhf =N∑α=1

JαS0 · Sα, (2.1)

where S0 and Sα are the spin operator of the electron spin and nucleus atsite rα, respectively, and,

S0 · Sα =(Sz0S

zα +

12

(S+0 S−α + S−0 S

+α )), (2.2)

The coupling strengths Jα between the electron spin and the nuclearspin at site rα is proportional to the square norm of electron wave functionat that site, i.e.,

Jα = J |ψ0(rα)|2, (2.3)

where J is a measure of the strength of the hyperfine interaction. Becausegenerally the electron wave function is inhomogeneous, Jα will typically

2.3. The Central Spin Model 7

Figure 2.2: The square modulus of the electron wave function ψ0, as a functionof the distance from the dot rα. Of in total Ntot spins in the dot, the electron onlyinteracts appreciably with the N nearest ones. We therefore neglect the nuclei atsites α > N . Figure taken from [7]

vary from site to site. Not much is known about the details of the couplingstrengths, and it will be interesting to study different possible distributionsfor them. For now, we will follow Coish and Loss [7]. They consider a systemof Ntot nuclear spins. However, the electron spin only interacts appreciablywith the N spins within a radius l0 from the electron (see Fig. 2.2). Thewave function then takes the form

ψ0(rα) = ψ0(0) exp

[−1

2

(rαl0

)2]. (2.4)

In d dimensions, (rα/l0)d = α/N . Furthermore, it is convenient to workin units where the coupling at the origin J0 = 2. Since we have a 2D electrongas, we obtain

Jα = 2 exp[− αN

]. (2.5)

Due to the coupling to the nuclear spins, the pure electron spin-state isnot an eigenstate of the system. An electron prepared in a pure spin-statewill evolve to a statistical mixture of several states.

2.3 The Central Spin Model

As mentioned, we are interested in the dynamics of an electron spin in aquantum dot under the influence of the hyperfine interaction with nuclearspin states in its environment. Since, in general, the timescales for theinteractions between nuclear spin states are much longer, we ignore thenucleus-nucleus couplings. We allow for an external magnetic field, h, butonly couple it to the electron spin. Any coupling to the nuclear spins can be

2.3. The Central Spin Model 8

formally eliminated by transforming to a rotating frame of reference, whichwill only cause a shift in h [7].

This system is described the Hamiltonian

HCS = hSz0 +N∑α=1

JαS0 · Sα (2.6)

where S and Jα are defined as in section 2.2. A very important propertyof this Hamiltonian is that a mapping between it and the BCS-Hamiltonianexist (see chapter 3 for details),

HBCS =N∑α=1

εαb†αbα − g

N∑α,β=1

b†αbβ (2.7)

where b†α = c†α↑c†α↓ creates a Cooper-pair, nα = 2b†αbα and they obey the

commutation relations

[bα, b†β] = δαβ(1− 2b†αbα),

[bα, bβ] = [b†α, b†β] = 0. (2.8)

In this Hamiltonian we do not allow unpaired electrons. We can do thisbecause of the so-called blocking effect [8]; Unpaired particles completelydecouple and behave as if they were free, and we can therefore safely discardthem.

In the pseudospin realization of the electron pairs:

Szα = b†αbα −12, S−α = bα, S+

α = b†α, (2.9)

the Hamiltonian takes the form, up to some constant

HBCS =N∑α=1

εαSzα − g

N∑α,β=1

S+α S−β , (2.10)

where the operators S±α and Szα obey the standard spin algebra. If weidentify,

Jα ≡ −g

εαh ≡ 1

g(2.11)

we have that[HCS, HBCS] = 0, (2.12)

which means that these two Hamiltonians share the same set of eigenstates.Cambiaggio, Rivas and Saraceno showed that the BCS model is integrable[9]. This means that the Algebraic Bethe Ansatz can be used to construct

2.4. Correlators 9

the full set of eigenstates |Φn〉 of this models. In spin language, the resultis:

|ΦNr({wi})〉 =Nr∏i=1

N∑α=1

S−αwi − εα

| ⇑0; ↑1, . . . , ↑N 〉. (2.13)

In this case, the fully polarized state | ⇑0; ↑1, . . . , ↑N 〉 is our reference state,i.e., the pseudo-vacuum. The Nr rapidities wi are solutions to the BetheAnsatz Equations:

0 =1g−

N∑α=1

1wi − εα

+Nr∑k 6=i

2wi − wk

, i = 1, . . . , Nr. (2.14)

This result was already obtained by Richardson in 1963 [8, 10] using differ-ent techniques, and these equation are therefore usually referred to as theRichardson equations.

In the central spin picture, the number of rapidities Nr corresponds tothe number of nuclear spins that are flipped with respect to the referencestate. In the pairing model, the reference state is actually the fully occupiedstate, and each of the Nr rapidities removes one pair with energy wi fromthe system1

2.4 Correlators

Obviously, there are many quantities of interest to study in this model. Wewill, however, discuss only the real time correlator [12, 13]

C(t) ≡ 〈Ψi|δSz0(t)Sz0 |Ψi〉 (2.15)

where |Ψi〉 is the initial combined electron and nuclear spin configuration,Sz0 acts on the electron spin only, and

δSz0 = Sz0(t)− Sz0 , (2.16)

withSz0(t) = eiHtSz0e

−iHt. (2.17)

If the electron spin is initially prepared in an eigenstate of Sz0 (but notnecessarily HCS), i.e. if Sz0 |Ψi〉 = ±1

2 |Ψi〉, then we clearly have

C(t) =12〈Ψi|Sz0(t)|Ψi〉 −

14, (2.18)

and our two-point function reduces to a one-point function.1We choose this rather strange convention to make contact to [11]. See chapter 3 for

more on this.

2.4. Correlators 10

It appears that we may, without any loss of generality, consider

D(t) ≡ 〈Ψi|Sz0(t)|Ψi〉 (2.19)

The |Ψi〉s are not necessarily eigenfunctions of the Hamiltonian. But, fol-lowing Bhaseen [14], we can of course expand them in these:

|Ψi〉 =∑n

cn|Φn〉. (2.20)

Furthermore, we can write

|Ψ(t)〉 ≡ e−iHt|Ψi〉, (2.21)

which givesD(t) = 〈Ψ(t)|Sz0 |Ψ(t)〉. (2.22)

Putting (2.20), (2.21) and (2.23) together we obtain the following expressionfor the decoherence function:

D(t) =∑n,m

c∗ncme−i(En−Em)t〈Φn|Sz0(0)|Φm〉. (2.23)

Besides the eigenstates, we also need to know the corresponding energies.These are given by:

E({wi}) = − 12g−

N∑α=1

1εα

+Nr∑i=1

1wi. (2.24)

From (2.14) it follows that the rapidities are either real, or come in complexconjugate pairs (CCPs). This guarantees that the energies are real, as theyshould be.

2.4.1 Fully Polarized Initial State

Despite the fact that we ignore the couplings between the nuclear spins, theyare still coupled to each other indirectly. Since the total spin of the systemis conserved, and the Hamiltonian contains spin-flip terms, we can flip theelectron spin by flipping a nuclear spin with opposite spin orientation. Ifwe then flip the electron spin back by repeating this flip-flop process witha different nuclear spin, it effectively is as though two nuclear spins flip-flopped. One notable exception is if the initial state is fully polarized, i.e., ifall nuclear spins have the opposite orientation as the electron spin. In thiscase, the electron spin can only be flipped back by flip-flopping with the samenuclear spin. This makes things considerably less complex, and although thisis not (yet) realizable experimentally, we will study it in somewhat moredetail to build some intuition. We will follow Bhaseen [14], although this

2.4. Correlators 11

result was previously obtained by Khaetskii, Loss and Glazman [13] usingdifferent techniques.

We prepare the system in the initial state

|Ψi〉 = | ⇓0; ↑1, . . . , ↑N 〉. (2.25)

Since the electron spin can only flip-flop with one nuclear spin, after timet the system is in a superposition of the initial state plus all possible singleelectron spin flips.

|Ψ(t)〉 = α(t)|Ψi〉+N∑k=1

βk(t)| ⇑, ↑, . . . , ↓k, . . . , ↑〉. (2.26)

The |Ψ〉s are eigenstates of the Sz0 operator, rather then of HCS. Thereforeit is readily seen that

Sz0 |Ψ(t)〉 = −12α(t)|Ψ0〉+

12

N∑k=1

βk(t)| ⇑, ↑, . . . , ↓k, . . . , ↑〉. (2.27)

from which it straightforwardly follows that

〈Ψ(t)|Sz0 |Ψ(t)〉 = −12|α(t)|2 +

12

N∑k=1

|βk(t)|2. (2.28)

The normalization condition gives

|α(t)|2 +∑k

|βk(t)|2 = 1 (2.29)

and we thus obtain

〈Ψ(t)|Sz0 |Ψ(t)〉 =12− |α(t)|2. (2.30)

From (2.26) it follows that

α(t) ≡ 〈Ψ0|Ψ(t)〉. (2.31)

Using (2.20) and (2.21) this can be written as

α(t) =∑n

e−iEntcn〈Ψi|Φn〉. (2.32)

However 〈Ψi|Φn〉 = c∗n and thus

α(t) =∑n

e−iEnt|cn|2 =∑n

e−iEnt|〈Ψi|Φn〉|2 (2.33)

2.4. Correlators 12

This is much simpler than the unpolarized case for several reasons. Firstof all, in (2.33) we have only one summation over the (usually very large)Hilbert space. Moreover, the eigenstates expansion of (2.26) all consist ofonly one rapidity, which makes them considerably simpler. Finally, thenumber of different eigenstates for this system is

(NNr

); In case Nr = 1, this

is equal to N , whereas if Nr ≈ N/2 this will be of order NNr , which typicallyis much, much larger.

