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. An interaction (or potential) on the quantum spin system, defining a quantum spin model, is a map Φ ∶ 픓(픾) → 프 with the properties that Φ(푋) ∈ 프푋 ∗ and Φ(푋) = Φ(푋) for each 푋 ∈ 픓(픾). It is called finite range if there is an 푅 > 0 such that for all finite sets 푋 with diam(푋) > 푅 one has Φ(푋) = 0. e Hamiltonian with a finite set Λ and an interaction Φ is given by
︾ 퐻︹ = 퐻︹ = ﾜ Φ(푋). 푋⊆︹
It is a self-adjoint element of 프︹.
Time evolution e time evolution (Heisenberg dynamics) of a quantum spin system correspond- ing to an interaction Φ (respectively, the Hamiltonian 퐻︹) is given by the one- ︹ parameter group (훼푡 )푡∈ℝ of automorphisms of 프︹,
︹ ︈푡퐻﹂ −︈푡퐻﹂ 훼푡 (퐴) = 푒 퐴푒 , 퐴 ∈ 프︹.
It is well-defined by the spectral theorem (퐻︹ being self-adjoint). ¹e notation 푥 ∈ 픾 is used throughout this thesis instead of 푥 ∈ 푉(픾) if there is no room for confusion. 1.2.
Example 1.1 (e XY model). Let ℌ푥 = ℂ for each 푥 ∈ 픾. e anisotropic XY model in an external field is defined by the one- and two-body interaction
⎧ ⎪ 휇푥[(1 + 훾푥)휎푥휎푦 + (1 − 훾푥)휎푥휎푦] if 푋 = {푥, 푦} and dist(푥, 푦) = 1 ⎪ {Φ(푋) = ⎨ 휈 휎 if 푋 = {푥 ⎪ 푥 푥 ⎪ 0 otherwise ⎩ where 0 1 0 −i 1 0 = 휎 = 휎 = 휎 1 0 i 0 0 −1
are the Pauli matrices, and the real-valued sequences {휇푥}, {훾푥} and {휈푥} describe the coupling strength, anisotropy and external magnetic field strength, respect- ively. e Hamiltonian for a finite subset Λ of 픾 is then given by
푋푌 퐻︹ = ﾜ 휇푥[(1 + 훾푥)휎푥휎푦 + (1 − 훾푥)휎푥휎푦] + ﾜ 휈푥휎푥 (1.1) 푥,푦∈︹ 푥∈︹ ︒︓(︗,︘)=︈︃
and, formally, the Hamiltonian of the full system may be wrien as
푋푌 퐻 = ﾜ 휇푥[(1 + 훾푥)휎푥휎푦 + (1 − 훾푥)휎푥휎푦] + ﾜ 휈푥휎푥 푥,푦∈픾 푥∈픾 ︒︓(︗,︘)=︈︃
|︹| |︹| -Elements in 프︹, Λ ∈ 픓(픾), are 2 × 2 -matrices corresponding to polyno mials in the Pauli matrices and the 2×2 identity matrix. asi-local observables are given by uniform limits of such polynomials. For completeness and later convenience, the most important properties of the Pauli matrices are stated below. (i) e commutator between two Pauli matrices is given by
훼 훽 훾 ￮휎 , 휎 = 2i휖훼훽훾휎
where 휖훼훽훾 is the Levi Cività tensor.
훼 훽 (ii) e anti-commutator between two Pauli matrices is 휎 , 휎 � = 2훿훼훽ퟙ. iii) (휎 훼) = ퟙ, Tr(휎 훼) = 0)
훼 훽 훾 (iv) 휎 휎 = 훿훼훽ퟙ + i휖훼훽훾휎 C 1. I Chapter 2
Jordan-Wigner transformation
Jordan Wigner transformation, introduced by J and W in T [WJ28], has proven to be a valuable tool in studying spin chains and their (quantum) phase transitions in one dimensional systems, most prominently in the application to the quantum XY model in one dimension by L et al. [LSM61].
2.1 e Jordan Wigner transformation in one dimen- sion
Consider the XY model on ℤ as defined above. In particular, the following section 푋푌 will be concerned with the Hamiltonians 퐻[−푁,푁] for 푁 ∈ ℕ and make use of an equivalence between the Pauli matrices and fermionic operators. is method, known as Jordan Wigner transformation, dates back to J and W [WJ28] and was used by L, S and M [LSM61] in the exact solution of the anisotropic XY chain. e infinite model was studied by A and M [AM85] aer the correspondence between the Pauli and CAR algebras had been given by A [Ara84].
2.1.1 Jordan Wigner transformation For 푁 ∈ ℕ let
푁− 푁 푋푌 (퐻푁 ∶= 퐻[−푁,푁] = ﾜ 휇푥[(1 + 훾푥)휎푥휎푥+ + (1 − 훾푥)휎푥휎푥+] + ﾜ 휈푥휎푥 (2.1 푥=−푁 푥=−푁 As a first step towards establishing the Jordan Wigner transformation, one introduces the set of raising and lowering operators (hard core bosons)
− + ∗ (푏푥 = 휎푥 = 휎푥 + i휎푥 , 푏푥 = 휎푥 = 휎푥 − i휎푥 (2.2
C 2. JW
for each 푥 ∈ [−푁, 푁] ∩ ℤ. ∗ ey are locally anticommuting, {푏푥, 푏푥} = ퟙ, commuting on different laice ∗ ∗ ∗ sites, [푏푥, 푏푦] = [푏푥, 푏푦] = [푏푥, 푏푦] = 0 for 푥 ≠ 푦, and satisfy a hard core condition, ∗ 푏푥 = (푏푥) = 0 (mixed algebra of raising/lowering operators). Furthermore, they have the properties
∗ ∗ [휎푥, 푏푥] = 2푏푥, [휎푥, 푏푥] = −2푏푥 ( ∗ ∗ ∗ ∗ (2.3 휎푥푏푥 = −푏푥, 푏푥휎푥 = 푏푥, 휎푥푏푥 = 푏푥, 푏푥휎푥 = −푏푥 By definition, the raising/lowering operators preserve the local structure, and satisfy the relations
∗ ∗ ∗ ∗ ∗ 휎푥 = 푏푥 + 푏푥, 휎푥 = i(푏푥 − 푏푥), 휎푥 = 2푏푥푏푥 − ퟙ = 푏푥푏푥 − 푏푥푏푥 (2.4) In particular, one has for 푦 ≠ 푥
∗ ∗ 휎푥휎푦 + 휎푥휎푦 = 2(푏푥푏푦 + 푏푦푏푥) ( ∗ ∗ (2.5 휎푥휎푦 − 휎푥휎푦 = 2(푏푥푏푦 + 푏푦푏푥)
ese can be used to express 퐻푁 in terms of the 푏 operators.
푁− 푁 ∗ ∗ ∗ ∗ ∗ (퐻푁 = ﾜ 2휇푥[푏푥푏푥+ + 푏푥+푏푥 + 훾푥(푏푥푏푥+ + 푏푥+푏푥)] + ﾜ 휈푥(2푏푥푏푥 − ퟙ) (2.6 푥=−푁 푥=−푁
∗ In order to construct fermionic creation and annihilation operators 푐푗 and 푐푗, respectively, obeying the canonical anti-commutation relations (CARs)
∗ ∗ ∗ {푐푥, 푐푦} = 훿푥푦ퟙ, {푐푥, 푐푦} = {푐푥, 푐푦} = 0, for all 푥, 푦 ∈ [−푁, 푁] ∩ ℤ (2.7) one needs to break the local structure of the bosonic operators 푏 defined above. (∗) (∗) One possible choice is to define 푐−푁 ∶= 푏−푁 and
⎞ 푥− ⎞ ⎛ 푥− ⎛ ⎟⎟ ∗ ∗ ⎜ ⎜ 푐푥 ∶= ﾟ 휎푦 푏푥, 푐푥 = 푏푥 ﾟ 휎푦 , 푦 = −푁 + 1, … , 푁 (2.8) ⎜푦=−푁 ⎟ ⎜푦=−푁 ⎟ ⎝ ⎠ ⎝ ⎠ -e operator ∏푥− is usually referred to as string, kink or soliton oper 푆푥 = 푦=−푁 휎푦 ator¹ (especially in the physics literature) [Fra13].
¹Let |±⟩ be the two eigenstates of 휎 . en 푆푥 creates a kink at laice site 푥,
푆푥 (|+⟩ ⊗ ⋯ ⊗ |+⟩) = |−⟩ ⊗ ⋯ ⊗ |−⟩ ⊗ |+⟩ ⊗ ⋯ ⊗ |+⟩ 2.1. T J W
Φ(푥) Φ(푦) . −푁 푥 푦 푁
Figure 2.1: Graphical representation of the laice sites contributing to the phases Φ(푥) and Φ(푦) in the one dimensional Jordan Wigner transformation. ey are directly related to the string operator (see remark).
Lemma 2.1 (Jordan Wigner Transformation). e operators defined in (2.8) sat- isfy the canonical anticommutator relations (2.7). Proof. Without loss of generality let 푥 < 푦. en by relation 2.3 and the commut- ativity of the 푏 operators on different laice sites one has (see also figure 2.1 for a graphical representation of which sites contribute to the product in the Jordan Wigner transformation)
⎞ 푥− ⎞ ⎛ 푦− ⎞ ⎛ 푦− ⎞ ⎛ 푥− ⎛ ∗ ⎟⎟ ∗ ⎜ ⎟ ⎜ ∗ ⎟ ⎜ ⎜ ∗ ￍ휎푥푏푥휎푥 ﾟ 휎푧 = −푐푦푐푥ﻰￌﻰ푐푥푐푦 = ﾟ 휎푧 푏푥 푏푦 ﾟ 휎푧 = 푏푦 ﾟ 휎푧 ￋ ⎜푧=−푁 ⎟ ⎜푧=−푁 ⎟ ⎜ 푧=−푁 ⎟ ⎜푧=−푁 ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ =−푏푥 ⎝ ⎠
-Since ∗ ∗ ∏푥− ∏푥− ∗ and the s are locally anti com 푐푥푐푥 = 푏푥 푧=−푁 휎푧 푧=−푁 휎푧 푏푥 = 푏푥푏푥 푏 muting the 푐 operators inherit this property. By the hard core condition and ∗ ∗ essentially the same calculation as above one proves that {푐푥, 푐푦} = {푐푥, 푐푦} = 0 for all 푥, 푦 ∈ {−푁, … , 푁}. ■
Remark. Another, equivalent way of constructing fermionic operators 푐 from the hard core bosons 푏 is to define (c.f. [LSM61] and the discussion in section 2.2)
︈︾(푥) ∗ ∗ −︈︾(푥) 푐푥 ∶= 푒 푏푥, 푐푥 = 푏푥푒 (2.9) with ∑푥− ∗ and inverse transformation Φ(푥) = 휋 푦=−푁 푏푦푏푦
−︈︾(푥) ∗ ∗ ︈︾(푥) 푏푥 = 푒 푐푥, 푏푥 = 푐푥푒 . (2.10)
∗ is is due to the fact that the operators (푏푦푏푦)푦=−푁,…,푁 are mutually commuting and thus
푥− ⎞ 푥− 푥− 푥− ⎛ ﾜ ∗ ⎟ ﾟ ∗ ﾟ ﾟ ⎜ (exp i휋 푏푦푏푦 = exp i휋푏푦푏푦 = exp i휋(휎푦 + ퟙ) = (−휎푦 ⎜ 푦=−푁 ⎟ 푦=−푁 푦=−푁 푦=−푁 ⎝ ⎠ ♦ which is unitarily equivalent to the string operator 푆푥. C 2. JW
∗ ∗ ∗ ∗ Using the relations 푐푥푐푥 = 푏푥푏푥, 푐푥푐푥+ = −푏푥푏푥+ and 푐푥푐푥+ = 푏푥푏푥+, one gets
푁− 푁 ∗ ∗ ∗ ∗ ∗ .(퐻푁 = ﾜ 2휇푥[−푐푥푐푥+ − 푐푥+푐푥 + 훾푥(푐푥푐푥+ + 푐푥+푐푥)] + ﾜ 휈푥(2푐푥푐푥 − ퟙ 푥=−푁 푥=−푁 (2.11)
2.1.2 Extension to the infinite ain e extension of the Jordan Wigner transformation to the (two-sided) infinite chain ℤ reveals some subtleties that were studied by A and M in their discussion of ground states of the 푋푌-model on ℤ [Ara84, AM85]. It turns out ∗ 퐶퐴푅 ∗ ∗ 푃 that the 퐶 -algebra 프 generated by {푐푥, 푐푥 ∶ 푥 ∈ ℤ}, and the 퐶 -algebra 프 훼 ∗ generated by the Pauli spin matrices {휎푥 ∶ 훼 = 1, 2, 3, 푥 ∈ ℤ} are different 퐶 - subalgebras of an enlarged 퐶∗-algebra 프ﾧ. Only their even parts with respect to a certain automorphism of 프ﾧ coincide. e problem is that the infinite product . is not a quasi-local observable² ∞−∏ 푥= 휎푥 푃 푃 Define the automorphism Θ− ∶ 프 ↦ 프 ,
⎛ −푁 ⎞ ⎛ −푁 ⎞ ⎟⎟ ⎜ ⎜ Θ−(퐴) ∶= lim ﾟ 휎푥 퐴 ﾟ 휎푥 , 푁→∞ ⎟ 푥= ⎟ ⎜ 푥= ⎜ ⎝ ⎠ ⎝ ⎠ which corresponds to a rotation of all the spins on the le half of the laice 푥 ≤ 0 by 휋 around the 휎-axis. Indeed, a simple application of the (anti-)commutation relations of the Pauli matrices shows that for 훼 = 1, 2, ⎛ −푁 ⎞ ⎛ −푁 ⎞ ⎛ −푁 ⎞ 훼 ⎜ ⎟ 훼 ⎜ ⎟ 훼 ⎜ ⎟ 훼 Θ−(휎푥 ) = lim ﾟ 휎푦 휎푥 ﾟ 휎푦 = 휎푥 lim ﾟ 휎푦 = 휎푥 for 푥 > 0, 푁→∞ 푁→∞ ⎟ 푦= ⎟ ⎜ 푦= ⎟ ⎜ 푥= ⎜ ⎠ ⎝ￍﻰﻰￌﻰﻰￋ ⎠ ⎝ ⎠ ⎝
=ퟙ
⎛ −푁 ⎞ ⎛ −푁 ⎞ 훼 ⎜ ⎟ 훼 ⎜ ⎟ 훼 Θ−(휎푥 ) = lim ﾟ 휎푦 휎푥 ﾟ 휎푦 = −휎푥 for 푥 ≤ 0, 푁→∞ ⎟ 푦= ⎟ ⎜ 푦= ⎜ ⎝ ⎠ ⎝ ⎠ and Θ−(휎푥) = 휎푥 for all 푥 ∈ ℤ. Further, one can easily see that Θ− is an involu- tion, Θ− = ퟙ. Now let 푇 be such that
∗ 푃 푇 = ퟙ, 푇 = 푇, 푇퐴푇 = Θ−(퐴) ∀퐴 ∈ 프
.²Indeed, one has ∏−푁 for all , thus it cannot converge in norm ￏ 푥= 휎푥ￏ = 1 푛 ∈ ℕ 2.1. T J W
and define 프ﾧ as the 퐶∗-algebra generated by 프푃 and 푇. It can be decomposed into a direct sum 프ﾧ = 프푃푇 + 프푃푇 = 프푃 + 프푃푇 and with this decomposition at ﾧ hand, the automorphism Θ− can be extended to 프 by seing
푃 . Θ−(퐴 + 퐴푇) ∶= Θ−(퐴) + Θ−(퐴)푇, 퐴, 퐴 ∈ 프
Remark. Introducing 푇 and the enlarged 퐶∗-algebra 프ﾧ was necessary to adapt the Jordan Wigner transformation 2.8 to the case of an infinite chain. If , then ∏−푁 ∏−푁 for all Λ = [−푁, 푁] ∩ ℤ Θ−(퐴) = 푥= 휎푥 퐴 푥= 휎푥 = 푇퐴푇 푃. In particular ∏−푁 푃, so ﾧ 푃, and the above reduces to 퐴 ∈ 프 푇 = 푥= 휎푥 ∈ 프 프 = 프 the discussion in section 2.1.1. ♦ Having introduced the enlarged algebra 프ﾧ it is possible to define the creation ∗ ﾧ and annihilation operators 푐푥, 푐푥 ∈ 프 by
휎 + i휎 푥− 푥 푥 ⎧∏ + ∗ 푐푥 ∶= 푇푆푥휎푥 = 푇푆푥 ⎪ 휎푦 , if 푥 ≥ 2 푦= ⎪ 2 . 푆푥 = ⎨ퟙ , if 푥 = 1 휎푥 − i휎푥 ⎪ 푥 − 푐푥 ∶= 푇푆푥휎푥 = 푇푆푥 ⎪∏ 휎푦 , if 푥 ≤ 0 푦= ⎩ 2
Here one can see that was used to substitute ∏ . In particular has 푇 푥=−∞ 휎푥 푇 ∗ all the necessary properties to make the family of operators 푐푥, 푐푥, 푥 ∈ ℤ, fulfil canonical anticommutation relations. is follows from [푆푥, 푆푦] = 0, 푆푥 = ퟙ and Θ−(푆푥) = 푆푥 for all 푥, 푦 ∈ ℤ. Also notice that for 푥 ≤ 푦 one has ⎧ ⎫ ⎧ ⎫ ∏푥− ± ± ∏푥− ⎪푧= 휎푧 휎푦 , 푥 ≥ 2⎪ ⎪ 휎푦 푧= 휎푧 , 푥 ≥ 2 ⎪ ± ⎪ ± ⎪ ⎪ ± ⎪ ± 푆푥휎푦 = ⎨ 휎푦 , 푥 = 1⎬ = ⎨ 휎푦 , 푥 = 1⎬ = Θ−(휎푦 )푆푥 ⎪ 푥 ± ⎪ ⎪ ± 푥 ⎪ ⎪∏ 휎푧 휎푦 , 푥 ≤ 0⎪ ⎪Θ−(휎푦 ) ∏ 휎푧 , 푥 ≤ 0⎪ ⎭ 푧= ⎭ ⎩ 푧= ⎩ and for 푥 > 푦 ⎧ ⎫ ⎧ ⎫ ∏푥− ± ± ∏푥− ⎪푧= 휎푧 휎푦 , 푥 ≥ 2⎪ ⎪−Θ−(휎푦 ) 푧= 휎푧 , 푥 ≥ 2 ⎪ ± ⎪ ± ⎪ ⎪ ± ⎪ ± 푆푥휎푦 = ⎨ 휎푦 , 푥 = 1⎬ = ⎨ 휎푦 , 푥 = 1⎬ = −Θ−(휎푦 )푆푥 ⎪ 푥 ± ⎪ ⎪ ± 푥 ⎪ ⎪∏ 휎푧 휎푦 , 푥 ≤ 0⎪ ⎪ 휎푦 ∏ 휎푧 , 푥 ≤ 0⎪ ⎭ 푧= ⎭ ⎩ 푧= ⎩ Hence,
+ − + − + − + − ∗ 푐푥푐푥 = (푇푆푥휎푥 )(푇푆푥휎푥 ) = 푆푥Θ−(휎푥 )푇 푆푥휎푥 = 푆푥Θ−(휎푥 )푆푥휎푥 = 푆푥Θ−(휎푥 )Θ−(휎푥 )푆푥 − + − + − + = 푆푥 (ퟙ − Θ−(휎푥 )Θ−(휎푥 )) 푆푥 = ퟙ − 푆푥Θ−(휎푥 )푆푥휎푥 = ퟙ − 푆푥Θ−(휎푥 )푇 푆푥휎푥 + − ∗ = ퟙ − 푇푆푥휎푥 푇푆푥휎푥 = ퟙ − 푐푥푐푥, C 2. JW
and for 푥 < 푦 one gets
∗ − + − + − + + − 푐푥푐푦 = 푇푆푥휎푥 푇푆푦휎푦 = 푆푥Θ−(휎푥 )푆푦휎푦 = −푆푥푆푦휎푥 휎푦 = −푆푦푆푥휎푦 휎푥 ∗ − + − + − + = −푆푦Θ−(휎푦 )푆푥휎푥 = −푆푦Θ−(휎푦 )푇 푆푥휎푥 = −푇푆푦휎푦 푇푆푥휎푥 = −푐푦푐푥.
