The zero-energy ground states of supersymmetric lattice models

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The zero-energy ground states of supersymmetric

lattice models

Ruben La

July 10, 2018

Bachelor thesis Mathematics and Physics & Astronomy Supervisor: prof. dr. Kareljan Schoutens, prof. dr. Sergey Shadrin

1

Institute of Physics

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

This thesis focusses on studying the ground states of a few supersymmetric lattice models: the one-dimensional zig-zag ladder [Hui10], the Nicolai model [Nic76] and the Z2 Nicolai model

[SKN17], and the two-dimensional triangular lattice model [HMM+_{12]. The main objective is}

to derive the ground state generating function of these models. Furthermore, we aim to study the behaviour of the ground states at different filling fractions and to give expressions for a few of the ground states.

The first part of the thesis introduces some of the preliminary concepts from homology theory that will be used to count the number of ground states. In particular, we formulate the homo-logical perturbation lemma [Cra04], which is one of the main ingredients for the computation of the number of ground states. Afterwards, the notion of supersymmetric lattice models will be introduced and some of its key features shall be discussed, especially the features concerning the ground states. In particular, a strong relation between the ground states of supersymmetric lattice models and the homology of the supercharge shall be discussed. Using techniques from homology theory, we can derive relations for the ground state generating functions of the lattice models mentioned above. For the zig-zag ladder, we shall give an expression in closed form for the ground state generating function, for which a recurrence relation was already known [Hui10]. For the Nicolai model, the Z2 Nicolai model and the triangular lattice model on a 3 × N lattice

with vertical periodicity and open horizontal boundaries, we shall derive a recurrence relation for the ground state generating function, as well as a closed formula for the Z2 Nicolai model

and the 3 × N triangular lattice model. Using these results about the generating functions, we shall study the ground states at different filling fractions and derive a few physical quantities such as the expectation of the filling fractions and the filling fraction for which the number of ground states is largest. Finally, we shall give expressions for some of these ground states. For the zig-zag ladder, we will present a few expressions for the ground states at quarter filling and for the Nicolai models, we shall identify all the classical ground states.

Title: The zero-energy ground states of supersymmetric lattice models Authors: Ruben La, ruben.la@student.uva.nl, 11039582

Supervisors: prof. dr. Kareljan Schoutens, prof. dr. Sergey Shadrin Second grader: prof. dr. Bernard Nienhuis, prof. dr. Jasper Stokman End date: July 10, 2018

Institute of Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1. Introduction

1_{In condensed matter physics, one of the main goals is to understand the physical properties of}

materials. One of the challenges that a condensed matter physicist encounters is that systems can become complex very quickly when the number of particles in the system is large. The physicist is then forced to make simplifications in order to be able to understand anything about the system. One such simplification is a lattice model. A lattice model is a physical model defined on a lattice that describes many-particle systems and their interactions. The particles of the systems are confined to the lattice sites and are not allowed to move freely in the continuum of space. Studying quantum many-body systems on a lattice was originally motivated by the tight-binding appoximation, which describes valence electrons in a crystal by approximating the electron orbitals by the atomic orbitals of the atoms involved. The associated binding model thus forms a lattice model, with the ions as its lattice sites. The tight-binding model is one of the simplest lattice models, as it does not take the interaction between the electrons into consideration. As such, the tight-binding model does its job quite well in describing materials in which the interactions between the electrons are weak. However, there exist materials for which the valence electrons are strongly interacting. Therefore, we want to have a model that includes the electrostatic repulsion of the electrons. One such model is the Hubbard model. The Hubbard model is described by a Hamiltonian consisting of a nearest-neighbour hopping term parametrised by t and an on-site interaction for electrons with opposite spin parameterised by some interaction strength U . Although the Hubbard model seems very simple, some of its properties are not yet completely understood. For instance, as mentioned in [Hui10], a precisely half-filled system with strong interactions U ≥ t describes a ferromagnet. However, upon doping away from half-filling, the anti-ferromagnetic order is destroyed. This regime of the Hubbard model is not well-understood and is one of the open problems concerning the Hubbard model.

One can propose a simpler model than the Hubbard model which still includes the strong interactions between the electrons. A very succesful type of lattice models was introduced in [FSdB03]: N = 2 supersymmetric lattice models. In a nutshell, supersymmetry is a theory originating from particle physics that describes a symmetry between fermions and bosons. The theory states that each elementary fermion forms a pair with a boson, and vice versa. These two particles together are called superpartners. Using this idea, one can introduce the theory of su-persymmetric lattice models, where the supersymmetry relates a many-particle state consisting of an odd number of lattice fermions (a fermionic state) with a many-particle state consisting of an even number of lattice fermions (a bosonic state). Supersymmetric lattice models are equipped with two operators Q and Q†_{, called supercharges, with the following properties. The}

operator Q (Q†) removes (adds) one fermion from (to) the system, or the other way around. Both operators square to zero and the Hamiltonian of the system reads H = {Q, Q†}. Su-persymmetric lattice models have a few key features. First of all, the Hamiltonian is positive definite, with which we mean that all the eigenenergies are non-negative. Furthermore, we can

1_{The information in [Hui10, Introduction] was used as a source for the historic overview of the lattice}

mod-els (the tight-binding model, the Hubbard model and the supersymmetric lattice modmod-els) presented in our Introduction.

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choose each fermionic energy eigenstate |f i with strictly positive energy in such a way that it forms a doublet with its superpartner |gi = Q|f i, which is a bosonic energy eigenstate whose particle number differs from the original state by precisely 1. The last feature is the most important one and relates the ground state structure to the homology of Q. It is interesting to study the ground state structure of the system, as it allows us to study and derive some interesting physical quantities. For instance studying the degeneracy of the ground states has many implications for the thermodynamics of the system as well as for other properties such as the transport of charge. Furthermore, as shown in [Mor16], there is a connection between the ground states and ergodicity breaking, which in turn is related to many-body localisation, which is one of the current hot topics in condensed matter physics. The last key feature of supersym-metric lattice models is that there exists an explicit isomorphism between the homology of the supercharge Q and the kernel of the Hamiltonian. This allows us to use results from homology theory to study the ground state structure.

1.1. Organisation of the thesis and main goals

Studying the ground state structure of supersymmetric lattice models is the main topic of this thesis. In order to study the ground state structure, we shall first establish the preliminaries needed to perform the required calculations. First, the thesis introduces some basic theory from homology theory [ML63]. We then proceed by introducing the homological perturbation lemma [Cra04], which is one of the main tools that will be used to calculate the homology of the supercharge. Afterwards, we will introduce the main concepts from supersymmetric lattice models for fermionic particles as in [Hui10] and discuss the main features of supersymmetric lattice models in detail. In particular, we shall rigorously establish the relation between the homology of the supercharge and the kernel of the Hamiltonian, the subspace containing all the ground states.

Having established the preliminaries, we proceed to the main goals of the project. The main objective of this project is to find the ground state generating function for a few particular supersymmetric lattice models. The ground state generating function is a power series (or a polynomial for supersymmetric lattice models with finite dimensional state spaces) for which each nth_{coefficient is the number of linear independent zero-energy}2_{ground states with fermion}

number n. The lattice models considered in this thesis are the one-dimensional zig-zag ladder [Hui10, Chapter 7], the Z2 Nicolai model [SKN17], the Nicolai model [Nic76], and the

two-dimensional triangular lattice model [vE05][GHP12]. We shall explicitly derive the recurrence relations for the ground state generating function of the Nicolai model, the Z2 Nicolai model,

and the triangular lattice model for the 3 × N lattice. The derivation of these results are presented in a ‘mathematical style’, as it is the main part for the mathematical component of the project. Building upon our results about the ground state generating function, we shall study a few of the many interesting physical properties of the models; one of our next goals is to study the ground states for different filling fractions. We shall also provide references to some recent articles that discuss the interesting related physics. Finally, the last goal of this project is to give the expressions for some of the ground states at different filling fractions. In particular, for the zig-zag ladder we give explicit expressions for the ground states at quarter filling and for the Nicolai models we identify all the classical states.

In the appendix, we shall discuss the theory of spectral sequences [ML63]. Initially, our idea was to use this theory to find the number of ground states. We came close to a proof, but this

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approach required checking a large number of details. Eventually, we abandoned this approach, as at some point, we discovered that the homological perturbation lemma gives a much more elegant and shorter proof.

1.2. Acknowledgements

I would like to thank prof. Kareljan Schoutens and prof. Sergey Shadrin for supervising me during this bachelor’s project. Many of the results presented in this thesis are thanks to great insights from both supervisors. For me, it has been a great experience to work within this field of research and to work on open problems in theoretical and mathematical physics. The various meetings and conversations have taught me a great deal and gave me a fairly good idea about what it is like to do research in mathematics and theoretical physics.