TheN+1 eigenstates (one for each nuclear spin and one from the electronspin) in the fully polarized case are, writing | ⇑0; ↑1, . . . , ↑N 〉 ≡ |0〉,

|φ1(w)〉 =N∑α=0

S−αw − εα

|0〉, (2.34)

were w is one of the N + 1 solutions to one-rapidity Richardson equations

0 =1g−

N∑α=0

1w − εα

. (2.35)

The norm of this state is then given by

(N1)−2 ≡ 〈φ1|φ1〉

=∑α,β

〈0|S+β S−α

(w − εβ)(w − εα)|0〉

=N∑α,β

δα,β(w − εβ)(w − εα)

〈0|0〉

=N∑α=0

1|w − εα|2

,

(2.36)

and the normalized eigenstates are

|Φ1〉 ≡ N1|φ1〉. (2.37)

Plugging (2.37) and (2.34) into (2.33), and setting ε0 = 0, gives

α(t) =∑{w}

e−iwt(N1

w

)2

(2.38)

where the sum is only over the N + 1 distinct solutions to the one-rapidityRichardson equation. We stress once more that this simple form cruciallydepends on the full polarization of the initial configuration. If a significantnumber of spins is flipped, things are far more complicated.

2.5. Semi-Classical Approximation 13

2.5 Semi-Classical Approximation

An alternative to the full quantum treatment of the problem is to take thesemi-classical limit of the central spin model. In this approximation, theensemble of nuclear spins is treated as an apparent magnetic field BN , calledthe Overhauser field ;

N∑α=1

JαSα = gµBBN . (2.39)

This magnetic field then assumes a random, a priory unknown, value, andunder its influence the electron spin will thus evolve in a random way, justlike in the quantum-mechanical description. Quite a lot of work has beendone in the semi-classical limit of the central spin problem (see e.g. [7],[12],[13], [15], [16], although this list is by no means exhaustive).

Although the semi-classical approximations is a considerable simplifi-cation as compared to the full quantum problem (e.g. the electron spininfluence on the nuclear spins is more or less eliminated), it is by no meanstrivial. In case the nuclear spins are fully polarized the field has a maximum;

BN,max =N∑α=1

1gµB

JαSα. (2.40)

Since typically Jα ∝ N−1, this value is independent of N . Distinctly dif-ferent is the (nearly) unpolarized case, were e.g. the spins are in thermalequilibrium. There is a small average polarization, caused by and orientedalong the direction of the external magnetic field h. Additionally, there isa statistical fluctuation about the average, analogous to the case of N cointosses. The root mean square (RMS) of this fluctuation is thus, at tempera-ture T , BN,max√

NT. Since at any given time the nuclear field assumes a random

and unknown value, it leads to randomness in the time evolution of theelectron spin. The timescale of the resulting dephasing of the electron spin,by convention called T ∗2 , is of the order of 10 ns. Were we to know exactlythe nuclear field at t = 0, our knowledge of the electron spin is lost due thesubsequent random evolution of the nuclear spin bath. The timescale T2

associated with this process is hard to obtain experimentally, but a lowerbound is given by approximately 1 µs. All the timescales are quoted from[3].

Besides the fully polarized situation we already discussed, Khaetskii,Loss and Glazman also studied the unpolarized case, treating it perturba-tively. They assume that at t = 0 the system is in an eigenstate of H0 =hSz0 +

∑JαS

zα and expand in the perturbation V = 1

2

∑α Jα(S+

0 S−α +S−0 S

+α ).

They conclude that the spin decay follows a power law

C(t)− C(∞) ∝(N

t

)3/2

(2.41)

2.5. Semi-Classical Approximation 14

for timescale larger than the timescale for decoherence

td =N

J. (2.42)

Note that this timescale increases with increasing N . This is consistent withthe observation that BN decreases with increasing N because fluctuationsbecome smaller for increasing N . Furthermore, td decreases with increas-ing J , which makes sense; reduce the interaction strength and coherence ismaintained.

The semi-classical limit gives an intuitive picture of the dynamics of thesystem, and it is sufficient to explain most of the experimental observationsto date. However, it does not incorporate the potentially important fullquantum behavior of the system. In order to analyze the microscopic corre-lations between the electron and nuclear spin states, e.g. the entanglementbetween them, we will need to solve the problem exactly.

Summary Quantum dots are a promising candidate to become a scalableq-bit realization. However, there are major challenges; most notable is theinteraction of the electron spin with its surroundings, which causes unwanteddecoherence. One of the more important of these interactions with the envi-ronment is the hyperfine coupling of the electron spin to the nuclear spins inthe host material. Because the central spin model that describes this inter-action is integrable, this problem is, in principle, exactly solvable using theAlgebraic Bethe Ansatz. In this way, all the eigenstates and energies, neededto calculate (dynamical) correlation functions, can be obtained. However,it involves solving a very large set of coupled, non-linear equations, one foreach spin. One possible simplification is to treat the ensemble of nuclearspins as an apparent magnetic field. The problem with this semi-classicallimit is that the potentially important microscopic correlations between theelectron and the nuclear spin states are not incorporated in this description.

In the next chapter we will use the Algebraic Bethe Ansatz to obtainthe full set of eigenstates of the central spin model.

3The Algebraic Bethe Ansatz

Our aim in this chapter is to derive the complete set of eigenstates of thecentral spin model using the Algebraic Bethe Ansatz (ABA). These can beused to compute correlation functions.

The main idea is that we assume that our model is integrable, otherwisenot specifying it. The fact that a model is integrable implies the existencesof a set of mutually commutative operators. We will generate these us-ing the transfer matrix, which in turn will be constructed by introducingthe monodromy matrix. From the transfer matrices we will construct theHamiltonian, and because the transfer matrices all commute with one an-other, they will also all commute with the Hamiltonian. Some entries of themonodromy matrix will act as raising and lowering operators on a pseudo-vacuum, provided they obey some constraints, the Bethe Ansatz equations.In this way we can create all eigenstates of the model. In this chapter wefollow the derivation in [17]. Other sources were [11], [18], [19] and [20].

3.1 The Transfer Matrix

Since we assume our model to be integrable, there must exist a complete setof commuting operators In in the Hilbert space H,

[In, Im] = 0 ∀n,m. (3.1)

These operators are the conserved charges of our (for now unknown) Hamil-tonian. In principle, there can be infinitely many of these operators. Inorder to generate the entire set In at once, we introduce the transfer matrix

15

3.1. The Transfer Matrix 16

t(u), acting in H, through the relation

ln t(u) =∞∑n=0

In(u− ε)n, (3.2)

which is a function of the spectral parameter u ∈ C and the inhomogeneityparameter ε, the interpretation of which will become clear later. The inverseof this relation can be obtained by differentiating ln t(u) n times with respectto u:

dn

dunln t(u) =

∞∑m=n

m!Im(u− ε)m−n. (3.3)

From this we easily see that

In =1n!

dn

dunln t(u)

∣∣∣∣u=ε

. (3.4)

Since all In commute with each other, so do the transfer matrices:

[t(u), t(v)] = 0, ∀u, v ∈ C. (3.5)

It is now convenient to introduce the monodromy matrix T (u), acting inthe tensor-product space of H with some auxiliary space V0, i.e., in H⊗ V0,and define the transfer matrix such that it is the trace of T (u) over V0, i.e.,

t(u) = tr0T (u), (3.6)

where the 0 subscript refers to V0.Condition (3.5) then imposes the commutation relation

[tr0T (u), tr0T (v)] = 0, ∀u, v ∈ C (3.7)

on the monodromy matrix.For tensor products, it is generally true that

tr(A)tr(B) = tr(A⊗B) (3.8)

andAB ⊗ CD = (A⊗ C)(B ⊗D). (3.9)

Introducing the notation (as in [11])

T1(u) = T (u)⊗ 10,

T2(u) = 10 ⊗ T (u), (3.10)

where T1 and T2 act in H⊗V0⊗V0, 10 is the identity in V0 and the subscriptrefers to the auxiliary space that it acts on non-trivially, we can write

T (u)⊗ T (v) = T1(u)T2(v) (3.11)

3.2. The Yang-Baxter Equation 17

The commutation relation (3.5) then simply tells us that

tr (T1(u)T2(v)) = tr (T2(v)T1(u)) ∀u, v ∈ C. (3.12)

This equation is satisfied if and only if T1 and T2 obey Yang-Baxteralgebra, i.e.,

R(u, v)T1(u)T2(v) = T2(v)T1(u)R(u, v), (3.13)

were the R-matrix R(u, v) acts in H⊗V0⊗V0. However, in H it always actsas the identity, so effectively it only relates the auxiliary spaces. Usually,the spectral parameter is a c-number such that the R-matrix is a functionof the difference of its arguments only:

R(u, v) ≡ R(u− v). (3.14)

3.2 The Yang-Baxter Equation

The commutation relations for the monodromy matrix can be straight-forwardly generalized to tensor product of n auxiliary vector spaces Vi,i = 1, . . . , n, with Ti acting on Vi and the matrix Rij relating Vi and Vj .

If we consider T (u)⊗T (v)⊗T (w) and impose associativity, repeated useof (3.13) leads to,

T1(u) (T2(v)T3(w)) = R−123 (v, w) (T1(u)T3(w))T2(v)R23(v, w)

= R−123 (v, w)R−1

13 (u,w)T3(w) (T1(u)T2(v))×R13(u,w)R23(v, w)

= R−123 (v, w)R−1

13 (u,w)R12(u, v)−1T3(w)T2(u)T1(v)×R12(u, v)R13(u,w)R23(v, w).

(3.15)

But, similarly,

(T1(u)T2(v))T3(w) = R−112 (u, v)T2(v)(T1(u)T3(w))R1(u, v)

= R−112 (u, v)R−1

13 (u,w)(T2(v)T3(w))T1(u)×R13(u,w)R12(u, v)

= R−112 (u, v)R−1

13 (u,w)R23(v, w)−1T3(w)T2(u)T1(v)×R23(v, w)R13(u,w)R12(u, v).

(3.16)

A necessary condition for (3.15) and (3.16) to be satisfied is that the R-matrix satisfies the Yang-Baxter equation, i.e.,

R12(u, v)R13(u,w)R23(v, w) = R23(v, w)R13(u,w)R12(u, v). (3.17)

3.3. The Bethe Ansatz Equations 18

3.3 The Bethe Ansatz Equations

Now, any matrix that satisfies (3.17) can be used for the Algebraic BetheAnsatz. We will take V0 ∼ C2, and use the gl(2) invariant form,

R(u− v) =

1 0 0 00 b(u− v) c(u− v) 00 c(u− v) b(u− v) 00 0 0 1

(3.18)

withb(u) =

u

u+ η, c(u) =

η

u+ η(3.19)

For convenience, we write the monodromy matrix as

T (u) =(A(u) B(u)C(u) D(u)

), (3.20)

where A,B,C and D are 2× 2 matrices. Note that for the transfer matrix,in this notation, we have t(u) = A(u) + D(u). Next, we define the pseudo-vacuum |0〉 through the following relations

A(u)|0〉 = a(u)|0〉〈0|B(u) = 0 (3.21)C(u)|0〉 = 0D(u)|0〉 = d(u)|0〉.