∗ ∗ Similarly it can be proven that {푐푥, 푐푦} = {푐푥, 푐푦} = 0. Let 프퐶퐴푅 denote the 퐶∗-subalgebra of 프ﾧ generated by those operators. It follows immediately, that
⎧ ∗ ⎧ ∗ ⎪푐푥 , if 푥 ≥ 1 ⎪푐푥 , if 푥 ≥ 1 Θ−(푐푥) = ⎨ Θ−(푐푥) = ⎨ ⎪−푐∗ , if 푥 ≤ 0 ⎪−푐∗ , if 푥 ≤ 0 ⎩ 푥 ⎩ 푥 Next, the relation between 프푃 and 프퐶퐴푅 is described. To do so, one has to introduce another involutive automorphism Θ of 프ﾧ, describing the rotation of all spins around the 휎-axis by 휋. By the construction of 프ﾧ it suffices to define
훼 훼 Θ(휎푥 ) = −휎푥 (훼 = 1, 2) Θ(휎푥) = 휎푥 Θ(푇) = 푇
for 푥 ∈ ℤ and extend it to an automorphism on 프ﾧ. Accordingly, the action on the fermionic operators is
∗ ∗ Θ(푐푥) = −푐푥, Θ(푐푥) = −푐푥 (푥 ∈ ℤ). Clearly, Θ = ퟙ, and by definition Θ leaves 프푃 and 프퐶퐴푅 invariant, i.e. for 퐴 ∈ 프푃/퐶퐴푅 also Θ(퐴) ∈ 프푃/퐶퐴푅. Θ can now be used to decompose both subalgebras into an even (Θ = 1) and 푃/퐶퐴푅 an odd part (Θ = −1). For an operator 퐴 ∈ 프 one has 퐴 = 퐴+ + 퐴− where 푃/퐶퐴푅 푃/퐶퐴푅 푃/퐶퐴푅 푃/퐶퐴푅 푃/퐶퐴푅 ∶ 퐴± = (퐴 ± Θ(퐴)), and 프 = 프+ + 프− with 프± = {퐴 ∈ 프 Θ(퐴) = ±퐴}. e relation between the spin and CAR algebras is given by
푃 퐶퐴푅 푃 퐶퐴푅 프+ = 프+ , 프− = 푇프− . (2.12) ﾪ ﾧ ﾧ Define another involutive automorphism Θ− ∶ 프 → 프 by
ﾪ 푃 Θ−(퐴 + 푇퐴) = 퐴 − 푇퐴, 퐴, ∈ 프 en ﾪ (Θ Θ−(퐴 + 푇퐴) = Θ (퐴 − 푇퐴) = Θ(퐴) − 푇Θ(퐴 ﾪ ﾪ ((Θ− (Θ(퐴) + 푇Θ(퐴)) = Θ− (Θ(퐴 + 푇퐴 = 2.2. G
ﾪ ﾪ ﾧ i.e. ΘΘ− = Θ−Θ on 프. Since 프ﾧ = 프푃 + 푇프푃 the extended algebra 프ﾧ can be decomposed into
ﾧ 퐶퐴푅 퐶퐴푅 퐶퐴푅 퐶퐴푅 프 = 프+ + 프− + 푇프+ + 푇프− 푃 퐶퐴푅 퐶퐴푅 프 = 프+ + 푇프−
e local Hamiltonian 퐻푁 (2.1) is quadratic in the spin operators, and there- fore lies in the positive subalgebra of 프푃. But the positive subalgebras of the spin and CAR algebras coincide, which concludes the discussion of the Jordan Wigner transformation in the case of an infinite chain.
2.2 Generalisations to graphs
e LSM ansatz 2.9 can be used to define a Jordan Wigner trans- formation on arbitrary (connected, finite) graphs. In this seing the Hamiltonian transforms to a more difficult one including statistical gauge fields which in gen- eral cannot be removed by gauge transformations.
Let 픾 = (푉, 퐸) be a finite, connected graph and {푐푥}푥∈픾 a fermionic field on 픾. Define the new operators
︈︾(푥) ∗ ∗ −︈︾(푥) 푎푥 = 푒 푐푥, 푎푥 = 푐푥푒 (2.13) with ∑ ∗ ∑ ∗ , for all . Φ(푥) = 휋 푧∈픾 휑(푥, 푧)푐푧푐푧 = 휋 푧≠푥 휑(푥, 푧)푐푧푐푧 휑(푥, 푧) ∈ ℝ 푥, 푧 ∈ 픾 e laer equality uses that, without loss of generality, one can assume 휑(푥, 푥) = ∗ ︈휙(푥,푥)푐푥푐푥 0, since 푒 푐푥 = 푐푥. ∗ Indeed, since 푐푥푐푥 is bounded, one has the norm-convergent series represent- ation
⎛ 푘 ⎞ ∗ (i훼) ︈훼푐푥푐푥 ⎜ ∗ 푘⎟ 푒 푐푥 = ퟙ + ﾜ (푐푥푐푥) 푐푥 = 푐푥 , 훼 ∈ ℝ ⎟ !푘≥ 푘 ⎜ ⎝ ⎠ ∗ ∗ ∗ ︈훼푐푥푐푥 due to the property 푐푥 = 0. e same applies to 푐푥푒 = 푐푥. By the canonical anti commutation relations the 푎 operators satisfy
∗ ︈︾(푥) ∗ −︈︾(푦) ︈︾(푥) ∗ −︈︾(푦) 푎푥푎푦 = 푒 푐푥푐푦푒 = 훿푥푦ퟙ − 푒 푐푦푐푥푒 ∗ ∗ ∗ ∗ ︈휋휑(푥,푦)푐푦푐푦 ∗ −︈휋휑(푥,푦)푐푦푐푦 ︈︾(푥) −︈︾(푦) ︈휋휑(푦,푥)푐푥푐푥 −︈휋휑(푦,푥)푐푥푐푥 ￍ푒 푐푥푒ﻰﻰﻰﻰﻰﻰￌﻰﻰﻰﻰﻰﻰￍ 푒 푐푦푒 푒 푒 ￋﻰﻰﻰﻰﻰﻰￌﻰﻰﻰﻰﻰﻰ훿푥푦ퟙ −ￋ =
︢휋휑(푥,푦) ∗ −︢휋휑(푦,푥) =푒 푐푦 =푒 푐푥 (2.14)
︈휋(휑(푥,푦)−휑(푦,푥)) ∗ −︈︾(푦) ︈︾(푥) = 훿푥푦ퟙ − 푒 푐푦푒 푒 푐푥 ︈휋(휑(푥,푦)−휑(푦,푥)) ∗ = 훿푥푦ퟙ − 푒 푎푦푎푥 C 2. JW for 푥, 푦 ∈ 픾. If 휑(푥, 푦) − 휑(푦, 푥) = 휗 mod 2 ∀푥 ≠ 푦, (2.15) then the 푎 operators are said to satisfy hard core anyonic statistics, i.e.
∗ ︈휋휗 ∗ 푎푥푎푦 = 훿푥푦ퟙ − 푒 푎푦푎푥 (2.16) e parameter 휗 ∈ [0, 1] interpolates between fermionic (휗 = 0) and hard-core (∗) bosonic (휗 = 1) statistics, and the hard-core property (푎푥 ) = 0 follows from the same property of the fermionic 푐 operators. In the calculation above the equalities
∗ ∗ ︈휋휑(푥,푦)푐푦푐푦 ∗ −︈휋휑(푥,푦)푐푦푐푦 ︈휋휑(푥,푦) ∗ 푒 푐푦푒 = 푒 푐푦 ∗ ∗ ︈휋휑(푦,푥)푐푥푐푥 −︈휋휑(푦,푥)푐푥푐푥 −︈휋휑(푦,푥) 푒 푐푥푒 = 푒 푐푥 have been used. ey follow from the following observation: in a basis where 1 0 0 0 0 1 푐∗ 푐 is diagonal, 푐∗ 푐 = , 푐 = , and 푐∗ = . en 푥 푥 푥 푥 0 0 푥 1 0 푥 0 0 푥 푥 푥
︈휋휑(푥,푦) ︈휋휑(푥,푦) ∗ ∗ ∗ 푒 0 0 1 0 푒 푒︈휋휑(푥,푦)푐푦푐푦 푐∗ 푒−︈휋휑(푥,푦)푐푦푐푦 = 푒︈휋휑(푥,푦)푐푦푐푦 푐∗ = = 푦 푦 0 1 0 0 0 0 푦 푦 푦 ︈휋휑(푥,푦) ∗ = 푒 푐푦. An analogous calculation yields the second equality. Unless stated otherwise it shall from now on be assumed that 휗 = 1, so that the 푎 operators describe hard-core bosons. Let 퐻 be the Hamiltonian of free fermions on 픾 with nearest neighbour hopping,
∗ ∗ 퐻 = ﾜ(푐푥푐푦 + 푐푦푐푥). 푥∼푦
en the transformation 2.13 yields the unitarily equivalent Hamiltonian
∗ ︈︾(푥) −︈︾(푦) ∗ ︈︾(푦) −︈︾(푥) ∗ −︈퐴(푥,푦) ∗ ︈퐴(푥,푦) 퐻 = ﾜ(푎푥푒 푒 푎푦 + 푎푦푒 푒 푎푥) = ﾜ(푎푥푒 푎푦 + 푎푦푒 푎푥) 푥∼푦 푥∼푦 (2.17) with a statistical gauge field
∗ ∗ 퐴(푥, 푦) = 휋 ﾜ ￮휑(푦, 푧) − 휑(푥, 푧) 푐푧푐푧 = 휋 ﾜ ￮휑(푦, 푧) − 휑(푥, 푧) 푎푧푎푧, (2.18) 푧≠푥,푦 푧≠푥,푦
∗ ∗ ∗ ︈훽푎푥푎푥 ∗ ︈훽푎푥푎푥 where it has been taken into account that 푎푥푒 = 푎푥 and 푒 푎푥 = 푎푥 for all (∗) 훽 ∈ ℝ due to the hard-core property (푎푥 ) = 0. 2.2. G
Statistical gauge transformations From calculation 2.14 it follows immediately that transformations of the form ̃ ︈︹(푥) with ∑ ∗ leave the particle statistics unchanged 푎푥 = 푒 푎푥 Λ(푥) = 휋 푧≠푥 휆(푥, 푧)푎푧푎푧 iff
휆(푥, 푦) = 휆(푦, 푥) mod 2 ∀푥 ≠ 푦. (2.19)
ey can be used to modify the statistical gauge field 퐴(푥, 푦) (2.18) while pre- serving particle statistics. Explicitly, the Hamiltonian 2.17 is unitarily equivalent to
∗ ︈︹(푥) −︈퐴(푥,푦) −︈︹(푦) ∗ ︈︹(푦) ︈퐴(푥,푦) −︈︹(푥) 퐻 = ﾜ(푎̃푥푒 푒 푒 푎̃푦 + 푎̃푦푒 푒 푒 푎̃푥) 푥∼푦 (2.20) ∗ −︈퐴(푥,푦)̃ ∗ ︈퐴(푥,푦)̃ = ﾜ(푎̃푥푒 푎̃푦 + 푎̃푦푒 푎̃푥) 푥∼푦 with ̃ ∗ 퐴(푥, 푦) = 휋 ﾜ ￮(휑(푦, 푧) − 휑(푥, 푧)) + (휆(푦, 푧) − 휆(푥, 푧)) 푎푧푎푧 푧≠푥,푦 (2.21) ∗ = 휋 ﾜ ￮(휑(푦, 푧) − 휑(푥, 푧)) + (휆(푦, 푧) − 휆(푥, 푧)) 푎̃푧푎̃푧 푧≠푥,푦
In principle one would like to use such statistical gauge transformations to get rid of the statistical gauge field in 2.17 altogether. is could be achieved by leing 휆(푤, 푧) = −휑(푤, 푧) for all 푤 ≠ 푧. But by condition 2.19 one would have
−휑(푤, 푧) = −휑(푧, 푤) mod 2 ∀푤 ≠ 푧 in contradiction to 2.15,
휑(푤, 푧) − 휑(푧, 푤) = 1 mod 2 ∀푤 ≠ 푧.
In the following section it will be shown that the occurrence of statistical gauge fields in Jordan Wigner transformations has to do with the structure of the underlying graph 픾.
2.2.1 Examples and the existence of special Jordan Wigner trans- formations It is an interesting question whether there are Jordan Wigner transformations on a given graph 픾 which map fermions to hard-core bosons without introdu- cing a statistical gauge field 퐴. Such fields correspond to higher order non-local C 2. JW interactions, ⎛ ⎞ ∗ ∗ −︈퐴(푥,푦) ∗ ⎜ −︈휋(휑(푦,푧)−휑(푥,푧))푎푧푎푧 ⎟ 푎푥푒 푎푦 = 푎푥 ﾟ 푒 푎푦 ⎜푧≠푥,푦 ⎟ ⎝ ⎠ ∗ −︈휋(휑(푦,푧)−휑(푥,푧)) ∗ = 푎푥 ﾟ ퟙ + (푒 − 1)푎푧푎푧푎푦 푧≠푥,푦
∗ ︈훼푎푧푎푧 ︈훼 ∗ ∗ since 푒 = 푒 푎푧푎푧 + (ퟙ − 푎푧푎푧), 훼 ∈ ℝ. Equation 2.18 shows that 퐴(푥, 푦) = 0 if and only if
휑(푥, 푧) = 휑(푦, 푧) mod 2 ∀푥 ∼ 푦, 푧 ≠ 푥, 푦. Together with the statistics condition 2.15, this yields the following system for the phases 휑(푥, 푦), 푥 ≠ 푦 ∈ 픾:
Adj(픾)(푥, 푦)(휑(푥, 푧) − 휑(푦, 푧)) = 0 mod 2 ∀푧 ≠ 푥, 푦 (2.22) 휑(푥, 푦) − 휑(푦, 푥) = 1 mod 2, with the adjacency matrix Adj(픾) of 픾. Leing 푒 denote the number of edges in 픾 and 푣 the number of vertices, there are in total 푣(푣 − 1) variables 휑(푥, 푦) and (푣(푣− equations defining the particle statistics together with 푒(푣 − 2) equations for the non-occurrence of a statistical gauge field. For the graph to be minimally connected, one has 푒 ≥ 푣 − 1, thus the system will in general be overdetermined for 푣 > 4, since 푣(푣 − 1) 푣(푣 − 1) − 푒(푣 − 2) ≤ − (푣 − 1)(푣 − 2) < 0 if 푣 > 4. 2 2 Definition 2.2 (Special Jordan Wigner transformations). A transformation of the above type satisfying the conditions 2.22 is henceforth called special Jordan Wigner transformation in this thesis.
