Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model Masterarbeit zur Erlangung des akademischen Grades Master of Science vorgelegt von Sebastian Fey geboren in Herborn Lehrstuhl fu¨r Theoretische Physik I Fakultät Physik Technische Universität Dortmund 2013 1. Gutachter : Dr. Kai P. Schmidt 2. Gutachter : Dr. Julien Vidal Datum des Einreichens der Arbeit: 30.September2013 Abstract In this thesis the phase transition between the non-Abelian topological phase of Kitaev’s honeycomb model and the polarized phase present at large fields is studied. To this end, perturbative Continuous Unitary Transformations (pCUT) about the high-field limit and exact diagonalization (ED) for several finite systems with both, open and periodic boundary conditions, are performed. Afterwards pCUT results for the thermodynamical limit are presented. For a positive interaction parameter the validity range of the series expansion cannot capture the phase transition. This is different for ED. A comparison to DMRG for systems up to 64 spins by Jiang et al. shows a deviation that might be due to finite-size effects. For a negative interaction parameter the ED spectra display very complex behavior. Series expansion of the one-particle gap and the second-derivative of the ground-state energy obtained in ED indicate a second-order phase transition out of the polarized phase. Nevertheless, a first-order phase transition and the presence of more than two phases in the phase diagram cannot be excluded. i Contents Contents Contents i List of Figures iii List of Tables vii 1 Introduction 1 2 Methods 7 2.1 Fourier transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Exact Diagonalization (ED & Lanczos algorithm) . . . . . . . . . . . . . 9 2.2.1 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.2 Lanczos algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Finite-size effects in the Kitaev model for h = 0 . . . . . . . . . . 15 2.3 Perturbative Continuous Unitary Transformations (pCUT) . . . . . . . 16 2.3.1 Continuous Unitary Transformations (CUT). . . . . . . . . . . . 17 2.3.2 Perturbative CUT . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Application to the model . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Extrapolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.1 Padé extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 dLog Padé extrapolation & critical parameters . . . . . . . . . . 27 3 Model Systems 31 3.1 Ground-state energy per site of small systems . . . . . . . . . . . . . . . 31 3.1.1 Isolated dimer in a uniform field . . . . . . . . . . . . . . . . . . 32 3.1.2 Three spins with x- and y-coupling . . . . . . . . . . . . . . . . . 35 3.1.3 Four spins with x-, y-, and z-coupling . . . . . . . . . . . . . . . 37 3.2 Gap for positive interaction parameter J . . . . . . . . . . . . . . . . . . 38 3.2.1 Isolated dimer in a uniform field . . . . . . . . . . . . . . . . . . 38 3.2.2 Three spins with x- and y-coupling . . . . . . . . . . . . . . . . . 40 3.2.3 Four spins with x-, y-, and z-coupling . . . . . . . . . . . . . . . 41 3.3 Gap for negative interaction parameter J . . . . . . . . . . . . . . . . . 42 3.3.1 Isolated dimer in a uniform field . . . . . . . . . . . . . . . . . . 42 3.3.2 Three spins with x- and y-coupling . . . . . . . . . . . . . . . . . 43 3.3.3 Four spins with x-, y-, and z-coupling . . . . . . . . . . . . . . . 43 Contents ii 3.4 Summary of the results for small open systems . . . . . . . . . . . . . . 44 4 Large Periodic Systems & Thermodynamic Limit 47 4.1 Phase transition for positive J . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.1 Periodic 8-site system . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.2 Periodic 18-site system . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1.3 Periodic 24-site system . . . . . . . . . . . . . . . . . . . . . . . . 53 4.1.4 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Phase transition for negative J . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.1 Periodic 8-site system . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.2 Periodic 18-site system . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.3 Periodic 24-site system . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.4 Thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Conclusion and Outlook 71 A Transformation from Antiferromagnetic to Ferromagnetic Interac- tion. 75 B Hamiltonian in Field Basis 77 C Eigenvalues of Plaquette Operators W 79 p D pCUT Results 81 D.1 Overview of pCUT series . . . . . . . . . . . . . . . . . . . . . . . . . . 