I would also like to thank Mike Daas, one of my fellow students, for helping me revise a few particular concepts from linear algebra.

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2. Preliminaries

The purpose of this chapter is to introduce the preliminary mathematics needed to understand the thesis and to formulate the homological perturbation lemma [Cra04], which we shall later use for computing the number of ground states for the supersymmetric lattice models. Throughout this chapter, it is assumed that the reader is familiar with undergraduate linear algebra, group theory and basic theory of modules over a ring.

The definitions and results in this chapter all involve modules. However, the only modules that we work with in the rest of this thesis are C-modules , i.e. C-vector spaces. Nevertheless, we state all the definitions and results in this chapter for modules instead of C-vector spaces, as they are presented in such a way in the literature.

2.1. Homology and cohomology

The notions of homology and cohomology were originally introduced in algebraic topology. It turned out to be interesting to study homology and cohomology in a purely algebraic setting, giving rise to homology theory. In this section, we shall introduce basic homology and coho-mology theory [ML63][Wei95].

Definition 2.1. Let R be a ring. A graded R-module is a family M = (Mn)n∈Z of R-modules

Mn.

Definition 2.2. Let R be a ring and (Mi₎

i∈Za collection of graded R-modules Mi= (Mni)n∈Z.

We define the direct sum L

i∈ZMi to be the graded R-module

M

i∈Z

Mi= (M

i∈Z

M_ni)_n∈Z.

Definition 2.3. Let R be a ring and let M and N be graded R-modules. A morphism of graded R-modules is a family f = (fn)n∈Z of morphisms of R-modules fn: Mn → Nn. We say that f

is an isomorphism of graded R-modules if fnis an isomorphism of R-modules for all n ∈ Z. We

say that the isomorphism of graded R-modules f−1:= (f_n−1) is the inverse of f .

Definition 2.4. Let R be a ring. A chain complex (K•, ∂) of R-modules consists of a graded

R-module K• and a collection ∂ = (∂n)n∈Z of R-module homomorphisms ∂n: Kn→ Kn−1such

that ∂n∂n+1 = 0. Abusing notation, we usually write ∂ = ∂n: Kn→ Kn−1.

Definition 2.5. Let K• be a chain complex. The homology H•(K•) of K• is defined to be the

graded module H•(K•) = (Hn(K•))n∈Z = (ker ∂n/im ∂n+1)n∈Z. The module Hn(K) is called

the nth _{homology module} _{of K} •.

Definition 2.6. Let R be a ring. A cochain complex (K•_{, ∂) of R-modules consists of a graded}

R-module K• = (Kn_{) and a collection ∂ = (∂}

n)n∈Z of R-module homomorphisms ∂n: Kn →

Kn+1 such that ∂n∂n+1 = 0. We shall usually write ∂ : Kn → Kn−1, omitting the subscript n

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Definition 2.7. Let K• _{be a cochain complex. The cohomology H}•_(K•_{) of K}• _{is defined to}

be the graded module H•_(K•_{) = (H}

n(K•)) = (ker ∂n/im ∂n−1). The module Hn(K•) is called

the nth _{cohomology module} _{of K}•_.

In homology theory, there is essentially no conceptual difference between the homology of a chain complex and the cohomology of a cochain complex. Given a chain complex (K•, ∂),

one naturally obtains a cochain complex by redifining the indices: Kn _{= K}

−n. Nevertheless,

it turns out that the indices of the (co)chain complexes that are studied in this thesis have a certain physical interpretation, hence it is convenient to know both definitions.

For each of the following definitions and results for chain complexes, there is a natural cor-responding ‘codefinition’ and ‘coresult’ for cochain complexes, which we shall not formulate in this thesis.

Definition 2.8. Let (K•, ∂) and (K•0, ∂0) be chain complexes. A chain map or a homomorphism

of chain complexes f: K• → K•0 is a family of module homomorphisms fn: Kn → Kn0, such

that ∂_n0 ◦ fn= fn−1◦ ∂n for all n ∈ Z. Abusing notation, we usually write f = fn: Kn→ Kn0 for

each n ∈ Z. If fn is an isomorphism for each n ∈ Z, we say that f is an isomorphism of chain

complexes with inverse f−1 _{= (f}−1

n )n∈Z. Furthermore, for each n ∈ Z, we have a well-defined

morphism of modules

Hnf: HnK• → HnK•0: c → fn(c).

The family H•f := (Hnf)n∈Z is called the graded module homomorphism induced by f .

Note that an isomorphism of chain complexes induces an isomorphism on the homologies. In other words, if two chain complexes are isomorphic, then their homologies are also isomorphic.

One can also consider direct sums of chain complexes. Proposition 2.1. Let ((Ki

•, ∂i))i∈I be a family of chain complexes. Then

M i∈I (K•i, ∂i) := ( M i∈I K•i, M i∈I ∂i) is a chain complex, where

M i∈I K_•i = (M i∈I K_ni)_n∈Z and M i∈I ∂i = (M i∈I ∂_ni)_n∈Z. Moreover, if (fi

•: (K•i, ∂i) → (Li•, di))i∈I is a collection of chain maps, then we have a chain map

M i f•i = ( M i f_ni)_n∈Z. Proposition 2.2. Let ((Ki

•, ∂i))i∈I be a family of chain complexes. Then we have an

isomor-phism of graded vector spaces H•( M i∈I K•i, M i∈I ∂i) ∼=M i∈I H•(K•i, ∂i). Definition 2.9. 1 _{Let (K}

•, ∂) be a chain complex and (L•, d) a cochain complex and fix k ∈ Z.

The chain complex (K[k]•, ∂[k]) and the cochain complex (L[k]•, d[k]) are defined by

K[k]n:= Kn−k and ∂n[k]:= (−1)k∂n−k.

L[k]n:= Ln+k and dn_[k]:= (−1)kdn+k.

1

The definition for chain complexes is taken from [nLa18]. The definition for cochain complexes is adapted from this definition, but note that both definitions are well-known in the literature.

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2.2. Homological perturbation lemma

The homological perturbation lemma is a very useful computational tool from homology theory. In this thesis, we shall mainly be concerned with one specific use of the homological perturbation lemma, which is to establish an isomorphism between the homologies of two (co)chain complexes. One version of the homological perturbation lemma is presented in [Cra04], called the main perturbation lemma, which we simply shall call the homological perturbation lemma.

Definition 2.10. Let (K•, ∂), (K•0, ∂) be chain complexes and f : K• → K•0 a chain map. We say

that f is a quasi-isomorphism or a q.i. if the graded module homomorphism H•(f ) : H•(K, ∂) →

H•(K0, ∂) induced by f is an isomorphism. Similarly, we define a q.i. of cochain complexes.

Definition 2.11. Let f, g : (K•, d) → (K•0, d0) be chain maps. A homotopy h from f to g is a

collection h = (hn: Kn→ Kn+10 )n∈Z of module homomorphisms such that

gn− fn= d0n+1hn+ hn−1dn,

for all n ∈ Z. We say that f and g are homotopic if there exists a homotopy from f to g. Usually, we write this condition as g − f = dh + hd. Similarly, we define a homotopy for cochain maps.

Definition 2.12. A homotopy equivalence data or HE data consists of 1. two (co)chain complexes (L, d), (M, ∂);

2. quasi-isomorphisms i : L → M and p : M → L;

3. a homotopy h between ip and the identity 1, i.e. a ip = 1 + ∂h + h∂. We usually represent a HE as a diagram:

(L, d) (M, ∂) i

p

h

Theorem 2.1 (Homological perturbation lemma). Consider the following HE data: (L, d) (M, ∂)

i p

h

and suppose δ is a differential on M (i.e. (M, δ) is a (co)chain complex) such that (∂ + δ)2_{= 0}

and such that (1 − δh) is invertible (an isomorphism of graded modules). The homomorphism δ is called a small perturbation. Now let A = (1 − δh)−1_δ _{and define}

i1 = i + hAi, p1 = p + pAh, h1 = h + hAh, d1 = d + pAi.

Then (L, d1) (M, ∂ + δ) i1 p1 h1 (2.1) is a HE data.

For a proof, we refer to [Cra04, Main Perturbation Lemma]. In this thesis, we are usually only interested in the existence of a quasi-isomorphism between the two (co)chain complexes, and not so much in the isomorphisms p1 and i1 themselves.

Corollary 2.1. Consider the HE data (2.1) in Theorem 2.1. Then we have an isomorphism H(L, d1) ∼= H(M, ∂ + δ) of graded modules (where H stands for H• or H•).

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3. Supersymmetric lattice models for

fermionic particles

In this chapter, the basic theory about supersymmetric lattice models for fermionic particles will be introduced. The definitions and main ideas discussed in this chapter are from [Hui10, Chapter 2].