Example 2.3 (Simple Paths). Let 픾 = 풫푁 be a simple path of length 푁 with ver- ,tices 푣, … , 푣푁 . en, as in the one dimensional Jordan Wigner transformation ⎧ ⎪1, if 푧 ≺ 푥 휑(푥, 푧) = Θ(푥, 푧) ∶= ⎨ ⎪0, if 푧 ⪰ 푥 ⎩ is a solution to 2.22. is can be seen by noting that
1, if 푥 ≺ 푧 Θ(푧, 푥) = = Θ(푥, 푧) + 1 − 훿 mod 2 (2.23) 0, if 푥 ⪰ 푧 푥푧 2.2. G
2 풞푁
푁 . 푌 풫 푁 푣푁 . 0 푣 . 1 2 3 1 (a) (b) (c)
Figure 2.2: e graphs considered in examples 2–4: (a) Simple path 풫푁 (b) Cycle graph 풞푁 (c) 푌-graph.
so Θ(푧, 푥) − Θ(푥, 푧) = 1 − 훿푥푧 = 1 mod 2 for 푥 ≠ 푧. Also, if 푥 ∼ 푦, one has
−훿푧푦, if 푥 ≺ 푦 Θ(푥, 푧) − Θ(푦, 푧) =  = 0 for 푧 ≠ 푥, 푦. 훿푧푥, if 푦 ≺ 푥
Hence there exists a special Jordan Wigner transformation on simple paths 풫푁 , which is of course not surprising, since this is the exact same case as in one dimension. ♦
Example 2.4 (Cycle Graph). Let 픾 be a cycle graph 풞푁 , 푁 ≥ 3, then the system of equations 2.22 determining the phases 휑(푥, 푦) is given by
휑(1, 푧) = 휑(2, 푧) mod 2 ∀푧 ≥ 3 휑(2, 푧) = 휑(3, 푧) mod 2 ∀푧 ≥ 4, 푧 ≤ 1 ⋮ 휑(푁 − 1, 푧) = 휑(푁, 푧) mod 2 ∀푧 ≤ 푁 − 2 휑(푁, 푧) = 휑(1, 푧) mod 2 ∀2 ≤ 푧 ≤ 푁 − 1 together with the statistics conditions
휑(푥, 푦) = 휑(푦, 푥) + 1 mod 2 ∀푥 ≠ 푦 ∈ {1, … , 푁}
In particular, one has
휑(1, 3) = 휑(2, 3) = 휑(3, 2) + 1 = ⋯ = 휑(푁, 2) + 1 = 휑(1, 2) + 1 = 휑(2, 1) = 휑(3, 1) = 휑(1, 3) + 1 meaning that one can construct the contradiction 0 = 1 out of part of the above conditions, so there exists no special Jordan Wigner transformation on 풞푁 for any 푁 ≥ 3. ♦ C 2. JW
Example 2.5 (푌-graph). Let 픾 be a 푌-graph (sometimes also called 3-legged star graph or claw), i.e. the graph with vertex set 푉 = {0, 1, 2, 3} and edge set 퐸 = {{0, 푗} ∶ 푗 = 1, 2, 3}. e system of equations 2.22 reads (everything mod 2)
휑(0, 2) = 휑(1, 2) 휑(0, 1) = 휑(2, 1) 휑(0, 1) = 휑(3, 1) 휑(0, 3) = 휑(1, 3) 휑(0, 3) = 휑(2, 3) 휑(0, 2) = 휑(3, 2)
휑(푥, 푦) = 휑(푦, 푥) + 1 ∀푥 ≠ 푦
But then
휑(0, 2) = 휑(1, 2) = 휑(2, 1) + 1 = 휑(0, 1) + 1 = 휑(3, 1) + 1 = 휑(1, 3) = 휑(0, 3) = 휑(2, 3) = 휑(3, 2) + 1 = 휑(0, 2) + 1
So 0 = 1, contradiction. is shows that there cannot be a special Jordan Wigner transformation on the 푌-graph. ♦
eorem 2.6. Let 픾 = (푉, 퐸) be a finite, connected graph. en there exists a special Jordan Wigner transformation in the above sense if and only if the graph is
.a simple path, 픾 = 풫|푉|−
Proof. If 픾 = 풫|푉|−, then the special Jordan Wigner transformation is given in example 2.3. For the converse assume that 픾 is not a simple path. en there are two cases:
(i) 픾 is a cycle graph 풞|푉| (ii) 픾 contains a 푌-graph Assume 픾 is a cycle graph. It was proved in example 2.4 that the system 2.22 has no solution, so there exists no special Jordan Wigner transformation.
erefore assume that 픾 ≠ 풫|푉|+, 풞|푉|. In this case it contains a 푌-subgraph .({픾푌 = ({푣, 푣, 푣, 푣}, {{푣, 푣푗} ∶ 푗 = 1, 2, 3 Assuming further that the system of equations 2.22 admits a solution on 픾, the equations in particular have to be true for 픾푌 . But as shown in example 2.5 these equations are enough to produce the contradiction
휑(푣, 푣) = 휑(푣, 푣) = 휑(푣, 푣) + 1 = 휑(푣, 푣) + 1 = 휑(푣, 푣) + 1
,휑(푣, 푣) = 휑(푣, 푣) = 휑(푣, 푣) = 휑(푣, 푣) + 1 = 휑(푣, 푣) + 1 = i.e. 0 = 1. Hence the system 2.22 cannot have solutions on 픾. ■ 2.2. G
2.2.2 e two-dimensional Jordan Wigner transformation
∗ Let Λ = [−퐿, 퐿] ∩ ℤ be a square in ℤ centred at 0, and {푐푥, 푐푥 ∶ 푥 ∈ Λ} be a family of fermionic annihilation/creation operators on Λ. e canonical basis .vectors are denoted by 푒 and 푒 As an immediate consequence of theorem 2.6 there cannot exist a special Jordan Wigner transformation on the two-dimensional laice Λ. Nevertheless, there do exist Jordan Wigner transformations leading to statistical gauge fields. ey are solutions to the condition 2.15 (for 휗 = 1),
휑(푥, 푦) − 휑(푦, 푥) = 1 mod 2 ∀푥 ≠ 푦, (2.24)
and shall be discussed in the following two subsections.
e FW solution Let Arg(푧) = Im Log 푧 ∈ [−휋, 휋) be the relative angle between 푧 (identifying 푧 = 푧 + i푧 with 푧 = (푧, 푧) ∈ Λ) and an arbitrary reference axis. en by the property
Arg(푧) = Arg(−푧) + 휋 mod 2휋
one can see that 휑(푥, 푧) = 휋 Arg(푧 − 푥) satisfies condition 2.24. e resulting Jordan Wigner transformation goes back to F and W [Fra89, Wan92]. Under this transformation a Hamiltonian of hard core bosons with nearest- neighbour hopping ﾜ ﾜ ∗ ∗ 퐻 = 푏푥푏푥+푒푗 + 푏푥+푒푗 푏푥 푥∈︹ 푗=,
transforms to the Hamiltonian
ﾜ ﾜ ∗ ︈퐴(푥,푥+푒푗) ∗ −︈퐴(푥,푥+푒푗) 퐻 = 푐푥푒 푐푥+푒푗 + 푐푥+푒푗 푒 푐푥 푥∈︹ 푗=,
of fermions with nearest-neighbour hopping coupled to a statistical gauge field 퐴(푥, 푦) defined on the edges (푥, 푦) of the laice, with
∗ 퐴(푥, 푥 + 푒푗) = 휋 ﾜ ￮휑(푥 + 푒푗, 푧) − 휑(푥, 푧) 푐푧푐푧. 푧≠푥,푥+푒푗
e statistical gauge field 퐴 generates a flux through elementary plaquees +푃 of the laice Λ. Let 푃 be such an elementary plaquee with corners 푥, 푥+푒, 푥 C 2. JW
Φ(푦) 푥 + 푒 푥 + 푒 + 푒 푦
푥 푃 휕푃 . 푥 + 푒 Φ(푥) 푥
. (a) (b)
Figure 2.3: Jordan Wigner transformation in two-dimensional laices: (a) e laice sites contributing to the phases in the A solution (b) Elementary plaquee in Λ.
푒 + 푒, and 푥 + 푒 (see figure 2.3). en the statistical flux through 푃 is given by ∑ , explicitly 퐵푃 = ℓ∈휕푃 퐴(ℓ)
(퐵푃 = 퐴(푥, 푥 + 푒) + 퐴(푥 + 푒, 푥 + 푒 + 푒) − 퐴(푥 + 푒, 푥 + 푒 + 푒) − 퐴(푥, 푥 + 푒 ∗ 휋 ￮휑(푥 + 푒, 푥) − 휑(푥 + 푒, 푥) 푐푥푐푥 = ∗ 휋 ￮휑(푥, 푥 + 푒) − 휑(푥 + 푒 + 푒, 푥 + 푒) 푐푥+푒 푐푥+푒 + ∗ 휋 ￮휑(푥 + 푒, 푥 + 푒 + 푒) − 휑(푥 + 푒, 푥 + 푒 + 푒) 푐푥+푒+푒 푐푥+푒++푒 + ∗ 휋 ￮휑(푥 + 푒 + 푒, 푥 + 푒) − 휑(푥, 푥 + 푒) 푐푥+푒 푐푥+푒 + 휋 푐∗ 푐 + 휋 푐∗ 푐 + 휋 푐∗ 푐 − 휋 푐∗ 푐 = 푥 푥 푥+푒 푥+푒 푥+푒+푒 푥+푒+푒 푥+푒 푥+푒
e A solution In his Ph.D. thesis M. A proposed another solution to the two dimensional Jordan Wigner transformation³ [Azz93]. It uses a more natural generalisation of the one-dimensional solution, with a phase function taking only the values 0 or
³Apparently unaware of this result, S published the same solution a few years later [Sha95]. His conclusions were wrong in several points, though [BA01]. 2.2. G
1,
(휑(푥, 푧) = Θ(푥 − 푧)(1 − 훿푥푧 ) + Θ(푥 − 푧)훿푥푧 . (2.25
Here, 푥 = (푥, 푥), 푧 = (푧, 푧) and Θ is the step function ⎧ ⎪1 if − 퐿 ≤ 푧푗 < 푥푗 Θ(푥푗 − 푧푗) = ⎨ . ⎪0 if 푥 ≤ 푧 ≤ 퐿 ⎩ 푗 푗 To check that condition 2.24 is indeed satisfied one only needs to recall the prop- erty 2.23 of the step function to see that
휑(푧, 푥) = (Θ(푥 − 푧) + 1 − 훿푥푧 )(1 − 훿푥푧 ) + (Θ(푥 − 푧) + 1 − 훿푥푧 )훿푥푧 Θ(푥 − 푧)(1 − 훿푥푧 ) + (1 − 훿푥푧 ) + Θ(푥 − 푧)훿푥푧 + (1 − 훿푥푧 )훿푥푧 =
휑(푥, 푧) + (1 − 훿푥푧 ) + (1 − 훿푥푧 )훿푥푧 =
휑(푥, 푧) + 1 − 훿푥푧 훿푥푧 = 휑(푥, 푧) + 1 if 푥 ≠ 푧 = en one has
⎞ 푥− 퐿 푥− ⎛ ﾜ ∗ ⎜ ﾜ ﾜ ∗ ﾜ ∗ ⎟ (Φ(푥) = 휋 휑(푥, 푧)푐푧푐푧 = 휋 푐(푧,푧)푐(푧,푧) + 푐(푥,푧)푐(푥,푧 푧≠푥 ⎜푧 =−퐿 푧 =−퐿 푧 =−퐿 ⎟ ⎠ ⎝ which results in a statistical gauge field of the form
⎞ 퐿 푥− ⎛ ⎜ ﾜ ∗ ﾜ ∗ ⎟ (퐴(푥, 푥 + 푒) = 휋 푐(푥,푧)푐(푥,푧) + 푐(푥+,푧)푐(푥+,푧 ⎟ 푧 =푥 + 푧 =−퐿⎜ ⎠ ⎝ 퐴(푥, 푥 + 푒) = 0 in the transformed fermionic Hamiltonian, and thus the statistical flux ∗ ∗ . 퐵푃 = 휋 푐푥+푒 푐푥+푒 − 푐푥+푒 푐푥+푒 It is interesting to note that even though the two Jordan Wigner transform-
ations produce a different flux 퐵푃, the two transformed Hamiltonians are still unitarily equivalent and the corresponding statistical gauge transformation (in the sense of 2.19) is given by Arg 휆(푥, 푦) = − 휋 (푧 − 푥) + Θ(푥 − 푧)(1 − 훿푥푧 ) + Θ(푥 − 푧)훿푥푧 Remark. A’s solution 2.25 has an immediate generalisation to higher di- mensional laices, for instance in three dimensions, Λ = [−퐿, 퐿] ∩ ℤ, the function 휑(푥, 푧) would read [BA01, Koc95, HZ93]
휑(푥, 푧) = Θ(푥 − 푧)(1 − 훿푥푧 ) + Θ(푥 − 푧)(1 − 훿푥푧 )훿푥푧
Θ(푥 − 푧)(1 − 훿푥푧 )훿푥푧 훿푥푧 + ♦ C 2. JW
푥 = (3, 5)
표 .
휃 = 1
Figure 2.4: Polar coordinates on the Cayley tree.
2.2.3 A Jordan Wigner transformation for tree graphs Having discussed the Jordan Wigner transformation for two-dimensional laices, it is straightforward to construct such a transformation for Cayley trees 퐵푧(푁) by representing each vertex in terms of suitable polar coordinates and using a modification of the A solution. Invoking theorem 2.6, also in this case there will be a statistical gauge field in the transformed Hamiltonian.
To this aim one assigns each vertex 푥 of the tree the coordinates 푥 = (푟푥, 휃푥), where 푟푥 = dist(푥, 표) ∈ {0, … , 푁} denotes the shell in which 푥 is situated and
휃푥 ∈ {1, … , 풩푟푥 } is the “angle” measured from some reference path (see figure 2.4). en, given a family of hard-core bosonic operators 푏 on 픹푧(푁), it is possible ︈︾(푥) to define Jordan-Wigner fermions via 푐푥 = 푒 푏푥 with ∗ Φ(푥) = 휋 ﾜ 휑(푥, 푧)푏푧푏푧 (2.26) 푧≠푥
⎞ 푟푥− 풩푟푥 풩푟푥 ⎛ ⎜ ∗ ﾜ ﾜ ∗ ﾜ ∗ ⎟ (2.27) = 휋 푏표푏표 + 푏(푟푧,휃푧)푏(푟푧,휃푧) + 푏(푟푥,휃푧)푏(푟푥,휃푧) , ⎟ 푟 = 휃 = 휃 =휃 ⎜ ⎠ 푧 푧 푧 푥+ ⎝ i.e. 휑(푥, 푧) = Θ(푟푥 − 푟푧)(1 − 훿푟푥푟푧 ) + Θ(휃푥 − 휃푧)훿푟푥푟푧 . e verification of condition 2.24 is done by a calculation analogous to the one in the two-dimensional case. e Hamiltonian of nearest neighbour hopping hard-core bosons is then unit- arily equivalent to
∗ ︈퐴(푥,푦) ∗ −︈퐴(푥,푦) 퐻 = ﾜ 푐푥푒 푐푦 + 푐푦푒 푐푥 푥∼푦 2.3. T WS with statistical gauge field (assuming 푥 ∼ 푦, 푥 ≺ 푦)
풩 휃푥− 푟푦 ﾜ ∗ ﾜ ∗ 퐴(푥, 푦) = 휋 푎(푟푥,휃푧)푎(푟푥,휃푧) + 휋 푎(푟푦,휃푧)푎(푟푦,휃푧). 휃푧= 휃푧=휃푦+
2.3 e WS approa
An alternative approach to a generalisation of the one-dimensional Jordan Wigner transformation⁴ is based upon the observation that the Pauli matrices 휎 and 휎 are generators of a 2×2 matrix representation of the Clifford algebra ℭ픩(2), since 훼 훽 {휎 , 휎 } = 2훿훼훽ퟙ.
Clifford algebras in a nutshell Let 푑 ∈ ℕ. e Clifford algebra ℭ픩(2푑) is the algebra generated by 2푑 elements satisfying the Clifford algebra relation
푎 푏 푎 푏 푏 푎 {Γ ,Γ } = Γ Γ + Γ Γ = 2훿푎푏ퟙ, 푎, 푏 ∈ {1, … , 2푑} (2.28)
us the Clifford algebra consists of
ퟙ, Γ 푎, 푎 = 1, … , 2푑 Γ 푎Γ 푏, 1 ≤ 푎 < 푏 ≤ 2푑 Γ 푎Γ 푏Γ 푐, 1 ≤ 푎 < 푏 < 푐 ≤ 2푑 ⋮ Γ ⋯ Γ 푑
e special element Γ 푑+ = (−i)푑Γ ⋯ Γ 푑 with the property (Γ 푑+) = ퟙ = {anticommutes with all the other generators of the Clifford algebra, {Γ 푎,Γ 푑+ 0 for all 푎 = 1, … , 2푑. One can show that the antisymmetric products of two Γ’s,
푎푏 푎 푏 푎 푏 (Γ = ︈ [Γ ,Γ ] = −iΓ Γ , 1 ≤ 푎 < 푏 ≤ 2푑 (2.29 generate a representation of SO(2푑), and for 1 ≤ 푎 < 푏 ≤ 2푑 + 1 a representation of SO(2푑 + 1) [Geo99].
⁴N [Nam50] used a similar argument in the one dimensional case, which is why W and S [Wos82, Szc85] refer to this method as the “Nambu trick”. C 2. JW
푑 푑 e Clifford algebra ℭ픩(2푑) can be represented by the algebra 픐푑 (ℂ) of 2 ×2 matrices, and generators satisfying the above relation for their anticommutator. For 푑 = 1 these are usually represented by the two Pauli matrices 휎 and 휎 (in this case, the element Γ is represented by 휎, since 휎 = −i휎휎), whereas in the case 푑 = 2 a representation of the ℭ픩(4) generators is given by the Dirac Gamma .matrices Γ ,…,Γ For later use, some elementary properties of the elements Γ 푎푏 are stated here: [Γ 푎,Γ 푏푐] = 0 for 푎 ≠ 푏, 푐 (2.30) {Γ 푎,Γ 푎푏} = 0 for 푎 ≠ 푏 (2.31) [Γ 푎푏,Γ 푐푑] = 0 for (푎, 푏) ≠ (푐, 푑) (2.32) {Γ 푎푏,Γ 푎푐} = 0 for 푏 ≠ 푐 and 푎 ≠ 푏, 푐 (2.33)
2.3.1 Link operators and their algebra Let 퐻 be the Hamiltonian of free fermions, described by fermionic creation and ∗ annihilation operators 푐푥, 푐푥 on a finite, symmetric digraph⁵ 핃 = (Λ, 퐸),
∗ ∗ ∗ 퐻 = 2휇 ﾜ 푐푥푐푦 + 푐푦푐푥 + ﾜ 휈푥(2푐푥푐푥 − ퟙ). (2.34) 푥푦∈퐸 푥∈︹
e parameter 휇 ∈ ℝ models the hopping strength to the nearest neighbour laice site, and {휈푥} ⊂ ℝ describe an on-site external potential in which the fermions move. It will later be assumed that the external potential is given by i.i.d. random variables (see chapter 3.2). e fermionic operators can be expressed in terms of two self-adjoint Major- ana operators for each site,
∗ ∗ 휉푥 = 푐푥 + 푐푥 and 휂푥 = −i(푐푥 − 푐푥), (2.35) with the properties 휁∗ = 휁 푥 푥 (2.36) {휁푥, 휍푦} = 2훿푥푦훿휁휍 with 휁, 휍 ∈ {휉, 휂}
that is, the operators generate a representation of ℭ픩(2|Λ|). In particular, 휁푥 = ퟙ for all 푥 ∈ Λ and 휁 = 휉, 휂. ∗ en, substituting 푐푥 = (휉푥 − i휂푥) and 푐푥 = (휉푥 + i휂푥) in the Hamiltonian 퐻 yields
퐻 = 휇 ﾜ i휉푥휂푦 − i휂푥휉푦 + ﾜ 휈푥i휉푥휂푥 푥푦∈퐸 푥∈︹
⁵at is, a directed graph 핃 = (Λ, 퐸) with the property that if (푥, 푦) ∈ 퐸, then also (푦, 푥) ∈ 퐸 2.3. T WS
휉 .
