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 Huse-Fisher: possibility of a chiral

transition between a Potts point and a Lifshitz point

n Intermediate (floating) critical phase

beyond Lifshitz point

ni=0,1 or 2

Huse-Fisher phase diagram

Huse and Fisher, 1982 Kosterlitz-Thouless Pokrovsky-Talapov

L

Difference between the 3 cases

n Δq: distance to 2π/3; ξ : correlation length n Δq 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 Δq 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 Fn+1

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? Golden ratio

Quantum Dimer Model

v/J

1 RK

+ Staggered states

Exclude staggered states

v/J

1 RK 2.67

Rung singlet Period 3

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

Flips plaquettes Counts vertical flippable plaquettes

Counts horizontal 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 Δq and ξ

Fit dimer-dimer correlations with Orstein- Zernicke

n Δq: extremely precise results (at least 3 digits) n ξ: up to thousand or so

à meaningful evaluation of Δq 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: Δq goes to zero with exponent 5/3, ξ

diverges with exponent 5/6 è Δq ξ à 0

n Vicinity of Potts: Single transition

è Δq goes to zero with exponent smaller than 1 è Δq 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 J1-J2 chains

Lavarelo, Roux, Laflorencie 2011

Quantum loop model I

• N. Chepiga, I. Affleck,

FM, PRB 2016

n Motivation: Ising transition in a frustrated

spin-1 chain inside singlet sector

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

Plaquette flipping Counts double bonds

Counts single bonds

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

àQuantum Dimer Model à Quantum Loop Model à Hard boson model à Fibonacci anyon chain: tricritical Ising

• r 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?