3.1. Supersymmetry

Supersymmetry is a theory in particle physics that relates bosonic and fermionic particles to each other. The theory of supersymmetry can also be extended to lattice models. Although there are many more interesting aspects from supersymmetry, such as the connections with quantum and conformal field theory, we shall only study the concepts needed to have a basic understanding of supersymmetric lattice models.

Before we give the definition of a supersymmetric lattice model, we first define what it means for a quantum mechanical system to be supersymmetric. A quantum mechanical system (H, H), where H is the state space1 _{and H is the Hamiltonian, is called N = 2 supersymmetric}2 _{if there}

exists a linear operator Q on H, called the supercharge, subject to the relations

{Q, Q} = {Q†, Q†} = 0 and H = {Q†, Q}, (3.1) where the anticommutator {·, ·} is defined by {A, B} = AB +BA and where Q†_{is the Hermitian}

conjugate of Q. Furthermore, the supercharge transforms a bosonic state into a fermionic state, and vice versa. Note that the commutation relations

[H, Q] = [H, Q†] = 0

are a direct consequence of (3.1). We say that a lattice model is N = 2 supersymmetric if it is described by an N = 2 supersymmetric quantum mechanical system. For lattice models with fermionic particles, a state with an even (odd) number of fermions is called a bosonic (fermionic) state. In particular, the supercharge transforms a state with an even number of fermions to a state with an odd number of fermions, and vice versa.

Supersymmetric systems have a few distinctive features. First of all, the Hamiltonian of the system is positive definite, with which we mean that for all states |ψi it holds that

hψ|H|ψi = hψ|(Q†Q+ QQ†)|ψi = |Q|ψi|2+ |Q†|ψi|2≥ 0. Hence an eigenstate with zero energy is necessarily a ground state of the system.

Secondly, there is a twofold degeneracy in the energy spectrum of the Hamiltonian. By this, we mean that all eigenstates of the Hamiltonian H with strictly positive energy form doublets,

1_{All state spaces considered in this thesis are finite dimensional complex Hilbert spaces.} 2

The origins of the terminology ‘N = 2’ shall not be elaborated in this thesis. One could think of N = 2 as saying that there are two supercharges. Furthermore, we usually drop the term ‘N = 2’ in this thesis.

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which are pairs of states |ai and |bi for which it holds that

Q|ai= 0 and Q†|ai =√E|bi, (3.2) Q|bi=√E|ai and Q†|bi = 0. (3.3) The singlets, which are states that are annihilated by both Q and Q†_{, form precisely all the}

zero-energy3 _{ground states of the system.}

To see that the non-zero energy eigenstates of H form doublets, first note that the eigenstates of the Hamiltonian (non-canonically) decompose into quadruplets (|s0_{i, Q|s}0_{i, Q}†_|s0_{i, QQ}†_|s0_i),

where |s0i is a eigenstate with positive energy Es. Now consider the (not necessarily normalised)

state

|si := |s0i − 1 Es

QQ†|s0i (3.4)

and note that

Q(Q|si) = 0, Q†(Q|si) = Es|si, Q†|si = 0, Q|si= Q|si.

In other words, we obtain doublets4 _{(|si, Q|si) and (Q}†_|s0_{i, QQ}†_|s0_{i). We say that the pair}

(|si, Q|si) (and the pair (Q†|s0i, QQ†|s0i)) is a pair of superpartners.

To see that the singlet states form all the groundstates, first consider a singlet |gi. Obviously, since Q|gi = Q†_{|gi = 0, we have H|gi = 0. Conversely, suppose |gi is a ground state. Since}

Q2 = (Q†)2 _{= 0, it holds that}

0 = H|gi = (QQ†+ Q†Q)|gi = (Q + Q†)2_|gi,

so

k(Q + Q†)|gik = hg|(Q + Q†)2|gi = 0.

It follows that (Q + Q†_{)|gi = 0 and thus we have QQ}†_{|gi = Q(Q + Q}†_{)|gi = 0 and Q}†_Q|gi ₌

Most of the supersymmetric lattice models considered in this thesis possess a particle-number symmetry. More precisely, the fermion particle-number operator F , which is an operator that ‘counts’ the number of fermions of a state, satisfies the following commutation relations

[F, Q†] = Q† and [F, Q] = −Q. (3.5) Note that F is an observable and hence a Hermitian operator. A direct consequence from (3.5) is that F commutes with the Hamiltonian and that superpartners differ in their fermion number by one: for f := hF |s|F i, it holds that

F Q|si= QF |si − Q|si = (f − 1)Q|si.

3_{Some terminology: whenever we say ‘ground state’, we always mean ‘zero-energy groundstate’, unless stated}

otherwise.

4_{Strictly speaking, from (3.2) and (3.3), it holds that the pair (|si,}_√1

EsQ|si are doublets, where we take

|ai =_√1

EsQ|si and |bi = |si. However, since the only difference is a factor 1 √

Es, we simply say (|si, Q|si) is

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In particular, we see that Q removes a fermion whereas Q†_{adds a fermion to the system. Note}

that that this condition is consistent with the requirement that Q maps a fermionic state to a bosonic state, and vice versa. Sometimes, the roles of Q and Q†are interchanged, in which case we have

[F, Q] = Q and [F, Q†] = −Q†. (3.6) Supersymmetry allows us to understand certain properties of lattice models, which without supersymmetry would be very difficult to derive. For instance, supersymmetry allows us to make predictions about certain properties of the ground states of particular supersymmetric lattice models (see Section 3.5).

3.2. Witten index

The Witten index is a physical quantity defined as

W = trh(−1)Fe−βHi,

where the trace runs over the entire state space and where β is the inverse temperature. Recall that all excited states form doublets with their superpartners, whose energy is the same and whose fermion-number differs by one from the original state. This implies that the contributions of the excited states to the trace cancel each other, hence we have

W = trGS[(−1)Fe−βH] = trGS[(−1)Fe−0·β] = trGS(−1)F. (3.7)

In other words, the Witten index is simply the number of bosonic ground states minus the number of fermionic ground states. Furthermore, we see that W is actually independent of the value of β, so an alternative way to evaluate W is by considering the limit β → 0, in which case W is the total number of linear bosonic states minus the total number of fermionic states.

Note that from (3.7), it follows that |W | is a lower bound for the total number of ground states. The Witten index is often used to determine whether ground states occur at all, as it is often easier to calculate than the total number of ground states. For instance, the Witten index looks very similar to the partition function, so in some cases, one is able to compute the Witten index using techniques similar to those used to compute the partition function. For example, in [vE05], van Eerten used the theory of transfer matrices to compute the Witten index for the supersymmetric triangular lattice model for spinless hard-core fermions (see Chapter 5).

The Witten index is the same as to the Euler characteristic (see Remark 3.2).

3.3. Lattice fermions

The description of supersymmetric lattice models can be made more concrete by defining the supercharges in terms of operators acting on the lattice particles. The particles considered in this thesis are spinless electrons, which we also call spinless fermions. An important property of the wavefunctions is that they are antisymmetric under the exchange of two fermions. Definition 3.1. Consider a supersymmetric lattice model whose lattice sites are labeled by 1, . . . , n. The annihiliation operator ci, which annihilates a fermion on site i and the creation

operator cj, which creates a fermion on site j, satisfy the following commutation relations.

{c†_i, cj} = δij and {ci, cj} = {c†i, c † j} = 0.

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Note that the anticommutation relations {ci, cj} = {c†i, c †

j} = 0 nicely display the fermionic

antisymmetry and is consistent with the slogan that ‘the exchange of two fermions leads to a minus sign’.

It should be clear that the particle-number operator for fermions is given by F = P

ic † ici,

where the sum is over all lattice sites. Furthermore, the supercharge is usually defined in terms of the creation and annihilation particles; see for instance the zig-zag ladder and the Nicolai models in Chapter 4.

3.4. Hard-core spinless fermions

One of the first type of supersymmetric lattice models was a supersymmetric lattice model for so-called hard-core spinless fermions, introduced in [FSdB03]. By the Pauli exclusion principle, each site on the lattice can be occupied by at most one fermion, as the fermions are spinless. Additionally, the hard-core character forbids two fermions from being nearest neighbours on the lattice. For a more in depth study, we refer to [FSdB03], [FNS03], [FS05], in which extensive research was done on a few specific two-dimensional models for hard-core spinless fermions. In this thesis, we shall study two models for hard-core spinless fermions: the zig-zag ladder and the triangular lattice model.

Lattice models for hard-core spinless fermions naturally carry supersymmetry. For each site i, consider the projection operator

Phii =

Y

j∼i

(1 − c†_jci),

where j ∼ i is notation for j being a neighbour of i. It is easy to show that P2

hii = Phii = Phii† ,

so Phii is indeed an orthogonal projection operator. We define the supercharges by

Q=X i c†_iPhii and Q†= X i ciPhii,

where the sum runs over the lattice sites i. It is easy to check that these operators satisfy the defining commutation relations in (3.1) for supercharges and the commutation relations with the particle-number operator in (3.6). Furthermore, the Hamiltonian is of the form

H= {Q†, Q}= Hkin+ Hpot, (3.8) where Hkin = X i X j∼i

Phiic†icjPhji and Hpot=

X

i

Phii.