81 D.1.1 Ground-state energy per site . . . . . . . . . . . . . . . . . . . . 81 D.1.2 Gap for J ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 D.1.3 Gap for J ≤ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 D.2 One-qp hopping amplitudes in the thermodynamic limit . . . . . . . . . 85 E ED Results 89 E.1 Eigenvalues of an isolated z-dimer in the (1,1,1)-magnetic field . . . . . 89 F Periodic Lattices 91 G Kitaev’s Three-Spin Interaction 93 H Phase Transition for Various J = J with Fixed J 95 x y z Bibliography 97 iii List of Figures List of Figures 1.1 Honeycomb and brickwall lattice for the Kitaev model. . . . . . . . . . . 2 1.1.(a) Honeycomb lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.(b) Brickwall lattice. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Phase diagram of the original Kitaev model. . . . . . . . . . . . . . . . . 3 1.3 Field after transformation to opposite interaction sign. . . . . . . . . . . 5 2.1 Lattice vectors and unit cell of the honeycomb lattice. . . . . . . . . . . 8 2.2 Difference between the energy spectrum of a finite system and the thermodynamical limit at the Kitaev point. . . . . . . . . . . . . . . . . . . 15 2.2.(a) Thermodynamic limit. . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.(b) Finite system.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The flow of the pCUT method. . . . . . . . . . . . . . . . . . . . . . . . 21 3.1 Dimer: Full ED spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Dimer: Convergence of the series coefficients of the ground state. . . . . 34 3.3 Dimer: Different orders of the series of the ground-state energy. . . . . . 34 3.4 3-site system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.5 3-site system: Full ED spectrum. . . . . . . . . . . . . . . . . . . . . . . 35 3.6 3-site system: Different orders of the series of the ground-state energy. . 36 3.7 4-site system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 4-site system: Different orders of the series of the ground-state energy. . 38 3.9 Dimer: Different orders of the series of the gap for J > 0. . . . . . . . . 39 3.10 3-site system: Different orders of the series of the gap for J > 0. . . . . 40 3.11 4-site system: Different orders of the series of the gap for J > 0. . . . . 41 3.12 Dimer: Different orders of the series of the gap for J < 0. . . . . . . . . 43 3.13 3-site system: Different orders of the series of the gap for J < 0. . . . . 44 3.14 4-site system: Different orders of the series of the gap for J < 0. . . . . 45 4.1 Two periodic lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.(a) 8 spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.(b) 18 spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 8-site system: Series and extrapolations for J > 0. . . . . . . . . . . . . 49 4.2.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.2.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 8-sitesystem: Lowpartofthespectrumandground-stateenergyderivatives. 50 List of Figures iv 4.3.(a) Spectrum, J = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3.(b) d E and d2 E , J = 1 . . . . . . . . . . . . . . . . . . . . . 50 dh 0 dh2 0 4.4 8-site system: Number of vortices for J > 0. . . . . . . . . . . . . . . . . 51 4.4.(a) J = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4.(b) h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.5 18-site system: Series and extrapolations for J > 0. . . . . . . . . . . . . 52 4.5.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.5.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 18-site system: 2nd derivative of ground-state energy for J > 0. . . . . . 52 4.7 18-site system: Number of vortices for J > 0. . . . . . . . . . . . . . . . 53 4.7.(a) J = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.(b) h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.8 Periodic 24-site system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.9 24-site system: Series and extrapolations for J > 0. . . . . . . . . . . . . 55 4.9.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.9.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.10 24-site system: 2nd derivative of ground-state energy for J > 0. . . . . . 55 4.11 24-site system: Number of vortices for J > 0. . . . . . . . . . . . . . . . 56 4.11.(a) J = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.11.(b) h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.12 Thermodynamic limit: Series and extrapolations for J > 0. . . . . . . . 