A tensor network approach to non-chiral topological order Carolin Wille Im Fachbereich Physik der Freien Universität Berlin eingereichte Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften. Berlin, 2019 Erstgutacher: Prof. Dr. Jens Eisert Zweitgutachter: Prof. Emil Bergholtz, Ph.D. Tag der Disputation: 15.5.2020 Peer-reviewed publications [1] C. Wille, O. Buerschaper, and J. Eisert, “Fermionic topological quantum states as tensor networks,” Phys. Rev. B 95, 245127 (2017). [2] C. Wille, R. Egger, J. Eisert, and A. Altland, “Simulating topological tensor networks with Majorana qubits,” Phys. Rev. B 99 (2019). Pre-prints [3] A. Bauer, J. Eisert, and C. Wille, “Towards a mathematical formalism for classifying phases of matter,” (2019), arXiv:1903.05413. ii Contents I. Introduction 1 1. Overview 3 2. Gapped local quantum phases 9 2.1. Entanglement ...................................... 10 2.1.1. Measures of entanglement ........................... 11 2.1.2. Entanglement vs. correlation ......................... 12 2.1.3. Area law ..................................... 12 2.2. Definition and classification .............................. 13 2.2.1. Short-range vs. long-range entangled states ................. 14 2.3. Tensor network states .................................. 16 2.3.1. Penrose notation ................................ 17 2.3.2. Matrix product states .............................. 19 2.3.3. Projected entangled pair states ........................ 21 2.3.4. Entanglement area law for tensor network states .............. 23 2.4. One dimension – A full classification via matrix product states .......... 24 2.4.1. Overview of the result ............................. 25 2.4.2. Injective MPS .................................. 25 2.4.3. G-injective MPS ................................. 28 2.4.4. Phase classification ............................... 31 2.4.5. Limitations of the approach for higher dimensions ............ 32 3. A conceptual introduction to topological order 35 3.1. The toric code ...................................... 35 3.1.1. The toric code as a quantum code ...................... 36 3.1.2. Topological ground state degeneracy .................... 37 3.1.3. Local indistinguishability of ground states ................. 37 3.1.4. Anyonic excitations of the toric code ..................... 38 3.2. G-injective PEPS ..................................... 39 3.2.1. Definition .................................... 39 3.2.2. Parent Hamiltonian ............................... 40 3.2.3. The toric code as a Z2-isometric PEPS .................... 42 3.2.4. Entanglement entropy ............................. 44 3.2.5. Local excitations ................................ 45 II. Non-chiral topological order 47 4. The algebraic theory of anyons 49 4.1. Fusion categories .................................... 50 4.1.1. Monoidal categories .............................. 51 iii Contents 4.1.2. Rigidity, semi-simplicity, linearity and unitarity .............. 52 4.2. Modular braided unitary fusion categories and anyons .............. 53 4.2.1. Anyons ...................................... 53 4.2.2. Modularity ................................... 55 4.2.3. Non-chiral anyon theories and the Drinfeld center ............. 56 5. MPO-isometric PEPS, string-nets and state-sums 59 5.1. Levin–Wen string net models ............................. 60 5.1.1. Deformation rules ............................... 62 5.1.2. Relation to fusion categories and remarks on tetrahedral symmetry ... 65 5.1.3. Anyons ...................................... 65 5.1.4. Tensor network construction ......................... 68 5.2. MPO-isometric PEPS .................................. 69 5.2.1. Translation invariant Hermitian projector MPOs and fusion categories . 71 5.2.2. MPO-isometric PEPS .............................. 73 5.2.3. Parent Hamiltonians and topological order ................. 74 5.2.4. String-net PEPS as MPO-isometric PEPS ................... 78 5.3. Example – The twisted quantum double models .................. 80 5.3.1. String-net description ............................. 81 5.3.2. MPO-isometric PEPS and the state sum construction ........... 84 5.4. General bosonic state sum constructions ....................... 88 5.4.1. Tensor lattice algebras ............................. 88 5.4.2. Chirality ..................................... 89 5.5. Fermionic non-chiral topological order ........................ 90 5.5.1. Fermionic tensor networks .......................... 91 5.5.2. Fermionic twisted quantum double models ................. 94 III. Synthetic topological quantum matter 105 6. Realizing topological order in mesoscopic systems 107 6.1. Low-energy effective theories ............................. 