The first term of the Hamiltonian is the nearest neighbour hopping term, which is also called the kinetic term. The fermions can hop between two neighbouring sites i and j provided that all neighbouring sites of i and all neighbouring sites of j are empty. The potential term contains a next-nearest neighbour repulsion, a chemical potential and a constant, of which all three depend on the lattice. This may not seem obvious, so for an example, we refer to the zig-zag ladder in in Section 4.1, for which the potential term has a more explicit expression. Furthermore, a simple example of a supersymmetric lattice model for hard-core fermions is the 6-site chain [Hui10, Subsection 2.1.4].

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3.5. Ground states and relation to homology

3.5.1. Ground state generating function

A very useful tool for studying the ground states of a lattice model is the ground state generating function P (z) = TrGS(zF), where the trace is over all the ground states of the system. If we

write an for the number of linear independent ground states with fermion number n, then we

have

P(z) = TrGS(zF) =

X

n∈Z

anzn.

If the state space is finite dimensional, then the sum runs over only a finite number of integers and P is a polynomial. Note that P (−1) = W is the Witten index and P (1) is the total number of ground states.

For finite dimensional state spaces, the ground state generating function is equivalent to the Poincar´e polynomial (see Remark 3.2).

3.5.2. Relation to the homology of Q

Before we can speak of the homology of Q, we must first give the state space the structure of a chain complex.

Proposition 3.1. Consider a supersymmetric lattice model with state space H and supercharge Q and suppose that Q and the particle-number operator F satisfy the commutation relations in (3.5)5_{. For any eigenvalue n ∈ Z of F , let V}

n be the eigenspace of F of the eigenvalue n. For

all other n ∈ Z, let Vn = 0. Then (V•, Q) = (Vn, Q|Vn: Vn → Vn−1)n∈Z

6 _{is a chain complex of}

vector spaces. Furthermore, it holds that H =L

n∈ZVn.

Similarly, we have a cochain (V•_{, Q}†_{) = (V}n_{, Q}†_|

Vn: Vn→ Vn+1) with Vn= V_nfor all n ∈ Z.

Proof. By assumption, Q : H → H is a linear endomorphism which squares to zero, hence it suffices to show that QVn⊆ Vn−1, such that the restriction and corestriction Q|Vn: Vn→ Vn−1

is well-defined. Suppose |ψi ∈ Vn. Using the commutation relations in (3.5), we find that

To see that H =L

n∈ZVnholds, note that H and F are Hermitian and hence diagonalisable.

Since H and F commute, there exists a basis of vectors which are eigenvectors of both H and F. Furthermore, since F is diagonalisable, its eigenspaces span the whole state space, and as all eigenvalues of F are integers, we have H =L

n∈ZVn.

Remark 3.1. Note that the proof only shows that (V•, Q) is a chain complex if Q and F satisfy

the commutation relations in (3.5) (or a cochain complex if Q and F satisfy (3.6)). There are supersymmetric lattice models for which these commutation relations are not satisfied, in which case (V•, Q) might not be a chain complex; see for instance the Z2 Nicolai model discussed in

Section 4.3).

5

If instead they satisfy (3.6), then we must interchange the roles of Q with Q† in this proposition.

6_{Abusing notation, we write Q for both the operator Q : H}

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Following the construction of the chain complex (V•, Q), we can now prove a result about

the homology of Q. The proof of Proposition 3.2 is adapted from the proof given in [Hui10, Chapter 2].

Proposition 3.2. Consider V• as in Proposition 3.1. For each n ∈ Z, we have a linear

isomor-phism

Φn: Vn∩ ker H → Hn(V•, Q) : |gi 7→ |gi + im Q. (3.9)

Proof. As shown in Section 3.1, it holds that the ground states of H are precisely all the singlets: H|gi = 0 if and only if Q|gi = Q†|gi = 0. It follows that Φn is well-defined. To show

that Φn is surjective, consider |gi ∈ ker(Qn: Vn → Vn−1). Recall that the state space can be

(non-canonically) decomposed into doublets and singlets: Hn= ( M i∈I span(ψi)) ⊕ ( M j∈J span(φj, Qφj)),

where each ψi is a singlet and φj is an eigenstate of H with positive energy and Q†φj = 0.

We can assume that ψi, φj and Qφj are also eigenstates of the particle-number operator F .

Namely, if we choose |s0i in (3.4) to be an eigenstate of both H and F (recall that there exists a basis of eigenvectors of both H and F , as they are commuting diagonalisable operators), then |si and Q|si in (3.4) are also eigenstates of both H and F . Since Q|gi = 0 and Qφj 6= 0 for

each j ∈ J, and since |gi ∈ Vn, |gi is of the form

|gi =X i∈I λiψi+ X j∈J µjQφj, λi, µj ∈ C,

where λi6= 0 if and only if ψi∈ Vn and µj 6= 0 if and only if φj ∈ Vn+1. Then |g0i :=P_iλiψi ∈

ker H ∩ Vn is a singlet and Φn(|g0i) = |gi. We conclude that Φn is surjective.

To show that Φn is injective, let |gi ∈ Vn∩ ker H and suppose Φn|gi = 0. Then |gi = Q|f i

for some |f i ∈ Vn+1. Recall that |gi is a singlet, so it holds that Q†Q|f i = Q†|gi = 0, so

kQ|f ik2 = hf |Q†Q|f i = 0, and hence |gi = Q|f i = 0, from which we conclude that Φn is

injective.

Note that Proposition 3.2 simply states that the number of linear independent ground states with fermion number n is equal to the dimension of the nth _{homology module of (V}

•, Q).

Remark 3.2. Following the construction of the chain complex (V•, Q) and the isomorphism

given in (3.9), we see that the Witten index is equal to the Euler characteristic of (V•, Q), which

is defined as

χ(V•) =

X

n∈Z

(−1)ndim Hn(V•),

and that the generating function (for finite dimensional state spaces) is equal to the Poincar´e polynomial of (V•, Q), which is defined as

PV•(X) =

X

n≥0

(dim Hn(V•)) · Xn.

This means that computing the homology H•(V•, Q), and in particular the dimensions of the

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4. One-dimensional supersymmetric models

In this chapter, we shall discuss a number of one-dimensional supersymmetric lattice models. Remember that whenever we say a state is a ground state, we mean that it is a zero-energy ground state, unless stated otherwise.

4.1. Zig-zag ladder

The zig-zag ladder is a hard-core lattice model defined in [Hui10, Subsection 7.2.2]. We consider a one-dimensional lattice which takes the form of a ladder. Schematically, the lattice looks like

. . .

0 2 4 6

1 3 5 7

Note that there is a periodicity ~v = (1, 2) in the vertical direction. In the horizontal direction, we can consider open boundary conditions or periodic boundary conditions with periodicity ~

u = (L, 0) (in which case N = 2L is even). Each site is labeled by i = 0, 1, 2, . . . , N − 1 as in the picture above. For the periodic zig-zag ladder, the projection operator is of the form

Phii = i+2 Y j=i−2 j6=i (1 − c†_jcj) = i+2 Y j=i−2 j6=i (1 − nj) where nj = c†_jcj.

The Hamiltonian for the zig-zag ladder is of the form (3.8). As mentioned in Section 3.4, Hpot

contains a next-nearest neighbour repulsion, a chemical potential and a constant. To illustrate this, we give a more explicit expression for Hpot than in (3.8):

Hpot= N

X

i=1

(1 − ni−2)(1 − ni−1)(1 − ni+1)(1 − ni+2)

= 2L − 4F +

N

X

i=1

(ni−2ni+1+ ni−1ni+2+ ni−2ni+2)

= 2L − 4F +

N

X

i=1

(2nini+3+ nini+4).

Note that due to the hard-core character, the terms nini+1and nini+2 cancel. Furthermore, as

mentioned in Section 3.4, Hpot consists of a next-nearest neighbour repulsion

PN

i=12nini+3+

nini+4 (next-nearest neighbours are precisely those sites whose indices differ by 3 or 4), a

chemical potential equal to 4 and a constant N = 2L. Heuristically, we can expect for a state that minimizes the potential energy (but not necessarily the total energy) that the fermions are exactly five sites apart. For example,

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. . .

0 2 4 6 8 10 12

1 3 5 7 9 11 13

is such a state. We shall not give an explicit expression for Hkin, as this does not give much

more insight about the properties of the Hamiltonian, other than what was already mentioned in Section 3.4.

The physics of this model is well-understood and is thoroughly discussed in [Hui10, Chapter 7]. In this section, we shall discuss a few of the results about the ground states found in [Hui10]. We shall also use some of these results to derive some of our own results.

