Floating phase vs chiral transition in classical and quantum models - PowerPoint PPT Presentation

Floating phase vs chiral transition in classical and quantum models F. Mila Ecole Polytechnique Fédérale de Lausanne Switzerland Natalia Chepiga Lausanne à Irvine

Scope n C-IC transition in 2D classical 3-state Potts model à Potts, chiral (Huse-Fisher), or intermediate critical phase n Hard-boson model of trapped alkali atoms in 1D à Similar physics à Efficient DMRG algorithm à Evidence for all 3 possibilities n Implications for quantum spin chains n Conclusions

Domain walls in commensurate phase A B C ≠ A C B Chiral perturbation Huse-Fisher, 1982

Asymmetric 3-state Potts in 2D n i =0,1 or 2 n Huse-Fisher: possibility of a chiral transition between a Potts point and a Lifshitz point n Intermediate (floating) critical phase beyond Lifshitz point

Huse-Fisher phase diagram Kosterlitz-Thouless L Pokrovsky-Talapov Huse and Fisher, 1982

Difference between the 3 cases n Δ q: distance to 2 π /3; ξ : correlation length n Δ q x ξ à 0 for Potts à cst > 0 for chiral à + ∞ for KT transition n Monte Carlo simulations in the eighties à systems too small to extract Δ q with sufficient precision Selke and Yeomans, 1982

MC simulations of asymmetric Potts model Selke and Yeomans, 1982

Field theory argument Intermediate phase away from Potts Haldane, Bak, Bohr, 1983; Schulz, 1983 Not necessarily true if dislocations are allowed Huse-Fisher, 1984

Hard boson model n Period-3 ordering of a chain of trapped alkali-metal atoms Bernien et al, Nature 2017 n Hard boson model n Two constraints Fendley, Sengupta, Sachdev 2004

Hard-bosons: phase diagram Fendley et al, 2004; Chepiga and FM, arXiv:1808.08990

Phase transitions n Transition out of period-2 phase: à Ising à Tricritical Ising point à First order n Disorder line: entirely inside the disordered phase n Transition out of period-3 phase: Commensurate-Incommensurate transition

Hard bosons – previous results n Fendley-Sengupta-Sachdev (2004) à Intermediate phase for U à - ∞ à Probable intermediate phase up to Potts n Samajdar, Choi, Pichler, Lukin, Sachdev (2018) à Evidence of non-integer dynamical exponent between Potts and V à + ∞ à Chiral transition between Potts and V à + ∞

Hilbert space n Constraints è dim = Fibonacci number F n+1 Golden ratio Fendley et al 2004 n Same Hilbert space dimension as Quantum Dimer Model on a ladder Sierra and Martin Delgado 1997 Are these models related?

Quantum Dimer Model Staggered states + RK v/J 1 0

Exclude staggered states Period 3 Rung singlet RK v/J 1 2.67 0 3-state Potts transition? Not clear à Gap closing, but complicated spectrum à Incommensurate short-range correlations right of RK point in rung singlet phase

General quantum dimer ladder Counts vertical flippable plaquettes Counts horizontal Flips plaquettes flippable plaquettes

QDM ó Hard bosons n Exclude the two staggered configurations Then N. Chepiga and FM, arXiv:1809:00746

DMRG algorithm n Implement constraint when building MPS à huge reduction of Hilbert space n For QDM, all tensors are block diagonal in label 0 or 1 à better than hard-boson model n Simulations up to 9’000 sites, routinely 4’800 à Bond dimension up to 2’200 to keep all states with Schmidt value larger than 10 -12

Extracting Δ q and ξ Fit dimer-dimer correlations with Orstein- Zernicke n Δ q: extremely precise results (at least 3 digits) n ξ : up to thousand or so à meaningful evaluation of Δ q ξ

Two-step fit ξ q

Hard-bosons: phase diagram Fendley et al, 2004; Chepiga and FM, arXiv:1808.08990

Three cases

Three cases n Potts: Δ q goes to zero with exponent 5/3, ξ diverges with exponent 5/6 è Δ q ξ à 0 n Vicinity of Potts: Single transition è Δ q goes to zero with exponent smaller than 1 è Δ q ξ à constant n Far from Potts: Two transitions : KT and PT, and intermediate critical phase in both directions

Nature of phase transition n U<-4.5 or V>6: Intermediate phase n U>-4.5 and V>6: chiral phase

Hard-bosons: phase diagram Fendley et al, 2004; Chepiga and FM, arXiv:1808.08990

Quantum Dimer Model N. Chepiga and FM, arXiv:1809:00746

Effective model for spin-1/2 ladder n Motivation: Ising transition in spin-1/2 ladders inside singlet sector (singlet-triplet gap does not close) Nersesyan and Tsvelik 1997 n Coupled J 1 -J 2 chains Lavarelo, Roux, Laflorencie 2011

Quantum loop model I n Motivation: Ising transition in a frustrated spin-1 chain inside singlet sector N. Chepiga, I. Affleck, FM, PRB 2016

Quantum loop model II n Defined on a zigzag chain n Spin-1 à Two valence bonds from each site à Loop model Some typical configurations

Quantum loop model III Counts double bonds Counts single bonds Plaquette flipping

QLM ó Hard bosons n Exclude states with double bond on leg n Exclude nearest-neighbor VBS and states connected to it (separate sector) Then N. Chepiga and FM, arXiv:1809:00746

Back to Quantum Loop Model N. Chepiga and FM, arXiv:1809:00746

Conclusions on 1D models n Constrained models: 4 equivalent models à Quantum Dimer Model à Quantum Loop Model à Hard boson model à Fibonacci anyon chain: tricritical Ising or Potts point depending on the sign of the density terms matrix elements

General conclusions n Phase diagram: very rich! à Ising à Tricritical Ising à 3-state Potts à Huse-Fisher chiral phase transition à Intermediate floating phase with KT and PT transitions

Perspectives n Locate the Lifshitz points, and investigate their properties n Check other consequences of chiral phase transition, e.g. dynamical exponent n Revisit the classical asymmetric Potts model. With tensor networks?