SLIDE 1 Entanglement spectrum of AKLT like states
- effective field theory approach
- A. Tanaka, National Institute for Materials Science
- w. S. Takayoshi, Geneva U.
NQS 2017 Oct 27, YITP, Kyoto
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) References
SLIDE 2 Contents of talk = belongs to ongoing project with following objective: Haldane-style mapping of AF spin systems in d-dimensions 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? 1980’s and 90’s: main target = stat-mech properties 1d: S=half-integer vs integer 2d: mod S=2 properties on square lattice QPT w.r.t. varying theta-value Question: e.g.
・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.
SLIDE 3 x τ
( ) [ ]
x Stop , τ φ
( ) [ ]
τ φ
edge
S
x τ τ x
( ) [ ]
x φ Ψ
Two sides of same coin
(bulk-boundary correspondence)
GS wave function Edge action Consequence of having total derivative topological term within effective action Surface effects arise depending on how you wrap up your space-time: (Gapped System) (path integral)
Suggests (and we confirm) that:
Effective action w. top-tem might be useful for studying GS entanglement properties!
One-slide-summary of our main message
SLIDE 4
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.
SLIDE 5 ( )
vac AKLT
S 1 1
∏
↑ + ↓ ↓ + ↑
− =
j j j j j
a a a a
† † † †
AKLT wave function for integer S (i.e. S valence bonds per link): Constraint on Schwinger boson (
)
phys S 2 phys = +
↓ ↓ ↑ ↑ j j j j
a a a a
† †
Spin coherent state basis { }
( )
vac v u
2S
∏
↓ ↑ +
≡ Ω
j j j j j j
a a
† †
( ) ( )
= Ω = + ∈
j j j j j j j j j
v u v , u , 1 v u , v , u
2 2 1
σ CP
{ }
( )
∏
+ +
= Ω = Ψ ⇒
j j j j j j
u v
u AKLT
S 1 1 AKLT
e.g. S=2 AKLT Arovas-Auerbach-Haldane PRL 1988 PBC
SLIDE 6 1 2 2 1 2 2 1 2 2 1 2 2
v ~ v ~ u ~ u ~ u v
u
+ + + +
+ =
j j j j j j j j
It is convenient to convert to the representation:
− ≡ ≡
+ + + + 1 2 1 2 1 2 1 2 2 2 2 2
u ~ v ~ v u , v ~ u ~ v u
j j j j j j j j
where: ≡
i
i i i i i
e φ θ θ 2 sin 2 cos v ~ u ~ Observe that now (
) ( )
− ≡ =
i i i i i i i i i i
n n θ φ θ φ θ σ cos , sin sin , cos sin v u v , u
←i=even ←i=odd So using this rep. just means: we are inverting the coordinate axes in spin space at the odd sites. Why are we doing this? Because: (overscores =CCs) So far all of this is just formal rewriting, and is completely general.
SLIDE 7 ( )
⋅ + = + ˆ , , 2 exp 2 1 v ~ v ~ u ~ u ~
2 1
z n n A i n n
j i j i j i j i
Formula for inner product of CP1 spinors: (familiar e.g. from “double exchange” in anomalous Hall effect)
z ˆ
Area of spherical triangle
⇒Motivates derivative expansion for
i i i i i i i
n n ≈ ⇔ Ω − ≈ Ω ⇒ Ω ⋅ Ω − = Ψ
+ + +
∏
1 1 S large S 1 2 AKLT
2 1 ). (x n
Accounting for fact that spin-space axes are inverted on odd sites, we find: [ ]
) (x n ω
continuum form
( ) ( )
, , , ,
3 2 1 3 2 1
n n
A n n n A − − − =
A j
n
i
n
behaves smoothly in continuum limit
SLIDE 8
c.f. N. Nagaosa
SLIDE 9 [ ]
[ ] ( )
∫ = Ψ
∂ − −
2
~ 2 1 ) ( 2 AKLT
) (
n dx g x n S i
x
e e x n
ω
AKLT wave function in continuum (large-S) form 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
[ ] ( )
∫ = ≡
∂ − − −
2
~ 2 1 ) ( 2 Feynman n dx g n S i S
e e e W
edge
τ
τ ω 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) .
