LMU M T U M . E B TMP eoretical and Mathematical Physics Master’s Thesis Jordan Wigner Transformations and antum Spin Systems on Graphs Tobias Ried Thesis Adviser: Prof. Dr. Simone Warzel . Submission Date: 31st August 2013 First Reviewer: Prof. Dr. Simone Warzel Second Reviewer: Prof. Dr. Peter Müller Thesis Defence: 9th September 2013 A e Jordan Wigner transformation is an important tool in the study of one- dimensional quantum spin systems. Aer reviewing the one-dimensional Jordan Wigner transformation, generalisations to graphs are discussed. As a first res- ult it is proved that the occurrence of statistical gauge fields in the transformed Hamiltonian is related to the structure of the graph. To avoid the difficulties caused by statistical gauge fields, an indirect transformation due to W and S is introduced and applied to a particular quantum spin system on a square laice – the constrained Gamma matrix model. In a random exterior magnetic field, a locality estimate on the Heisenberg evolution – a zero-velocity Lieb Robinson type bound in disorder average – is proved for the constrained Gamma matrix model, using localisation results for the two-dimensional Ander- son model. A Firstly, I would like to thank Professor Simone Warzel for supervising this thesis. roughout my studies of physics and mathematical physics, her enthu- siasm for those subjects has been a constant source of inspiration. I am deeply grateful for her guidance and support. I am also indebted to Professor Peter Müller who agreed to second review my thesis. For leing me use one of his group’s offices, which enormously facilitated the writing process, I am thankful to Professor Jan von Del. e TMP programme would probably not be what it is if it were not for Robert Helling. His commitment is gratefully acknowledged. My thanks and appreciations also go to Alessandro Michelangeli and every- one else who enriched this final phase of my studies not only on an academic, but also on a personal level. Contents 1 Introduction 5 1.1 antum mechanics on combinatorial graphs .......... 6 1.2 antum spin systems ....................... 7 2 Jordan-Wigner transformation 11 2.1 e Jordan Wigner transformation in one dimension ...... 11 2.1.1 Jordan Wigner transformation ............... 11 2.1.2 Extension to the infinite chain ............... 14 2.2 Generalisations to graphs ...................... 17 2.2.1 Examples and the existence of special Jordan Wigner trans- formations .......................... 19 2.2.2 e two-dimensional Jordan Wigner transformation .. 23 2.2.3 A Jordan Wigner transformation for tree graphs .... 26 2.3 e WS approach .................. 27 2.3.1 Link operators and their algebra ............. 28 3 e Gamma matrix model (GMM) 37 3.1 Jordan Wigner transformation for the constrained GMM .... 38 3.2 Dynamical Localisation and Lieb-Robinson bounds ....... 45 3.2.1 e Anderson model on ℤ푑 ................ 45 3.2.2 Lieb Robinson bounds and exponential clustering .... 49 4 Conclusion and Outlook 61 A Some missing calculations in S’s theorem 63 Bibliography 67 C Chapter 1 Introduction spin systems are an important playground for studying many-body Q quantum phenomena. Due to the fact that the Hilbert space at each laice site is finite-dimensional they are relatively simple models, yet still capture some important physical features that would otherwise be too difficult to treat in the general framework. is includes the study of quantum dynamics, thus focussing on many-body scaering effects. In the study of one-dimensional quantum spin systems, a transformation between spins (or hard-core bosons) and fermions first introduced by J and W [WJ28] has proven to be a valuable tool. It can be used to transfer results from (non-interacting) fermionic systems to spin systems with nearest neighbour interactions (Heisenberg model, 푋푌-model, 푋푋푍-model et cetera). Recent developments in the theory of random Schrödinger operators, in par- ticular the Anderson model, make it possible to analyse random quantum spin systems and look for analogues of dynamical localisation (the notion of spec- tral or eigenfunction localisation does not make sense any longer in many-body systems). is thesis has the following objectives: (i) Review the one-dimensional Jordan Wigner transformation and discuss gen- eralisations to general (finite, connected) simple graphs. In particular it will be proven that the occurrence of statistical gauge fields in the transformed Hamiltonian is related to the structure of the graph. (ii) Introduce an indirect Jordan Wigner