Imposing the commutation relations (3.13) then leads to 16 commutation

3.3. The Bethe Ansatz Equations 19

relations, one for each entry:

[A(u), A(v)] = [B(u), B(v)] = [C(u), C(v)] = [D(u), D(v)] = 0

[D(u), A(v)] =c(u− v)b(u− v)

(B(v)C(u)−B(u)C(v)

),

[A(u), D(v)] =c(u− v)b(u− v)

(C(v)B(u)− C(u)B(v)

),

[B(u), C(v)] =c(u− v)b(u− v)

(A(v)D(u)−A(u)D(v)

),

[C(u), B(v)] =c(u− v)b(u− v)

(D(v)A(u)−D(u)A(v)

),

A(v)B(u) =1

b(u− v)B(u)A(v)− c(u− v)

b(u− v)B(v)A(u),

B(v)A(u) =1

b(u− v)A(u)B(v)− c(u− v)

b(u− v)A(v)B(u), (3.22)

A(u)C(v) =1

b(u− v)C(v)A(u)− c(u− v)

b(u− v)C(u)A(v),

C(u)A(v) =1

b(u− v)A(v)C(u)− c(u− v)

b(u− v)A(u)C(v),

D(u)B(v) =1

b(u− v)B(v)D(u)− c(u− v)

b(u− v)B(u)D(v),

B(u)D(v) =1

b(u− v)D(v)B(u)− c(u− v)

b(u− v)D(u)B(v),

D(v)C(u) =1

b(u− v)C(u)D(v)− c(u− v)

b(u− v)C(v)D(u),

C(v)D(u) =1

b(u− v)D(u)C(v)− c(u− v)

b(u− v)D(v)C(u).

Next, we define

|ΦNr({u})〉 =Nr∏j=1

B(uj)|0〉, (3.23)

which, when we impose some constrains, will turn out to be an eigenfunctionof t(u) = A(u) +D(u). To see this, let’s act with A(u) and D(u) on |ΦNr〉.Using the commutation relations above this gives:

A(u)Nr∏j=1

B(vj)|0〉 = ΛNr∏j=1

B(vj)|0〉 −Nr∑n=1

ΛnB(u)Nr∏j 6=n

B(vj)|0〉 (3.24)

and

D(u)Nr∏j=1

B(vj)|0〉 = ΛNr∏j=1

B(vj)|0〉 −Nr∑n=1

ΛnB(u)Nr∏j 6=n

B(vj)|0〉. (3.25)

3.4. Construction of the Hamiltonian 20

The coefficients are:

Λ = a(u)Nr∏j=1

1b(u− vj)

= a(u)Nr∏j=1

u− vj + η

u− vj, (3.26)

Λ = d(u)Nr∏j=1

1b(vj − u)

= d(u)Nr∏j=1

u− vj − ηu− vj

, (3.27)

Λn = a(vn)c(vn − u)b(vn − u)

Nr∏j 6=n

1b(vj − vn)

, (3.28)

Λn = d(vn)c(u− vn)b(u− vn)

Nr∏j 6=n

1b(vn − vj)

. (3.29)

We see that |ΦNr(u)〉 is an eigenfunction of t(u), with eigenvalues Λ+Λ,if Λn = Λn (since this is the condition for the B(u)|0〉 terms to cancel). Thisrequirement leads to the Bethe Ansatz equations

a(vn)d(vn)

Nr∏j 6=n

b(vn − vj)b(vj − vn)

= 1, n = 1, . . . , Nr. (3.30)

In the next section we will find an explicit expression for them.

3.4 Construction of the Hamiltonian

Anticipating that the Hilbert space of the BCS model can be decomposedin N sub-Hilbert spaces Hα, one for each unblocked energy level α,

H =N⊗α=1

Hα, (3.31)

we can express the monodromy matrix as a product of N so-called L-operators defined in V0 ⊗Hα:

T (u) = LN (u, εN )⊗ · · · ⊗ L1(u, ε1). (3.32)

Were the εαs are the inhomogeneity parameters of (3.2).The only restriction on L is that (3.13) is satisfied. Setting

Lj(u, εα) = g0R0α(u− εα), (3.33)

withgα = e−2ηSzα/gN ∈ su(2) (3.34)


does this, as can easily be checked by explicitly plugging (3.33) and (3.32)into (3.13); this gives

R0102T1(u)T2(v) = R0102

(g01R01N (u− εN )⊗ · · · ⊗ g01R011(u− ε1)

)×(g02R02N (v − εN )⊗ · · · ⊗ g02R021(v − ε1)

)=(g02R02N (v − εN )⊗ · · · ⊗ g02R021(v − ε1)

)×(g01R01N (u− εN )⊗ · · · ⊗ g01R011(u− ε1)

)R0102

= T2(v)T1(u)R0102 ,

where the subscript 0i refers to the ith auxiliary space and we have used(3.17).

The R-matrix (3.18) can be more compactly written as

Rαβ(u) = b(u)1α ⊗ 1β + c(u)Pαβ, (3.35)

where Pαβ is the permutation matrix;

Pαβ =

1 0 0 00 0 1 00 1 0 00 0 0 1

. (3.36)

Note that Rαβ(0) = Pαβ.The transfer matrix, evaluated at εα, then is explicitly given by,

t(εα) = tr0(g0R0N (εα − εN ) . . . g0P0α . . . g0R01(εα − ε1))= tr0(P0αg0R0α−1(εα − εα−1) . . . g0R01(εα − ε1)× g0R0N (εα − εN ) . . . g0R0α+1(εα − εα+1))g0)

= tr0(gαRαα−1(εα − εα−1) . . . gαRα1(εα − ε1)× gαRαN (εα − εN ) . . . gαRαα+1(εα − εα+1)gαP0α)

= 2gαRαα−1(εα − εα−1) . . . gαRα1(εα − εα)× gαRαN (εα − εN ) . . . gαRαα+1(εα − εα+1)gα

(3.37)

where we have used that the trace is invariant under cyclic permutation andthat, after we have moved P0α through the Rs, only a trace over Pα had tobe taken, which simply gives a factor 2.

In the limit η → 0 we have R(u) = 1 which allows us to expand in η,i.e. we go to the quasi-classical limit. Since, to first order in η we then haveb(u) = 1− η/u, c(u) = η/u and g = 1− ηSzα/gN we have

Rαβ(u) = 1α ⊗ 1β − ηRαβ(u) +O(η2), (3.38)


whereRαβ(u) =

1u

(1α ⊗ 1β − Pαβ) . (3.39)

The transfer matrix then becomes

t(εα) = 2(

1− η2SzαgN

)(1α⊗α−1 − ηRαα−1(εα−1 − εα)

). . .

× . . .(

1− η2SzαgN

)(1α⊗1 − ηRα1(ε1 − εα)

)

×(

1− η2SzαgN

)(1α⊗N − ηRαN (εN − εα)

). . .

× . . .(

1− η2SzαgN

)(1α⊗α+1 − ηRαα+1(εα+1 − εα)

= 2(

1− η2SzαgN

)N11⊗N + η

N∑β 6=αRαβ(εβ − εα)

(3.40)

Using the identity 2Pαβ − 1α1β = σα · σβ = 4Sα · Sβ, and dropping anuninteresting constant term we can write this as

t(εα) = 11⊗···⊗N + ητα (3.41)

where

τα = −2gSzα + 2

N∑β 6=α

Sα · Sβεα − εβ

. (3.42)

For future reference, the elements of the monodromy matrix in the quasi-classical limit are given by:

A(u) =−1g

+N∑α=1

Szαu− εα

, B(u) =N∑α=1

S−αu− εα

, (3.43)

C(u) =N∑α=1

S+α

u− εα, D(u) =

1g−

N∑α=1

Szαu− εα

.

Choosing the pseudo-vacuum to be the tensor product of highest weightstates, i.e.,

|0〉 ≡ | ⇑0; ↑1, . . . ↑N 〉, (3.44)

gives

Szα|0〉 =12|0〉, (3.45)


and

a(u) = e−η/gNr∏α=1

u− εα − η/2u− εα

, (3.46)

d(u) = eη/gNr∏α=1

u− εα + η/2u− εα

. (3.47)

Plugging this into (3.26), and expanding in η gives the eigenvalues λα of τα;

λα =−12g

+N∑β 6=α

1(εα − εβ)

+Nr∑i=1

1vi − εα

. (3.48)

We can now define the central spin Hamiltonian through

HCS ≡ −12τ0, (3.49)

setting ε0 = 0 and making the identification

Jα ≡g

εα, h ≡ 1

g(3.50)

we obtain the Hamiltonian

HCS = hSz0 +N∑α=1

JαS0 · Sα, (3.51)

and the interpretation of the inhomogeneity parameters becomes clear; inthe central spin model they are proportional to the inverse of the hyperfinecouplings between the central spin and the nuclear spins. The eigenvaluesof HCS are given by λ0:

ECS({vi}) =−12g−

N∑α=1

1εα

+Nr∑i=1

1vi

(3.52)

Plugging (3.46) into (3.30) gives the explicit expressions for the BetheAnsatz equations Fi. They read:

0 = Fi =1g−

N∑α=1

1vi − εα

+Nr∑j 6=i

2vi − vj

, i = 1, . . . , Nr. (3.53)

We have indeed recovered the Richardson equations (2.14).To be complete, the pairing model (2.7) is constructed by,

HBCS = −gN∑α=1

(εα −g

2)τα +

g3

4

N∑α,β=1

τατβ. (3.54)

3.5. Correlation functions 24

In this case, the inhomogeneity parameters are just the energy levels. Theeigenvalues are:

EBCS({vi}) =N∑α=1

εα − 2Nr∑α=1

vi. (3.55)

The minus sign in from of the second term is due to the fact that in our setupthe reference state is fully occupied an each rapidity “removes” a particlefrom the system.

