Integer quantum Hall effect for bosons: A physical realization

Transcription

1 Integer quantum Hall effect for bosons: A physical realization T. Senthil (MIT) and Michael Levin (UMCP). (arxiv: ) Thanks: Xie Chen, Zhengchen Liu, Zhengcheng Gu, Xiao-gang Wen, and Ashvin Vishwanath.

2 An obsession in modern condensed matter physics ``Exotic Phases of Matter - Gapped phases with ``topological quantum order, fractional quantum numbers (eg, fractional quantum Hall state, gapped quantum spin liquids) - phases with gapless excitations not required by symmetry alone (eg, fermi and non-fermi liquid metals, gapless quantum spin liquids) Emergent non-local structure in ground state wavefunction: Characterize as ``long range quantum entanglement

3 How ``simple can ``interesting be? Long range entangled phases have many interesting properties. Some of these interesting things may not actually require the long range entanglement. A sub-obsession: how exotic can a phase with short ranged entanglement be? Dramatic progress in the context of topological band structures of free fermion models in recent years.

4 Integer quantum Hall effect of electrons: the original topological insulator

5 Modern topological insulators Key characterization: Non-trivial surface states with gapless excitations protected by some symmetry

6 Interactions? Current frontier: interaction dominated generalizations of the concept of topological insulators? Move away from the crutch of free fermion Hamiltonians. Useful first step: study possibility of topological insulators of bosons Necessitates thinking more generally about these phases without the aid of a free fermion model.

7 Integer Quantum Hall Effect (IQHE) for bosons? Integer quantum Hall effect of fermions: Often understand in terms of filling a full Landau level. For bosons this obviously fails (no Pauli). Can bosons be in an IQHE state with (1) a quantized integer Hall conductivity (2) no fractionalized excitations or topological order (unique ground state on closed manifolds) (3) a bulk gap Yes! (according to recent progress in general classification of short ranged entangled phases) 1. Cohomology classification (Chen, Liu, Gu, Wen, 2011) 2.?? (Kitaev, unpublished) 3. Chern-Simons classification (Lu, Vishwanath, 2012)

8 This talk A physical realization of an integer quantum Hall state of bosons Simple, possibly experimentally relevant, example of the kind of state the formal classification shows is allowed to exist.

9 Two component bosons in a strong magnetic field Two boson species bi each at filling factor ν = 1 H = I H I = H int = H I + H int (1) d 2 xb I i A 2 2m µ b I (2) d 2 xd 2 x ρ I (x)v IJ (x x )ρ J (x ) (3) External magnetic field B = A. ρ I (x) =b I (x)b I(x) = density of species I

10 Symmetries and picture Number of bosons N1, N2 of each species separately conserved: two separate global U(1) symmetries. Total charge = N1+N2 Call N1 - N2 = total ``pseudospin Species 2 Charge current Species 1 B-field Pseudospin current Later relax to just conservation of total boson number

11 Guess for a possible ground state If interspecies repulsion V 12 comparable or bigger than same species repulsion V 11,V 22, particles of opposite species will try to avoid each other. Guess that first quantized ground state wavefunction has structure ψ =... i.j (z i w j ) e i z i 2 + w i 2 4l 2 B (1) z i,w i : complex coordinates of the two boson species.

12 Flux attachment mean field theory i.j (z i w j ): particle of each species sees particle of the other species as a vortex. Flux attachment theory: Attach one flux quantum of one species to each boson of other species. Mutual composite bosons 1 2 ν = 1 => on average attached flux cancels external magnetic flux. Mutual composite bosons move in zero average field.

13 Chern-Simons Landau Ginzburg theory Reformulate in terms of mutual composite bosons. Implement flux attachment through Chern-Simons gauge fields. L = I L I + L int + L CS L I = b I( 0 ia I0 + iα I0 ) b I b I i( A I α I ) b I 2 + µ b I 2 L int = V IJ b I 2 b J 2 L CS = 1 4π µνλ (α 1µ ν α 2λ + α 2µ ν α 1λ ) (1) 2m

14 Chern-Simons Landau Ginzburg theory Reformulate in terms of mutual composite bosons. Implement flux attachment through Chern-Simons gauge fields. L = I L I + L int + L CS L I = b I( 0 ia I0 + iα I0 ) b I b I i( A I α I ) b I 2 2m + µ b I 2 L int = V IJ b I 2 b J 2 Chern-Simons gauge L CS = 1 fields 4π µνλ (α 1µ ν α 2λ + α 2µ ν α 1λ ) (1) Mutual Chern-Simons implements flux attachment