휂
Figure 2.5: Example of the double graph for a two-dimensional square laice
To allow a more compact description of the products of two Majorana fer- mions on neighbouring vertices and an easier framework for the proof of his theorem, S used the notion of a double laice (see figure 2.5) in his pa- per [Szc85].
Definition 2.7. e (directed) double laice/graph 핃ﾪ is the directed graph with ﾪ ̃ ̃ vertex set Λ = Λ × {휉, 휂} and edge set 퐸, where (푥휁, 푦휍) = ((푥, 휁), (푦, 휍)) ∈ 퐸 if one of the following holds: (i) 푥푦 ∈ 퐸, or (ii) 푥 = 푦 and 휁 ≠ 휍.
Operators defined on the edges of the (double) graph are usually called link ̃ operators. Given a family of link operators {푆(ℓ) ∶ ℓ ∈ 퐸}, and a path 훾 = ℓ∘⋯∘ℓ푛 of length 푛 ∈ ℕ, one defines the associated path operator as
푛− (푆(훾) = (−i) 푆(ℓ) ⋯ 푆(ℓ푛
(for a degenerate path consisting of only one vertex 푣 one sets 푆(푣) = iퟙ).
Definition 2.8 (Link algebra). Let {푆(ℓ) ∶ ℓ ∈ 퐸}̃ be a family of operators defined on the edges of the double laice (link operators). ey satisfy the link algebra if they have the following properties: i) 푆(ℓ)∗ = 푆(ℓ), (푆(ℓ)) = ퟙ for all ℓ ∈ 퐸̃)
(ii) {푆(ℓ), 푆(ℓ′)} = 0 if the edges ℓ, ℓ′ ∈ 퐸̃ have one common vertex, and [푆(ℓ), 푆(ℓ′)] = 0 otherwise
(iii) ∏ Tr 푥∈︹ 푆(푥휉, 푥휂) = 0 (iv) If 훾 is closed, then 푆(훾) = iퟙ.
ﾪ Example 2.9. e operators on 핃 defined by 푆(푥휁, 푦휍) = i휁푥휍푦, where 휁, 휍 ∈ {휉, 휂} and 휉푥, 휂푥 being a family of Majorana fermions on Λ as above, satisfy the link algebra. C 2. JW
e Hamiltonian 퐻, expressed in terms of these link operators, is given by
퐻 = 휇 ﾜ 푆(푥휉, 푦휂) − 푆(푥휂, 푦휉) + ﾜ 휈푥푆(푥휉, 푥휂). 푥푦∈퐸 푥∈︹
e first two defining properties (i)-(ii) follow immediately from the Major- ana algebra 2.36. To see that (iv) holds, let 훾 = ℓ ∘ ⋯ ∘ ℓ푁 be a closed path of ﾪ 푖 푖+ 푁 .(( length 푁 in 핃, ℓ푖 = ((푥푖, 휁 ), (푥푖+, 휁 )), 푖 = 1, … , 푁 − 1, ℓ푁 = ((푥푁 , 휁 ), (푥, 휁 en
푁− 푁− 푁 푁 ( 푆(훾) = (−i) (i휁푥 휁푥 )(i휁푥 휁푥 ) ⋯ (i휁푥푁− 휁푥푁 )(i휁푥푁 휁푥 푁− 푁 푁 i) i 휁푥 (휁푥 ) ⋯ (휁푥푁 ) 휁푥 = iퟙ−) = Finally, one has
ﾟ ﾟ ﾟ ∗ ﾟ ∗ Tr 푆(푥휉, 푥휂) = Tr (i휉푥휂푥) = Tr (2푐푥푐푥 − ퟙ) = Trℌ푥 (2푐푥푐푥 − ퟙ) = 0 푥∈︹ 푥∈︹ 푥∈︹ 푥∈︹
∗ since the operators {2푐푥푐푥 − ퟙ}푥∈︹ are mutually commuting with eigenvalues ±1 of the same multiplicity. ♦
S now proved in [Szc85] that the properties of the link operators (definition 2.8) determine the link algebra uniquely up to unitary isomorphisms. is result can be used to prove an implicit version of the Jordan Wigner trans- formation by finding a family of operators satisfying the link algebra. e main result is a representation of the link algebra in terms of higher-dimensional gamma matrices with appropriate constraints that reduce the dimensionality to the re- quired two per laice site describing fermionic degrees of freedom. e precise statement of the theorem and its proof are presented in the following. e next chapter will then deal with an application of this method to a square laice.
eorem 2.10 (Szczerba [Szc85]). Let 핃 = (Λ, 퐸) be a directed graph and 핃ﾪ = (Λ,̃ 퐸)̃ its associated double graph. en, given a set {푆′(ℓ) ∶ ℓ ∈ 퐸}̃ of link operators on a finite-dimensional Hilbert space ℌ satisfying the link algebra, there exists a family of Majorana operators {휉푥, 휂푥 ∶ 푥 ∈ Λ} on ℌ obeying the relations 2.36, such that
′ ̃ 푆 (ℓ) = i휁푥휍푦 , ℓ = (푥휁, 푦휍) ∈ 퐸, 휁, 휍 ∈ {휉, 휂} (2.37)
If {푆(ℓ) ∶ ℓ ∈ 퐸}̃ denotes the link operators from example 2.9, there is a unitary transformation 푈 with
푈푆(ℓ)푈∗ = 푆′(ℓ) for all ℓ ∈ 퐸̃ (2.38) 2.3. T WS
e proof of theorem 2.10 proceeds in two steps by reduction of the problem to a rooted spanning tree of 핃, where the natural ordering on rooted trees can be used to construct the required operators 휉 and 휂, and later extended to the whole graph. ̃ ̃ ̃ Lemma 2.11. Let 핋 = (Λ, 퐸핋) be a directed tree on Λ with an arbitrary vertex ′ ̃ fixed as root 표̃ = 표휉, and let {푆 (ℓ) ∶ ℓ ∈ 퐸핋} be a family of link operators on a finite-dimensional Hilbert space ℌ with properties (i)-(ii) and (iii) replaced by the condition ′ ﾟ ′ ′ Tr푆 (훾표휂 ) 푆 (훾푥휉 )푆 (훾푥휂 ) = 0, (2.39) 푥≠표 ﾪ where 훾푥휁 denotes the unique path from the root of 핋 to the vertex 푥휁 (property (iv) holds automatically, since a tree does not contain any cycles). en there
exists a family of operators {휉푥, 휂푥 ∶ 푥 ∈ Λ} obeying the Clifford relations 2.36, such that ′ ̃ 푆 (ℓ) = i휁푥휍푦 , ℓ = (푥휁, 푦휍) ∈ 퐸핋, 휁, 휍 ∈ {휉, 휂} (2.40) ′ ̃ Proof [Szc85]. Consider the family of path operators {푆 (훾푥̃ ) ∶ 푥̃ ≠ 표̃ ∈ 퐸핋}. ey have the properties ′ ∗ ′ 푆 (훾푥̃ ) = 푆 (훾푥̃ ) ′ ′ (2.41) {푆 (훾푥̃ ), 푆 (훾푦̃ )} = 2훿푥̃푦̃ for 푥,̃ 푦̃ ≠ 표̃ A proof of this and some other useful properties of the path operators are presen- ted in appendix A. Defining 푆 = i |︹|−푆′(훾 ) ∏ 푆′(훾 )푆′(훾 ), by means of 표휂 푥≠표 푥휉 푥휂 2.41 one has ′ ′ ﾟ ′ ′ ′ |︹|− ′ ﾟ−|︹| ∗ 푆 = (−i) 푆 (훾푥휂 )푆 (훾푥휉 )푆 (훾표휂 ) = (−i) 푆 (훾표휂 ) 푆 (훾푥휂 )푆 (훾푥휉 ) 푥≠표 푥≠표 ′ ′ ′ |︹|− ﾟ−|︹| = (−i) 푆 (훾표휂 )(−1) 푆 (훾푥휉 )푆 (훾푥휂 ) = 푆, 푥≠표 and ′ ′ |︹|− ′ ﾟ ′ ′ |︹|− ′ ﾟ 푆 = i 푆 (훾표휂 ) 푆 (훾푥휉 )푆 (훾푥휂 )i 푆 (훾표휂 ) 푆 (훾푥휉 )푆 (훾푥휂 ) 푥≠표 푥≠표 ′ ′ ′ ﾟ−|︹| = (−1) 푆 (훾표휂 ) 푆 (훾푥휉 )푆 (훾푥휂 ) 푥≠표 ′ ′ ′ ′ ﾟ−|︹| = (−1) 푆 (훾푥휉 )푆 (훾푥휂 )푆 (훾푥휉 )푆 (훾푥휂 ) 푥≠표 |︹|− ﾟ ′ ′ −|︹| = (−1) (−1) 푆 (훾푥휉 ) 푆 (훾푥휂 ) = ퟙ 푥≠표 C 2. JW
Further, by assumption 2.39, Tr푆 = 0. erefore there exists a unitary auto- morphism 푊 of ℌ that diagonalises 푆,
−ퟙ 0 푊푆푊 ∗ = 0 ퟙ
e corresponding eigenspaces are denoted by ℌ±. Now for each 푥̃ ≠ 표̃ the path ′ ′ operators 푆 (훾푥̃ ) commute with S, [푆, 푆 (훾푥̃ )] = 0, so that in the ONB where 푆 is diagonal the path operators are block diagonal,
− ′ ∗ 푇푥̃ 0 푊푆 (훾푥̃ )푊 = + , 푥̃ ≠ 표.̃ 0 푇푥̃
By the properties 2.41 it follows immediately that for all 푥,̃ 푦̃ ≠ 표̃ one has
# ∗ # (푇푥̃ ) = 푇푥̃ # # , # ∈ {+, −}. (2.42) {푇푥̃ , 푇푦̃ } = 2훿푥̃푦̃ It is a basic result in the theory of operator algebras (cf. eorem 5.2.5 in [BR02]) # that the operators 푇푥̃ are uniquely determined up to unitary transformations. Hence, there exists a unitary isomorphism 푉 ∶ ℌ− → ℌ+ such that
− ∗ + 푉푇푥̃ 푉 = 푇푥̃ for all 푥̃ ≠ 표휉, 표휂 (2.43)
Here it is important that from 2.42 it does not follow the existence of such a unitary transformation for 푥̃ = 표휂. One rather gets from
− − − −ퟙ 0 푇표 ∏ 푇푥 푇푥 0 푊푆푊 ∗ = i |︹|− 휂 푥≠표 휉 휂 = 0 ퟙ 0 푇+ ∏ 푇+ 푇+ 표휂 푥≠표 푥휉 푥휂 푇− ∏ 푇− 푇− 0 i |︹|− 표휂 푥≠표 푥휉 푥휂 = 0 푇+ 푉 ∏ 푇 + 푇+ 푉 ∗ 표휂 푥≠표 푥휉 푥휂 the relations
− − − ﾟ−|︹| i 푇표휂 푇푥휉 푇푥휂 = −ퟙ 푥≠표 ∗ + + + ﾟ−|︹| i 푇표휂 푉 푇푥휉 푇푥휂 푉 = ퟙ 푥≠표
# and thus by (푇표휂 ) = ퟙ,
+ − ∗ 푇표휂 = −푉푇표휂 푉 . (2.44) 2.3. T WS
is is the key observation in the proof of this lemma, since this unitary iso- morphism can now be used to construct the desired operators. Also, the above step reveals one drawback of the WS method: it is only used that such a 푉 exists, but not how it is constructed (which will be quite difficult if the number of laice sites |Λ| is very high). Now define the operator
0 푉 ∗ 휂 = , (2.45) 표 푉 0 with the properties
∗ ′ (휂표) = 휂표 (휂표) = ퟙ ′ ∗ (2.46) {휂표, 푊푆 (훾표휂 )푊 } = 0 ′ ∗ [휂표, 푊푆 (훾푥̃ )푊 ] = 0 for all 푥̃ ≠ 표휉, 표휂
e first two equalities follow immediately from the definition of 휂표, for the third equality one uses ∗ 푇− 0 0 푉 ∗푇+ ′ ∗ 0 푉 표휂 표휂 휂표푊푆 (훾표휂 )푊 = + = − 푉 0 0 푇표휂 푉푇표휂 0 0 −푇− 푉 ∗ −푇− 0 ∗ 표휂 표휂 0 푉 ′ ∗ = + = + = −푊푆 (훾표휂 )푊 휂표 −푇표휂 푉 0 0 −푇표휂 푉 0
′ by means of 2.44. e same calculation for 푆 (훾푥̃ ) (푥̃ ≠ 표휉, 표휂) and 2.43 yields the fourth equality. Finally, define for 푥 ≠ 표 the operators
′ ∗ ′ ∗ 휉표 = −i푊푆 (훾표휂 )푊 휂표 = i휂표푊푆 (훾표휂 )푊 ′ ∗ 휉푥 = −i휉표푊푆 (훾푥휉 )푊 (2.47) ′ ∗ 휂푥 = −i휉표푊푆 (훾푥휂 )푊
∗ ′ ∗ ′ Since 휉표 = −i푊푆 (훾표휂 )푊 휂표i휂표푊푆 (훾표휂 )푊 = ퟙ, the laer two equalities can ′ ∗ be used to represent a path operator as 푊푆 (훾푥휁 )푊 = i휉푥휁푥 for 휁 = 휉, 휂, and, collecting all the previous properties, one gets
∗ (휁푥) = 휁푥
{휁푥, 휍푦} = 2훿푥푦훿휁휍 with 휁, 휍 ∈ {휉, 휂} It remains to show that each link operators can be represented as product ̃ of two of these operators. ereto let ℓ = (푥휁, 푦휍) ∈ 퐸 and assume first that C 2. JW
′ ′ ∗ 푥휁 = 표휉. en 푆 (ℓ) = 푆 (훾푦휍 ) = 푊 i휉표휍푦푊. For all other edges with 푥휁 ≠ 표휉 one has 훾푥휁 ∘ (푥휁, 푦휍) = 훾푦휍 and therefore, by definition of the path operators, ′ ′ ′ ′ 푆 (훾푦휍 ) = −i푆 (훾푥휁 )푆 (푥휁, 푦휍). Since 푆 (훾푥휁 ) = ퟙ, it follows that ′ ′ ′ ∗ ∗ 푆 (푥휁, 푦휍) = i푆 (훾푥휁 )푆 (훾푦휍 ) = i푊 i휉표휁푥푊 푊 i휉표휍푦푊 ∗ ∗ ∗ = 푊 (−i휉표휁푥휉표휍푦)푊 = 푊 i휉표휁푥휍푦푊 = 푊 i휁푥휍푦푊 is concludes the proof. ■ With lemma 2.11 at hand, it is straightforward to prove the theorem for gen- eral graphs. ﾪ Proof of S’s theorem 2.10 [Szc85]. Fix an arbitrary point 푥휉 ≡ 표, and let 핋 be a directed spanning tree of 핃ﾪ with root 표. By lemma 2.11, applied to the family ′ ̃ {푆 (ℓ) ∶ ℓ ∈ 퐸핋} of link operators on the spanning tree, there exists a family of ﾪ ′ ∗ Majorana operators defined on all the vertices of 핃 such that 푊푆 (ℓ)푊 = i휁푥휍푦 ̃ for any ℓ = (푥휁, 푦휍) ∈ 퐸핋. Indeed, the only property that needs to be checked in order to be able to apply this lemma is the trace equality 2.39. is follows directly from property (iii) (definition 2.8) of operators satisfying the link algebra ′ ′ ′ since 푆 (푥휉, 푥휂) = i푆 (훾푥휉 )푆 (훾푥휂 ) as argued at the end of the proof of lemma 2.11. is reduces point (iii) of the link algebra to 2.39. Using these Majorana fermions, one can define new link operators on the whole double laice 퐿̃ by ̃ ̃ 푆(ℓ) = i휁푥휍푦, ℓ = (푥휁, 푦휍) ∈ 퐸. ′ ̃ ese operators are unitarily equivalent to the link operators 푆 (ℓ) for ℓ ∈ 퐸핋. Furthermore, for any edge in 핃 one has ′ ′ ′ 푆 (푥휁, 푦휍) = i푆 (훾푥 )푆 (훾푦 ) 휁 휍 (2.48) ̃ ̃ ̃ 푆(푥휁, 푦휍) = i푆(훾푥휁 )푆(훾푦휍 ) where 훾푥휁 and 훾푦휍 denote the unique paths from the root 표 to 푥휁 (푦휍 respectively) along the directed spanning tree 핋. erefore the corresponding path operators ̃ are products of link operators to edges in 퐸핋 only, which implies ′ ∗ ̃ 푊푆 (훾푥휁 )푊 = 푆(훾푥휁 ). By means of 2.48 it then follows that ′ ′ ′ ∗ ̃ ∗ ̃ ∗ ̃ ̃ 푆 (ℓ) = i푆 (훾푥휁 )푆 (훾푦휍 ) = i푊 푆(훾푥휁 )푊푊 푆(훾푦휍 )푊 = 푊 i푆(훾푥휁 )푆(훾푦휍 )푊 = 푊 ∗푆(ℓ)푊̃ for all edges ℓ ∈ 퐸̃ . is concludes the proof that the link algebra (definition 2.8) defines the link operators uniquely up to unitary transformations (푈 = 푊 ∗). ■ 2.3. T WS
e reason why this result can be thought of as a generalisation of the usual Jordan Wigner transformation is explained in context of the “Gamma Matrix Model” in the following chapter. C 2. JW Chapter 3
e Gamma matrix model (GMM)
order to demonstrate the applicability of the WS method, the -I quantum spin system on a square laice Λ = [1, 퐿] ∩ℤ with periodic bound ary conditions and Hamiltonian (Gamma matrix model)
ﾜ ﾜ ︶ (퐻 = 휇 Γ푥 Γ푥+푒 + Γ푥 Γ푥+푒 − Γ푥 Γ푥+푒 − Γ푥 Γ푥+푒 + 휈푥Γ푥 , (3.1 푥∈︹ 푥∈︹
with nearest-neighbour interaction strength 휇 ∈ ℝ and in an external mag- netic field described by {휈푥}푥∈︹ ⊂ ℝ, is discussed. e Hamiltonian acts on .ℌ = ⨂ ℂ 푥∈︹ 퐻︶ can be interpreted physically as the Hamiltonian of spin-3/2 particles fixed at the sites of the laice Λ with short-range quadrupole-octupole interac- tions. A similar Hamiltonian (Gamma matrix model) has been discussed recently by Y et al. [YZK09] and W et al. [WCF12] in the context of a spin- generalisation of the Kitaev model on a honeycomb laice [Kit06] to laices with connectivity greater than three. It is argued in [YZK09] that the ground state of their Hamiltonian is an algebraic spin liquid, i.e. a spin liquid whose fermionic excitations (spinons) are gapless. If one defines the matrices ([MNZ04])
⎞ ⎞ ⎛ √√ ⎛ ⎞ i ⎛ − ⎟ ⎜ ⎟ ⎜ ⎟ 1 i √ −i ⎜ √ 푆 = ⎜ ⎟ , 푆 = ⎜ ⎟ , 푆 = ⎟ √ ⎜ √ i −i ⎟ − ⎜ ⎟ 1 ⎜ ⎟ ⎜ − ⎠ i √ ⎟ ⎝ ⎜ ⎟ √ ⎜ ⎠ ⎠ ⎝ ⎝
the Gamma matrices can be represented by symmetric bilinear combinations of
C 3. T G (GMM)
the SU(2) spin- matrices, namely Γ = 푄 = {푆, 푆} = 휎 ⊗ 휎 Γ = 푄 = {푆, 푆} = 휎 ⊗ ퟙ √√ Γ = 푄 = {푆, 푆} = 휎 ⊗ 휎 Γ = 푄() = (푆) − (푆) = 휎 ⊗ ퟙ √√ () Γ = 푄 = (푆 ) − ퟙ = 휎 ⊗ 휎 푎 -Hence the Γ matrices may be interpreted as spin- quadrupole (or nematic) op ,erators 푄 (the Γ 푎 operators corresponding to spin octupole operators) [MNZ04 ,Wu06, TZY06, WCF12] or, alternatively, as two-orbit spin- operators [Wen03 WCF12].