4.1.1. Ground states of the zig-zag ladder

In [Hui10], extensive research was done on the ground states of particular type of two-dimensional hard-core fermion supersymmetric lattice models, of which the zig-zag model is a simple special case. The key ideas used in [Hui10] to study the ground states were (co)homology theory and the relation between the ground states and the number of rhombus and square tilings on the lattice, which was inspired by the results from [Jon06] and [Jon10]. We shall not go into detail about how these results were derived. Instead, we shall use the results to study a few properties of the ground states of the zig-zag ladder.

It was shown numerically in [Hui10] that the generating function PN(z) = TrGS(zF) of the

ground states of the zig-zag ladder with open boundaries is given by the recursion relation PN(z) = zPN −4(z) + zPN −5(z)

with initial values P0(z) = 1, P1(z) = 0, P2(z) = z, P3(z) = 2z and P4(z) = 2z. Using this

recurrence relation, we can give an expression for PN(z) in closed form.

Proposition 4.1. We have PN(z) = X f ∈Z f N −4f + 2 + f N −4f + 1 + f N −4f zf. (4.1) Proof. We can prove this by induction. It is a trivial exercise to check that P0(z) = 1, P1(z) = 0,

P2(z) = z, P3(z) = 2z and P4(z) = 2z. Now suppose PN −4(z) and PN −5(z) are of the form

(4.1) for some N ∈ Z≥5. Then it holds that

PN(z) = X f ∈Z f N −4f + 2 + f N −4f + 1 + f N −4f zf =X f ∈Z f −1 N −4f + 2 + 2 f −1 N −4f + 1 + 2 f −1 N −4f + f N −4f − 1 zf = zPN −4(z) + zPN −5(z).

Proposition 4.1 allows us to determine a few interesting properties about the ground states. First of all, note that the f(th)_{coefficient a}

f of PN(z) is non-zero if and only if 0 ≤ N − 4f ≤ f .

In other words, there exist ground states if and only if the filling fraction is between 1 5 and

1 4,

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400 420 440 460 480 500 f 1010 1030 1050 1070 1090 10110 10130 af

Figure 4.1.: The number of ground states af with fermion number f plotted

against the fermion number f for the zig-zag ladder with 2000 sites.

Furthermore, we can investigate the value of f for which af is maximal. In [Hui10], it was

conjectured that the largest number of ground states occurs at a filling of fmax= 2₉N. However,

using our closed formula for PN(z), we can show that this cannot generally be true. Namely,

for N = 1800, we find that

log(a400) ≈ 275.127 < log(a401) ≈ 275.763

where we note that 400 = 2₉N, which disproves the fact that fmax= 2₉N.

To study the behaviour of fmax, we expand the domain of the binomial coefficients to the real

numbers using gamma functions: gN,0(f ) := Γ(f + 1) Γ(N − 4f + 1)Γ(5f − N + 1) =: f N −4f gN,1(f ) := Γ(f + 1) Γ(N − 4f + 2)Γ(5f − N ) =: f N −4f + 1 gN,2(f ) := Γ(f + 1) Γ(N − 4f + 2)Γ(5f − N − 1) =: f N −4f + 2 .

Hence fmax = dxNe or fmax = bxNc, where xN is the maximum of the function g = gN,0+

gN,1+ gN,2 on the interval [N₅,N₄].

Proposition 4.2. We have limN →∞x_NN = α, where α ≈ 0.2241145 is the unique real solution

to the polynomial equation (5x − 1)5 _{= x(1 − 4x)}4_.

Proof. It is a simple but tedious exercise to show that gN,iare continuous functions on [N₅,N₄],

that they have precisely one maximum at some xN,i ∈ (N₅,N₄) and that these functions are

increasing and decreasing on the left and right of this maximum, respectively. We first study the asymptotic behaviour of these maxima. From the above it follows that if we find an f ∈ [N

5, N

4]

such that _{N −4f}f = f −1

N −4(f −1), then it holds that f − 1 ≤ xN,0≤ f and thus f −1 N ≤ xN,0 N ≤ f N.

For such an f , we must have

1 = y(1 − 4y + 4/N )(1 − 4y + 3/N )(1 − 4y + 2/N )(1 − 4y + 1/N ) (5y − 1 − 1/N )(4y − 1 − 2/N )(5y − 1 − 3/N )(5y − 1 − 4/N ) ,

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where y = _Nf. We immediately see that y must exist as it is the solution of a polynomial equation with odd degree 5, which always has a real root. As N goes to infinity, the solution for y converges to the (unique) real solution of the equation

1 = x(1 − 4x)

4

(5x − 1)5 ,

which is equal to α. Since y − 1 N ≤

xN,0

N ≤ y, it follows that limN →∞ xN,0

N = α. Analogously,

we can show that xN,1

N and xN,2

N also converge to α. Now note that each gN,i is monotone on

the left of xN,i and on the right of xN,i. Hence the maximum xN of g = gN,0+ gN,1+ gN,2 lies

between mini(xN,i) and maxi(xN,i). It follows immediately that limx_NN = α.

4.1.2. Ground states at quarter filling

For the zig-zag ladder with N = 4n sites and horitzontal periodicity (2n, 0), there are four ground states at quarter filling of which we have an explicit expression [Hui10, Section 7.4]:

Ψα= A n−1

Y

j=0

(c†_4j+α _{mod 4n}− c†_4j+α+1 _{mod 4n})|0i, α= 0, 1, 2, 3.

Recall that the sites on the upper (lower) rung are labeled by odd (even) integers. These ground states are particularly interesting, as they seem to exhibit so-called charge-ordering, that is, the expectation of the charge (the fermions that are considered are electrons and have a charge of 1) of each site i is not uniform. For instance, for Ψα, the expected number of fermions hnki = 1₂

for k ≡ α, α + 1 mod 4, whereas hnki = 0 for k ≡ α + 2, α + 3 mod 4. For ground states with

filling smaller than 1/4 we did not manage find analytical expressions for the ground states. The ground states for a ladder with open boundaries are very similar to the ground states of the periodic chain. For the chain with length N = 4n − 2 and N = 4n − 1 with filling f = n, which is the maximal filling for which there exist ground states, we have ground state

|ψi = A

n

Y

j=0

(c†_4j − c†_4j+1)|0i.

From the ground state generating function, we can see that this is the unique ground state at this filling for N = 4n − 2. For the chain with length 4n, the state

|ψi = A

n

Y

j=0

(c†_4j+1− c†_4j+2)|0i

is a ground state. The ground state generating function tells us that there are n + 1 and

n

2 + n + 1 ground states for the chains with length 4n − 1 and 4n, respectively. However, we

were unable to find the remaining ground states. We were also unable to find any ground states for the chain with length 4n − 3.

4.2. State space structure for one-dimensional chains

Up until now, we have only viewed the state space of a supersymmetric lattice model on a one-dimensional chain in an abstract manner, without knowing what the elements of Hn exactly

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the commutation relations that they satisfy and a few of the properties that follow from these commutation relations. In this subsection, we shall give Hn for a one-dimensional chain a

particular structure on which we can explicitly define the annihilation and creation operators. We can depict an empty one-dimensional chain as

and a classical state as for instance

where a black (white) ball represents an occupied (empty) site. Any state is a linear combination of states of this form. Alternatively, for the chain with length n, we can regard each state as a formal linear combination of elements of {0, 1}n_{, where a 1 denotes the presence of a}

fermion on a site and a 0 denotes the absence of a fermion on a site. In other words, we write H_n= span_C{0, 1}n_{as the 2}n_{dimensional C-vector space consisting of formal linear combinations}

of elements of {0, 1}n_{. In particular, a classical state is just an element of {0, 1}}n_{. We can also}

use bra-ket notation |x1, . . . , xni = (x1, . . . , xn). However, in the rest of this chapter, it turns

out that the notation (x1, . . . , xn) is slightly more favourable, so we stick with the latter, but

this is purely formal.

Our state spaces are required to be inner product spaces. We define the inner product on H_n to be the inner product h·, ·i is uniquely defined by hx, yi = δxy for all classical states

x, y ∈ {0, 1}n_.

The annihilation and creation operators are the linear maps defined by ci(x1, . . . , xn) = c(n)_i (x1, . . . , xn) = (−1) Pi−1 j=1xj_δ xi,1(x1, . . . , xi−1,0, xi+1, . . . , xn), c†_i(x1, . . . , xn) = c(n)†_i (x1, . . . , xn) = (−1) Pi−1 j=1xj_δ xi,0(x1, . . . , xi−1,1, xi+1, . . . , xn), (4.2)

for all (x1, . . . , xn) ∈ {0, 1}n, and the particle-number operator F =Pic † ici is given by F(x1, . . . , xn) = n X i=1 xi ! · (x1, . . . , xn).