SLIDE 10 x τ ( ) [ ]
∫ =
∂ × ∂ ⋅ n n n dx d i top
x
e x n S
4
,
τ
τ π θ
τ
WZ edge
S S =
x τ τ x
( ) [ ]
x n Ψ
Two sides of same coin
(bulk-boundary correspondence)
GS wave function Edge action (Gapped for integer S) (path integral)
Suggests that:
Effective action w. top-tem might be useful for studying GS entanglement properties!
consistent with existence of an underlying effective action
- w. top-term (in this case, this is just Haldane’s NLσ model+θ term)
( )
S 2π θ =
cf T.K. Ng PRB 1994
SLIDE 11 top kinetic eff
S S S + =
We now make contact with our proposed action (ST, Pujol, AT, PRB 2016) for detecting SPT states
NLσ total derivative
General form:
SLIDE 12 top kinetic eff
S S S + =
( ) [ ] ( ) ( )
{ }
2 1 ,
2 2
∫
∂ + ∂ = φ φ τ τ φ
τ x kinetic
dx d g x S
Proposed action (ST, Pujol, AT, PRB 2016) for d=1
( ) ( ) { }
2 1 Q
v
∫
∂ ∂ − ∂ ∂ = φ φ τ π
τ τ x x
dx d
1+1d: easy-plane Haldane gap phase
NLσ total derivative
O(2) NLσ vortex Berry phase term
( ) [ ]
, Q S ,
v
π τ φ i x Stop =
General form: Planar version of Haldane’s well-known mapping to “O(3) NLσ+top-term” More tractable than the original O(3) model.
τ x
τ τ ∆ −
φ
x
∆
φ
τ
∆
τ τ ∆ +
τ
φ
x
∆
φ
τ
∆ φ
τ
∆ φ
τ
∆
φ
x
∆ φ
x
∆
× S i
) even (= j 1 − j 1 + j 2 + j
Reduces to counting space-time vorticity
Sachdev 2001
( )
, sin , cos φ φ ≡ n
SLIDE 13 Digress using 1+1d case. First rewrite in manifest total-derivative form: Not a pure gauge : c.f. Surface effect on spatial edge: φ τ τ
τ τ
∫ ∫
∂ ± = ± = 2 S S d i a d i Sedge Fractional spin at end of spin chain (T.K. Ng 1994) Surface effect on temporal edge:
( ) [ ] ( ) ( )
Z ∈ ∂ = − = ∫ ∝ = Ψ
∫ ∫
− −
φ π τ φ φ
φ φ x x a dx i S
dx e e x D x
x x eff init
2 1 Q , 1 ,
Q S S
Topology-sensistive/nonsensitive when S=odd/even. BBC: consistent w. fact that S=odd is SPT state under TR-symmetry.
φ
µ φ µ φ µ
∂ = ∂ − ≡
−
) ( ) (
i i
e e i a
x τ τ x
BBC
( )
S 2 , , 2
2 1
π θ φ τ π θ
µ µ τ τ
= ∂ ≡ ∂ − ∂ =
∫
a a a dx d i S
x x top
S: integer
SLIDE 14 Planar limit is accessed by putting
. cos ≡ θ
( ) [ ]
( )
∫
∈ ∂ ≡ ∫ = Ψ
∂ − −
Z φ π φ
φ π x g dx iS
dx e e x
x
2 1 Q ,
x ~ 2 1 Q AKLT
2 x
The continuum form of he AKLT wave function gives consistent results.
SLIDE 15 top kinetic eff
S S S + =
Proposed action (ST, Pujol, AT, PRB 2016) for d=2
( )
[ ]
, Q 2 S ,
m
π τ
µ
i r a Stop =
NLσ total derivative
2+1d: “Haldane gap phase”(=AKLT-like state) on square lattice
( )
[ ] ( )
2 1 ,
2 2
∫
∂ =
ν µ λµν µ
ε τ τ a r d d g r a Skinetic , 2 z iz a z z n
µ µ
σ ∂ ≡ =
† †
O(3) NLσ =CP1
) ( 2 1 Q
2 m ν µ λ λµν
ε τ π a r d d ∂ ∂ =
∫
monopole Berry phase term
General form: 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.
time
( )
τ
xy
Q
( )
τ τ ∆ +
xy
Q
xy monopole
Q Q ∆ = monopole
SLIDE 16 ( ) [ ]
( )
4 1 Q ,
xy ~ 2 1 Q 2
2 xy
Z ∈ ∂ × ∂ ⋅ ≡ ∫ ∝ Ψ
∫
∂ − −
n n n dxdy e e y x n
y x n dxdy g S i
π
α
π
Wave functional which follows from the effective action has the form
) , ( y x = α
Suggesting that the GS is sensitive to topology only when S=4×integer (we are restricting in 2d to the case where S=even.) It is clear that the same wave functional follows immediately via the AKLT wave function approach through a simple coupled wire treatment (Amounts to replacing ‘τ’ with ‘y’ Haldane’s derivation of 1+1d effective action).