3.1 Jordan Wigner transformation for the constrained GMM
In the discussion of the Gamma matrix model it is useful to introduce a family of (elementary) plaquee (flux) operators. For a vertex 푥 ∈ Λ the elementary .plaquee 푃 ≡ 푃푥 is defined as the square with corners 푥, 푥 + 푒, 푥 + 푒 + 푒, 푥 + 푒 Its boundary 휕푃 will always be supposed to be positively oriented. Given such a plaquee, one can define ( 푊푃 = (Γ푥 Γ푥+푒 )(Γ푥+푒 Γ푥+푒+푒 )(Γ푥+푒+푒 Γ푥+푒 )(Γ푥+푒 Γ푥 ( (3.2 . Γ푥 Γ푥+푒 Γ푥+푒+푒 Γ푥+푒− =
Additionally, for a fixed value of 푥 (respectively 푥), one can define two global (flux) operators
퐿 ﾟ (푊푋 ≡ 푊푋 (푥) = Γ(푧,푥)Γ(푧+,푥 푧= (3.3) 퐿 ﾟ (푊푌 ≡ 푊푌 (푥) = Γ(푥,푧)Γ(푥,푧+ 푧= e collection of the plaquee and the global (flux) operators will be denoted
by {푊훼 ∶ 훼 ∈ {푋, 푌, 푃 ∶ 푃 elementary plaquee}}. ey have the following important properties:
Lemma 3.1. e family {푊훼}훼∈{푋,푌,푃∶푃 elementary plaquee} satisfies the algebra ∗ 푊훼 = 푊훼, 푊훼 = ퟙ ︶ [푊훼, 푊훽] = 0, [푊훼, 퐻 ] = 0
Tr(푊훼 ⋯ 푊훼푘 ) = 0 3.1. J W GMM
for all 훼, 훽 and 훼푖 ≠ 훼푗 whenever 푖 ≠ 푗.
In physics language, the operators 푊훼 correspond to ℤ fluxes through the elementary plaquees 푃 and two global ℤ fluxes [Wen03, YZK09]. To make this more apparent, one may write 푊훼 = − exp(iΦ훼) with Φ훼 taking the values 0 and 휋. Proof. e self-adjointness of the plaquee and global flux operators follows im- mediately from the self-adjointness of Γ 푎 and Γ 푎푏 for 푎, 푏 = 1, … , 5, the identity 푎 푎푏 푊훼 = ퟙ from (Γ ) = (Γ ) = ퟙ for 푎, 푏 = 1, … , 5. Given two elementary plaquees 푃 and 푄 that have no vertex in common, the property [푊푃, 푊푄] = 0 is trivial. If they have one common vertex, the com- mutator is zero as a consequence of equation 2.32, e.g. if 푃 and 푄 have their lower le respectively upper right corner 푥 in common, then
( 푊푃푊푄 = (−Γ푥 Γ푥+푒 Γ푥+푒+푒 Γ푥+푒 )(−Γ푥−푒−푒 Γ푥−푒 Γ푥 Γ푥−푒 .Γ푥−푒−푒 Γ푥−푒 Γ푥 Γ푥−푒 )(−Γ푥 Γ푥+푒 Γ푥+푒+푒 Γ푥+푒 ) = 푊푄푊푃−) = In case they have two vertices in common, then the commutation property fol- lows from equation 2.33. For example, if the boom two vertices of 푃, 푥 and 푥 + 푒, and the top two vertices of 푄 coincide, then
( 푊푃푊푄 = (−Γ푥 Γ푥+푒 Γ푥+푒+푒 Γ푥+푒 )(−Γ푥−푒 Γ푥+푒−푒 Γ푥+푒 Γ푥 .Γ푥−푒 Γ푥+푒−푒 Γ푥+푒 Γ푥 )(−Γ푥 Γ푥+푒 Γ푥+푒+푒 Γ푥+푒 ) = 푊푄푊푃−) (−1) = As for the global flux operators, one has
퐿 퐿 ﾟ ﾟ 푊푋 푊푌 = Γ(푧,푥)Γ(푧+,푥) Γ(푥,푧)Γ(푥,푧+) = 푊푌 푊푋 푧= 푧=
since the only non-trivial commutators are those between Gamma matrices at the intersection point 푥 = (푥, 푥) of the two global loops
(훾푋 = 훾푋 (푥) = (1, 푥) → (2, 푥) → ⋯ → (퐿, 푥) → (1, 푥
,(훾푌 = 훾푌 (푥) = (푥, 1) → (푥, 2) → ⋯ → (푥, 퐿) → (푥, 1 where the commutation relation follows from the short calculation
Γ푥 Γ푥 Γ푥 Γ푥 = Γ푥 Γ푥 Γ푥 Γ푥 .
Given 푊푋 and a plaquee operator 푊푃, there are three possible situations:
(i) 푃 and the loop 훾푋 have no common vertex, in which case [푊푋 , 푊푃] = 0. C 3. T G (GMM)
. Figure 3.1: Sketch of the assignment of the Gamma matrices to the edges of the square laice Λ.
ii) e boom edge of 푃, (푥, 푥+푒), lies in 훾푋 . en [푊푋 , 푊푃] = 0 follows from) Γ푥 Γ푥 Γ푥+푒 Γ푥+푒 ,Γ푥 Γ푥+푒 ] = −[Γ푥 Γ푥+푒 ,Γ푥 Γ푥+푒 ] = 0, these terms being] the only operators in 푊푋 and 푊푃 that do not commute trivially. Indeed, one has by (2.33)
Γ푥 Γ푥+푒 Γ푥 Γ푥+푒 = Γ푥 Γ푥 Γ 푥+푒 Γ 푥+푒 = −Γ푥 Γ푥 −Γ 푥+푒 Γ 푥+푒 Γ푥 Γ푥+푒 Γ푥 Γ푥+푒 =
(iii) In case the top edge of 푃 lies in 훾푋 , then [푊푋 , 푊푃] = 0 can be shown by a similar calculation to the one in (ii).
Analogously, it follows that [푊푌 , 푊푃] = 0 for all elementary plaquees 푃. In the same manner, using (2.30)-(2.33), one can prove that the flux operators commute with the Hamiltonian 퐻︶, by showing that the commutators of each ︶ summand in 퐻 with 푊훼 are zero, distinguishing the cases where they have none, one or two vertices in common. e proof of the trace property is based on the identities TrΓ 푎 = 0 and TrΓ 푎푏 = 0 (Γ 푎푏 being antisymmetric), as well as the tensor product structure of the flux operators. Since the trace property is not essential in the remainder of this thesis, its rather lengthy proof is omied here. ■
In the laice Λ there are |Λ| = 퐿 elementary plaquees 푃, which give rise to |Λ| − 1 independent plaquee operators 푊푃, since there is the constraint
ﾟ 푊푃 = ퟙ 푃 elem. plaq.
which follows from the assumption of periodic boundary conditions and the fact 푎 that (Γ ) = ퟙ. As for the global flux operators, defining 푊푋 ≡ 푊푋 (푥) and .푊푌 ≡ 푊푌 (푥) for fixed 푥 and 푥 determines 푊푋/푌 (푦/) for any other 푦 and 푦 is follows from 푊푋 (푦) = 푊푋 ∏ 푊푃 where 풮 (푥, 푦) denotes the set (푃∈풮 (푥,푦 .{[(of all elementary plaquees in the strip {푧 ∈ Λ ∶ 푧 ∈ [min(푥, 푦), max(푥, 푦 3.1. J W GMM
Similarly, 푊푌 (푦) = 푊푌 ∏ 푊푃. erefore, there are altogether |Λ| + 1 (푃∈풮 (푥,푦 independent elements in {푊훼}. Since they all commute with the Hamiltonian, eigenstates of 퐻︶ can be chosen to be eigenstates of the {푊훼} as well, such that the Hilbert space ℌ splits into sec- tors ℌ{푤훼} of fixed flux configurations, i.e. 푊훼휓{푤훼} = 푤훼휓{푤훼} for 휓{푤훼} ∈ ℌ{푤훼} and for each 훼,
ℌ = ﾘ ℌ{푤훼} (3.4) +|﹂| {푤훼}∈{±}
e orthogonal projections onto the respective subsectors are given by
ퟙ + 푤 푊 ퟙ + 푤 푊 ퟙ + 푤 푊 푥 푋 푦 푌 ﾟ 푃 푃 Ξ{푤훼} = 2 2 푃⊂핃 2 푃 elem. plaq.
e projection onto the sector with 푤훼 = 1 ∀훼 will be of particular import- ance and is henceforth just denoted by Ξ = Ξ{}. In particular, one has ℌ = Ξℌ ⊕ (ퟙ − Ξ)ℌ.
In view of the previous chapter, the connection to link operators is described next. Let 핃 = (Λ, 퐸) be the symmetric digraph corresponding to the square laice, and let 핃ﾪ be its directed double laice. For an edge ℓ ∈ 퐸̃ one defines the family
︶ ︶ 푆 (푥휉, (푥 + 푒)휉) = −푆 ((푥 + 푒)휉, 푥휉) = Γ푥 Γ푥+푒 ︶ ︶ 푆 (푥휉, (푥 + 푒)휉) = −푆 ((푥 + 푒)휉, 푥휉) = Γ푥 Γ푥+푒 ︶ ︶ (푆 (푥휂, (푥 + 푒)휂) = −푆 ((푥 + 푒)휂, 푥휂) = Γ푥 Γ 푥+푒 (3.5 ︶ ︶ 푆 (푥휂, (푥 + 푒)휂) = −푆 ((푥 + 푒)휂, 푥휂) = Γ푥 Γ푥+푒 ︶ ︶ 푆 (푥휉, 푥휂) = −푆 (푥휂, 푥휉) = Γ푥 of link operators. In terms of these operators, the Hamiltonian can be wrien as
︶ ︶ ︶ (퐻 = 휇 ﾜ 푆 (푥휉, (푥 + 푒)휂) + 푆 (푥휉, (푥 + 푒)휂 푥∈︹ (3.6) ︶ ︶ ︶ (푆 (푥휂, (푥 + 푒)휉) − 푆 (푥휂, (푥 + 푒)휉) + ﾜ 휈푥푆 (푥휉, 푥휂 − 푥∈︹
e link operators 3.5 clearly satisfy properties (i) and (ii) of the link algebra .see definition 2.8). Also, 푆︶(ℓ−) = −푆︶(ℓ) for all ℓ ∈ 퐸̃ by definition) C 3. T G (GMM)
For an elementary plaquee 푃 in Λ × {휉} one has
︶ ︶ ︶ ⋅(푆 (휕푃) = (−i) 푆 (푥휉, (푥 + 푒)휉)푆 ((푥 + 푒)휉, (푥 + 푒 + 푒)휉 ︶ ︶ (푆 ((푥 + 푒 + 푒)휉, (푥 + 푒)휉)푆 ((푥 + 푒)휉, 푥휉 ⋅ = (−i) 푊푃 = i푊푃.
Furthermore, for the loop
(훾푥 = (푥휉, (푥 + 푒)휉) → ((푥 + 푒)휉, (푥 + 푒)휂) → ((푥 + 푒)휂, 푥휂) → (푥휂, 푥휉
with arbitrary corner 푥 ∈ Λ,
︶ ︶ ︶ ︶ ︶ (푆 (훾푥) = (−i) 푆 (푥휉, (푥 + 푒)휉)푆 ((푥 + 푒)휉, (푥 + 푒)휂)푆 ((푥 + 푒)휂, 푥휂)푆 (푥휂, 푥휉 i) Γ푥 Γ 푥+푒 Γ 푥+푒 (−Γ푥 Γ 푥+푒 )(−Γ푥 ) = (−i) Γ푥 Γ푥 Γ푥 iΓ푥+푒 Γ푥+푒−) = i) (−Γ푥 Γ푥 Γ푥 )ퟙ푥+푒 = i(Γ푥 ) ퟙ푥+푒 = iퟙ−) =
︶ Substituting (1 ↔ 3) and (2 ↔ 4) proves the identity 푆 (훾̃ 푥) = iퟙ for the loop
.(훾̃ 푥 = (푥휉, (푥 + 푒)휉) → ((푥 + 푒)휉, (푥 + 푒)휂) → ((푥 + 푒)휂, 푥휂) → (푥휂, 푥휉
For the global loop 훾푋 = (1, 푥)휉 → (2, 푥)휉 → ⋯ → (퐿, 푥)휉 → (1, 푥)휉 (with 푥 fixed) one has
︶ ︶ ︶ 퐿− ︶ (푆 (훾푋 ) = (−i) 푆 ((1, 푥)휉, (2, 푥)휉) ⋯ 푆 ((퐿 − 1, 푥)휉, (퐿, 푥)휉)푆 ((퐿, 푥)휉, (1, 푥)휉 퐿− 퐿− i) Γ(,푥)Γ(,푥) ⋯ Γ(퐿−,푥)Γ(퐿,푥)Γ(퐿,푥)Γ(,푥) = (−i) 푊푋−) =
퐿− ︶ .Analogously, 푆 (훾푌 ) = (−i) 푊푌 for a global loop in 푥-direction at fixed 푥 erefore, the link operators 푆︶(ℓ) satisfy property (iv) of the link algebra if 푊푃 = ퟙ for all elementary plaquees 푃, 푊푋 = 푊푌 = ퟙ, and 퐿 ∈ 4ℕ. In this case, 푆︶(훾) = iퟙ for all closed loops in the double laice 핃ﾪ, since any such loop operator can be decomposed into a product of loop operators corresponding to
elementary plaquees or either of the two global loop operators 푊푋/푌 . e identities 푊훼 = ퟙ ∀훼 ∈ {푋, 푌, 푃} hold exactly in the sector ℌ{} of the Hilbert space.
Lemma 3.2. e projection operator Ξ commutes with all link operators, [Ξ, 푆︶(ℓ)] = 0 for all ℓ ∈ 퐸̃ .