3.5 Correlation functions

As stated in chapter 2, the ultimate goal is to calculate (dynamical) corre-lation functions; we are particularly interested in the decoherence function

D(t) =∑n,m

c∗ncme−i(En−Em)t〈Φn|Sz0(0)|Φm〉. (3.56)

However, this is a formidable task. First of all, it involves a double sumover all the

(NNr

)eigenstates of the system. Moreover, the form factors,

〈Φn|Szα|Φm〉 are also far from trivial to compute, even when the solutionsto the Richardson equations are available. Therefore, actually computingthese kinds of correlation functions is beyond the scope of this project. Tobe complete, though, we will give some relevant expressions. For moredetails, see [21]. To make explicit that each eigenstate corresponds to oneparticular solution of the Richardson equation, in this section we will adoptthe notation |Φn〉 = |{u}〉.

In order to calculate the form factors, 〈{w}|Szα|{v}〉, we must first obtainan expression for the scalar product

〈{w}|{v}〉 = 〈0|Nr∏b=1

C(wb)Nr∏a=1

B(va)|0〉, (3.57)

where B(u) and C(u) are defined as in (3.43).These scalar functions are given by the Slavnov formula [22] which, for

our model, reads

〈{w}|{v}〉 =

∏Nri6=j(vj − wi)∏

j<i(wj − wi)∏i<j(vj − vi)

detNr

G({v}, {w}), (3.58)

where

Gij =vj − wjvj − wk

N∑α=1

1(wi − εα)(vj − εα)

−Nr∑k 6=i

2(wi − wk)(vj − wk)

.

(3.59)


The norm of a state is then obtained by simply setting {v} = {w}, i.e.,|{v}|2 = detNrJ , with

Jij =

{∑Nα=1

1(vi−εα)2

− 2∑Nr

k 6=i1

(vi−vk)2, i = j

2vi−vj , i 6= j.

(3.60)

Note that this is the Jacobian of the the Richardson equations.Now, to compute the form factors of Szα, we note that

Szα = − limu→εα

D(u), (3.61)

which leads to

〈{w}|Szα|{v}〉 = −∏Nri=1(wi − εα)∏Nr

k=1(vk − εα)∏Nrk>l(vk − vl)

∏Nri<j(wi − wj)

×det(F −Q(εα))

(3.62)

where

Fij =

N∑α=1

1(wi − εα)(vj − εα)

−∑k 6=i

2(wi − wk)(vj − wk)

∏l 6=i

(vj − wl)

(3.63)

and

Qij(u) =2∏k 6=j(vj − vk)(u− wi)2

(3.64)

Note that (3.62) is a quite complicated expression, containing several prod-ucts over all rapidities. This means that we have to take a very large doublesum over highly non-trivial expressions. Obviously, in practice this cannotbe done unless a way can be found to significantly simplify the double sum.Our hope is for each state only a small fraction of all possible form factorswill give a significant contribution. Further study is needed to shed morelight on this subject.

Summary In order to calculate correlation functions of the central spinmodel, we need all the eigenstates of the Hamiltonian. Fortunately, there isa mapping between the central spin model and the well known BCS modelfor superconductivity, and these models therefore share the same set of eigen-states. The BCS model (and thus the central spin model as well) is inte-grable, which means that we can use the Algebraic Bethe Ansatz to find theeigenstates. The main idea is to construct a so called monodromy matrixsuch that the Hamiltonian can be constructed from the trace of this matrix.The other to elements of this monodromy matrix than act as raising andlowering operators on a pseudo-vacuum. The states obtained by acting with


the raising operator on the pseudo-vacuum are eigenstates of the Hamilto-nian, provided the Bethe Ansatz Equations are satisfied. Each eigenstatethen corresponds to a solution to these equations.

Unfortunately, for the case at hand, the Bethe Ansatz Equations (alsoknown as the Richardson equations), are a very large set of coupled, non-linear equations, one for each of the N spins. Moreover, for Nr excitations,these equations have

(NNr

)solutions. No method is know to obtain all of

these analytically. We therefore have to resort to numerical solutions. Inthe next chapter we will discuss a numerical method to systematically obtainall solutions to the Richardson equations.

4Numerical Procedure

In order to construct the eigenstates of the central spin Hamiltonian, thesolutions to the Richardson equations need to be obtained. Unfortunately,these are very complicated set of equations, and it is not generally possible tosolve them analytically. We must therefore content ourselves with numericalsolutions. This, however, is also easier said than done. The main aim of thismasters project is to write a computer program that solves the Richardsonequations numerically for general distributions of the energy levels and atarbitrary strength of the coupling constant.

In this chapter we will discuss the details of the numerical procedure weused to solve the Richardson equations.

4.1 General Scheme

For a system of N energy levels εα, containing Nr rapidities wi, the Richard-son equations form a set of Nr coupled, non-linear equations:

0 = Fi =1g−

N∑α=1

1wi − εα

+Nr∑k 6=i

2wi − wk

, i = 1, . . . , Nr, (4.1)

with(NNr

)different sets of solutions {wi}. For any system of reasonable size,

the number of different solutions will be very, very large. For any quantityof physical interest, all these solutions, or at least some systematic sampleof the these, should be obtained.

Fortunately, for vanishing g the solutions are trivial. In the BCS pic-ture, at g = 0 we only have the free theory, which means that each of the

27

4.1. General Scheme 28

rapidities should be strictly equal to one of the energy levels, i.e. wi(g =0) ∈ {εα}, ∀i . The

(NNr

)different solutions to (4.1) then correspond to

different ways in which Nr particles can be distributed over N energy levels.We can now use these solutions to find the solutions at arbitrary values

of the coupling constant g. The idea is very simple; use a solution at g = g∗

as an 0th order approximation for a solutions at g = g∗ + δg, then use asuitable numerical method to obtain the actual solution. We will use theNewton-Raphson method (see 4.1.1 for details). Since it is straightforwardto systematically obtain all solutions at g = 0, by repeated use of this stepa complete set of solutions can, in principle, be found at arbitrary g.

However, things are not that simple. As was already noted by Richardsonhimself [23, 24], for certain “critical” values of the coupling constant, gc,we have that simultaneously wi → εγ and wi → wj for some i,j and γ.This creates two diverging and mutually (almost) canceling terms in (4.1).Exactly at g = gc, we have that wi = wj = εγ and the equations are notproperly defined. However, we know that the two rapidities “collapse” ontoeach other to form a complex conjugate pair (CCP). Similarly, a CCP canalso split up again.

This poses a problem because in order to calculate correlation functionswe need all the solutions to the Richardson equations. As stated, we obtainthem by constructing each solution iteratively starting from the known solu-tions at g = 0. Other, simpler methods might find some solutions (althoughwe do not know of any), but we hold it for unlikely that a method existsthat can generate all solutions without dealing with these critical points inone way or another. Now, in order to “cross” the critical point, we need anapproximation for the rapidities after the collapse. Moreover, this approxi-mation must be sufficiently accurate because otherwise the Newton-Raphsonmethod might converges to “wrong” solution, i.e., one corresponding to an-other initial state. If this is the case, and we know from experience that ithas this tendency when we are not careful, we will not get the full set ofsolutions.

In the literature several ways to deal with this are available, all basedon a change of variables for the collapsing rapidities [21, 25, 26]. We will,however, use a somewhat more simple approach, which might be less elegant,but, hopefully, will be more generally applicable. See section 4.1.2 for details.

4.1.1 Newton-Raphson Method

As mentioned above, for the Richardson equations, we can use a solutionat g = g∗ as an 0th order approximation for the roots at g = g∗ + δg.The Newton-Raphson method can then be used to find the actual rootsat g = g∗ + δg. See [27] for a detailed discussion of the Newton-Raphsonmethod.

This is a simple, general and quite efficient technique to find a solution


to a set of N coupled (nonlinear) equations

Fi(x1, . . . , xN ) = 0 i = 1, . . . , N, (4.2)

provided some sufficiently good initial approximation is known (see [27]).Suppose we have such a zeroth order approximation x(0) for the solutionand want to improve this. We can expand it in a Taylor series:

Fi

(x(0) + δx

)= Fi

(x(0)

)+

N∑j=1

∂Fi(x(0)

)∂xj

δxj +O(δx2), (4.3)

or, in matrix notation

F(x(0) + δx

)= F

(x(0)

)+ J

(x(0)

)· δx +O

(δx2). (4.4)

Setting F(x(0) + δx) = 0, we obtain, to first order in δx,

δx = −J(x(0)

)−1· F(x(0)

). (4.5)

This equation can be solved using e.g. LU -decomposition. We can then addthis correction to our initial approximation to obtain,

x(1) = x(0) + δx. (4.6)

Repeated use of this procedure could, in principle, land us arbitrarily closeto the root. However, for the convergence of this procedure it is of eminentimportance that the initial guess is good enough. If not, this procedurehas the unfortunate tendency to grossly misbehave. There are algorithmsthat have global convergence, but experience teaches us that these are tooinefficient, and it turns out that to put some effort in making good initialguesses, especially close to the critical points, is a wiser approach (see 4.1.2).

Now, in our case the set of equations reads:

0 = Fi =1g−

N∑α=1

1wi − εα

+Nr∑k 6=i

2wi − wk

, i = 1, . . . , Nr, (4.7)

which means that for the Jacobian matrix we have,

Jij ≡∂Fi∂wj

=

∑N

α=1

(1

wi−εα

)2−∑Nr

k 6=i

(2

wi−wk

)2if i = j(

2wi−wj

)2if i 6= j

(4.8)

As for the initial guess. Suppose we have a solution Fi (g∗, {wi}) = 0 (e.g.at g = 0) and want to find one at g = g∗+δg. We can then use this solutionas our initial guess. (In practice, we will use a linear extrapolation of thewis as our initial guess.)


1

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5 3 3.5

Re(w

)

g

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5

Im(w

)

g

Figure 4.1: Two critical points. The real part (left panel) and imaginary part(right panel) of two rapidities. At g = 0 the two rapidities are equal to the energylevel ε1 = 1 and ε2 = 2. As we increase g, the rapidities first move away from eachother. They collapse at g ≈ 0.75 exactly at the energy level ε2 and form a complexconjugate pair. As the real part of the rapidities approaches ε3 = 3 we see theimaginary parts moving toward each other, and at g ≈ 2.7 the rapidities becomereal again and split up.

4.1.2 Dealing with the Critical Points

In principle, it should now be straightforward to obtain all solutions. Unfor-tunately, the critical points discussed above significantly complicate things.