15 Chern-Simons Landau Ginzburg theory Reformulate in terms of mutual composite bosons. Implement flux attachment through Chern-Simons gauge fields. L = I L I + L int + L CS L I = b I( 0 ia I0 + iα I0 ) b I b I i( A I α I ) b I 2 2m + µ b I 2 L int = V IJ b I 2 b J 2 Chern-Simons gauge L CS = 1 fields 4π µνλ (α 1µ ν α 2λ + α 2µ ν α 1λ ) (1) Mutual Chern-Simons implements flux attachment Probe gauge fields Mutual composite fermions see zero average field: condense them. Internal Chern-Simons gauge fields lock to probe gauge fields.

16 Physical properties: Hall transport Effective probe Lagrangian L eff = 1 4π µνλ (A 1µ ν A 2λ + A 2µ ν A 1λ ) (1) New probe gauge fields that couple to charge and pseudospin currents A c = A 1 +A 2 2,A s = A 1 A 2 2. L eff = 1 2π µνλ (A cµ ν A cλ A sµ ν A sλ ) (2) Electrical Hall conductivity σ xy =2 Pseudospin Hall conductivity σ s xy = 2. Integer quantum Hall effect

17 Edge states Charge current Pseudospin current Comments: 1. Counterpropagating edge states but only one branch transports charge. 2. Thermal Hall conductivity = 0

18 Symmetry protection of edge states Species 2 Include interspecies tunneling: Pseudospin not conserved, only total particle number conserved. Counterpropagating edge modes cannot backscatter due to charge conservation. Species 1 Edge modes are preserved so long as total charge is conserved. B-field ``Symmetry Protected Topological Phase of bosons

19 Effective topological field theory Two component Chern-Simons theory L = 1 4π (a 1da 2 + a 2 da 1 )+ 1 2π (da 1 + da 2 ) A c (1) K-matrix = σ x Unique ground state on closed manifolds as DetK =1 Connect to general discussion of Lu, Vishwanath (Ashvin talk)

20 Ground state wavefunction (ignore interspecies tunneling) Naive guess Ψ ({z i,w j })= i,j (z i w j ) e i z i 2 + w i 2 4 (1) Problem: Unstable to phase separation (see using Laughlin plasma analogy) Fix, for example, using ideas initiated by Jain (1993) for some fermionic quantum Hall states Ψ flux = P LLL z i z j 2 w i w j 2 i<j (z i w j ) e i i,j i<j z i 2 + w i 2 4 (1) P LLL : projection to lowest Landau level

21 Pseudospin properties The edge theory for this state is identical to the SU(2) 1 WZW conformal field theory. Edge theory has (emergent) pseudospin SU(2) rotation symmetry. Suggests state itself can be stabilized for a pseudospin SU(2) invariant Hamiltonian. Can show wavefunction of previous slide is actually a pseudospin singlet

22 Microscopics: a simple Hamiltonian H = I H I = H int = H I + H int (1) d 2 xb I i A 2 2m µ b I (2) d 2 xd 2 x ρ I (x)v IJ (x x )ρ J (x ) (3) Simple and realistic interaction: V II (x) = g s δ (2) (x) V 12 (x) = g d δ (2) (x) g s = g d : Pseudospin SU(2) invariance

23 Possible Phase diagram 0 1 gd/gs Decoupled: Pfaffian + Pfaffian (non-abelian) SU(2) symmetric point: Phase separated: Possibly non-abelian k= 4 Read-Rezayi state Near SU(2) symmetric point, recent exact diagonalization work show an incompressible, spin singlet state (Grass et al, 2012arXiv G, Furukawa, Ueda 2012arXiv F) Candidates: 1. Boson IQHE 2. A non-abelian spin singlet state (Ardonne, Schoutens 1999)

24 Prospects - experiments Obvious place to look is in ultracold atoms in strong artificial magnetic fields. The delta function repulsion is realistic and controllable. Challenge: get fields high enough to be in the quantum Hall regime

25 Prospects - future theory Dramatic progress in classification of ``Symmetry Protected Topological and other Short Ranged Entangled Phases. Future: 1. Better understanding of their universal physical properties, and their physical/experimental realizations. 2. Interplay between symmetry and ``long range entanglement.