Entanglement spectrum of AKLT like states - effective field theory - PowerPoint PPT Presentation

NQS 2017 Oct 27, YITP, Kyoto Entanglement spectrum of AKLT like states - effective field theory approach A. Tanaka, National Institute for Materials Science w. S. Takayoshi, Geneva U. References AT Nov. 2017 issue of 「数理科学」（ in Japanese ） ; AT and ST, in preparation; ST, P. Pujol and AT, Phys. Rev. B 94 235159 (2016); ST, K. Totsuka and AT, Phys. Rev. B 91 155136 (2015); K.-S. Kim and AT, Mod. Phys. Lett. B 29 1540054 (2015)

Contents of talk = belongs to ongoing project with following objective: Haldane-style mapping of AF spin systems in d-dimensions 1980’s and 90’s: main target = stat-mech properties e.g. 1d: S=half-integer vs integer 2d: mod S=2 properties on square lattice QPT w.r.t. varying theta-value Question: Can we extract the topological protection (STP vs non-STP) and entanglement properties of the ground state of disordered (Haldane gap/AKLT-like) phases in this language? ・ Complementary to MPS/tensor-network schemes. ・ A personal motivation: demystify path integral approach to SPT by X. Chen et al, Science 2012 via somewhat more conventional language.

One-slide-summary of our main message Consequence of having total derivative topological term within effective action Surface effects arise depending on how you wrap up your space-time: [ ( ) ] φ τ S τ edge Edge action τ x (Gapped [ ] ( ) Two sides of same coin φ τ System) S top , x x (bulk-boundary correspondence) [ ] ( ) τ Ψ φ (path integral) x GS wave function Suggests (and we confirm) that: Effective action w. top-tem x might be useful for studying GS entanglement properties!

In previous work we started by deriving, Haldane style, effective actions, to arrive at the above mentioned bulk-boundary correspondence. In this talk, I will take an easier and perhaps more accessible (for many people) route: I will start directly with the well-known AKLT wavefunction and extract, in the large-S limit,the same topological information. Then I will move on to discuss entanglement properties in light of this.

AKLT wave function for integer S (i.e. S valence bonds per link): e.g. S=2 AKLT ( ) ∏ S † † † † = − AKLT a a a a vac ↑ + ↓ ↓ + ↑ PBC j j 1 j j 1 Constraint on Schwinger boson ( j ) + = † † a a a a phys 2 S phys ↑ ↑ ↓ ↓ j j j j ( ) Spin coherent state basis { }  ∏ 2S Ω ≡ ↑ + † † u a v a vac ↓ j j j j j j    ( ) ( ) u  2 2   ∈ + = Ω = σ j 1 u , v CP , u v 1 , u , v   j j j j j j j v   j { }  ( ) ∏ Arovas-Auerbach-Haldane ⇒ Ψ = Ω = S AKLT u v - v u PRL 1988 + + AKLT j j j 1 j j 1 j

~ ~  −       u u u v It is convenient to convert to (overscores +         + ≡ ≡ 2 j 2 j 2 j 1 2 j 1 ,         ~ ~ the representation: =CCs) v v v       u   + + 2 j 2 j 2 j 1 2 j 1 θ     i ~ cos   u where:     ≡ 2 i   θ ~     v e φ  i  i sin i i   2  ( ) ← i=even ≡ θ φ θ φ θ     u n sin cos , sin sin , cos Observe that now ( )   σ = i i i i i i  u , v    ← i=odd − i i    v n i i So using this rep. just means: we are inverting the coordinate axes in spin space at the odd sites. So far all of this is just formal rewriting, and is completely general. ~ ~ ~ ~ = + Why are we doing this? Because: u v - v u u u v v + + + + 2 j 2 j 1 2 j 2 j 1 2 j 2 j 1 2 j 2 j 1

Formula for inner product of CP 1 spinors: (familiar e.g. from “double exchange” in anomalous Hall effect)   1 + ⋅   ( )    1 n n   2 i ~ ~ ~ ~ + =  i j ˆ u u v v   exp A n , n , z ˆ z i j i j i j     2 2 A  Area of spherical triangle  n on unit sphere   n j S     − Ω ⋅ Ω i large S 1 ∏   Ψ 2 = ⇒ Ω ≈ − Ω + i i 1   + AKLT i 1 i   2 i   ⇔ ≈ n n behaves smoothly in continuum limit + i 1 i        ( ) ( ) = − − − ⇒ Motivates derivative expansion for A n , n , n A n , - n , n n ( x ). 1 2 3 1 2 3 Accounting for fact that spin-space axes are inverted on odd sites, we find:  [ ] continuum form ω n ( x )

c.f. N. Nagaosa

AKLT wave function in continuum (large-S) form  ( ) 1 ∫  [ ] S − ∂ 2  [ ] − ω dx n i n ( x ) ~ x Ψ = 2 g 2 n ( x ) e e AKLT Replace in above “x” with imaginary time “ τ ”. This is then identical to the Feynman weight for a single spin S/2 object (quantum rotor) at the end of open AKLT chains. e.g. S=2 AKLT  1 ∫ ( )  [ ] S − ∂ 2 − ω τ dx n i n ( ) τ − ~ ≡ = S 2 g 2 W e edge e e Feynman spin Berry phase for fractional spin S/2 object This demonstrates that : Same quantum number fractionalization as edge state is inherent in the bulk AKLT wave function even under PBC ! (also note similarities with X. Chen et al, Science 2012) .