The first problem we encounter is how to find the critical points. Un-fortunately, there is no (known) general method that a priory gives all thecritical points (or even the number of them) for a given system. We willtherefore have to identify them as we go along. As mentioned, a criticalpoint is characterized by a divergence in some of the terms of (4.7) due tothe conditions wc1 → wc2 and wc1 → εγ (see Fig 4.1). A quick glance atthe Jacobian matrix (4.8) shows us that it will have quadratically divergentterms, at least on the diagonal. Since, as g → gc, ∂Fi

∂wj→∞, we will in gen-

eral have a quite large initial deviation in the Richardson equations. Thisneed not be a problem because the convergence of the Newton-Raphson de-pends on the distance of the initial guess to the true root, and not on thedistance of the equations to 0. In other words, as long as w(0)

i −wi is small,it need not be a problem if Fi({w(0)

i }) is large. However, it will turn out that

as g approaches gc,∣∣∣dwidg ∣∣∣ → ∞ as well. This is one of the main difficulties

in dealing with the critical points, for it forces us to make very small stepsin g when approaching gc.

A look at (4.7) shows us that, in general, a rapidity can either be real, orit must form a complex conjugate pair (CCP) with another rapidity. This isbecause the imaginary part of the rapidities only have each other to cancel,whereas the real parts should exactly cancel the 1/g term. Therefore, thereare two types of critical points: One where two real rapidities collapse toform a CCP, and one where a CCP splits up in two real rapidities. Obviously,


1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

Re(w)

g

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

Im(w)

g

Figure 4.2: The real part (red lines, left panel) and imaginary part (red lines, rightpanel) of two rapidities collapsing on εγ = 2 (green line, right panel). Althoughthe derivative of the individual rapidities diverges as g approaches gc, the “centerof mass” wc of the two rapidities (right panel, blue line) remains perfectly smooth.This means that we can use wc, together with its distance to εγ , ∆εγ , to approximategc. If we also know the distance of the rapidities to their center of mass, ∆wc, therapidities are completely defined. In case the rapidities form a CCP , ∆wc is justthe imaginary part of the rapidities (right panel).

both types of collapses need to happen exactly on an energy level, for weneed the (wc− εγ)→ 0 term as a means to cancel the other divergent term.Note that this implies that an unpaired rapidity that is not above the highestor below the lowest energy level, is trapped between two energy levels. Thatis, the only way it can cross an energy level is by collapsing onto anotherrapidity.

Despite the numerical instabilities near g = gc, by taking small enoughsteps in g we can still get quite close to gc. In fact, we are only limited bythe precision we work in, which, in principle, can be arbitrarily large (see4.2.6). However, since the equations are not well defined exactly at g = gc,the actual collapse of the rapidities poses some problems. Our approach willbe to get sufficiently close to gc and then “jump over” the critical point byhand. In order to do this, we must find a way to approximate wc1 and wc2 atg = gc+δg, given {wi} at g = gc−δg. We will use a method inspired on theones used in [21] and [26], but instead of preforming a change of variableson the collapsing rapidities, we will only try to approximate them close tothe critical point. The reason for this will become clear later.

Although near a critical point some rapidities have divergent derivatives,nothing special happens to the physical quantities. For instance, EBCS =∑

α εα − 2∑Nr

i wi, is perfectly smooth at g = gc. We can use this to ouradvantage, for it means that sum of the two collapsing rapidities must besmooth as well (see Fig. 4.2). Consider the center of mass wc of twocollapsing rapidities wc1 and wc2,

wc ≡12

(wc1 + wc2) . (4.9)


The imaginary part of wc always vanishes, since wc2 = w∗c1, so we cannotget any information from that. However, the fact that wc behaves smoothlymeans that we can use it to estimate what gc will be. We know that atg = gc, wc = εγ ; therefore

gc ≈ g +εγ − wcdwc/dg

, (4.10)

provided we are close enough to gc. In this case, close enough means that wcis approximately linear in g. Furthermore, wc will also remain smooth afterthe collapse, and we can thus approximate wc after the collapse by linearextrapolation;

wc(gc + δg) ≈ wc(gc − δg) + 2δgdwcdg

∣∣∣∣wc=gc−δg

. (4.11)

Now that we have an approximation for wc, we only need to find anapproximation for the distance between wc1, wc2 and wc,

∆wc ≡12|wc1 − wc2| (4.12)

In case of two real rapidities collapsing into a complex conjugate pair, ∆wcmust be purely imaginary, and if a complex conjugate pair splits up, it mustbe purely real. Furthermore, it is convenient to rewrite wc in terms of theenergy level on which the collapse, εγ and the distance between wc and εγ ,∆εγ , i.e.,

wc ≡ εγ + ∆εγ . (4.13)

We then have, at g = g + δg

wc1 = εγ + ∆εγ + ∆wcwc2 = εγ + ∆εγ −∆wc (4.14)

Near a critical point there are three different relevant scales. The firstis set by the system, and it is the spacing between the energy levels. Inthe case that the energy levels are not homogeneously spaced, it is set bythe distance between εγ and the nearest other level, |εγ − εγ±1|. The otherscales are set by ∆wc and ∆εγ . Since dwc1/dg � 1, the hierarchy is:

∆εγ � ∆wc � |εγ − εγ±1| . (4.15)

We now split up the Richardson equations in the divergent and the non-divergent terms:

0 =1g− 1wc1 − εγ

+2

wc1 − wc2−

N∑α6=γ

1wc1 − εα

+Nr∑

k 6=c1,c2

2wc1 − wk

(4.16)


Plugging this in (4.14), we obtain,

0 =1g− 1

∆εγ + ∆wc+

1∆wc

−N∑γ 6=α

1εγ − εα + ∆εγ + ∆wc

+∑

k 6=c1,c2

2εγ − wk + ∆εγ + ∆wc

.

(4.17)

Expanding the last two terms in ∆εγ + ∆wc gives:

N∑γ 6=α

1εγ − εα + ∆εγ + ∆wc

+∑

k 6=c1,c2

2εγ − wk + ∆εγ + ∆wc

≈ S1 − S2(∆εγ + ∆wc)

(4.18)

where

Sn =N∑α6=γ

(1

εγ − εα

)n−

∑k 6=c1,c2

(2

εγ − wk

)n. (4.19)

Writing Fc1 and Fc2 in these terms gives

Fc1 ≈1g− S1 −

1∆εγ + ∆wc

+1

∆wc+ S2(∆εγ + ∆wc)

Fc2 ≈1g− S1 −

1∆εγ −∆wc

− 1∆wc

+ S2(∆εγ −∆wc) (4.20)

If we now look at 12(Fc1 + Fc2), and set this equal to zero, we obtain

− 1g

+ S1 ≈(

−1∆ε2γ −∆w2

c

+ S2

)∆εγ , (4.21)

and since we already know how to approximate ∆εγ we easily see that

∆wc ≈

√−∆εγ

1/g − S1 + S2∆εγ+ ∆ε2γ , (4.22)

which, indeed, can only be either purely real or purely imaginary.Obviously, there is no guarantee that this will work. As mentioned, we

obtain our approximation for ∆εγ by linear extrapolation. If this somehowis inaccurate, the Newton method will not converge. Furthermore, our ap-proximation depends on all the other rapidities, and we do not know whatthey are. We therefore linearly extrapolate these, but we cannot know howaccurate this is.

Especially unstable are points where several pairs of rapidities collapseat almost exactly the same gc. This happens more often than one wouldexpect, because, for some reason, certain kinds of configurations lead to

4.2. Implementation 34

a situation where a number of rapidities seem to attract each other (seeFig. 5.1, bottom panels). Several pairs split up and re-pair with otherrapidities in some complicated way. There are a number of reasons why thisis problematic. First of all, ∆εγ tends to deviate from linearity strongly,which makes it necessary to get extremely close to gc before a collapse willwork. It also means that it is hard to predict which pair will collapse first,and the linear approximation of the other rapidities is inaccurate, since theother rapidities are collapsing rapidities as well, which makes their slope ofthe same order. Again, the only way to deal with this is by making verysmall steps in g and getting extremely close to gc. This is also the reason whya change of variables becomes more or less useless. All of these approachesrequire that the critical point is identified beforehand and none of them iscapable of dealing with several critical points at once.

4.2 Implementation

In principle, we now have all the ingredients to solve the Richardson equa-tions. In this sections we will discuss some of the details, and challenges, ofthe actual implementation.

4.2.1 Young-Diagram Representation of States

The system itself is defined by three things only. We have the numberof energy levels N , their distribution, and the number of rapidities Nr.However, each system has a very, very large number of solutions, and it istherefore necessary to find a way to systematically label them. Since, in ourapproach, each set of solutions is obtained from a solution at g = 0, it isconvenient to label them according to these states. We thus need to find away to label all the

(NNr

)ways in which Nr particles can be distributed over

N levels.It is convenient to use Young-diagrams (YD) to graphically represent

the different states. To start with, we take the Ground State (GS) as ourpoint of reference, which is represented by an empty diagram. Now, eachexcitation of the rapidity closest to the Fermi Level (FL) is represented byan entry in the first row of the YD. Each excitation of the next rapidityis an entry in the second row, and so on. We always interpret the highestexcited rapidity as coming from directly below the FL; this is even the caseif that level is itself occupied. In that case, we interpret that as a lowerlying rapidity excited to just below the FL. Note that this guarantees thata lower row cannot be longer than a higher one. We can easily translatea YD back to the state it was constructed from: Consider an YD with Rrows whose lengths are given by the numbers ri, i = 1, . . . , R. Since eachrow can contain a at most N − Nr entries, we have 1 ≤ r ≤ N − Nr. The


lowest Nr − R levels are occupied. The next lowest rapidity lies at levelNr −R+ rR, the next one at Nr −R+ rR−1, and so on.

Figure 4.3: A Young-Diagram representation of a state with 8 rapidities (filledcircles) distributed over 16 energy levels. There are 4 particle hole pairs. The leftdiagram represents the particles below the Fermi-level (gray vertical line), the rightthe particles above it.

For a computer implementation it is more convenient to translate eachYD to a number. This can be straightforwardly done. The empty YD getslabeled 0. One single box 1, two boxes in the first row 2, and so on. In thefirst row, there can be a maximum of N −Nr boxes. The next one is a YDwith one box in the first an second row, labeled N −Nr + 1. Continuing inthis fashion, it is easy to associate a number to each YD.