transformation due to W and S [Wos82, Szc85] and apply it to a particular quantum spin system – the constrained Gamma Matrix model. (iii) Present zero-velocity Lieb Robinson bounds (in disorder average), whose validity can be used to define localisation of quantum spin systems in ran- C 1. I dom magnetic fields similar to strong dynamical localisation in the Ander- son model. ese can be used to prove exponential decay of (averaged) ground state correlations [HSS12], and therefore provide a simplified ver- sion of the mobility gap concept introduced by H [Has10]. (iv) Prove a zero-velocity Lieb Robinson type bound for the constrained Gamma Matrix Model, thereby providing another example (next to the isotropic ran- dom 푋푌 chain discussed by H et al. [HSS12]) of the above concepts. In the remainder of this introductory chapter, some basic notions of quantum mechanics on combinatorial graphs and quantum spin systems are reviewed. 1.1 antum meanics on combinatorial graphs Combinatorial graphs are the underlying laices in tight binding models of elec- trons in solid state physics, hard core bosons in optical laices, and spin systems (just to name a few applications). A combinatorial graph 픾 = (푉, 퐸) consists of a set of vertices 푉 = 푉(픾) and edges (or links) 퐸 = 퐸(픾) ⊂ {{푥, 푦} ∶ 푥, 푦 ∈ 푉} connecting them. Two vertices 푥 and 푦 are said to be neighbouring, 푥 ∼ 푦, if {푥, 푦} ∈ 퐸. e graph 퐺 is assumed to be simple, i.e. two vertices are at most connected by one edge and vertices are not connected to themselves. Let Adj(픾) be the adjacency matrix of 픾, i.e. the symmetric |푉| × |푉|-matrix with elements ⎧ ⎪1, if 푥 ∼ 푦 Adj(픾)(푥, 푦) = ⎨ ⎪0, otherwise ⎩ A simple path 풫푁 is a graph with vertex set 푉 = {푣, … , 푣푁 } and edges .{퐸 = {{푣, 푣}, {푣, 푣}, … , {푣푁−, 푣푁 }} such that 푣푘 ≠ 푣푗 for all 푘, 푗 ∈ {1, … , 푁 ere is a natural ordering on 풫푁 , namely 푣푗 ⪯ 푣푘 if and only if 푗 ≤ 푘, i.e. 푣푗 lies .on the path from 푣 to 푣푘 ,If additionally the first and the last vertex 푣 and 푣푁 are connected by an edge .the resulting graph is called a cycle or simple loop 풞푁+ of length 푁 + 1 Bethe lattice and Cayley tree e Bethe laice 픹푧 with coordination number 푧 ≥ 2 is the (infinite) connected, cycle-free graph with the property that each vertex has exactly 푧 neighbours. Fix- ing one particular vertex as root 표, the collection of its 푧 neighbours is referred to as the first shell. Each vertex in the first shell has another (푧 − 1) child vertices, 1.2. 핋푘(푁) 픹푧(푁) 표 . 표 . (a) (b) Figure 1.1: (a) Cayley tree 픹푧(푁) and (b) its forward part 핋푘(푁). combined forming the second shell. Proceeding along up to 푁 shells, the result- 푛− ing tree 픹푧(푁) is called Cayley tree (of size 푁). e 푛th shell has 풩푛 = 푧(푧−1) vertices and the Cayley tree 픹푧(푁) has a total number of 푁 (푧 − 1)푁 − 1 푧(푧 − 1)푁 − 2 풩 = 1 + ﾜ 풩푛 = 1 + 푧 = 푛= 푧 − 2 푧 − 2 vertices. ere is a natural (partial) ordering on 픹푧, defined by 푥 ⪯ 푦 ∶⇔ x lies on the unique path from 표 to 푦. Given two vertices 푥, 푦 ∈ 픹푧 their distance dist(푥, 푦) is given by the length of the unique path connecting them. In particular, the 푛th shell is the set of all vertices 푥 with dist(푥, 표) = 푛, and 픹푧(푁) = {푥 ∈ 픹푧 ∶ dist(푥, 표) ≤ 푁}. + Sometimes it is customary to consider only the forward part 핋푘(푁) = 픹푧 (푁) of a Cayley tree, i.e. the regular rooted tree graph with branching number 푘 = 푧 − 1. 1.2 antum spin systems is section is intended to describe the usual mathematical framework when discussing quantum spin systems [BR02, Nac06]. Let 픾 be a connected (finite or C 1. I 푑 infinite) graph, e.g. the 푑-dimensional laice ℤ , 푑 ≥ 1, or the Bethe laice 픹푧 with coordination number 푧 ≥ 2 (see section 1.1). For each 푥 ∈ 픾 let ℌ푥 = ℌ{푥} be a finite dimensional Hilbert space¹. For simplicity it is assumed that all the Hilbert spaces ℌ푥 have the same dimension 푛 .푛 ≥ 2, such that ℌ푥 ≅ ℂ ∀푥 ∈ 픾. Denote by 픓(픾) the set of finite subsets of 픾 For Λ ∈ 픓 (픾) define ℌ = ⨂ ℌ . ︹ 푥∈︹ 푥 ∗ Observables of the system at site 푥 are elements of the 퐶 -algebra 프{푥} = 픅(ℌ푥) of bounded linear operators on ℌ푥, which is isomorphic to the algebra of 푛×푛 complex matrices 픐푛(ℂ). e algebra of observables for a system in a finite set Λ is 프 = 픅(ℌ ) = ⨂ 프 . ︹ ︹ 푥∈︹ {푥} , Since for two disjoint finite sets Λ ∩ Λ = ∅ one has ℌ︹∪︹ = ℌ︹ ⊗ ℌ︹ there is a natural embedding of 프︹ into 프︹∪︹ given by , 프︹ ∋ 퐴 ↦ 퐴 ⊗ ퟙ︹ ∈ 프︹∪︹ ∗ i.e. 프︹ ≅ 프︹ ⊗ ퟙ︹ ⊆ 프︹∪︹ . e 퐶 -algebra of local observables is defined as ,(the union of the increasing family (프︹)︹∈픓(픾 프︋︎︂ = ﾌ 프︹ , (픓(픾∋︹ ∗ and its norm closure is the 퐶 -algebra of quasi-local observables 프 = 프︋︎︂.