︶ Proof. By the very same argumentation as in lemma 3.1 it follows that [푊훼, 푆 (ℓ)] = ퟙ+푊훼 ︶ ̃ ︶ and thus [ , 푆 (ℓ)] = 0 for all 훼 and ℓ ∈ 퐸. is implies [Ξ, 푆 (ℓ)] = 0 for ,0 all ℓ ∈ 퐸̃ . ■ 3.1. J W GMM
.By lemma 3.2 the link operators leave the space ℌ{} = Ξℌ = ranΞ invariant It follows that the family of constrained link operators
︶ ︶ ̃ 푆︺(ℓ) = Ξ푆 (ℓ)Ξￖ , ℓ ∈ 퐸 (3.7) ︑︀︍︺ inherits properties (i), (ii) and (iv) from the corresponding ones of the operators 푆︶(ℓ). Any other closed path in 핃ﾪ can be decomposed into a product of the above elementary loops. erefore, property (iv) holds for the family of constrained ︶ ̃ operators {푆︺(ℓ) ∶ ℓ ∈ 퐸}. e constrained Hamiltonian is given by
︶ ︶ ︶ (퐻︺ = 휇 ﾜ 푆︺(푥휉, (푥 + 푒)휂) + 푆︺(푥휉, (푥 + 푒)휂 푥∈︹ (3.8) ︶ ︶ ︶ .(푆︺(푥휂, (푥 + 푒)휉) − 푆︺(푥휂, (푥 + 푒)휉) + ﾜ 휈푥푆︺(푥휉, 푥휂 − 푥∈︹
It still remains to check the trace property (iii). By the identity
|︹| ﾟ Γ푥 = ﾟ(−i) Γ푥 Γ푥 Γ푥 Γ푥 = (−1) ﾟ Γ푥 Γ푥 ﾟ Γ푥 Γ푥 푥∈︹ 푥∈︹ 푥∈︹ 푥∈︹
퐿 퐿 퐿 퐿 (퐿−) ﾟ ﾟ ﾟ ﾟ |︹| (Γ (푥,푥)Γ(푥+,푥) Γ(푥,푥)Γ(푥,푥+ (−1) (−1) = 푥= 푥= 푥= 푥=
퐿 퐿
ﾟ 푊푌 (푥) ﾟ 푊푋 (푥) = 푥= 푥= it follows that, on the subspace , one has ∏ and thus Ξℌ Ξ 푥∈︹ Γ푥 Ξￗ = ퟙ ︑︀︍︺ ⎛ ⎞ ⎛ ⎞ ⎟ |︹|− ⎜ ⎟ ︶ ⎜ Tr︺ℌ ﾟ 푆︺(푥휉, 푥휂) = Tr︺ℌ Ξ ﾟ Γ푥 Ξￗ = Tr︺ℌ ퟙ︺ℌ = rkΞ = 2 ⎜ 푥∈︹ ⎟ ⎜ 푥∈︹ ︺ℌ⎟ ⎝ ⎠ ⎝ ⎠ Hence, the family of constrained link operators does not fulfil the link algebra and S’s theorem is not directly applicable. erefore an auxiliary Hilbert space ℌ ≅ ℂ and an operator Γ on ℌ with the properties
∗ Γ) = Γ, (Γ) = ퟙ, Trℌ Γ = 0)
needs to be introduced (e.g. Γ = 휎), and the definition of the link operators has to be slightly modified. C 3. T G (GMM)
̃ ﾪ ,Let ℌ = ℌ ⊗ ℌ and fix a vertex 푣휉 in the double laice 핃. For concreteness the choice 푣휉 = (1, 1)휉 is made here. en the modified link operators are defined as
⎧ ︶ ⎪ퟙ ⊗ 푆 (ℓ) if 푣휉 ∉ ℓ 푆̃︶(ℓ) = ⎨ (3.9) ⎪Γ ⊗ 푆︶(ℓ) if 푣 ∈ ℓ 휉 ⎩
︶ ︶ By the identities i푊푃 = 푆 (휕푃) and i푊푋/푌 = 푆 (훾푋/푌 ) the plaquee and global flux operators can be naturally extended to ℌ̃ . Since in a closed loop con- taining 표휉 there are exactly two adjoining edges, and Γ = ퟙ, this extension is ﾪ trivial, 푊훼 = ퟙ ⊗ 푊훼. ̃︶ ̃ ̃ ̃ Further, the restrictions 푆︺(ℓ) to Ξℌ = (ퟙ⊗Ξ)ℌ = ℌ ⊗Ξℌ are link operators satisfying (i), (ii) and (iv) of the link algebra. Property (iii) now does hold due to
,the introduction of the Γ factor. Indeed ⎛ ⎞ ⎛ ⎞ ︶ ︶ ̃ ⎜ﾟ ̃ ⎟ ⎜ﾟ ̃ ⎟ .Tr︺ℌ 푆︺(푥휉, 푥휂) = Trℌ Γ ⋅ Tr︺ℌ 푆︺(푥휉, 푥휂) = 0 ⎜ 푥∈︹ ⎟ ⎜ 푥∈︹ ⎟ ⎝ ⎠ ⎝ ⎠ S’s theorem 2.10 hence applies to the Hamiltonian
ﾪ ︶ ̃︶ ̃︶ (퐻 = 휇 ﾜ 푆 (푥휉, (푥 + 푒)휂) + 푆 (푥휉, (푥 + 푒)휂 푥∈︹
̃︶ ̃︶ ̃︶ (푆 (푥휂, (푥 + 푒)휉) − 푆 (푥휂, (푥 + 푒)휉) + ﾜ 휈푥푆 (푥휉, 푥휂 − 푥∈︹
︶ ퟙ ⊗ 퐻 + 휇(Γ − ퟙ) ⊗ Γ푣 Γ 푣+푒 + Γ푣−푒 Γ푣 + Γ푣 Γ푣+푒 + Γ푣−푒 Γ푣 = Γ푣 Γ푣+푒 − Γ푣−푒 Γ푣 + Γ푣 Γ푣+푒 + Γ푣−푒 Γ푣 + 휈푣(Γ − ퟙ) ⊗ Γ푣 −
.{constrained to the flux sector ℌ ⊗ ℌ{ ︶ Since the operator Γ ⊗ ퟙ commutes with ퟙ ⊗ 퐻 as well as with the other ﾪ ︶ ﾪ ︶ = terms in 퐻 , eigenstates 휓 of 퐻 can be classified by the property (Γ ⊗ ퟙ)휓 휓. In particular, in the sector of the Hilbert space where Γ ⊗ ퟙ = ퟙ ⊗ ퟙ one± has 퐻ﾪ ︶ = ퟙ ⊗ 퐻︶. is may help in the understanding of the physics of the extended Gamma matrix model, is however not essential in the further discussion, which is based on S’s theorem and therefore relies crucially on the validity of the link algebra. Corollary 3.3 (Jordan Wigner transformation for the Gamma matrix model). e (extended) Gamma matrix model 퐻ﾪ ︶ with constraints 3.2. D L LR
(i) 푊푃 = ퟙ for all elementary plaquees 푃 and
(ii) 푊푋 = 푊푌 = ퟙ is unitarily equivalent to free fermions on the square laice 핃 = (Λ, 퐸) (with peri- odic boundary conditions),
∗ ∗ ∗ 퐻 = 2휇 ﾜ(푐푥푐푦 + 푐푦푐푥) + ﾜ 휈푥(2푐푥푐푥 − ퟙ). 푥푦∈퐸 푥∈︹
In particular, there exists a unitary transformation 푈 such that
ﾪ ︶ ∗ 퐻︺ = 푈퐻푈 (3.10)
ﾪ ︶ ﾪ ︶ where 퐻︺ = Ξ퐻 Ξￖ . ︑︀︍︺ Remark. As illustrated by S in his paper [Szc85], the above construction with link operators expressed as products of Dirac Gamma matrices, also works on a laice with different vertex degrees, at least if they are all even. In this case, the Dirac Gamma matrices have to be replaced by generalised Gamma matrices (generators of ℭ픩(deg(푥))). ♦
3.2 Dynamical Localisation and Lieb Robinson Bounds for the GMM
e correspondence between the constrained Gamma matrix model and free fer- mions on the laice Λ can be used to get insights in the behaviour of the system when the on-site magnetic field is not deterministic, but given by i.i.d. random variables. e theory of free fermions in a random on-site potential on a planar laice is well-understood in terms of localisation properties, which imply certain locality bounds for the constrained Gamma matrix model.
3.2.1 e Anderson model on ℤ푑
Let {푣푥}푥∈ℤ푑 (푑 ≥ 1) be a family of real-valued independent and identically dis- tributed (i.i.d.) random variables on a probability space (Ω, 픉, ℙ) with common distribution 푃
푑 ℙ{푣푥 ∈ 퐴} = 푃(퐴) for all Borel sets 퐴 ⊂ ℝ and 푥 ∈ ℤ . It is further assumed that 푃 is absolutely continuous with respect to Lebesgue measure on ℝ with bounded density and compact support, 푃( d휈) = 휌(휈) d휈 ∞ .(with 휌 ∈ 퐿 (ℝ C 3. T G (GMM)
e Anderson model on ℤ푑 is the random Hamiltonian 푑 퐻휔 = 퐾 + 휆푉휔 on ℓ (ℤ ) (3.11) where 퐾 = −Δ−2푑 is up to a constant shi the graph Laplacian on ℤ푑, explicitly ,(for 푓 ∈ ℓ(ℤ푑 (퐾푓)(푥) = − ﾜ 푓(푦), 푦∈ℤ푑 dist(푥,푦)= the random potential 푉휔 is the multiplication operator by the i.i.d. random vari- ables,
(푉휔푓)(푥) = 푣푥푓(푥), and 휆 ≥ 0 models the disorder strength. It is a classical result in the theory of random Schrödinger operators, that the spectrum of 퐻휔 (as well as its spectral components) is almost surely deterministic, # # 휎(퐻휔) = Σ, 휎 (퐻휔) = Σ (# ∈ {푝푝, 푎푐, 푠푐}). is follows from the ergodicity of 퐻휔 with respect to laice translations. In the i.i.d. case considered here, one has 휎(퐻휔) = [−2푑, 2푑] + supp푃 [Pas80, KS80, KM82]. Definition 3.4 (Localisation Types). One says that the Anderson model exhibits (i) spectral localisation in the energy regime 퐼 ⊂ ℝ if there is ℙ-almost surely only pure point spectrum in 퐼, i.e. 퐼 ∩ Σ 푎푐 = 퐼 ∩ Σ 푠푐 = ∅ a.s. (ii) exponential eigenfunction localisation in 퐼 away from some random centres of localisation 휉휔, if for eigenfunctions 휓휔 of 퐻휔
−휂|푥−휉휔 | |휓휔(푥)| ≤ 퐶휔푒 ℙ − almost surely with inverse localisation length 휂 = 휂(퐼) > 0 (iii) (strong) dynamical localisation in the energy interval 퐼 if
⎤ ⎡ −︈푡퐻휔 −휂|푥−푦| 피 ⎢sup ￖ⟨훿푦, 푒 푃퐼 (퐻휔)훿푥⟩ￖ ⎥ ≤ 퐶푒 (3.12) ⎣ 푡∈ℝ ⎦ with 퐶 = 퐶(퐼) < ∞ and inverse localisation length 휂 = 휂(퐼) > 0. Here, 푑 {훿푥}푥∈ℤ푑 denotes the canonical orthonormal basis of ℓ (ℤ ), 훿푥(푦) = 훿푥푦 for 푑 all 푥, 푦 ∈ ℤ , and 푃퐼 (퐻휔) is the spectral projection of 퐻휔 onto 퐼. In this definition, dynamical localisation is the strongest notion of localisa- tion, and if it holds in some energy interval 퐼, then the spectrum in 퐼 is a.s. pure point. Dynamical localisation also implies that for energies in 퐼 there is no quantum transport in the sense that all moments of the position operator¹ |푋|
¹(|푋|휓)(푥) ∶= |푥|휓(푥) for 푥 ∈ ℤ푑 3.2. D L LR are finite for all times,
푝 −︈푡퐻휔 sup ‖|푋| 푒 푃퐼 (퐻휔)휓‖ < ∞ ℙ − almost surely 푡∈ℝ
.[for all 푝 > 0 and 휓 ∈ ℓ(ℤ푑) compactly supported [Sto11 One way of proving strong dynamical localisation in the Anderson model on ℤ푑 is via the fractional moments method (or AM method [AM93]), which proceeds via exponential bounds on the fractional moments of the Green’s function 퐺휔 of the random Hamiltonian 퐻휔, 1 퐺휔(푥, 푦; 푧) = ⟨훿푥, 훿푦⟩, 푧 ∈ ℂ ⧵ ℝ 퐻휔 − 푧
Restrictions of the Hamiltonian or the Green’s function to subsets Λ ⊂ ℤ푑 are ︹ ︹ denoted by 퐻휔 and 퐺휔(푥, 푦; 푧). By a rank-two perturbation argument using Kreĭn’s formula one can prove for all 푠 ∈ (0, 1) the a priori bound²
푠 퐶 (휌) 피 ￖ퐺 (푥, 푦; 푧)ￖ ≤ 푠 (3.13) 푥,푦 휔 휆푠
푑 for some constant 퐶푠(휌) < ∞ and all 푥, 푦 ∈ ℤ , 푧 ∈ ℂ ⧵ ℝ, 휆 > 0 [Sto11, AW13]. Making use of the resolvent identity
︹ {푥} ︹ ︹⧵{푥} 퐺휔(푥, 푦; 푧) = 퐺휔 (푥, 푥; 푧)훿푥푦 + 퐺휔(푥, 푥; 푧) ﾜ 퐺휔 (푢, 푦; 푧) 푢∶푑(푢,푥)=
︹⧵{푥} and noting that the Green’s function 퐺휔 does not depend on 푉(푥), one can show that for 푥 ≠ 푦
︹ 푠 ︹ 푠 ︹⧵{푥} 푠 피 ￖ퐺휔(푥, 푦; 푧)ￖ ≤ ﾜ 피 ￮ￖ퐺휔(푥, 푥; 푧)ￖ ￖ퐺휔 (푢, 푦; 푧)ￖ 푢∶푑(푢,푥)=
퐶 푠 푠 ﾜ ︹⧵{푥} ≤ 푠 피 ￖ퐺휔 (푢, 푦; 푧)ￖ . 휆 푢∶푑(푢,푥)=
e first inequality is a straightforward application of Jensen’s inequality, whereas in the second step first a disorder average over 푉(푥) was taken, using the a pri- ︹ 푠 −푠 ori bound 피푥 ￖ퐺휔(푥, 푥; 푧)ￖ ≤ 퐶푠휆 . Iterating this expansion, one can prove the following theorem [AW13] due to A and M [AM93]
²피푥,푦[⋅] = 피 ￮ ⋅ ￖ {푉(푢)}푢∉푥,푦 denotes the conditional expectation with {푉(푢)}푢∉푥,푦 fixed C 3. T G (GMM)
eorem 3.5 (Complete localisation for high disorder). For any disorder strength 휆 satisfying
/푠 휆 > (2푑퐶푠) (3.14)
for some 푠 ∈ (0, 1) there is a constant 퐶 < ∞ such that for all Λ ⊂ ℤ푑 and 푥, 푦 ∈ Λ
︹ 푠 −휂푑(푥,푦) sup 피 ￖ퐺휔(푥, 푦; 푧)ￖ ≤ 퐶푒 (3.15) 푧∈ℂ⧵ℝ
푠 uniformly in Λ with a localisation length 휂 < log( |휆| ). 푑퐶푠 e connection to the definition of dynamical localisation is established by
the eigenfunction correlator 푄휔(푥, 푦; ⋅) of 퐻휔, which is defined as the total vari- ation of the spectral measure associated with two sites 푥, 푦 ∈ ℤ푑. at is, for any Borel set 퐼 ⊂ ℝ,
푄휔(푥, 푦; 퐼) = sup ￖ⟨훿푥, 푃퐼 (퐻휔)퐹(퐻휔)훿푦⟩ￖ . 퐹∈풞 (ℝ) 퐹‖∞≤‖
It has the important property that
−︈푡퐻휔 −︈푡퐻휔 sup ￖ⟨훿푦, 푒 푃퐼 (퐻휔)훿푥⟩ￖ ≤ sup ￖ⟨훿푦, 푒 푃퐼 (퐻휔)훿푥⟩ￖ ≤ 푄휔(푥, 푦; 퐼) 푡∈ℝ 푡∈ℝ
−︈푡퐻휔 where the first inequality simply follows from ￖ⟨훿푦, 푒 푃퐼 (퐻휔)훿푥⟩ￖ ≤ 1 and the −︈푡퐻휔 second one by the particular choice 퐹(퐻휔) = 푒 . Under the present assumptions on the single-site distribution, for each 푠 ∈ (0, 1) there is a constant 푐푠(휌) < ∞ such that for any bounded open set 퐼 ⊂ ℝ the eigenfunction correlator, averaged over the disorder, satisfies [AW13]
푠 피푄휔(푥, 푦; 퐼) ≤ 푐푠(휌) sup ﾙ 피 ￖ퐺휔(푥, 푦; 퐸 + i휂)ￖ d퐸. (3.16) 휂|> 퐼|
/푠 Puing everything together, it follows that for large disorder 휆 > (2푑퐶푠) the Anderson model exhibits strong dynamical localisation throughout the entire spectrum. Of course, the above is just a sketch of the most basic localisation result in the discrete Anderson model. Much more could be said about different disorder regimes, and far more general methods than the one used here are nowadays at hand. A more detailed account on the topic can be found in the notes by K [Kir07] and S [Sto11], as well as in the book by A and W [AW13], which is still in preparation at the time of writing of this thesis. 3.2. D L LR
3.2.2 Lieb Robinson bounds and exponential clustering Consider a quantum spin system on a finite vertex set Λ as described in section
, en for two local observables (cf. the definition in section 1.2) 퐴 ∈ 프︵ .1.2
퐵 ∈ 프︵ (Ω and Ω finite, disjoint, Ω ∪ Ω ⊂ Λ) one has
퐴, 퐵] ∶= [퐴 ⊗ ퟙ︹⧵︵ , 퐵 ⊗ ퟙ︹⧵︵ ] = 0]
Lieb-Robinson bounds are bounds on the commutator aer time evolution of one of the operators of the form
(|휂(︃︈︒︓(︵,︵)−푣퐿푅|푡− ︹ (ￏ ￮훼푡 (퐴), 퐵 ￏ ≤ 퐶︵,︵ ‖퐴‖‖퐵‖푒 (3.17
e velocity 푣퐿푅 depends on the specific interaction Φ and is finite for a fairly general class of interactions, which are essentially finite-range [LR72]. Physic- ally, 푣퐿푅 is the group velocity with which information or excitations in Ω can .propagate under the Heisenberg evolution to Ω Lieb Robinson bounds have been successfully used to prove exponential clus- tering in presence of a spectral gap, i.e. a spectral gap implies exponential decay of ground state correlations [Has04, NS06]. Since the converse implication is in general false, and the assumption of a spectral gap quite strong, one would like to find weaker requirements on quantum spin systems that allow one to infer exponentially decaying correlation functions. An interesting step in this direction is a result due to H, S and S [HSS12]. In their paper they prove that zero-velocity Lieb Robinson bounds imply exponentially decaying ground state correlations up to a logarithmic correction in the gap size. e validity of a zero-velocity Lieb Robinson bound has been proposed by the above authors as a simplified version of H’ definition of a mobility gap, i.e. a gap to propagating excitations [Has10].