It is useful to check that ci and c†_i are indeed creation and annihilation operators.

Proposition 4.3. The operators ci and c†i from (4.2) are annihilation and creation operators,

respectively, that is, they satisfy the relations {ci, cj} = {c†_i, c_j†} = 0 and {c†_i, cj} = δij.

Proof. For all i, j ∈ {1, . . . , n} with i < j and (x1, . . . , xn) ∈ {0, 1}n it holds that

c2_i(x1, . . . , xn) = ci[±δxi,1(x1, . . . , xi−1,0, xi+1, . . . , xn)] = 0, hence c2 i = 0, and cicj(x1, . . . , xn) = ci[(−1) Pj−1 k=1xk_δ xj,1(x1, . . . , xj−1,0, xj+1, . . . , xn)] = (−1)Pj−1k=1xk+ Pi−1 k=1xk_δ xj,1δxi,1(x1, . . . , xi−1,0, . . . , xj−1,0, xj+1, . . . , xn), cjci(x1, . . . , xn) = cj(−1)xk+ Pi−1 k=1xi_δ xi,1(x1, . . . , xi−1,0, xi+1, . . . , xn) = (−1)Pi−1k=1xk+ Pj−1 k=1xk−1δ_x j,1δxi,1(x1, . . . , xi−1,0, . . . , xj−1,0, xj+1, . . . , xn) = −cicj(x1, . . . , xn),

hence {ci, cj} = cicj+ cjci= 0. By similar arguments, it follows that {c†i, c †

j} = 0. Similarly, we

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One easily checks that c†_i is indeed the Hermitian conjugate of ci, i.e. for all x, y ∈ {0, 1}n it

must hold that

hx, c†_iyi= hcix, yi.

4.3. Z

2

Nicolai model

In this section, we shall study the Z2 Nicolai model, introduced in [SKN17] and named after the

German physicist Hermann Nicolai. The Z2 Nicolai model is a supersymmetric fermion lattice

model on a one dimensional chain of length n:

The chain can by chosen to be either periodic or with open boundaries. Its 2n_{dimensional state}

space H is equipped with a supercharge given by Q=X

j

(gcj+ cj−1cjcj+1), (4.3)

where j runs over the lattice sites and where g ≥ 0 is some fixed real number. For the periodic chain, j runs from 1 to n, and for the chain with open boundaries, j runs from 1 to n − 2. Note that c†_ic†_j = −c†_ic†_j, so the Hermitian conjugate Q† of the supercharge is given by

Q†=X j (gcj + cj−1cjcj+1)†= X j gc†_j+ c†_j+1c†_jc†_j−1=X j gc†_j− c†_j−1c†_jc†_j+1.

The Hamiltonian H = {Q, Q†} is of the form

H = Hfree+ H1+ H2+ g2n, where Hfree = g n X j=1 (2cjcj+1− cj−1cj+1− 2c†jc † j+1+ c † j−1c † j+1) (4.4) H1 = n X j=1 (1 − 3nj+ 2njnj+1+ njnj+2), (4.5) H2 = n X j=1 c†_jc†_j−1cj+2cj+3− cjcj−1c † j+2c † j+3 + n X j=1 h (nj−1+ nj− 1)c†_j+1cj−2− (nj−1+ nj− 1)cj+1c†_j−2 i . (4.6) The definition and physical interpretation of Hfree, H1 and H2 are taken from to [SKN17,

Chapter II]. The term Hfreedescribes the pairing terms of nearest and next-nearest neighbouring

particles 2cjcj+1 and −cj−1cj+1, respectively, and their Hermitian conjugates. The H1 term

consists of an on-site potential −3njand a repulsive interaction between two particles on nearest

and next-nearest neighbouring sites, given by the terms 2njnj+1 and njnj+2, respectively. The

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third-neighbour hopping term, whose amplitude nj−1+ nj− 1 is determined by the occupation

of the two sites j − 1 and j.

The Z2 Nicolai model differs from most of the supersymmetric models in that the

particle-number operator F =Pn i=1c

†

iciis not a conserved quantity (for g > 0), as it does not satisfy the

commutation relations given in (3.5) or (3.6). However, there is a Z2 symmetry: [H, (−1)F] = 0.

The main goal of [SKN17] was to study spontaneous supersymmetry breaking. For finite-sized lattices, supersymmetry is said to be spontaneously broken if there are no zero-energy ground states. In [SKN17], it was shown that supersymmetry is spontaneously broken for all values g > 0. Since our goal is to study the zero-energy ground states, we shall only consider the case g = 0 in this thesis. Note that in this case that, although F and Q do not satisfy the commutation relations in (3.5), they do satisfy

[Q, F ] = 3Q and [Q†, F] = −3Q, from which we can see that F does commute with the Hamiltonian.

The following proposition shall verify that both the periodic and non-periodic Z2 Nicolai

model are N = 2 supersymmetric.

Proposition 4.4. Let n ∈ N and let c1, . . . , cn be annihilation operators and let Q be as in

(4.3). Then for both the periodic and the non-periodic chain, the following relations hold {Q, Q} = {Q†, Q†} = 0.

Proof. We shall only give a proof for the non-periodic chain, as the proof for the periodic chain is completely analogous. Recall the commutation relations for the creation and annihilator operators

{c†_i, cj} = δij and {ci, cj} = {c†i, c † j} = 0.

Using that cjci = −cicj, we find

{Q, Q} = 2Q2 _{= 2} n−2 X j=1 cjcj+1cj+2 n−2 X i=2 cici+1ci+2 = 2X i,j cjcj+1cj+2cici+1ci+2 = 2X i<j

cjcj+1cj+2cici+1ci+2+ cici+1ci+2cjcj+1cj+2+

X

i

cici+1ci+2cici+1ci+2

= 2X

i<j

cjcj+1cj+2cici+1ci+2+ (−1)9cjcj+1cj+2cici+1ci+2+ 0 = 0.

Similarly, we show that {Q†_{, Q}†_{} = 0.}

4.3.1. Ground state generating function of the Z2 Nicolai model

In [SKN17, Appendix A], the numerical calculations for the number of ground states an of the

non-periodic chain with up to 16 sites were given. According to the On-Line Encyclopedia of Integer Sequences, it appeared that an satisfies the recurrence relation

an= 2an−2+ 2an−3 with a0 = 1, a1 = 2, a2 = 4. (4.7)

It was conjectured in [SKN17] that this recurrence relation holds, but no proof was given. In this subsection, we shall derive the ground state generating function, from which (4.7) immediately follows by evaluating z = 1.

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Theorem 4.1. The ground state generating function for the Z2Nicolai model with open

bound-ary conditions satisfies the recurrence relation

Pn(z) = 2zPn−2(z) + (z + z2)Pn−3(z),

with initial values P0(z) = 1, P1(z) = 1 + z and P2(z) = 1 + 2z + z2.

Recall the identification Hn = span_C{0, 1}n and the notation introduced in Section 4.2. As

in Proposition 3.1, we can grade the state space by the fermion number. For each q ∈ Z, let Vq⊆ Hn be the subspace containing all states with fermion number q (the eigenspace of F for

the eigenvalue q). In particular, it is spanned by all classical states with fermion number of q: Vq:= span_C(Bq), where Bq := {x ∈ {0, 1}n:

n

X

i=1

xi = q}.

Note that if q /∈ {0, . . . , n}, then Vq = 0 and that Hn=L_q∈ZVq. Define the linear operators

d= d(n)_:= n−2

X

i=1

cici+1ci+2 and ∂ = ∂(n):= c1c2c3= Q − d.

Clearly, ∂2 _{= 0 and analogous to the proof of Proposition 4.4, it follows that d}2_{= 0. Note that}

dVk⊆ Vk−3, ∂Vk⊆ Vk−3 and QVk⊆ Vk−3. In a picture, we have:

V0 V1 V2 V3 V4 V5 V6 V7 V8 · · · · Q Q Q Q Q

In particular, the graded vector space V• = (Vk)k∈Z does not form a chain complex with these

differentials, but we are able to ‘split’ V• into three chain complexes.

Proposition 4.5. Define the graded vector spaces

V•0 = (Vk0)k∈Z = (V3k)k∈Z, V•00= (Vk00)k∈Z = (V3k+1)k∈Z, V•000= (Vk000)k∈Z= (V3k+2)k∈Z

and write d0 = (d3k)k∈Z, d00 = (d3k+1)k∈Z, d000 = (d3k+2)k∈Z, ∂0 = (∂3k)k∈Z, . . . , Q000 =

(Q3k+2)k∈Z. Then (V0, d0), (V0, ∂0), (V0, Q0), (V00, d00), (V00, ∂00), (V00, Q00), (V000, d000), (V000, ∂000),

and (V000_{, Q}000_{) are chain complexes. In a picture, we have}

V0 V3 V6 V9 V12 (V0 •, ∂0) : · · · · ∂0 ∂0 ∂0 ∂0 ∂0 ∂0 V1 V4 V7 V10 V13 (V•00, ∂00) : · · · · ∂00 ∂00 ∂00 ∂00 ∂00 ∂00 V2 V5 V8 V11 V14 (V000 • , ∂000) : · · · · ∂000 ∂000 ∂000 ∂000 ∂000 ∂000 etc.

