1. Dualities in multi-channel Luttinger Liquids Igor Yurkevich

2. Outline 1.LL - classic example of duality (KF 1992) 2.Some recent history 3.Interaction-protected topological insulators 4.More history (quantum Brownian motion) 5.Origin of dualities 6.Mixtures: peculiarities.

3. Duality (def): the mathematical equivalence of two seemingly different theoretical descriptions of a physical system Duality (in this talk): relations between seemingly different physical systems

4. Luttinger Liquid (LL) 1D fermions (m = odd) or bosons (m = even) Low-energy Hamiltonian Commutations: Luttinger parameter: K = 1 non-interacting fermions or infinite strongly repulsive bosons 0 < K < 1 repulsive fermions K > 1 attractive fermions or repulsive bosons

5. Luttinger Liquid (LL) Lagrangian: 0 Non-linear perturbation: RG – integrating out energies Perturbative in coupling eq-n: Here d – physical dimension, Δ - scaling dimension: Solution: Δ > d - perturbation irrelevant; Δ 0

6. Local scatterer in LL (Kane, Fisher ‘91) Weak backscattering potential - backscattering irrelevant: ideal conductor ? Weak tunneling - tunneling irrelevant: ‘ideal’ insulator

7. insulator conductor Local scatterer in LL (phases) v t Duality:

8. Local scatterer in LL (duality) Duality: 1 1 Sufficient information to predict phases that are stable: 1 1 C-phase I-phase Self-duality point Either c-phase (conductor) or i-phase (insulator) and nothing else

9. Weak backscattering IVY, Galda, Yevtushenko, Lerner 2013 LL + 1D phonons (duality) Weak tunneling Weak backscattering Weak tunneling Parameters Luttinger constant: Fermi and sound Velocities: El-phonon Coupling Constant g:

10. IVY, Yevtushenko 2014 Generic bath LL + multi-channel bosons (duality)

11. insulator conductor LL + bath (duality) No new phases: either conductor or insulator Duality c c i i

12. X=0 scatterer ‘phase’ - ordered set of continuous (conducting) and disjoint (insulating) wires (LLs) Generic multi-channel LL Whether this phase is stable should be tested by all scaling dimensions of all possible perturbations Boundary conditions Lagrangian disjoint continuous

13. Generic multi-channel LL Transformationwith matrix is diagonalising Lagrangian but deforms BCs In chiral fields it means new scattering matrix:

14. Instead of Luttinger parameter we have ‘Luttinger matrix’ now All scaling dimensions can be written in a universal compact form Multi-channel LL: Luttinger K-matrix Perturbation: N bs backscattered and N tun tunneled particles where T and R are projectors onto subspaces of conducting and insulating channels.

15. Weak scatterer vs weak link in a n-th channel (arbitrary phase) - Stands for either continuous wire or disjoint wires 1 : n N : Dualities in multi-channel LL

16. cc-phaseic-phase ci-phaseii-phase Four dualities between scaling dimensions of 8 perturbations Dualities in 2-channel LL Scal. Dim’s vs parameters

17. ic ci 1 Prediction w/o calculation: Mixtures (FF, FB or BB) - can exist mixed phase (Line of unstable Fixed Points) - no new (on the top of ‘expected’) phases (line of stable FPs) cc ii cc/ii N=2 LL generic phase diagram ‘Luttinger’ matrix Moderate inter-channel Interaction: 1

18. ic ci 1 cc ii cc/ii N=2 LL generic phase diagram ‘Luttinger’ matrix Intermediate inter-channel interaction: destruction of mixed state 1

19. ic ci 1 cc ii N=2 LL generic phase diagram ‘Luttinger’ matrix Strong inter-channel interaction: 1 new stable FP emerges inside mixed state: both channels have finite conductance 0 < G < 1

20. Interaction-protected Topological Insulator Top Insulator: – scattering between edges - interaction intra-edges Regrouped: -interaction between channels -scattering intra-channels Inter-channel interaction

21. ic ci 1 cc ii cc/ii ‘Luttinger’ matrix 1 Interaction-protected Topological Insulator No inter-edge interaction (single point on phase diagram) Weak intra-edge interaction: - Not fully protected (mixed state) Strong intra-edge interaction: -Swallowed by 2-particle SFP with finite conductance 0 < G < 1

22. Origin of dualities Leggett (1980) Two-level system + bath -> loss of quantum coherence A ‘particle’ in double-well potential: Localisation in one of them = loss of quantum coherence If environment is quadratic (set of oscillators) with Ohmic spectral function and linearly coupled to the particle: Caldeira-Leggett model (1981): Friction constant (dissipation) Overdamped particle in periodic potential: diffusion vs localisation

23. Origin of dualities (1D QBM) A. Schmid (1983) Expansion wrt periodic potential Expansion wrt instantons (strong potential) Duality between weak and strong dissipation limits: Neutral Coulomb gas with log repulsion (fugacity)

24. Origin of dualities Anderson, Yuval (1971) RG for Coulomb gas Diffusion Localisation Both Coulomb gases are generated from two actions Original action Dual action mobility

25. Origin of dualities Rescaling to absorb friction constant from dissipation term and introducing lattice constant and reciprocal vector: Quantum Brownian motion on - original lattice - reciprocal lattice -Short lattice constant = diffusion; - Long lattice constant, i.e. short reciprocal lattice constant = localisation Scaling dimensions:

26. Quantum Brownian motion (QBM) In higher dimensions Generalisation: QBM in higher dimensions Scaling dimensions: - Direct lattice - Reciprocal lattice Yi, Kane (1998) Basis vectors

27. QBM in arbitrary dimension QMB on direct lattice QMB on reciprocal lattice Most relevant (dangerous) scaling dimensions 1 1 1 1 1 1

28. QBM in 1D 1D Bravais lattice: v t Two phases only! No other either stable or unstable fixed points.

29. QBM in 2D Hexagonal Bravais lattice: v t v t Honeycomb lattice (non-Bravais):

30. N-channel LL Lagrangian: Transformation: -- QBM on N-dim lattice

31. 1 and 2-channel LL Number of LL channels = dimension of QBM - Single LL + arbitrary bath = 1D QBM - universal duality! - 2-channel LL + arbitrary bath = 2D QBM – non-universal duality - Oblique lattice -- Luttinger K-matrix is a lattice structure matrix: inequivalent channels (mixtures): Commensurate/incommensurate – multi-particle scattering contributions win over!

32. Conclusions Multi-channel LL phase diagram is controlled by a single symmetric NxN matrix -- ‘Luttinger’ matrix All dualities are encoded in this matrix Dualities can be understood by relating Luttinger matrix to ‘lattice structure matrix ‘of equivalent QBM model