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

115 Pages·2013·4.09 MB·English

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
Fakult¨at Physik
Technische Universit¨at 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 Pad´e extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 dLog Pad´e 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 ther-
modynamical 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