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The Zero-Energy Ground States of Supersymmetric Lattice Models

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The Zero-Energy Ground States of Supersymmetric Lattice Models 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 University of Amsterdam 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, [email protected], 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 2 Contents 1. Introduction 5 1.1. Organisation of the thesis and main goals . .6 1.2. Acknowledgements . .7 2. Preliminaries 8 2.1. Homology and cohomology . .8 2.2. Homological perturbation lemma . 10 3. Supersymmetric lattice models for fermionic particles 11 3.1. Supersymmetry . 11 3.2. Witten index . 13 3.3. Lattice fermions . 13 3.4. Hard-core spinless fermions . 14 3.5. Ground states and relation to homology . 15 3.5.1. Ground state generating function . 15 3.5.2. Relation to the homology of Q ........................ 15 4. One-dimensional supersymmetric models 17 4.1. Zig-zag ladder . 17 4.1.1. Ground states of the zig-zag ladder . 18 4.1.2. Ground states at quarter filling . 20 4.2. State space structure for one-dimensional chains . 20 4.3. Z2 Nicolai model . 22 4.3.1. Ground state generating function of the Z2 Nicolai model . 23 4.3.2. Ground states at different filling fractions . 30 4.3.3. Classical ground states and ground states at 1=3rd and 2=3rd filling . 31 4.3.4. Supersymmetric SYK model . 32 4.4. Nicolai model . 32 4.4.1. Relation to localisation . 33 4.4.2. Ground state generating function of the Nicolai model . 33 4.4.3. Ground states of the Nicolai model at different filling fractions . 35 4.4.4. Classical ground states . 36 5. Two-dimensional supersymmetric models 37 5.1. Triangular lattice model . 37 5.1.1. Current results about the ground states . 37 5.2. Ground states of the 3 × N triangular lattice . 38 5.3. M × N triangular lattice with periodic vertical boundary conditions . 41 6. Conclusion 42 Bibliography 44 3 Populaire samenvatting 46 A. Spectral sequences 48 A.1. Spectral sequences . 48 A.2. Filtrations . 48 A.3. Bicomplexes . 49 A.4. Application to the Nicolai models . 50 4 1. Introduction 1In 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 tight-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 Qy, called supercharges, with the following properties. The operator Q (Qy) 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 = fQ; Qyg. 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 1The 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 models) presented in our Introduction. 5 choose each fermionic energy eigenstate jfi with strictly positive energy in such a way that it forms a doublet with its superpartner jgi = Qjfi, 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.
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