57 4.12.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.12.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.13 18- and various 24-site systems: 2nd derivative of ground-state energy for J > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.14 Curves of 24, 32, 36, and 64 spin systems from [JGQT11]. . . . . . . . . 59 4.15 8-site system: Series and extrapolations for J < 0. . . . . . . . . . . . . 60 4.15.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.15.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.16 8-site system: Low part of the spectrum with pCUT curves. . . . . . . . 61 4.17 8-site system: 2nd derivative of ground-state energy for J < 0 . . . . . . 62 4.18 8-site system: Number of vortices for J < 0. . . . . . . . . . . . . . . . . 62 4.18.(a) J = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.18.(b) h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.19 18-site system: Series and extrapolations for J < 0. . . . . . . . . . . . . 63 4.19.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.19.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.20 18-site system: 2nd derivative of ground-state energy and vortex number for J < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.20.(a) d2E/dh2, J = −1 . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.20.(b) d2E/dJ2, h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.20.(c) Vortex number, J = −1. . . . . . . . . . . . . . . . . . . . . . . 64 4.20.(d) Vortex number, h = 1. . . . . . . . . . . . . . . . . . . . . . . 64 v List of Figures 4.21 24-site system: Series and extrapolations for J < 0. . . . . . . . . . . . . 65 4.21.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.21.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.22 24-site system: 2nd derivative of ground-state energy and vortex number for J < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.22.(a) d2E/dh2, J = −1 . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.22.(b) d2E/dJ2, h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.22.(c) Vortex number, J = −1. . . . . . . . . . . . . . . . . . . . . . . 66 4.22.(d) Vortex number, h = 1. . . . . . . . . . . . . . . . . . . . . . . 66 4.23 Thermodynamic limit: Series and extrapolations for J < 0. . . . . . . . 67 4.23.(a) pCUT series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.23.(b) Extrapolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.24 Comparison of the gap calculated with ED and pCUT. . . . . . . . . . . 69 4.25 18- and various 24-site systems: 2nd derivative of ground-state energy for J < 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.25.(a) J = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.25.(b) h = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A.1 Transformation of the interaction sign of the Kitaev model. . . . . . . . 75 A.2 Application of the transformation that changes the interaction sign on the Kitaev model in a uniform field. . . . . . . . . . . . . . . . . . . . . 76 A.2.(a) Transformation pattern. . . . . . . . . . . . . . . . . . . . . . . 76 A.2.(b) Transformed field. . . . . . . . . . . . . . . . . . . . . . . . . . 76 C.1 Plaquette of the honeycomb lattice for definition of W . . . . . . . . . . 79 p D.1 9-site system: Series for J > 0. . . . . . . . . . . . . . . . . . . . . . . . 83 D.2 9-site system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 D.3 9-site system: Series for J < 0. . . . . . . . . . . . . . . . . . . . . . . . 84 D.4 Illustration of the site labeling. . . . . . . . . . . . . . . . . . . . . . . . 85 F.1 Overview of periodic lattices I. . . . . . . . . . . . . . . . . . . . . . . . 91 F.1.(a) Periodic 8-site system. . . . . . . . . . . . . . . . . . . . . . . 91 F.1.(b) Periodic 18-site system. . . . . . . . . . . . . . . . . . . . . . . 91 F.1.(c) Periodic 24-site system. . . . . . . . . . . . . . . . . . . . . . . 91 F.2 Overview of periodic lattices II. . . . . . . . . . . . . . . . . . . . . . . . 92 F.2.(a) Periodic 24-site system (diamond). . . . . . . . . . . . . . . . 92 F.2.(b) Periodic 24-site system (4x3 plaquettes). . . . . . . . . . . . . 92 H.1 Illustration of the path in the phase diagram. . . . . . . . . . . . . . . . 95 H.2 18-site system: 2nd derivative of ground-state energy for various J = J . 96 x y H.3 24-site system: 2nd derivative of ground-state energy for various J = J . 96 x y

Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model

Similar Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model