We might be interested in states with a specific number of particle-hole-pairs Nph, i.e. with a specific number of rapidities Nph above the FL. Nph isnot immediately obvious from the Young Diagrams we constructed. Whatwe can do, then, is first specify Nph and then use one diagram to representthe distribution of the Nr − Nph particles over the Nr levels beneath theFL and one diagram to represent the distribution of the Nph holes abovethe FL. See figure 4.3. Another advantage of splitting up the YD in twois that, for larger systems, the number of possible configurations (and thusof different YD) becomes enormous. By splitting the YD we can keep thenumbers labeling them somewhat more manageable.

All in all, we now need three numbers to uniquely label each state. Thenumber of particle-hole pairs, the label of the YD representing the particlesbelow the FL, and the label of the particles above it.

4.2.2 General Structure of the Program

As mentioned before, our strategy will be to obtain solutions iteratively,starting from the known solutions at g = 0. We keep on repeating thefollowing procedure until we have found a solution at our desired value forg.


• First, we check if there are any rapidities that are about to collapse.For this, we use a number of criteria such as their slopes and thedistance to the nearest energy level. Although a single collapse mightseem quite easy to identify, there are a number of pesky problem. See4.2.3 for details.

• If no collapse is to be expected, we will increase g by ∆g and try to findthe corresponding set of rapidities. Because of efficiency, it is usefulput some effort in finding a sensible ∆g. On the one hand, we want tominimize the number of steps we need to take. However, the larger thesteps we make in g, the worse our initial approximation will be. Thiscan either mean that we need (somewhat) more steps in the Newtoniteration, or it can mean that we won’t be able to find a solution at all.Our strategy will be to opt for quite large ∆g and keep monitoring ifthe rapidities are well behaved (see 4.2.4). If we see that the Newtonprocedure fails, we try a smaller step, say ∆g → 1

2∆g . In this way, wecan maximize our step size, while still minimizing unnecessary New-ton iteration steps. Furthermore, these checks drastically reduce thechances of converging on another solution.

• If a collapse is expected, a special routine is called to deal with thecollapse (see 4.2.5). It might happen that, in order for our approxima-tion (4.22) to be sufficiently good, i.e. to be able to use the first orderapproximation of ∆εγ , we need to get extremely close to gc. In thatcase, the (wc1−wc2)−2 terms in the Jacobian become extremely large,and machine precision is not accurate enough anymore. We then calla routine that uses arbitrary precision (see 4.2.6), and therefore, inprinciple, should be able to handle it. The problem is that workingin arbitrary precision is relatively slow, so we will try to avoid this asmuch as possible.

Using this procedure, we should be able to obtain a solution of the Richard-son equations for any given system at any g. However, there are some limita-tions. First of all, we will generally be interested in obtaining a solution for(a representative sample of) the full Hilbert space. This becomes extremelylarge extremely fast. Apart from the obvious problem this poses for com-putation times, it also means that the number of possible different difficultsituations is sheer infinite. Especially since the collapsing, splitting-up andre-collapsing of rapidities can happen in quite complex ways (see Fig. 5.1,bottom panels). We therefore cannot expect our implementation to find asolution for all possible different initial configurations of a system. In generalsomewhere between 95% and 99% should be possible.

There is one more small problem; exactly at g = 0, the Richardsonequations are not well defined. However, we do know that wi(g = 0) ∈{εα} ∀i. We therefore expand the set of equations around g = 0, writing


wi(δg) = εwi + δwi, to obtain

0 =1δg− 1δwi−O(

1εα − εβ

). (4.23)

So, we see thatdwidg

∣∣∣∣g=o

= 1. (4.24)

Provided δg � |εα − εβ|, we can thus use δwi = δg as a first order approxi-mation and expect the Newton procedure to converge.

4.2.3 Identifying Collapsing Rapidities

One of the more important, and surprisingly difficult, aspects in successfullysolving the Richardson equations is to find a set of criteria to identify ifrapidities are about to collapse, and, if so, on what energy level. On facevalue, this seems quite straightforward, for the slopes of the rapidities go toinfinity as g → gc. However, there is a very large variety of conditions underwhich collapses occur. An example is the case where a CCP approachesan energy level from below, crosses it, and then almost immediately “turnsaround” and collapses on it form above. It is difficult to discriminate betweenthis case and a CCP that collapses immediately, especially when trying tokeep the algorithm as general and flexible as possible.

Let us start by defining some obvious criteria for a collapse:

• Two rapidities that are about to collapse should be each others nearestrapidity.

• They should either be each others complex conjugate, or they shouldboth be purely real. That is, an unpaired an a paired rapidity cannotcollapse on each other.

• They should be moving toward each other.

• Their slope should increase.

• For unpaired rapidities, there should be exactly one energy-level be-tween them.

It is possible that more than one set of rapidities meets these criteria. Inthat case, we check which set is expected to collapse first. The criterion weuse is which of the rapidities have the largest slope. Although this will notalways give the correct result, it has the advantage that eventually, whenwe get close enough to collapse, it will always identify the correct pair.


4.2.4 Checks and Step Size

To save time, want to make as large as possible steps in g. However, there isthe risk that our first approximation is not sufficiently accurate, causing theNewton iteration to diverge, or worse, run into another solution. Luckily,this can be dealt with quite easily. The main idea is to try a large step andsee if it works. After every step in the Newton iteration, we check if nothingis going wrong. Signs of something going wrong are:

• An unpaired rapidity crossing an energy level. This should not happensince, as long as it does not collapse onto another rapidity, unpairedrapidities are “trapped” between energy level. Obviously, this is notthe case when an unpaired rapidity is either above the highest or belowthe lowest energy level. In that case, the rapidity will usually go toinfinity as g goes to infinity.

• The imaginary part of a paired rapidity changes sign. This also shouldonly happen when two rapidities collapse. (And even then the imagi-nary part does not change sign, it becomes 0.)

• A rapidity becomes unreasonably large. It is self evident that thisindicates some problem with the Newton iteration.

• The Newton iteration does not converge within a reasonable numberof iteration steps. Obviously, what is reasonable is a bit vague. Expe-rience should teach us this.

There is one other risk; it is possible that the Newton iteration walks into a“wrong” solution, i.e. one corresponding to a different initial state. We canquite easily check for this by checking if the total energy is smooth in g. Wetherefore check if

−1Nr

dEBCS

dg≡ 1Nr

∑i

dwidg

(4.25)

is small enough. Usually this should be somewhere in between 0 and 5. Asa bound on dEBCS

dg we use its previous value, i.e., we check if

− αlowerdEBCS(g)

dg<dEBCS(g + δg)

dg< −αupper

dEBCS(g)dg

(4.26)

where αlower < 1 and αupper > 1. This actually is equivalent to a check ond2EBCSdg2

. We choose αlower = 0.9 and αupper = 1.1, but other choices might bebetter, depending on the system under consideration. Note that this is nota watertight check, but it usually is good enough. Since any set of solutionsshould be unique, when we are sampling the entire Hilbert space we canalways check afterward to see if there are some degeneracies. These canthen be dealt with individually, e.g., by setting αlower and αupper tighter.


4.2.5 Approaching gc

If two rapidities are identified as about to collapse, a function is called todeal with this. To begin with, using (4.10) we try to estimate what gcapproximately is. We now need to get close enough to gc in order for ourapproximation (4.22) to be accurate enough. It is important to keep checkingif the rapidities are actually collapsing, and if there are no other rapiditiesthat collapse sooner. This is because it is a priory not possible to decide forsure what is going to happen. Possibilities are e.g. two rapidities movingtoward each other, getting quite close, and then move away from each otheragain (often to collapse on some other rapidity) or two pairs of rapiditiesthat collapse at almost the same gc, making it very hard to figure out whichones are to collapse first.

If we are close enough to gc, we try to collapse the rapidities. If this fails,we just try to get closer to gc (using the same procedure) and try it again.In principle, we can get arbitrarily close to gc, thus this procedure shouldalways succeed. In practice, however, there will always be situations wherethe implementation fails for one reason or another. When two rapiditiesget so close that machine precision is not accurate enough, and thereforethe Newton procedure becomes numerically unstable, we resort to arbitraryprecision (see section 4.2.6).

4.2.6 Arbitrary Precision

If two rapidities get very close to each other, some terms in the Jacobianget very large, and machine precision is not accurate enough anymore. Wehave to find something more precise. We use the ARPREC package, whichallows us to work in arbitrary precision. In principle, we can now get ar-bitrarily close to any critical point, and we should thus be able to handleany situation. However, we still encounter situations where this does notwork. In these case usually something went wrong with the identificationof collapsing rapidities. We either somehow fail to identify a collapse or weidentify two rapidities as collapsing while in fact they do not (yet). Sincewe will get increasingly close to the point where this happens, we break theroutine if the size of the steps in g will get below some minimal value.

Summary Because the full set of solutions to the Richardson equationscannot be obtained analytically, a numerical approach has to be used. Fortwo reasons, even this is very complicated:

First of all, the number of different solutions is enormous, and all so-lutions are needed to calculate quantities of physical interest. This can bedealt with by constructions all solutions iteratively from the known solu-tions of the free theory. Step by step the interaction strength is increased,at each step using the previous solution as a zeroth order approximation.


The Newton-Raphson method is then used to find the actual solution. Thisis straightforward but, because there are so many solutions, and becauseeach solution takes many iterations, it takes a lot of time,

The second complication are the so called critical points. As the inter-action strength is being increased, points will be encountered where pairs ofrapidities will “collapse” on each other, causing some terms in the Richard-son equations to diverge. These collapses can either be two real valuedrapidities that collapse to form a complex conjugate pair, or it can be a pairthat splits up into two real rapidities. Right at the critical point the equa-tions are not well defined. Fortunately, all physical quantities remain wellbehaved around and at the critical points. This fact can be used to obtainan approximation of the rapidities after the collapse. It might be that thisapproximation is not accurate enough for the Newton-Raphson method toconverge. In that case, we need to find a solution closer to the critical point.

In this way, it is in principle possible to obtain all solutions. In practice,however, there will always be situations where the implementation somehowfails. Also, the number of solutions becomes extremely large for systems ofmore than about ten rapidities. In the next section we will discuss someresults, mainly to test the program, and to begin building some intuition forthe general behavior of the rapidities.