consistent with existence of an underlying effective action w. top-term (in this case, this is just Haldane’s NLσ model+θ term) = S S τ edge WZ Edge action τ cf T.K. Ng PRB 1994 θ    ∫  [ ] ( ) τ ⋅ ∂ × ∂ i d dx n n n τ = τ π x S n , x e 4 x (Gapped top Two sides of same coin for integer S) x (bulk-boundary correspondence) ( )  [ ] ( ) θ = 2 π τ Ψ (path integral) S n x GS wave function Suggests that: Effective action w. top-tem x might be useful for studying GS entanglement properties!

We now make contact with our proposed action (ST, Pujol, AT, PRB 2016) for detecting SPT states = + General form: S S S eff kinetic top total derivative NLσ

Proposed action (ST, Pujol, AT, PRB 2016) for d=1 = + General form: S S S eff kinetic top total derivative NLσ  1+1d: easy-plane Haldane gap phase ( ) ≡ φ φ n cos , sin , 0 Planar version of Haldane’s well -known mapping to “O(3) NLσ+top -term” { } [ ( ) ] ( ) ( ) 1 ∫ φ τ = τ ∂ φ + ∂ φ O(2) NLσ 2 2 S , x d dx τ kinetic x 2 g [ ] { ( ) ( ) } ( ) 1 ∫ = τ ∂ ∂ φ − ∂ ∂ φ φ τ = π vortex Berry phase term Q d dx S top , x i S Q , τ τ π v x x v 2 τ + ∆ τ τ ∆ φ ∆ φ Reduces to counting τ τ ∆ φ ∆ φ × space-time vorticity x x τ i S of shaded region. ∆ φ ∆ φ τ τ ∆ φ ∆ φ Sachdev 2001 x x τ − ∆ τ − ( = + + j 1 j even ) j 1 j 2 More tractable than the original O(3) model. x

Digress using 1+1d case. First rewrite in manifest total-derivative form: θ ( ) ∫ = τ ∂ − ∂ ≡ ∂ φ θ = π S: integer 1 S i d dx a a , a , 2 S τ τ µ µ π top x x 2 2 − φ φ ≡ − ∂ = ∂ φ Not a pure gauge : c.f. i i a i ( e ) ( e ) µ µ µ Surface effect on spatial edge: τ S ∫ ∫ = ± τ = ± τ ∂ φ S edge i S d a i d BBC τ τ 2 x Fractional spin at end of spin chain (T.K. Ng 1994) Surface effect on temporal edge: τ ∫ [ ] φ ( ) ( ) ( ) − ∫ − i S dx a Ψ φ = φ τ ∝ = − S S Q x x D , x e eff e 1 x , φ init 1 x ∫ Topology-sensistive/nonsensitive = ∂ φ ∈ Z Q dx π x x when S=odd/even. 2 BBC: consistent w. fact that S=odd is SPT state under TR-symmetry.

The continuum form of he AKLT wave function gives consistent results. θ ≡ Planar limit is accessed by putting cos 0 . ( ) ∫ 1 − ∂ φ 2 [ ] ( ) dx 1 ∫ ~ − π x Ψ φ = ≡ ∂ φ ∈ iS Q 2 g x e e , Q dx Z x π AKLT x x 2

Proposed action (ST, Pujol, AT, PRB 2016) for d=2 = + General form: S S S eff kinetic top total derivative NLσ 2+1d: “Haldane gap phase”(=AKLT-like state) on square lattice  σ [ ] ( )    ( ) 1 ∫ 2 = ≡ ∂ O(3) NLσ τ = τ ε ∂ † † 2 n z z , a iz z S kinetic a , r d d r a µ µ µ λµν µ ν =CP 1 2 2 g [ ] π   1 ( ) S ∫ = τ ε ∂ ∂ τ = 2 Q d d r ( a ) S top a , r i Q , λµν λ µ ν µ π monopole Berry phase term m m 2 2 ( ) τ + ∆ τ Q xy time = ∆ Q Q monopole xy monopole ( ) τ Q xy A suitably coarse- grained version of Haldane’s monopole Berry phases (1988) . Can be derived systematically as “coupled wires” of 1+1d WZW models. Provides field theory representation for “weak” SPTs.