109 6.1.1. Self-energy expansion and the Kitaev honeycomb model ......... 110 6.1.2. Bloch method and Jordan–Farhi gadgets ................... 113 6.1.3. The Schrieffer–Wolff method ......................... 117 6.1.4. Perturbative parent Hamiltonians ...................... 119 6.2. Majorana Cooper box networks ............................ 125 6.2.1. Majorana Cooper box ............................. 125 6.2.2. Tunneling Hamiltonian ............................ 127 6.2.3. Effective low-energy theory .......................... 129 6.2.4. Destructive interference mechanisms .................... 131 6.2.5. Engineering multi-qubit operators ...................... 134 6.2.6. Hierarchical designs .............................. 137 6.3. String-net phases in MCB networks .......................... 142 6.3.1. General design ................................. 143 6.3.2. D = 2 – The double semion blueprint .................... 145 6.3.3. D = 5 – A double Fibonacci candidate .................... 147 7. Summary and discussion 149 iv Contents A. Technical details on fMPO-symmetric fPEPS 153 A.1. Concatenation stability ................................. 153 A.1.1. Hamiltonian gauge of the fermionic toric code model ........... 154 B. Perturbation analysis of the double semion MCB network 157 Bibliography 159 v Part I. Introduction 1 1. Overview This Thesis discusses how non-chiral topological order in two-dimensions can be accessed within a tensor network framework. The approach is two-fold. On the one hand, we extensively discuss how tensor networks can be used as a tool to characterize topological order [1, 4–11] and how this approach blends in with other mathematical frameworks, such as string-net models [12], state sum constructions [3, 13–17] and the abstract algebraic theory of anyons formulated using category theory [18, 19]. On the other hand we use the tensor network description as a conceptual tool in the design of synthetic topological quantum matter built from mesoscopic devices, concretely networks of tunnel coupled Majorana Cooper boxes [2]. Topological order Topological order is undoubtedly one of the most intriguing subjects studied in modern con- densed matter physics. Its discovery and theoretical investigation throughout the last decades has profoundly changed our understanding of phases of matter and the mechanisms underlying their formation [20] – an achievement only recently rewarded by the Nobel prize [21]. States of matter baptized “novel” or “exotic” due to their puzzling properties in the early 80’s are nowadays understood well enough to attempt harvesting their fundamental “quantumness” for applications in quantum computation [22] – a technology with revolutionary potential [23–26]. At the heart of this remarkable development is the insight, that the formation of different phases of matter can not only be attributed to the breaking of different physical symmetries – as captured beautifully by Landau’s theory of symmetry breaking [27, 28] – but also to a purely quantum mechanical properties captured by the entanglement structure of the system [20]. Entanglement characterizes the information shared between subsystems which exceeds the information captured by classical correlations [29]. As such it is a genuine quantum many body phenomenon and relies on strong interactions. Topological ordered states of matter are characterized by long-range entanglement [30] which means that information is shared in a particular non-local fashion. As a consequence a system with topological order is sensitive to global properties of its environment which can not be detected locally – more precisely it is sensitive to the topology of the manifold on which it resides. The simplest example of such a dependency is the topological ground state degeneracy, i.e., the dimension of the ground state space depends on topological manifold invariants, e.g. the genus of a two-dimensional surface [31]. Topological order in two dimensions is of particular interest, because of the theoretically predicted point-like quasi-particle excitations which behave neither as bosons nor as fermions when exchanging the position of identical particles. These so-called anyons [31–34] feature the curious property that the wave-function is not uniquely defined by the position of the particles and instead there is a certain Hilbert space compatible with the presence of a number of such particles. For certain systems, exchanging the position of two such particles has the effect of performing a unitary rotation in this Hilbert space[18, 35, 36]. This unconventional property can again be attributed to the systems sensitivity to topological properties. An intuitive 3 1. Overview interpretation of the latter statement is to consider the particle trajectory as knotted world-lines.