5Results

We have built a computer program in order to numerically solve the Richard-son equations. In this chapter we will discuss some first results. Althoughwe are not yet in a position to obtain physically relevant quantities, it isnonetheless useful to consider some of the results we obtained. First of all,it is good to double-check whether or not the solutions we obtain make anysense. Furthermore, it is of course interesting to test the performance ofthe program. Finally, to calculate actual correlation functions of reasonablysized systems, it will not be practical to solve the Richardson equations forall initial configurations. We therefore will need to built some intuition onthe general properties of the solutions, so that hopefully a small sample ofall solutions will be sufficient.

5.1 Some Individual States

Let us start building intuition for the behavior of the rapidities by consid-ering the evolution in g of some configurations. The top panels of figure 5.1show the rapidities of the ground state, i.e. the state with only the highestNr levels filled at g = 0. The reason that this is the ground state is thatthe larger the rapidities are, the smaller (central spin model), or the morenegative (BCS model), their contributions to the total energy are. We usedN = 32, Nr = N/2 and εα = 2 exp(α/N). As we increase g, at some pointthe lowest two rapidities will collapse. Further increasing g, the two nextlowest collapse, and so on, until all rapidities form complex conjugate pairs(CCP). The real as well as the imaginary part of the rapidities increasesapproximately linearly in g. This is qualitatively the same behavior as in

41

5.2. The Full Hilbert Space 42

the case of homogeneous level spacing (not plotted). The ground state isthe only state for which all rapidities go to infinity.

In general, it need not be the case that all rapidities form CCPs. In fact,if Nr is odd, this is not possible. Furthermore, it also is very well possiblethat some rapidities stay confined between the energy levels, while others gooff to infinity. In general, a complex sequence of fusing and splitting up ofrapidities leads to a some configuration at g →∞, and the relation betweenthe configuration at g = 0 and g � 1 is highly nontrivial (middle panels).To complicate things even further, for some states, there are points whereat almost (but not exactly) the same value for g, many collapses occur (seeFig. 5.1, bottom panels).

5.2 The Full Hilbert Space

To check the performance of the implementation, and to build intuition, itis useful to scan the entire Hilbert space for a small system. We used asystem of N = 14 energy levels at half filling, i.e., Nr = 7. We checked itfor two different distributions of the energy levels. Besides the exponentialdistribution discussed in 2.3, εα = 2 exp[α/N ], we also considered equallyspaced energy levels: εα = 2 + 0.1α.

5.2.1 Performance

It is not realistic to expect that our implementation will find a correct so-lution in all cases. When we scan over the entire Hilbert space, there willtherefore be some states for which no correct solution is obtained. Basi-cally, there are two possibilities of what can go wrong. First, the algorithmmight get stuck at some point, e.g. a particularly complex collapse. Also,the Newton-Raphson method might converge to a “wrong” solution. Webuilt in checks against this (see section 4.2.4), but these are not watertight.However, because each solution is unique, we can always check afterward ifthere are any degeneracies.

A system of 14 energy levels at half filling has(147

)= 3432 distinct solu-

tions. For the system with exponentially spaced energy levels, the programfailed in 11 instances. Furthermore, 6 pairs of solutions were the same. Ofeach of these pairs, we assume that one of the two solutions is correct. Thismeans that in total 17 solutions are missing. We thus have 99.6% of allsolutions. For the system with homogeneous level spacing, the performancewas somewhat less good. It failed in 142 cases and there where 7doublesolutions, meaning 95.6% of all solutions were obtained.

The reason why the performance for the homogeneous case is signifi-cantly less good is not entirely clear. There are reasons to believe thatfor homogeneous level spacing the situations where many rapidities collapsearound the same value of g are somewhat more complex, in the sense that


3

3.5

4

4.5

5

5.5

6

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Re(w

)

g

-3

-2

-1

0

1

2

3

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Im(w

)

g

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Re(w

)

g

-6

-4

-2

0

2

4

6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Im(w

)

g

2

3

4

5

6

7

8

9

10

11

12

0 1 2 3 4 5 6

Re(w

)

g

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

Im(w

)

g

Figure 5.1: Real part (left panels) and imaginary part (right panels) of the ra-pidities {wi} as a function of g, for different initial configurations. The top tworows are systems of N = 32 energy levels and Nr = 16 rapidities. The bottomrow shows a system of N = 14 and Nr = 7. For all systems, the level spacingof the energies was exponential, i.e, εα = 2 exp(α/N). For the ground state (toppanels), at g = 0 the highest Nr levels are occupied, as g increases, first the lowesttwo rapidities will form a CCP, then the next lowest two, and so forth, until allrapidities (Nr even), or all but one (Nr odd, not shown), form CCPs. For a typicalstate, the forming and splitting up of CCPs is somewhat more complicated (middlepanels). Some states show particularly complex behavior; in a very short intervalin g, many collapse happen (bottom panels). Typically, a number of CCPs splitsup and re-collapses in another configuration.


the interval in g where this happens is even smaller. Another possibility isthat we mainly used exponential level spacing when developing the program,and that therefore some values are more optimal for this case.

5.2.2 Distributions

Figure 5.2 shows a scan of the full Hilbert space, at different values of g. Theplots show the locations of the rapidities in the complex plane. We plottedall rapidities of all states. These plots thus consist of

(NNR

)sets of solutions,

each of which consists of Nr rapidities, making a total of 7 ×(147

)≈ 2.44

data points. It does not show which rapidities belong together. However, itteaches us a lot about overall restrictions on, and properties of, the rapidities.All plots are for exponentially spaced energy levels, except for the bottomright panel, which for homogeneous spacing.

At g = 0 (top left panel) we clearly see that the rapidities are all equalto the energy levels, i.e., wi ∈ {εα}. There therefore appear to be only 14data points.

Increasing the coupling constant (g = 0.25, top right panel and g = 1,middle left panel), we see that rapidities begin to form CCPs. Arcs betweenthe energy levels begin to form and some rapidities move away from theenergy levels. The symmetry with respect to y=0 is because all rapiditiesare either real or form a complex conjugate pair with another rapidity.

At very large g as compared to the average level spacing (g = 1000,middle right panel), we see that some rapidities are moving away from theenergy level (approximately) linearly in g. Moreover, we see that they onlygo to certain specific points in the complex plane. We will studies thissomewhat more in depth in section 5.3.

If we zoom in on the rapidities that stay around the energy levels forlarge g, we see a lot of structure (bottom panels). The rapidities that goto infinity become almost equal for the exponential and the homogeneoussystem (not shown). The rapidities that stay finite behave qualitatively inthe same way. Obviously, exactly at the εs, there are no rapidities, but wesee that arcs have formed between them. Note that a quite a lot of therapidities are complex conjugate pairs. Although a CCP, in contrast to apurely real rapidity, could cross the energy levels and go off to infinity, aconsiderable number of them stay “trapped” in between the energy levels.The details of the energy levels thus remains important at large g. Wenote that no CCPs appear in between the lowest two and in between thehighest two energy levels, the reason and physical interpretation of this isstill unclear. One difference between the exponentially and homogeneouslyspaced systems is that the latter is almost symmetric with respect to themiddle of the energy levels. The reason for this is a particle-hole symmetryat g = 0 that somehow is preserved in the large g limit. We did not studythe details of this, but we point it out because these kinds of symmetries


-1

-0.5

0

0.5

1

2 2.5 3 3.5 4 4.5 5 5.5

Im(w

)

Re(w)

-3

-2

-1

0

1

2

3

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

Im(w

)Re(w)

-15

-10

-5

0

5

10

15

2 4 6 8 10 12 14

Im(w

)

Re(w)

-15000

-10000

-5000

0

5000

10000

15000

0 2500 5000 7500 10000

Im(w

)

Re(w)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2 2.5 3 3.5 4 4.5 5

Im(w

)

Re(w)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2 2.2 2.4 2.6 2.8 3 3.2 3.4

Im(w

)

Re(w)

Figure 5.2: A scan if the full Hilbert space for a system of N = 14 energy levelsat half filling. The energy level distribution is exponential, i.e. εα = 2eα/N , exceptfor the bottom right panel, where it was homogeneous: ε+ α = 2 + 0.1i. At g = 0,all wi ∈ {ε} (top left). As g increases, some rapidities move away from the εswhile others remain relatively close to them (top right, middle). There is a lot ofstructure in the rapidities that stay finite (bottom). The homogeneously spacedsystem appears to be symmetric with respect to the middle of the energy levels(bottom right)

5.3. Large g Limit 46

are know to cause numerical instabilities when calculating correlators forsimilar models.

Summarizing, the general behavior of the rapidities is as follows. Atg = 0, all rapidities are strictly equal to an energy level. As we increase gCCPs will begin to form, as well as some structure in the rapidities aroundthe energy levels. As g gets large in comparison with the average levelspacing, some rapidities will stay near the energy levels, while others willgo to infinity linearly in g. This will be in a highly non random way (seesection 5.3).

5.3 Large g Limit

In the g → ∞ limit, i.e. the h → 0 limit, the Central Spin Hamiltonianreduces to

HCS =N∑α=1

JαSo · Sα, (5.1)

which has global su(2) invariance. This implies that[HCS, S

±tot

]= 0, (5.2)

meaning that if |ΨNr〉 is an eigenstate of Hcs, so is S−tot|ΨNr〉. We can thusgo to a different spin representation of the system. It is known that in thislimit the rapidities either stay in the order of ε or go to infinity linearly.In [28], the authors derive an algorithm that gives for each configuration atg = 0 the number of rapidities that stay finite as g goes to infinity. Therapidities that go to infinity decouple, and we can therefore divide the statesinto two sectors

|ΨNr〉 = |{wi}Nfr

; {∞}N∞r 〉 (5.3)

where Nfr is the number of rapidities that stay finite and N∞r the number

of them that go to infinity. Since the latter decouple, this state can be builtby applying the raising operator on a state with Nf

r rapidities, i.e.,

|ΨNr〉 =(S−tot

)N∞r |{wi}Nfr〉. (5.4)

In this representation, the state with Nfr = Nr = N/2 (i.e. N∞r = 0) is the

highest weight state, and the states with N∞r 6= 0 form multiplets. A morephysical interpretation of the decoupled rapidities remains difficult.