Let 퐻︹ be the Hamiltonian
푑 ﾜ ﾜ ∗ ∗ ﾜ ∗ 퐻︹ = 2휇 푐푥푐푥+푒푘 + 푐푥+푒푘 푐푥 + 휈푥(2푐푥푐푥 − ퟙ), 휇, 휈푥 ∈ ℝ ︹∋푥∈︹ 푘= 푥 on ℌ = ⨂ ℂ, Λ ⊂ ℤ푑 finite and connected. Using the anticommutation 푥∈︹ relations it can be brought into a symmetric form,
푑 ﾜ ﾜ ∗ ∗ ∗ ∗ ﾜ ∗ ∗ 퐻︹ = 휇 푐푥푐푥+푒푘 − 푐푥푐푥+푒푘 + 푐푥+푒푘 푐푥 − 푐푥+푒푘 푐푥 + 휈푥(푐푥푐푥 − 푐푥푐푥), ︹∋푥∈︹ 푘= 푥 C 3. T G (GMM)
∗ ∗ 푡 -and introducing the vector 퐶 = (푐, … , 푐|︹|, 푐, … , 푐|︹|) this can be wrien com pactly as
A(︹) 0 |︹|퐻 = 퐶∗M (︹)퐶 with M (︹) = ∈ ℂ|︹|× ︹ 0 −A(︹) and A(︹) ∈ ℂ|︹|×|︹| given by
(︹) (|︹|A = 휇Adj(Λ) + diag(휈, … , 휈
Here, Adj(Λ) stands for the adjacency matrix of Λ ⊂ ℤ푑 considered as (un- directed) combinatorial graph. Since Adj(Λ) is symmetric, A(︹) is self-adjoint, thus also (M (︹))∗ = M (︹). In particular, by the spectral theorem for self-adjoint matrices, there is a unitary matrix U that diagonalises A(︹), UA(︹)U ∗ = D = (︹) , diag(휆, … , 휆|푉|). en the unitary matrix W = U ⊕ U diagonalises M
D 0 WM (︹)W ∗ = 0 −D
is fact can be used to introduce creation and annihilation operators for new ∗ fermions 퐵 = (푏, … , 푏|︹|, 푏, … , 푏|︹|), namely
퐵 = W퐶
It follows from a straightforward computation that the canonical anticom- mutation relations are in this vector-valued formalism equivalent to
0 ퟙ 퐶퐶∗ + 핁(퐶퐶∗)푡핁 = ퟙ, 핁 = . ퟙ 0
Since 핁 commutes with W, it follows that
퐵퐵∗ + 핁(퐵퐵∗)푡핁 = W (퐶퐶∗ + 핁(퐶퐶∗)푡핁) W ∗ = ퟙ hence the 푏-operators indeed are fermionic operators. In terms of these operators the Hamiltonian 퐻︹ can be wrien as
D 0 퐻 = 퐶∗M퐶 = 퐵∗푊M푊 ∗퐵 = 퐵∗ 퐵 ︹ 0 −D ∗ ∗ ∗ (︹) = ﾜ 휆푥(푏푥푏푥 − 푏푥푏푥) = ﾜ 2휆푥푏푥푏푥 − 퐸 ퟙ 푥∈︹ 푥∈︹ where (︹) ∑ . 퐸 = 푥∈︹ 휆푥 3.2. D L LR
In this form it is particularly easy to calculate the time evolution of the fer- mionic operators. For any 푥 ∈ Λ one has
[퐻︹, 푏푥] = −2휆푥푏푥 and thus, by definition of the Heisenberg dynamics, d 훼︹(푏 ) = i훼︹([퐻 , 푏 ]) = −2i휆 푏 . d푡 푡 푥 푡 ︹ 푥 푥 푥
︈휆푥푡− ︹ ︹ Together with 훼 (푏푥) = 푏푥 it follows that 훼푡 (푏푥) = 푒 푏푥, and consequently
푒−︈﹕푡 0 훼︹(퐵) = 퐵 푡 0 푒︈﹕푡
is result can be used to obtain the time evolution of 퐶, for
푒−︈﹕푡 0 푒−︈﹕푡 0 훼︹(퐶) = 훼︹(W ∗퐵) = W ∗훼︹(퐵) = W ∗ 퐵 = W ∗ W퐶 푡 푡 푡 0 푒︈﹕푡 0 푒︈﹕푡 ︈﹒(﹂)푡− ︈﹞(﹂)푡 푒 0− = 푒 퐶 = (﹂) 퐶. 푒︈﹒ 푡 0
(︹) −︈﹒(﹂)푡 Introducing the short-hand notation A푥푦 (푡) = 푒 , one can immediately 푥푦 write down the Heisenberg evolution of the 푐-operators,
︹ (︹) (︹) ∗ 훼푡 (푐푥) = ﾜ A푥푦 (−2푡)푐푦 + ﾜ A푥푦 (2푡)푐푦. (3.18) 푦∈︹ 푦∈︹
It shall from now on be assumed that the family {휈푥}푥∈︹, describing the on-site potential in the fermionic Hamiltonian, is a family of i.i.d. random variables as
in section 3.2.1. en 퐻︹ is essentially the second quantisation of the Anderson model on ℓ (Λ) with disorder strength 휆 = 휇 . In particular it is reasonable to define Definition 3.6. e matrices M (︹) are dynamically localised if there exist 퐶, 휂 > 0 such that for all Λ ⊂ ℤ푑 and 푥, 푦 ∈ Λ
⎡ ⎤ (︹) −휂푑(푥,푦) 피 ⎢sup ￗA푥,푦(푡)ￗ⎥ ≤ 퐶푒 . (3.19) ⎣ 푡∈ℝ ⎦ It has been proved by H et al. [HSS12], that dynamical localisation of the matrices M (︹) implies zero-velocity Lieb-Robinson bounds aer disorder average in one dimension. C 3. T G (GMM)
Example 3.7 (Dynamical localisation implies zero-velocity Lieb-Robinson bounds 푁 -aer disorder average in one dimension). Let ℌ = ⨂ ℂ and consider the 푋푌 푥= model on ℌ. Looking back at section 2.1.1 a short calculation shows that in this case
⎞ 휈 −휇 ⎛ ⎜−휇 ⋱ ⋱ ⎟ A(푁) = ⎜ ⋱ ⋱ ⋱ ⎟ ⎜ ⋱ ⋱ −휇⎟ ⎜ ⎟ ⎝ −휇 휈푁 ⎠ e following holds: eorem 3.8 (Hamza, Sims, Stolz [HSS12]). M (푁) is dynamically localised and there exist 퐶′, 휂 > 0 such that for all 1 ≤ 푥 < 푦, and any 푁 ≥ 푦 one has
⎡ ⎤ 푁 ′ −휂푑(푥,푦) 피 ⎢sup ￎ[훼푡 (푋), 푌]ￎ⎥ ≤ 퐶 ‖푋‖‖푌‖푒 (3.20) ⎣ 푡∈ℝ ⎦ for all 푋 ∈ 프{푥} and 푌 ∈ 프[푦,푁].³ , Furthermore, leing 휓 be the almost sure (normalised) ground state of 퐻푁 there is a constant 퐶 < ∞ and a inverse correlation length 휂′ > 0 such that
−휂′푑(푥,푦) (피 ￮ ￖ⟨휓, 푋푌휓⟩ − ⟨휓, 푋휓⟩⟨휓, 푌휓⟩ￖ ≤ 퐶‖푋‖ ‖푌‖푁푒 . (3.21 ♦
If one instead considers the constrained Gamma matrix model on ℌ ⊗ Ξℌ in a random exterior magnetic field {휈푥}푥∈︹, which is assumed to satisfy the as- sumptions of subsection 3.2.1, it would be interesting to see if similar results also hold. Using a Jordan Wigner transformation in the sense of Szczerba (theorem 2.10) it has been proven in corollary 3.3 that the GMM-Hamiltonian is unitarily equi- valent to the second quantised version of the Anderson model on ℓ(Λ). In a regime of high disorder 휆 ≫ 1 (cf. condition 3.14) or, equivalently, low kinetic energy 휇 ∼ 1/휆 ≪ 1, one has complete dynamical localisation, which corres- (︹) .(|︹|ponds to dynamical localisation of the matrix A = 휇Adj(Λ)+diag(휈, … , 휈 ∗ ̃︶ Definition 3.9. For Ω ⊂ Λ let 픖︵ be the 퐶 algebra generated by the set {푆︺(ℓ) ∶ ℓ ∈ Ω}ﾪ of link operators corresponding to edges ℓ in the (doubled) set Ωﾪ⁴.
³In the context of this theorem, 프︵ refers to the local algebra generated by the Pauli spin matrices in Ω. ⁴Ω being identified in a natural way as subgraph of 핃. 3.2. D L LR
Ω
. Ι
Figure 3.2: On the definition of the relation Ι < Ω.
Lemma 3.10. For fixed 푥 ∈ Λ let 푐푥 be a fermionic annihilation operator and 푌 ∈ 픖︵ such that 푥 ∉ Ω (assuming for simplicity that Ω is a box subset of Λ with the property 푥푖 < 푦푖 for all 푦 ∈ Ω, 푖 = 1, 2)⁵. en dynamical localisation of the matrix 퐴(︹) implies
휂 − ⎤ ⎡ ︹ ∗ 1 + 푒 −휂︃︈︒︓(푥,︵) 피 ⎢sup ￏ ￮훼푡 (푐푥), 푈 푌푈 ￏ⎥ ≤ 4퐶 −휂 ‖푌‖푒 (3.22) ￍ1 − 푒ﻰﻰﻰￌﻰﻰﻰ푡∈ℝ ⎦ ￋ ⎣
=∶퐶′ with 퐶 being the constant from definition 3.6.
Proof. Fix the vertex (1, 1) as root 표 of the (directed) spanning tree of Λ depicted in figure 3.3 (directed away from the root) and complete it to a spanning tree of ﾪ 핃 by adding the (directed) edges (푥휉, 푥휂) for 푥 ∈ Λ. e assumptions on Ω imply that 표 ∉ Ω. By equation 3.18 one has
︹ ∗ (︹) ∗ (︹) ∗ ∗ [훼푡 (푐푥), 푈 푌푈] = ﾜ A 푥푧(2푡)[푐푧, 푈 푌푈] + ﾜ A 푥푧(−2푡)[푐푧, 푈 푌푈]. (3.23) 푧∈︹ 푧∈︹
Since
( ∗ ̃︶ ̃︶ for (3.24 ,푐푧 = (휉푧 + i휂푧) = (−i휉표)푈 (푆︺(훾푧휉 ) + i푆︺(훾푧휂 ))푈 푧 ≠ 표 (푐 = (휉 + i휂 ) = (−i푈∗푆̃︶ (훾 )푈휂 + i휂 ) (3.25 표 표 표 ︺ 표휂 표 표
⁵To shorten notation, this shall be wrien as {푥} < Ω. More generally, for sets Ι, Ω ⊂ ℤ푑 the .relation Ι < Ω is defined by 푥 < 푦 and 푥 < 푦 for all 푥 ∈ Ι and 푦 ∈ Ω. See figure 3.2 C 3. T G (GMM) by the construction 2.47 in S’s theorem (푈 = 푊 ∗ here as remarked at the end of the proof of theorem 2.10), the commutators in 3.23 can be examined more closely.
Firstly, the properties of 휂표 (2.46) imply that for any 푣, 푤 ≠ 표 the commutator ∗ ̃︶ [휂표, 푈 푆︺(푣휁, 푤휍)푈] is zero. is follows from the observation that
∗ ̃︶ ∗ ∗ ̃︶ ￮휂표, 푈 푆︺(푣휁, 푤휍)푈 = 푈 ￮푈휂표푈 , 푆︺(푣휁, 푤휍) 푈 ∗ ∗ ̃︶ ̃︶ = 푈 ￮푈휂표푈 , i푆︺(훾푣휁 )푆︺(훾푤휍 ) 푈 ∗ ∗ ̃︶ ̃︶ ∗ ̃︶ ∗ ̃︶ = i푈 ￮푈휂표푈 , 푆︺(훾푣휁 ) 푆︺(훾푤휍 )푈 + i푈 푆︺(훾푣휁 ) ￮푈휂표푈 , 푆︺(훾푤휍 ) 푈 = i ￮휂 , 푈∗푆̃︶ (훾 )푈 푈∗푆̃︶ (훾 )푈 + i푈∗푆̃︶ (훾 )푈 ￮휂 , 푈∗푆̃︶ (훾 )푈 ￍ표 ︺ 푤휍ﻰﻰﻰﻰￌﻰﻰﻰﻰￍ표 ︺ 푣휁 ︺ 푤휍 ︺ 푣휁 ￋﻰﻰﻰﻰￌﻰﻰﻰﻰￋ
( by (2.46) = by (2.46= = 0.
e identity in the second line was proved at the end of the proof of lemma 2.11, and third line is a simple application of Leibniz’s rule for commutators. erefore, 푌 being in the local algebra generated by the link operators corresponding to ﾪ ∗ edges in Ω, it follows that [휂표, 푈 푌푈] = 0 (by repeated application of Leibniz’s rule if necessary). Further, one has
∗ ∗ ̃︶ ∗ [휉표, 푈 푌푈] = ￮−i푈 푆︺(훾표휂 )푈휂표, 푈 푌푈 ∗ ̃︶ ∗ ∗ ̃︶ ∗ = −i ￮푈 푆︺(훾표휂 )푈, 푈 푌푈 휂표 − i푈 푆︺(훾표휂 )푈 ￮휂표, 푈 푌푈 = −i푈∗ ￮푆̃︶ (훾 ), 푌 푈휂 − i푈∗푆̃︶ (훾 )푈 ￮휂 , 푈∗푌푈 ￍ표ﻰﻰￌﻰﻰￍ︺ 표휂 표 ︺ 표휂 ￋﻰﻰￌﻰﻰￋ
since 표∉︵ == = 0,
∗ concluding the prove that [푐표, 푈 푌푈] = 0. For 푧 ≠ 표, the representation 3.24 yields
∗ ︈ ￮ ∗ ̃︶ ̃︶ ∗ 푐푧, 푈 푌푈] = − 휉표 푈 푆︺(훾푧휉 ) + i푆︺(훾푧휂 ) 푈, 푈 푌푈] ︈ ∗ ∗ ̃︶ ̃︶ ￍ[휉표, 푈 푌푈] 푈 푆︺(훾푧휉 ) + i푆︺(훾푧휂 ) 푈ﻰﻰￌﻰﻰ ￋ −
= = − ︈ 휉 푈∗ ￮푆̃︶ (훾 ) + i푆̃︶ (훾 ), 푌 푈. 표 ︺ 푧휉 ︺ 푧휂
Denote by Ω the set of all vertices in the strip to the right of Ω and of the same height as Ω (not including Ω, see figure 3.3). en by definition of the path 3.2. D L LR
Ω Ω
. 표 = (1, 1)
Figure 3.3: e spanning tree with root of 핃 is shown, together with the defini- tion of the set Ω.
operators, the assumed spanning tree and 푌 ∈ 픖︵, it follows that
̃︶ ̃︶ [푆︺(훾푧휉 ) + i푆︺(훾푧휂 ), 푌] = 0 for all 푧 ∉ Ω ∪ Ω, and therefore
∗ [푐푧, 푈 푌푈] = 0 for all 푧 ∉ Ω ∪ Ω.
∗ Since 푐푦 = (휉푧 − i휂푧) for all 푧 ∈ Λ, analogous considerations show that ∗ ∗ [푐푧, 푈 푌푈] = 0 for all 푧 ∉ Ω ∪ Ω. Taking this into account, equation 3.23 becomes
︹ ∗ (︹) ∗ (︹) ∗ ∗ [훼푡 (푐푥), 푈 푌푈] = ﾜ A 푥푧(2푡)[푐푧, 푈 푌푈] + ﾜ A 푥푧(−2푡)[푐푧, 푈 푌푈]. 푧∈︵∪︵ 푧∈︵∪︵
By dynamical localisation and the bound ‖ [퐴, 퐵] ‖ ≤ 2‖퐴‖ ‖퐵‖ on the commut- C 3. T G (GMM)
ator of two operators, one gets
⎡ ⎤ ︹ ∗ 피 ⎢sup ￏ ￮훼푡 (푐푥), 푈 푌푈 ￏ⎥ ⎣ 푡∈ℝ ⎦ ⎡ ⎤ ⎡ ⎤ (︹) ∗ (︹) ∗ ∗ ≤ ﾜ 피 ⎢sup ￗA 푥푧(2푡)ￗ⎥ ￏ [푐푧, 푈 푌푈] ￏ + ﾜ 피 ⎢sup ￗA푥푧 (−2푡)ￗ⎥ ￏ [푐푧, 푈 푌푈] ￏ 푧∈︵∪︵ ⎣ 푡∈ℝ ⎦ 푧∈︵∪︵ ⎣ 푡∈ℝ ⎦ ⎡ ⎤ ⎡ ⎤ ∗ (︹) ∗ (︹) ≤ 2‖푈 푌푈‖ ﾜ 피 ⎢sup ￗA 푥푧(2푡)ￗ⎥ + 2‖푈 푌푈‖ ﾜ 피 ⎢sup ￗA 푥푧(−2푡)ￗ⎥ 푧∈︵∪︵ ⎣ 푡∈ℝ ⎦ 푧∈︵∪︵ ⎣ 푡∈ℝ ⎦ ≤ 4‖푌‖ ﾜ 퐶푒−휂푑(푥,푧) ≤ 퐶′‖푌‖푒−휂︃︈︒︓(푥,︵) 푧∈︵∪︵
■ . with ′ +푒−휂 퐶 = 4퐶 −푒−휂
︹ ∗ Clearly, the same estimate holds for 훼푡 (푐푥). By the homomorphism property ︹ of 훼푡 and Leibniz’s rule,
︹ ∗ ︹ ︹ ∗ [훼푡 (푐푥푐푦), 푈 푌푈] = [훼푡 (푐푥)훼푡 (푐푦), 푈 푌푈] ︹ ︹ ∗ ︹ ∗ ︹ = 훼푡 (푐푥)[훼푡 (푐푦), 푈 푌푈] + [훼푡 (푐푥), 푈 푌푈]훼푡 (푐푦)
it follows that (assuming 푥, 푦 ∉ Ω)
⎡ ⎤ ︹ ′ −휂︃︈︒︓(푥,︵) ′ −휂︃︈︒︓(푦,︵) 피 ⎢sup ￎ [훼푡 (푐푥푐푦), 푌]ￎ⎥ ≤ 퐶 ‖푌‖푒 + 퐶 ‖푌‖푒 ⎣ 푡∈ℝ ⎦ ≤ 퐶′‖푌‖푒−휂︃︈︒︓({푥,푦},︵)
∗ ∗ ∗ ∗ e inequality remains valid if 푐푥푐푦 is exchanged by 푐푥푐푦, 푐푥푐푦, 푐푥푐푦 etc. us, since the link operators 푆(푥휁, 푦휍) can be expressed in terms of no more than four of the above terms, e.g.
