Nonlinear Sigma Model(NLSM) and its Topological Terms

Transcription

1 Nonlinear Sigma Model(NLSM) and its Topological Terms Dec 19, MIT

2 NLSM and topological terms Motivation - Heisenberg spin chain 1+1-dim AFM spin-z and Haldane gap 1+1-dim AFM spin-z odd /2 and gapless quasi-long range order Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview

3 Why it is interesting? Explain the physics of... FM(ω k 2 ), spin- 1 AFM(ω k), spin-1 AFM(gap) 2

4 Three Take-Home Messages: (1). NLSM+Topological Terms are useful to describe perturbation from symmetry breaking(v.e.v or ground state). (2). Topological(Topo) Terms will emerge sometimes - interesting physics shows up. (3). Topo Terms enhance ground state degeneracy(degen.) or make excitation gapless.

5 NLSM and topological terms Motivation - Heisenberg spin chain What is NLSM? It describes an action, expanding fields around a symmetry breaking(v.e.v) phase. Take O(N) vector model as example, S 0 = dt 1 g ( µ n) 2,with n(x) = ( π(x), 1 π(x) 2 ), then S NLSM = dt 1 g [( µ π) 2 + ( π µ π)2 1 π 2 1 a d π ]. We will discuss O(N) NLSM and SU(N) NLSM. What are topological terms? Give mapping like π n (S m ), with quantized winding number, written as action S top is imaginary number. It gives quantization condition for spin s, such as e i4πs = 1. We will discuss Wess-Zumino(WZ) and Θ-term.

6 Motivation - Heisenberg spin chain H XXX = J ij S i Sj J ij ( Si x S x j NLSM and topological terms Motivation - Heisenberg spin chain + S y i S y j ) + J z ij Si z Sj z, generically related to spin-charge separation in 1+1-dim, and the large U limit of Hubbard model (half-filling Mott insulator) by S i = 1 2 c iα ( σ αβ)c iβ. i,α,β

7 NLSM and topological terms Motivation - Heisenberg spin chain H XXX continue... Moreover, we already have: Holstein-Primakoff(so FM has magnon ω k 2 and AFM has ω k for spin-1/2 chain). Lieb-Schultz-Mattis theorem: spin-1/2 AFM must have either gapless or 2-fold degeneracy, other examples: AKLT and Majumdar-Ghosh. Jordan-Wigner transformation, Bosonization, Bethe ansatz...

8 NLSM and topological terms Motivation - Heisenberg spin chain H XXX continue... Moreover, we already have: Holstein-Primakoff(so FM has magnon ω k 2 and AFM has ω k for spin-1/2 chain). Lieb-Schultz-Mattis theorem: spin-1/2 AFM must have either gapless or 2-fold degeneracy, other examples: AKLT and Majumdar-Ghosh. Jordan-Wigner transformation, Bosonization, Bethe ansatz... Why bother to use NLSM and Topo terms? (1) NLSM can extend to higher (d+1)-dimensions. (2) Haldane conjecture: spin-1 AFM chain is gapped! Now use NLSM+Topo to understand H XXX AFM phase.

9 for a single spin S E = s j log( 1+ n(t j+1) n(t j ) 2 ) is j Φ( n(t j+1), n(t j )) dt( t n) 2 + S Berry S 0 + iss WZ 1 g O(3) NLSM: S 0 (non-degenerate gd state) O(3) NLSM+WZ ((2s+1)-degenerate state) S Berry = dt n t n = is dt du ɛ abc ɛ µν n a ( µ n b ν n c ) O(2) NLSM+Θ term = particle on a ring S Berry = is(1 cosθ) dt t φ = is A t φdt = is dudt(ɛ µν F µν ) O(3) O(2)

10 : S 0 = d d x 1 2g ( µ n) dim FM is not described by NLSM, no Topo terms. 1+1-dim AFM: NLSM +Topo= S 0 + S Berry,n expand around Neel state: N i = ( 1) i n 1 (a d /S) 2 L 2 + (a d /S) L Θ S Berry,n i 4π dt dx n ( x n t n) = iθw. Θ = 2πs, nontrivial topological term as Θ = π when s = Z odd /2. expecting enhance ground state degeneracy or gapless. 1+1-dim AFM spin-z chain(spin-1), there is no Topo. 1+1-dim AFM spin-z odd /2 chain(spin-1/2), there is Topo.

11 1+1-dim AFM spin-z and Haldane gap 1+1-dim AFM spin-z odd /2 and gapless quasi-long range order 1+1-dim AFM spin-z chain and Haldane gap ans1: topological objects? (not rigorous... ) instanton or skyrmion from nontrivial π 2 (S 2 ) mapping generates a length scale, short range correlation length ξ = ae Sn/2 = ae 2π/g, a is the cut-off(lattice size), this induces a mass gap scale m 1/ξ ans2: RG β(g) is weak coupled asymptotic free in UV and strong coupled confinement in IR, ex: like QCD. There is a confinement mass gap scale: m a 1 e 2π/g. It is the conjectured Haldane gap phase for spin-z 1+1-dim AFM Neel state.

12 1+1-dim AFM spin-z and Haldane gap 1+1-dim AFM spin-z odd /2 and gapless quasi-long range order 1+1-dim AFM spin-z odd /2 and gapless quasi-long range order ans1: RG - O(3)NLSM+WZ SU(2)NLSM+WZ ( SU 1 (2) WZW), compute β(g) SU1(2)WZW find IR stable critical fixed point. ans2: Quantum Interference: S top = e iθw = e i2πsw = ( 1) W. Topological sectors for s= Z odd /2 cancel with each other. ans3: Map to XY model low T phase:(affleck) S easy plane = S XY, spin wave + S vortices = d 2 x[ 1 2g ( φ)2 (γ/2)cos(θ/2)(e 2πiφ/g + e 2πiφ/g ). By Coleman-Mermin-Wagner theorem, no Goldstone boson in dim, so it is quasi-long Range(quasi-LR) gapless ω k.

13 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview Why it is interesting? Explain the physics of... FM(ω k 2 ), spin- 1 2 AFM(ω k), spin-1 AFM(gap)

14 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview Why it is interesting? Explain the physics of... FM(ω k 2 ), spin- 1 2 AFM(ω k), spin-1 AFM(gap)

15 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview Three Take-Home Messages: (1). NLSM+Topological Terms are useful to describe perturbation from symmetry breaking(v.e.v or ground state). (2). Topological(Topo) Terms will emerge sometimes - interesting physics show up. (3). Topo Terms enhance ground state degeneracy(degen.) or make excitation gapless. and...

16 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview

17 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview

18 Why NLSM+Topo is interesting? 3 Take-Home Messages Take-Home Pic: Overview THANK YOU FOR YOUR ATTENTION.