However, it teaches us quite a lot about the behavior of the solutions.First of all, it implies that subsets of solutions of different systems coincide,at least for the finite rapidities. The subset of states of a systemN∗r rapiditiesand N∞r = 0 and a system with N∗r + n rapidities and N∞r = n have thesame solutions for their finite rapidities. Figure 5.3 shows the overlap of twosets of solutions for a system of N = 14 exponentially spaced energy levels,


-1.5

-1

-0.5

0

0.5

1

1.5

2 2.5 3 3.5 4 4.5 5 5.5

Im(w

)

Re(w)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

3.4 3.45 3.5 3.55 3.6 3.65 3.7

Im(w

)

Re(w)

Figure 5.3: Overlap of the solutions to a system with Nr = 7, N∞r = 1 (+) andone with Nr = 6, N∞r = 0 (×) at g = 100. We used N = 14 levels εα = 2 exp(α/N).Only the rapidities near the εs are shown (right panel). Clearly, the rapidities areall (almost) equal. This is especially clear if we zoom in (right panel).

i.e. εα = 2 exp(α/N), with Nfr = 5. The red points correspond to the finite

solutions for Nr = 7, N∞r = 1 and the green points to the solutions forNr = 6, N∞r = 0. Clearly, the match is very good. We can thus concludethat, at large g, the solutions for the finite rapidities only depend on theenergy level and Nf

r , and are independent of N∞r . This means that if wehave all solutions of a system for Nr = N/2, we automatically know all thesolutions (for the finite rapidities) of the same system with Nr < N/2 aswell. Furthermore, we can now also easily deduce how many solutions Nsol

there are for N∞r :

Nsol(N∞r = n) = Nsol(N∞r ≥ n)−Nsol(N∞r > n)

=(

N

Nr − n

)−(

N

Nr − n− 1

),

(5.5)

i.e. it is the number of solutions for a system of Nr −n rapidities minus thenumber of one with Nr − n − 1 rapidities. Note that, although at large gthe solutions are the same, they are not at small g. If we start constructingthe states from g = 0, generally, the way a state with Nf

r = n and N∞r = mrapidities “arrives” at a certain solution in quite a different way than a statewith Nf

r = n and N∞r 6= m.As mentioned earlier, the rapidities that go to infinity will form an arc

in the complex plane. This is clearly visible in figure the left panel of figure5.4.

If we consider the behavior of the N∞r that go to infinity, we can ap-proximate

wi ={zig i = 1, . . . , N∞rwi = 0 others

(5.6)


-1500

-1000

-500

0

500

1000

1500

250 500 750 1000

Im(w

)

Re(w)

2 4 6

10 12

18 20

28 30

40 42

54 56

0 1 2 3 4 5 6 7

∑i z

i

Nr

∞

Figure 5.4: The left panel shows the rapidities that go to infinity linearly in gfor a system of N = 14 exponentially spaced energy levels and Nr = 7 at g = 100.Each color represents solutions with specific N∞r , between 1 (black dots) and 7 (reddots). We see that these rapidities form an arc in the complex plane. The locationof the arc depends only N∞r , Nr and N , and not on the details of the energy levels.The right panel shows the sum of the slopes of all rapidities for a system withN = 16, Nr = 8 (+) and Nr = 6 (×), at g=1000, for all states. Clearly, the datamatches equation (5.9) extremely well.

The N∞r Richardson equations for these rapidities then become

0 =1g

1− N

zi+

2Nfr

zi+

N∞r∑k 6=i

2zi − zk

. (5.7)

If we sum over these, we see that the antisymmetric last term drops out, sowe are left with:

N∞r∑i=1

1zi

=N∞r

N − 2Nfr

. (5.8)

In the BCS model, dEdg is given by -

∑N∞ri=1 zi. Although this is not clear

from the equations, our numerical solutions show that this always will bean integer value. The sum of the slopes of the rapidities is given by

N∞r∑i=1

zi = N∞r (N∞r +N − 2Nr + 1)

= N∞r

(N −N∞r − 2Nf

r + 1).

(5.9)

The right panel of figure 5.4 shows the sum of the slope of rapidities for asystem with N = 16 exponentially distributed energy level, Nr = 8 (+) andNr = 6 (×). They fit equation (5.9) extremely well. Note that the datasets in this plot contain 8×

(168

)≈ 105 respectively 6×

(166

)≈ 5× 104 data

points. This of course is no actual proof, but rather a strong hint that (5.9)is indeed true. Why this is the case, we do not yet know. A possibility is


that there is a mapping between the g →∞ limit of the pairing model andthe free limit of some other model, and that these energies map onto theelementary excitations of that model.

Summary In order to calculate correlation functions of the central spinmodel, all its eigenstates need to be known. These can be obtained usingthe Algebraic Bethe Ansatz, which involves solving the Richardson equa-tions. There is no general analytic solution to them, so we must resort to anumerical approach.

The numerical implementation discussed in chapter 4 is able to deal withcomplicated situations. As was to be expected, a scan of the full Hilbertspace shows that it will not be successful in all situations. However the vastmajority of the solutions, between 95% and 99%, can be obtained by thisprogram.

For large values of the coupling constant g, some rapidities will staybetween the energy levels, while others will go to infinity linearly with g.The rapidities that stay finite do not “feel” the infinite rapidities in the sensethat the finite solutions only depend on the energy levels and the other finiterapidities. On the other hand, the solutions that go to infinity depend ontotal number of rapidities as well as the number of energy levels. It turnsout that the total slope of the infinite rapidities will always be an integer.The physical interpretation of this, if there is any, is not yet clear.

6Conclusion and Prospects

For this project, we set out to write a numerical implementation that solvesthe Richardson equations for an arbitrary distribution of the energy levels.The reason we want to do this is because we need them to construct theeigenstates of the central spin Hamiltonian. These in turn are needed tocalculate the decoherence function, which describes the decoherence of anelectron spin in a quantum dot, due to hyperfine interactions with nuclearspin states in its environment.

Because the program had to be as flexible as possible, we used a some-what unusual way to deal with the critical points. Instead of performingsome change of coordinates on the collapsing rapidities, we only used thefact that their sum remains smooth at the critical point. This means that,in some situations, we need to get very close to the critical point. If, in thesesituations machine precision is not good enough, we can resort to routinesthat work in arbitrary precision. The main drawback of this is that theseroutines tend to be much slower. Although this method might be consideredless elegant, it is much more simple, making it more suitable to deal withmany different situations.

In this way, it is possible to obtain the full set of solutions to the Richard-son equations for systems up to about ten rapidities at half filling. For largersystems, the number of solutions simply becomes too large. Although thisprocedure will still fail every now and then, it did find the vast majority(95% to 99%) of the solutions for the different systems it was tested on.When our implementation fails, it is mainly due to problems with the iden-tification of the collapsing rapidities. If the performance of the programneeds to be improved, we would recommend starting here.

We made some interesting observations about the behavior of the ra-

50

51

pidities at large g. In this limit, the rapidities can be divided in those thatincrease linearly in g and those that stay in between the energy levels. Thelatter do not “feel” the former, in the sense that their solutions only dependon the other rapidities that stay finite and the energy levels. As a conse-quence, if we have all solutions for a system of Nr rapidities, we can easilyconstruct all finite rapidities for systems with N∗r < Nr rapidities. Theseare then given by the finite solutions of the system with Nr rapidities andn ≥ Nr −N∗r , where n is the number of rapidities that go to infinity. Fur-thermore, for the rapidities that go to infinity, it appears that sum of theirslopes is always equal to an integer that only depends on n and Nr. Whythis is the case is not yet clear.

The next step from here would be to start calculating form factors ofsmall systems. This will hopefully help gain insight into which pairs ofstates have relevant contributions to correlators and which do not. Forlarger systems then, a smart sample of the full set of solutions need to bepicked; the results obtained for smaller systems might help establish what asmart sample would be.

Now, in this fashion, it might be possible to go up to systems of a coupleof dozen of spins. Obviously, this still is very small compared to the numberof nuclei the electron actually interacts with. Here, again, the hope is that arepresentative sample of the spins will suffice to capture the essential physicsof the system.

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[2] D. Loss and D. P. DiVincenzo. Quantum Computation with QuatumDots. Physical Review A, 57(1):120–126, 1998.

[3] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K.Vandersypen. Spins in few-electron quantum dots. Review of ModernPhysics, 2007.

[4] F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. Willemsvan Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwen-hoven, and L. M. K. Vandersypen. Control and Detection of Singlet-Triplet Mixing in a Random Nuclear Field. Science, 2005.

[5] I. T. Vink, K. C. Nowack, F. H. L. Koppens, J. Danon, Y. V. Nazarov,and L. M. K. Vandersypen. Locking electron spins into magnetic res-onance by electron-nuclear feedback. Nature Physics, 2009, AdvancedOnline Publication.

[6] T. Fujisawa, D. G. Austing, Y. Tokura, Y. Hirayama, and S. Tarucha.Allowed and forbidden transitions in artificial hydrogen and heliumatoms. Nature, 419:278–281, September 2002.

[7] W. A. Coish and D. Loss. Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics. Physical Review B, 70(19):195340,November 2004.

[8] R. W. Richardson and N. Sherman. Exact eigenstates of the pairing-force hamiltonian. Nuclear Physics, 52:221 – 238, 1964.

[9] M. C. Cambiaggio, A. M. F. Rivas, and M. Saraceno. Integrabilityof the pairing hamiltonian. Nuclear Physics A, 624:157–167, February1997.

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[12] A. V. Khaetskii, D. Loss, and L. Glazman. Electron spin decoherence inquantum dots due to interaction with nuclei. Physical Review Letters,88(18), May 2002.

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[14] J. Bhaseen. Working notes: Decoherence in quantum dots. Privatecommunication.

[15] M Bortz and J Stolze. Spin and entanglement dynamics in thecentral-spin model with homogeneous couplings. Journal of Statisti-cal Mechancis: Theory and Experiment, 6:18, June 2007.

[16] M Bortz and J Stolze. Exact dynamics in the inhomogeneous central-spin model. Physical Review B, 76(1):014304, July 2007.

[17] H.-Q. Zhou, J. Links, R. H. McKenzie, and M. D. Gould. Superconduct-ing correlations in metallic nanoparticles: Exact solution of the BCSmodel by the algebraic Bethe ansatz. Physical Review B, 65(6):060502,February 2002.

[18] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin. Quantum InverseScattering Method and Correlation Functions. Cambridge UniversityPress, 1993.

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The Central Spin Problem and the Richardson Equations