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Note that V0

k= V3k, Vk00= V3k+1, etc. so it may be useful to consider the following notation.

Write K• = (Kq)q∈Z, where Kq =      K_k0 := Hk(V•0, ∂0) if q = 3k, K_k00:= Hk(V•00, ∂00) if q = 3k + 1, K_k000:= Hk(V•000, ∂000) if q = 3k + 2. In particular, we have Kq= ker(∂ : Vq→ Vq−3)/ im(∂ : Vq+3 → Vq).

Proposition 4.6. Let h := c†₁c†₂c₃†. There exist family of maps i = (iq: Kq → Vq)q∈Z and

p= (pq: Vq→ Kq)q∈Z such that

i0 := (i3k)k∈Z: (K•0,0) → (V•0, ∂0), p0:= (p3k)k∈Z: (V•0, ∂0) → (K•0,0),

i00:= (i3k+1)k∈Z: (K•00,0) → (V•00, ∂00), p00:= (p3k+1)k∈Z: (V•00, ∂00) → (K•00,0),

i000 := (i3k+2)k∈Z: (K•000,0) → (V•000, ∂000), p000:= (p3k+2)k∈Z: (V•000, ∂000) → (K•000,0)

are chain maps and such that (K•0,0) (V•0, ∂0) i0 p0 h0 (K•00,0) (V•00, ∂00) i00 p00 h00 (K•000,0) (V•000, ∂000) i000 p000 h000 (4.8) are HE data, where h0= (h3q), h00= (h3q+1) and h000 = (h3q+2) are homotopies.

Proof. We will show that the three conditions given in Definition 2.12 are satisfied. First of all, it is obvious that all six objects in (4.24) are chain complexes. Next, let q ∈ Z and define

pq: Vq → Kq: v 7→

(

[v] := v + ∂Vq+3 if v ∈ ker ∂,

0 else.

For all v ∈ Vq, we have

pq−3∂qv= [∂qv] = 0 = 0pqv.

Considering the three cases q ≡ 0, 1, 2 mod 3 gives that p0, p00 and p000 are chain maps. Now note that βq := {[x] : x ∈ {0, 1}n, n X i=1 xi= q, x1+ x2+ x3∈ {0, 3}}/ (4.9)

is a basis for Kq and that different classical states are in different homology classes. Hence there

exists a unique well-defined linear map iq: Kq → Vq that maps each basis element [x] ∈ βq to

x. For each basis element [x] ∈ βq of Kq, we have (x1, x2, x3) 6= (0, 0, 0), so

(∂i)[x] = ∂x = 0 = (i ◦ 0)x, hence ∂i = i ◦ 0, and thus i0_{, i}00 _{and i}000 _{are chain maps.}

To show that h0, h00 and h000 are homotopies from 1 to i0p0, i00p00 and i000p000, respectively, we will show that ip = 1 + ∂h + h∂. Consider a classical state x ∈ Hn (recall that a classical state

is an element of {0, 1}n _{and a basis element of the state space H}

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1. If x1 = x2 = x3 = 0, then x ∈ im ∂, so we have ip(x) = 0 and h∂(x) = 0. We also have

∂h(x) = ∂(1, 1, 1, x4, . . . , xn) = −x, hence (1 + ∂h + h∂)(x) = x − x = 0 = ip(x).

2. If x1 = x2 = x3 = 1, then ip(x) = ∂h(x) = 0, h∂(x) = −h(0, 0, 0, x4, . . . , xn) = −x, and

hence (1 + ∂h + h∂)(x) = x − x = 0 = ip(x).

3. In all other cases, we have h(x) = ∂(x) = 0, hence ip(x) = i([x]) = x = 1(x) = (1 + ∂h + h∂)(x).

We conclude that ip = 1 + ∂h + h∂.

The idea is now to apply the homological perturbation lemma on the three HE data from Proposition 4.6, using the differentials d0, d00 and d000 as our small perturbations for the three HE data. We will now show that d is indeed a small perturbation and we will give an expression for (1 − d0_h0₎−1_{, (1 − d}00_h00₎−1 _{and (1 − d}000_h000₎−1_.

Lemma 4.1. The endomorphism of graded vector spaces 1−dh is invertible with inverse 1+dh. It immediately follows that this also holds if we take d0 _{and h}0 _{instead of d and h; if we take d}00

and h00 instead of d and h; and if we take d000 and h000 instead of d and h. Proof. Note that (c†₁)2 _{= 0 and c}†

1cj = −cjc†1 for all j > 1, so hdh= n−2 X j=2 c†₁c†₂c†₃cjcj+1cj+2c†₁c†₂c†₃= n−2 X j=2 (−1)5c†₁c†₁c†₂c₃†cjcj+1cj+2c†₂c†₃= 0, so (1 − dh)(1 + dh) = 1 − dhdh = 1.

Lemma 4.2. It holds that p(1 + dh)di = pdi. It immediately follows that this also holds if we take p0, d0, h0 and i0 instead of p, d, h and i; if we take p00, d00, h00 and i00 instead of p, d, h and i; and if we take p000, d000, h000 and i000 instead of p, d, h and i.

Proof. We will show that p(1 + dh)di − pdi = pdhdi = 0. It suffices to show that pdhdi([x]) = 0 for x ∈ {0, 1}n _{with x}

1+ x2+ x3 ∈ {0, 3}, since for each q ∈ Z, the set β/ q defined in (4.9) is a

basis for Kq. Consider the following cases

1. If x1 = 1, then di([x]) = dx is either zero or a linear combination of classical states whose

first entry is 1, so hdi([x]) = hd(x) = 0.

2. If x1 = 0, x2 = x3 = x4 = 1, then di([x]) = ±(0, 0, 0, 0, x5, . . .) +Pjyj where each yj is

of the form ±(0, 1, . . . ), so hdi(x) = ±(1, 1, 1, 0, x5, . . . , xn). Then dhdi([x]) is either zero

or a linear combination of classical states whose first three entries are 1. In both cases, we have p(dhdi([x])) = 0.

3. If x1 = x2 = 0 and x3 = x4 = x5 = 1, we find by similar reasoning as in step 2

that hdi([x]) = ±(1, 1, 1, 0, 0, x6, . . . , xn), and that dhdi([x]) = 0 or dhdi([x]) is a linear

combinations of classical states whose first three entries are 1, so p(dhdi([x])) = 0. 4. In all remaining cases, we have x2+ x3+ x4 6= 3 6= x3+ x4 + x5 so, so di([x]) = dx is

either zero or a linear combination of classical states whose first three entries are not all equal to 0, so h(di([x])) = 0, hence pdhdi(x) = 0.

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Notation: We wish to find a recurrence relation that depends on the length m of the chain, so it is convenient to label the objects with m: we write V0

• = V 0(m) • , V•00= V 00(m) • and V•000 = V 000(m) • .

To avoid overloading notation, we do not label the objects p, p0, p00, d, d0, d00, Q, Q0, . . . etc. with m, as it should be clear from the context which chain length they are associated with.

Lemma 4.3. Recall Definition (2.9). Let n ≥ 6. There exist isomorphisms of chain complexes (K•0, p0d0i0) ∼= 2(V 000(n−2) • [1], Q000[1]) ⊕ (V 000(n−3) • [1], Q000[1]) ⊕ (V 00(n−3) • [1], −Q00[1]), (4.10) (K_•00, p00d00i00) ∼= 2(V•0(n−2), −Q0) ⊕ (V•0(n−3), −Q0) ⊕ (V•000(n−3)[1], −Q000[1]), (4.11) (K•000, p000d000i000) ∼= 2(V 00(n−2) • , −Q00) ⊕ (V•00(n−3), −Q00) ⊕ (V•0(n−3), Q0). (4.12)

Remark 4.1. The overloading notation here (and in Corollary 4.1 and Theorem 4.2) may lead to confusion, but it is useful to keep in mind and write out the original grading of the graded objects V• and K•: we have K_k0 = K3k, V

000(n−2) k [1] = V 000(n−2) k−1 = V (n−2) 3(k−1)+2= V (n−2) 3k−1 , etc.

Proof of Lemma 4.3. Let q ∈ Z and recall that βq in (4.9) is a basis for Kq. Define the graded

vector spaces K1 •, K•2, K•3 and K•4 by K_q1 = span{[x] ∈ βq: x ∈ {0, 1}n, x1= 0, x2 = 0, x3 = 1}, K_q2 = span{[x] ∈ βq: x ∈ {0, 1}n, x1= 1, x2 = 1, x3 = 0}, K_q3 = span{[x] ∈ βq: x ∈ {0, 1}n, x1= 1, x2 = 0}, K_q4 = span{[x] ∈ βq: x ∈ {0, 1}n, x1= 0, x2 = 1}.