∗ ∗ ∗ ∗ ∗ ∗ 푆(푥휉, 푦휉) = i휉푥휉푦 = i(푐푥 + 푐푥)(푐푦 + 푐푦) = i(푐푥푐푦 + 푐푥푐푦 + 푐푥푐푦 + 푐푥푐푦) ∗ ∗ ∗ ∗ 푆(푥휉, 푥휂) = i휉푥휂푥 = i(푐푥 + 푐푥)i(푐푥 − 푐푥) = 푐푥푐푥 − 푐푥푐푥
one has
⎡ ⎤ ︹ ′ −휂︃︈︒︓({푥,푦},︵) ′ 휂 −휂︃︈︒︓(푥,︵) ￍ 4퐶 (1 + 푒 ) ‖푌‖푒ﻰﻰￌﻰﻰ피 ⎢sup ￎ [훼푡 (푆(푥휁, 푦휍)), 푌]ￎ⎥ ≤ 4퐶 ‖푌‖푒 ≤ￋ ⎣ 푡∈ℝ ⎦ =∶퐶″ 3.2. D L LR and
⎡ ⎤ ︹ ′ −휂︃︈︒︓(푥,︵) 피 ⎢sup ￎ [훼푡 (푆(푥휉, 푥휂)), 푌]ￎ⎥ ≤ 2퐶 ‖푌‖푒 , ⎣ 푡∈ℝ ⎦ respectively. ︹ ﾪ ︶ Let 훽푡 be the Heisenberg evolution associated with the Hamiltonian 퐻︺(Λ),
ﾪ ︿ ﾪ ︿ ︹ ︈푡퐻﹃ −︈푡퐻﹃ 훽푡 (퐴) = 푒 퐴푒 .
Lemma 3.11. Let 푥, 푦 ∈ Λ, and 푌 ∈ 픖︵ such that 푥, 푦 ∉ Ω, 푥, 푦 < Ω. Assume further that the matrix 퐴(︹) is dynamically localised in the sense of definition 3.6. en
⎡ ⎤ ︹ ̃︶ ″ −휂︃︈︒︓(푥,︵) 피 ⎢sup ￎ [훽푡 (푆︺(푥휁, 푦휍)), 푌]ￎ⎥ ≤ 퐶 ‖푌‖푒 (3.26) ⎣ 푡∈ℝ ⎦ and
⎡ ⎤ ︹ ̃︶ ′ −휂︃︈︒︓(푥,︵) 피 ⎢sup ￎ [훽푡 (푆︺(푥휉, 푥휂)), 푌]ￎ⎥ ≤ 2퐶 ‖푌‖푒 . (3.27) ⎣ 푡∈ℝ ⎦
̃︶ ∗ ﾪ Proof. By Szczerba’s theorem, 푆︺(ℓ) = 푈푆(ℓ)푈 for all (directed) edges ℓ ∈ 핃. Since
︹ ̃︶ ∗ ︹ ̃︶ ￏ ￮훽푡 (푆︺(ℓ)), 푌 ￏ = ￏ 푈 ￮훽푡 (푆︺(ℓ)), 푌 푈 ￏ ﾪ ︿ ﾪ ︿ ﾪ ︿ ﾪ ︿ ∗ ︈푡퐻﹃ ∗ ̃︶ ∗ −︈푡퐻﹃ ∗ ∗ ∗ ︈푡퐻﹃ ∗ ̃︶ ∗ −︈푡퐻﹃ = ￏ푈 푒 푈 푈 푆︺(ℓ)푈 푈 푒 푈 푈 푌푈 − 푈 푌푈 푈 푒 푈 푈 푆︺(ℓ)푈푈 푒 푈ￏ = ￎ 푒︈푡퐻﹂ 푆(ℓ)푒−︈푡퐻﹂ 푈∗푌푈 − 푈∗푌푈 푒︈푡퐻﹂ 푆(ℓ)푒−︈푡퐻﹂ ￎ ︹ ∗ = ￏ ￮훼푡 (푆(ℓ)), 푈 푌푈 ￏ it follows by lemma 3.10 that
⎡ ⎤ ⎡ ⎤ ︹ ̃︶ ︹ ∗ 피 ⎢sup ￎ [훽푡 (푆︺(푥휁, 푦휍)), 푌]ￎ⎥ = 피 ⎢sup ￎ [훼푡 (푆(푥휁, 푦휍)), 푈 푌푈]ￎ⎥ ⎣ 푡∈ℝ ⎦ ⎣ 푡∈ℝ ⎦ (3.28) ≤ 퐶″‖푌‖푒−휂︃︈︒︓(푥,︵)
̃︶ ■ and analogously for 푆︺(푥휉, 푥휂). Having lemmata 3.10 and 3.11 at hand, the following zero-velocity Lieb Robin- son bound in disorder average can be proved: C 3. T G (GMM)
ﾪ ︶ Proposition 3.12. Assume that the constrained Gamma Matrix Model 퐻︺ is dy- namically localised in the sense of definition 3.6. en there exist constants 푐, 휂 > 0 such that for any two sets Ι, Ω ⊂ Λ with Ι < Ω one has
︹ −휂︃︈︒︓(﹚,︵) 피 ￮ ￎ [훽푡 (푋), 푌]ￎ ≤ 푐 min(1, |푡|) ‖푋‖ ‖푌‖ 푒 (3.29)
for all 푋 ∈ 픖﹚ and 푌 ∈ 픖︵.
︹ −휂︃︈︒︓(﹚,︵) . Proof. Part 1: 피 ￮ ￎ [훽푡 (푋), 푌]ￎ ≤ 푐 |푡|‖푋‖‖푌‖푒
Let 푋 ∈ 픖﹚, 푌 ∈ 픖︵, and define the function
︹ 푓(푡) = [훽푡 (푋), 푌]
with derivative
′ ︹ ﾪ ︶ ︹ ﾪ ︶ ︹ ﾪ ︶ ︹ 푓 (푡) = i ￮훽푡 ([퐻︺, 푋]), 푌 = i ￮훽푡 ([퐻︺(Ι), 푋]), 푌 = i ￮[훽푡 (퐻︺(Ι)), 훽푡 (푋)], 푌 ︹ ︹ ﾪ ︶ ︹ ﾪ ︶ ︹ = −i ￮[훽푡 (푋), 푌], 훽푡 (퐻︺(Ι)) − i ￮[푌, 훽푡 (퐻︺(Ι))], 훽푡 (푋)
ﾪ ︶ ﾪ ︶ Here, 퐻︺(Ι) only contains the terms in 퐻︺ that do not commute with 푋. e last line is an application of the Jacobi identity for commutators. us, 푓 satisfies the differential equation
′ ︹ ﾪ ︶ ︹ ﾪ ︶ ︹ 푓 (푡) = −i ￮푓(푡), 훽푡 (퐻︺(Ι)) + i ￮[훽푡 (퐻︺(Ι)), 푌], 훽푡 (푋)
Lemma 3.13. Let 퐴(푡), 푡 ∈ ℝ, be a family of linear, norm preserving operators⁶ in some Banach space 픅. For any function 퐵 ∶ ℝ → 픅, the solution of
휕푡푌(푡) = 퐴(푡)푌(푡) + 퐵(푡)
with boundary condition 푌(0) = 0, satisfies the bound
푡 ‖푌(푡)‖ ≤ ﾙ ‖퐵(푠)‖d푠.
Proof. A proof of this can be found in Appendix A of N et al. [NOS06] (Lemma A.1 therein). ■
⁶i.e. the mapping 훾푡 ∶ 픅 → 픅 which maps 푥 ∈ 픅 to the solution 푋(푡) of the differential equation 휕푋(푡) = 퐴(푡)푋(푡) with initial datum 푋(0) = 푥 is an isometry, ‖훾푡(푥)‖ = ‖푥‖ for all 푡 ∈ ℝ. 3.2. D L LR
By the above lemma, one obtains |푡| ︹ ︹ ﾪ ︶ ︹ ‖푓(푡)‖ = ￏ ￮훽푡 (푋), 푌 ￏ ≤ ﾙ ￏ ￮[푌, 훽푠 (퐻︺(Ι))], 훽푠 (푋) ￏ d푠 |푡| ︹ ﾪ ︶ ≤ 2‖푋‖ ﾙ ￏ ￮훽푡 (퐻︺(Ι)), 푌 ￏ d푠 ﾪ ︶ Since 퐻︺(Ι) is given by ﾪ ︶ ̃︶ ̃︶ (퐻︺(Ι) = 휇 ﾜ 푆︺(푥휉, (푥 + 푒)휂) + 푆︺(푥휉, (푥 + 푒)휂 푥∈﹚−
̃︶ ̃︶ ̃︶ (푆︺(푥휂, (푥 + 푒)휉) − 푆︺(푥휂, (푥 + 푒)휉) + ﾜ 휈푥푆︺(푥휉, 푥휂 − 푥∈﹚
− where Ι = [Ι ∪ (Ι − 푒 − 푒))] ∩ Λ, one has |푡| ︹ ︹ ﾪ ︶ 피 ￏ ￮훽푡 (푋), 푌 ￏ ≤ 2‖푋‖ ﾙ 피 ￏ ￮훽푡 (퐻︺(Ι)), 푌 ￏ d푠 ≤ 8휇‖푋‖ ﾜ 퐶″ ‖푌‖푒−휂︃︈︒︓(푥,︵)|푡| 푥∈﹚− ′ −휂︃︈︒︓(푥,︵) + 2 max |휈푥|‖푋‖ ﾜ 2퐶 ‖푌‖푒 |푡| 푥∈︹ 푥∈﹚ −휂︃︈︒︓(﹚,︵) 푐 |푡|‖푋‖‖푌‖푒 ≥ ″ ′ .|Explicitly, the constant is 푐 = 8휇퐶 + 4퐶 max푥∈︹ |휈푥 ︹ −휂︃︈︒︓(﹚,︵) .Part 2: 피 ￮ ￎ [훽푡 (푋), 푌]ￎ ≤ 푐 ‖푋‖‖푌‖푒 for all 푡 ∈ ℝ
̃︶ By lemma 3.11, the inequality holds for 푋 being a link operator, 푋 = 푆︺(푥휁, 푦휍), with 푥, 푦 ∈ Ι. If 푋 is the sum of such link operators, the claim follows by triangle inequality. If 푋 is the product of link operators corresponding to links with vertices in Ι, a straightforward application of Leibniz’s rule yields the inequality. It follows that for general 푋 ∈ 픖﹚ there exists a constant 푐 < ∞ such that ︹ −휂︃︈︒︓(﹚,︵) .피 ￮ ￎ [훽푡 (푋), 푌]ￎ ≤ 푐 ‖푋‖‖푌‖푒 holds for all 푡 ∈ ℝ ■ .Part 3: Choosing 푐 = max(푐, 푐) concludes the proof of inequality 3.29 is is a first step in proving zero-velocity Lieb Robinson bounds for the con- strained, extended Gamma matrix model. e proof still relies on the additional assumption of the relative positions of the sets Ι and Ω. It is also an open ques- tion, whether or not the local algebras generated by the link operators cover all physically interesting observables. Nevertheless, it has been demonstrated how the methods in [HSS12] may be applied in the context of an approach via the generalised Jordan Wigner transformation due to W and S. C 3. T G (GMM) Chapter 4
Conclusion and Outlook
this thesis the Jordan Wigner transformation has been discussed as an im- I portant tool in studying properties of one-dimensional quantum spin systems. Various possible extensions of this transformation have been introduced. It turned out that the appearance of statistical gauge fields aer the transformation is re- lated to the structure of the underlying graph. In particular, one (surprising) result was that only simple path graphs allow for a special Jordan Wigner trans- formation in the sense of definition 2.2. An alternative approach, going back to N in the 1950’s and reformu- lated in terms of link operators is based upon the observation that the Pauli spin matrices are generators of a Clifford algebra. Using higher dimensional Clifford algebras in expressing the link operators is the main idea behind the W S approach. One of the most basic applications of S’s theorem 2.10 – the Gamma matrix model – was introduced, and its connection to the link operators estab- lished. Aer slight modifications, the constrained Gamma matrix model allows for a description by Jordan Wigner fermions. is relation allowed one to study the constrained GMM in a random external field and prove localisation bounds on the Heisenberg evolution – a zero-velocity Lieb Robinson type bound in dis- order average (3.29). Even though the bound presented here relies on additional assumptions, it is nevertheless interesting in that it provides another example of a quantum spin system where some of the techniques developed by H, S and S can be applied. As hinted at the end of the general discussion of S’s theorem and the Gamma matrix model, these methods can also be applied to appropriate sub- spaces of higher dimensional quantum spin systems or quantum spin systems on general graphs (keeping in mind the restriction on the vertex degrees), using higher dimensional generalisations of the Gamma matrices.
C 4. C O Appendix A
Some missing calculations in S’s theorem
properties of the link and path operators that were le out in the present- S ation of the proof of theorem 2.10 and lemma 2.11 are presented here for completeness, since they mainly involve calculations and cannot be found in the original paper [Szc85].
Lemma A.1 (Path operators). Let {푆(ℓ) ∶ ℓ ∈ 퐸}̃ be a family of link operators satisfying properties (i) and (ii) of definition 2.8, and 푆(ℓ−) = −푆(ℓ). en the path operators 푆(훾) (see definition 2.8, (iv)) have the following properties: i) 푆(훾 ∘ 훾) = −i푆(훾)푆(훾) if the vertex at the end of 훾 is the first vertex of) ,훾 and 훾 ∘ 훾 denotes their concatenation ii) {푆(훾), 푆(훾)} = 0 if 훾 and 훾 are simple paths with exactly one common) edge-point (beginning or end),
(iii) [푆(훾), 푆(ℓ)] = 0 for all closed paths 훾 and ℓ ∈ 퐸̃ .
,iv) 푆(훾)∗ = 푆(훾) and (푆(훾)) = ퟙ if 훾 is not a closed path)
,v) 푆(훾)∗ = −푆(훾) and (푆(훾)) = −ퟙ if 훾 is a closed path)
.vi) 푆(훾−) = −푆(훾)− for all paths 훾)
Proof. roughout the proof let 훾 = 훾 = ℓ ∘ ⋯ ∘ ℓ푚, 훾 = ℓ푚+ ∘ ⋯ ∘ ℓ푚+푛 for ﾪ 푚, 푛 ∈ ℕ, be paths in 핃 with the property that ℓ푚 and ℓ푚+ are neighbouring edges.
A A. S S’
i) en 훾 ∘ 훾 = ℓ ∘ ⋯ ℓ푚+푛 and)
푚+푛− (푆(훾 ∘ 훾) = (−i) 푆(ℓ) ⋯ 푆(ℓ푚)푆(ℓ푚+) ⋯ 푆(ℓ푚+푛 푚− 푛− i) (−i) 푆(ℓ) ⋯ 푆(ℓ푚) (−i) 푆(ℓ푚+) ⋯ 푆(ℓ푚+푛)−) =
.(i푆(훾)푆(훾− =
= {(ii) Assume additionally that 훾 and 훾 are simple paths. en {푆(ℓ푚), 푆(ℓ푚+) 0, implying
푚+푛− (푆(훾)푆(훾) = (−i) 푆(ℓ) ⋯ 푆(ℓ푚)푆(ℓ푚+) ⋯ 푆(ℓ푚+푛 푚+푛− (i) 푆(ℓ) ⋯ 푆(ℓ푚+)푆(ℓ푚) ⋯ 푆(ℓ푚+푛−)− = 푚+푛− ,(i) 푆(ℓ푚+) ⋯ 푆(ℓ푚+푛)푆(ℓ) ⋯ 푆(ℓ푚) = −푆(훾)푆(훾−)− = the laer equality being a consequence of the fact that all other permuta- tions involved in this calculation involve only link operators to non-neighbouring edges, which commute by assumption on the link operators. iii) Suppose that 훾 is a closed path, i.e. ℓ and ℓ푚 are neighbouring edges, and) let ℓ ∈ 퐸̃ . If ℓ and 훾 have no common vertex, [푆(훾), 푆(ℓ)] = 0 is trivial. In case they share exactly one common vertex, there are two adjoining edges
.(to ℓ in 훾, say ℓ푘 and ℓ푘+ for some 푘 ∈ {1, … , 푚} (identifying 푚 + 1 ≡ 1 Hence
푚− (푆(훾)푆(ℓ) = (−i) 푆(ℓ) ⋯ 푆(ℓ푘)푆(ℓ푘+) ⋯ 푆(ℓ푚)푆(ℓ 푚− (푆(ℓ)(−i) 푆(ℓ) ⋯ 푆(ℓ푚) = 푆(훾)푆(ℓ (−1) =
If 훾 and ℓ have two common vertices, i.e. ℓ ≡ ℓ푘 ∈ 훾, then
푚− (푆(훾)푆(ℓ) = (−i) 푆(ℓ) ⋯ 푆(ℓ푘−)푆(ℓ푘)푆(ℓ푘+) ⋯ 푆(ℓ푚)푆(ℓ 푚− .(푆(ℓ)(−i) 푆(ℓ) ⋯ 푆(ℓ푚) = 푆(훾)푆(ℓ (−1) =
(iv) Assume 훾 is not a closed path. en ∗ 푚− 푚− ∗ (푆(훾) = (−i) 푆(ℓ) ⋯ 푆(ℓ푚) = i 푆(ℓ푚) ⋯ 푆(ℓ 푚− 푚− 푚− (i (−1)푆(ℓ)푆(ℓ푚) ⋯ 푆(ℓ) = i (−1) 푆(ℓ)푆(ℓ) ⋯ 푆(ℓ푚 = = 푆(훾) and
푚− (푆(훾) = (−1) 푆(ℓ) ⋯ 푆(ℓ푚)푆(ℓ) ⋯ 푆(ℓ푚 푚− (푆(ℓ) 푆(ℓ) ⋯ 푆(ℓ푚)푆(ℓ) ⋯ 푆(ℓ푚 (−1)− = 푚− 푚− .푆(ℓ) 푆(ℓ) ⋯ 푆(ℓ푚) = ퟙ (−1) (−1) =
(v) If 훾 is a closed path, then the steps in (iv) stay essentially the same, apart from the fact that {푆(ℓ), 푆(ℓ푚)} = 0 instead of [푆(ℓ), 푆(ℓ푚)] = 0, leading to an additional minus sign in both calculations.
(vi) Now let 훾 be arbitrary.
푚− 푚− − −− ( 푆(훾)푆(훾 ) = (−i) 푆(ℓ) ⋯ 푆(ℓ푚) ⋅ (−i) 푆(ℓ푚 ) ⋯ 푆(ℓ 푚− 푚 .푆(ℓ) ⋯ 푆(ℓ푚)푆(ℓ푚) ⋯ 푆(ℓ) = −ퟙ (−1) (−1) =
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I hereby declare that I have wrien this thesis without any help from others and without the use of documents and aids other than those stated above. I have mentioned all used sources and cited them correctly according to established academic citation rules.
Hiermit versichere ich, die vorliegende Arbeit selbstständig und lediglich unter Zuhilfe- nahme der genannten ellen verfasst zu haben.
M, 31 A 2013 T R