Constructing Landau Formalism for Topological Order: Spin Chains and Ladders
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1 Constructing Landau Formalism for Topological Order: Spin Chains and Ladders Gennady Y. Chitov Laurentian University Sudbury, Canada Talk at Washington University in St. Louis, October 20, 2016
2 Collaborators: Toplal Pandey (PhD student) Earlier related work: A. Kalz R. Meyer M. Azzouz Students: S.J. Gibson B. Ramakko Ref.: Supported by:
3 Outline:! Motivation: Phase Transitions without Local Order Parameters --! Examples! Dimerized XY chain: Exact Results! Dimerized Two-Leg Ladders: Mean-Field (Free-fermionic)! Approximation!! Local vs Topological String Order Parameters; Winding Numbers! (Topological indices)!! Conclusions & Future work!
4 Motivation: Phase transitions without local order parameters Conventional Landau Theory 1. Berezinskii-Kosterlitz-Thouless Transition. 2D classical XY model S = 0 No long-range order i finite T C SS i j exponential power-law Binding-Unbinding of vortices Dual (nonlocal) order parameter (Wiegmann, 1977)
5 2. Solid-on-Solid Models. Surface Roughening Transitions Den Nijs & Rommelse PRB 1989 No long-range order Mapping: SOS 6-vertex model Ising spin More Mapping quantum spin model String Order Parameter (SOP) Hidden (Topological) Order
6 3. Frustrated 2D nn+nnn Ising Model G.Y.C. and C. Gros, Low Temp. Phys. (2005) FP = Floating Phase A. Kalz and G.Y.C., PRB (2013) PM to FP transition = BKT transition -> NO local Order Parameter Power-Law Correlation function Analogue: 2D ANNNI model P. Bak, Rep. Prog. Phys. (1982)
7 4. Kitaev Model (exactly solvable, free Majorana fermions) Kitaev, Ann. Phys, 2003, 2006 Feng, Zhang & Xiang, PRL 2007 Kitaev Model (SOP) Jordan Wigner Majorana Fermions XY spin chain in transverse field (local Order Parameter) α = 0, β=1/8, ν=1, η=1/4 Critical indices of the 2D Ising Model
8 Intermission:
9 Dimerized XY chain: Refs: Perk, et al, Physica (75) Ye, et al, Comm. Theor. Phys. (02,03) Jordan-Wigner Transformation: Gap: Lines of Quantum Phase Transitions:
10 String Order Parameter: Refs: den Nijs and Rommelse, PRB (89) Berg, et al, PRB (08) GYC and TP, (16) String Operator: String-String Correlation Function: String Order Parameter:
11 Dimerized XY chain: (contd) Duality Transformation (E.g., Fradkin & Susskind, PRB(78)): Dimerized XY Chain (h=0) Two Decoupled Ising Chains in Transverse Field (residing on even/odd dual sites) Nonlocal SOP of XY Chain Local (Landau) OP for Ising Chains in Transverse Field
12 Local and Nonlocal (String) Order Parameters: String Order Parameters of the XY chain: Magnetization: Topological winding number:
13 Dimerized Two-Leg Ladders: Refs: Martin-Delgado, et al (96-08) For more refs, Jordan-Wigner Transformation + Mean-Field Approximation: Staggered Ladder: Columnar Ladder: Gap: Lines of Quantum Phase Transitions: See also GYC, et al, PRB (08) Gap: Lines of Quantum Phase Transitions-None (Always Gapped)
14 String Order Parameters (Staggered Ladders): Effective Free-Fermionic Hamiltonian (mean-field) in the Majorana representation : Inverse Jordan-Wigner transformation from Majorana to Dual Spins:
15 Two different paths used for String Operators: String Order Parameters (mutually exclusive): Critical indices of the 2D Ising Model Topological winding number:
16
17 Model Hamiltonians: Similarities & Extensions Dimerized XY chain Staggered 2-leg ladder (approximate) Bogoliubov-de Gennes Hamiltonian of topological SC (Class DIII) TI/SC Hamiltonians, Kitaev ladder
18 String Order Parameters Calculations: 2-leg ladder Transverse Ising chain with three-spin interaction Generalization: n-leg ladders/tubes Ising chains with multi-spin interactions (work in progress)
19 Conclusions & Future work: 1. Dimerized XY chain: Exact SOPs and Landau (local) OPs calculation. Winding numbers. 2. Dimerized Ladders: Mean-Field approximation -> Similar Program. Analytical results for SOPs (!!) 3. Unifying framework: local and nonlocal OPs, hidden symmetries. 4. Similar Hamiltonians: chains, ladders, topological insulators/ superconductors. 5. Work in progress: spin tubes and n-leg ladders.
20 THE END THANK YOU!
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