It is easy to show that K`

•0 = (K3k` )k∈Z, K•`00 = (K3k+1` )k∈Z and K•`000 = (K3k+2` )k∈Z form

chain complexes with differential p0d0i0, p00d00i00 and p000d000i000, respectively, for ` = 1, 2, 3, 4. Note that (K0, p0d0i0) ∼= L4

`=1(K•`0, p0d0i0), (K00, p00d00i00) ∼=

L4

`=1(K•`00, p00d00i00) and (K000, p000d000i000) ∼=

L4

`=1(K•`000, p000d000i000). We will show that the right hand sides of these equations are isomorphic

to the right hand sides of (4.10), (4.11) and (4.12), respectively. For each q ∈ Z, define the linear map f1

q: V (n−3)

q−1 → Kq1 by defining the image of the basis vectors of V (n−3) q−1 :

f_q1: V_q−1(n−3)→ K_q1: (y1, . . . , yn−3) 7→ [(0, 0, 1, y1, . . . , yn−3)].

Note that V_q−1(n−3) = 0 if and only if K1

q = 0, in which case fq1 is clearly a linear isomorphism.

In the case that K1

q 6= 0 6= V (n−3)

q−1 , then fq1 is also clearly a linear isomorphism. Now consider

a classical state y ∈ V_q−1(n−3) (so y ∈ {0, 1}n−3_{). Note that if}

1y= (0, 0, 1, y), so we have

p(cjcj+1cj+2(0, 0, 1, y)) = −fq1cj−3cj−2cj−1y, for j = 4 . . . , n − 2. (4.13)

Now note that since p(0, 0, 0, . . . ) = 0, we have pc2c3c4(0, 0, 1, y) = pc3c4c5(0, 0, 1, y) = 0. Using

this together with (4.13), we obtain pdif_q1y= pd(0, 0, 1, y) = n−2 X j=4 pcjcj+1cj+2(0, 0, 1, y) = n−2 X j=4 −f1 q−3cj−3cj−2cj−1y= −fq−31 Qq−1y, so (pqdqiq) ◦ fq1 = fq−31 ◦ (−Qq−1), hence f10= (f_3k1 )_k∈Z: (V•000(n−3)[1], Q000[1]) → (K•1 0 , p0d0i0), f100= (f_3k+11 )k∈Z: (V 0(n−3) • , Q0) → (K•1 00 , p00d00i00), f1000 = (f1 3k+2)k∈Z: (V 00(n−3) • , Q00) → (K•1 000 , p000d000i000)

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are isomorphisms of chain complexes. Again, keep in mind Remark 4.1 to help with the notation. By the same reasoning, we can show that the linear maps

f_q2: V_q−2(n−3)→ K_q2: (y1, . . . , yn−3) 7→ [(1, 1, 0, y1, . . . , yn−3)],

f_q3: V_q−1(n−2)→ K_q3: (y1, . . . , yn−2) 7→ [(1, 0, y1, . . . , yn−2)],

f_q4: V_q−1(n−2)→ K4

q: (y1, . . . , yn−2) 7→ [(0, 1, y1, . . . , yn−2)]

define isomorphisms of chain complexes f20 _{= (f}2

3k), f200 = (f3k+12 ), etc. The isomorphisms

f30⊕ f40_{⊕ f}10 _{⊕ f}20_{, f}300 _{⊕ f}400 _{⊕ f}100_{⊕ f}200 _{and f}3000 _{⊕ f}4000 _{⊕ f}1000 _{⊕ f}2000 _{give the desired}

isomorphisms in (4.10), (4.11) and (4.12), respectively.

Corollary 4.1. We have the following isomorphisms of Z-graded vector spaces:

H•(K•0, p0d0i0) ∼= 2H•(V•000(n−2)[1], Q000[1]) ⊕ H•(V•000(n−3)[1], Q000[1]) ⊕ H•(V•00(n−3)[1], Q00[1]),

H•(K•00, p00d00i00) ∼= 2H•(V•0(n−2), Q0) ⊕ H•(V•0(n−3), Q0) ⊕ H•(V•000(n−3)[1], Q000[1]),

H•(K•000, p000d000i000) ∼= 2H•(V•00(n−2), Q00) ⊕ H•(V•00(n−3), Q00) ⊕ H(V•0(n−3), Q0).

Note in particular that the minus signs in (4.10), (4.11) and (4.12) disappear. Theorem 4.2. We have the following isomorphisms of Z-graded vector spaces:

H•(V•0(n), Q0) ∼= 2H•(V•000(n−2)[1], Q000[1]) ⊕ H•(V•000(n−3)[1], Q000[1]) ⊕ H•(V•00(n−3)[1], Q00[1]),

(4.14) H•(V•00(n), Q00) ∼= 2H•(V•0(n−2), Q0) ⊕ H•(V•0(n−3), Q0) ⊕ H•(V•000(n−3)[1], Q000[1]), (4.15)

H•(V•000(n), Q000) ∼= 2H•(V•00(n−2), Q00) ⊕ H•(V•00(n−3), Q00) ⊕ H(V•0(n−3), Q0). (4.16)

Proof. Consider the leftmost HE data in (4.24) with d0 as a small perturbation. By Lemma 4.1 and Lemma 4.2, we have p0_{(1 − d}0_h0₎−1_d0_i0 _{= p}0_{(1 + d}0_h0_)d0_i0 _{= p}0_d0_i0_{, so we have the following}

isomorphisms of Z-graded vector spaces: H•(V•0(n), Q0) = H•(V•0(n), ∂0+ d0) ∼ = H•(H•(V•0(n), ∂0), p0d0i0) = H•(K•0, p0d0i0) ∼ = 2H•(V•000(n−2)[1], Q000[1]) ⊕ H•(V•000(n−3)[1], Q000[1]) ⊕ H•(V•00(n−3)[1], Q00[1]),

where the first isomorphism follows from Corollary 2.1 and Proposition 4.6 and the second isomorphism follows from Corollary 4.1. The proof for the other two isomorphisms is completely analogous.

We are now ready to prove Theorem 4.1.

Proof of Theorem 4.1. Write a(n)_f for the number of ground states with fermion number f on the chain with length n. Then we have

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Namely by Proposition 3.2 and (4.14), (4.15) and (4.16), this result holds for f = 0, 1, 2 mod 3, respectively. Therefore, the ground state generating function is given by

Pn(z) = 2zPn−2(z) + (z + z2)Pn−3(z). (4.17)

The initial values (n = 0, 1, 2) can be found by counting the ground states at different filling fractions for n = 3, 4, 5, which together with (4.17) uniquely determines Pn(z) for all n.

We can explicitly give an expression for the ground state generating function in closed form. Theorem 4.3. The ground state generating function PN(z) of the Z2 Nicolai model on the

chain with length N is given by PN(z) = X f ∈Z (aN,f + aN −1,f + aN −1,f −1+ aN −2,f + aN −2,f −2) zf, (4.18) where aN,f = X m∈Z f − m N −2f + 2m N − 2f + 2m m 23f −N −3m. = X m∈Z f − m m, N −2f + m, 3f − 3m − N 23f −N −3m.

Proof. The idea of this proof is to give a solution to (4.17) by relating our problem of counting the number of ground states to another counting problem. Consider the following ‘tiles’1_:

We can concatenate these tiles to form a chain of length N . The idea is now to count the number of possible tilings of a chain with length N . For each f ∈ Z, the number of possible tilings containing f fermions is given by

X m1,m2,n1,n2∈Z: 3m1+3m2+2n1+2n2=N, m1+2m2+n1+n2=f m1+ m2+ n1+ n2 m1+ m2 m1+ m2 m2 n1+ n2 n1 ,

where m1 is the number of tiles of the form , m2 is the number of tiles of the form

, n1 is the number of tiles of the form , n2 is the number of tiles of the form .

Given the two constraints on m1, m2, n1 and n2, we can write this sum as a sum over m := m2

and n := n1: X m,n∈Z f − m N −2f + 2m N − 2f + 2m m 3f − N − 3m n =X m∈Z f − m N −2f + 2m N − 2f + 2m m X n∈Z 3f − N − 3m n =X m∈Z f − m N −2f + 2m N − 2f + 2m m 23f −N −3m =X m∈Z f − m m, N −2f + m, 3f − 3m − N 23f −N −3m= aN,f.

1_{The author of this thesis does not know what the intuition is behind the decision to count these tilings. The}

idea of looking at these tilings is presented to the author by prof. Kareljan Schoutens; more information shall be presented in [LSS].

The zero-energy ground states of supersymmetric lattice models