SLIDE 17
Whichever approach (effective action or AKLT wavefunction) we start with, we have a continuum wave functional which can be put to test, to see whether they give information on the entanglement properties of the GS. We now turn to this.
SLIDE 18 ( ) [ ]
x
Q S
) (
π
φ φ
i
Ae GS x x
−
= = Ψ
Indicators of SPT Observables
GS O GS O ˆ ˆ =
: cancellation of phase factor. UCSB group (Y.-Z. You et al): “strange correlator” as SPT indicator
( ) ( )
GS GS GS GS Cstr ˆ cos ˆ cos φ τ φ ≡
GS :Topologically trivial state (i.e. w.o. top-term) We find that
( ) ( ) ( )
( ) [ ]
( )
( ) [ ]
, cos cos
∫ ∫
′ − ′ −
′ ′ ≡
x S x S str
e x D e x x D C
φ φ
φ φ φ φ
( ) [ ] ( ) ( )
∫
∂ + ∂ = φ φ φ
x x
i g dx x S 2 S ~ 1
2
‘x’ ⇒ ‘τ’: Correlator of a 0+1d action! (next slide) →can show: LRO/SRO for odd/even S
SLIDE 19 Euclidean action in 0+1d
( )
S , 2 ~ 1
2
∫
≡ ∂ + ∂ = π θ φ π θ φ τ
τ τ
i g d S 2nd term = θ-term 2 1 Q , Q Z ∈ ∂ ≡ =
∫
φ τ π θ
τ τ τ θ
d i S Partition function: topological sectors
( ) [ ] ( )
( ) [ ]
Q Q Q τ φ θ
σ τ τ τ
τ φ τ φ
NL
S i
e D e Z
− ∈ −
∫ ∑
=
Z θ
S
σ NL
S
( ) ( ) ( )
τ τ τ
θ θ
π θ θ
Q Q Q
1 S 1 S − = ⇒ = = ⇒ =
i i
e
e even Expect suppression of large phase fluctuations when θ＝π.
AB-like phase interference
2 π θ = Φ Φ
Odd S: θ=π Even S: θ=0
SLIDE 20 Phase correlation
( ) ( ) ( ) ( ) ( )
) 1 ( 4 1 2 1 ˆ cos ˆ cos
2 ~ 4 ~
= + = = ≡
− −
π θ θ φ τ φ τ
τ τ g g
e e C
( ) ( )
( )
ˆ ˆ ˆ
2
G O n e O O
n E E
G n
∑
− −
=
τ
τ
2 ˆ 4 ~ ˆ
2
− = π θ N g H Hamiltonian
[ ]
, ˆ , ˆ , ˆ Z ∈ = = n n n n N i N φ Coming back to our problem (“τ”→“x”), this implies that: strange correlator has LRO/SRO for odd/even S.
SLIDE 21 Entanglement spectrum GS GS = ρ Pure state density matrix Partition system into subsystems A and B.
tr
ent
ˆ B H A
e− ≡ = ρ ρ
GS : Time evolution from to
−∞ = τ . − = τ
GS : Time evolution from to
∞ = τ . + = τ
∴Path-integral representation of
A A A
φ ρ φ ′
: discontinuity at
. = τ
BC:
( ) ( ) ( ) ( )
,
2 2 1 1
x x x x
A A A A
φ φ φ φ = ′ = ′
( )
( )
S
II I 2
2 g ~ 2 1
π θ φ φ ρ φ
φ π θ φ
= ∫ = ′
+
∂ + ∂ −
∫
Surf Surf x x
i dx A A A
e x D
Reduces to same 0+1d problem as in the strange-correlator study!
( )
x
A
φ
( )
x
A
φ′ τ x
B B
( )
x
B
φ
( )
x
B
φ
1
x
2
x
A
I II
SLIDE 22
n=0 n=-1 n=1 n=0 n=1 n=-1 n=2 n n ES ES S : even S : odd Entanglement spectra=energy spectra for our previous 0+1d problem θ=0 θ=π 2-fold degenerate/nondegenerate entanglement spectrum when S=odd/even. (Consistent w. Pollman et al.)
SLIDE 23 For BA (instead of BAB) type partition, “Rindler coordinate” approach leads to identical results (due to rotational symmetry).
( )
x
A
φ
( )
x
A
φ′
τ
x
B
( )
x
B
φ
( )
x
B
φ
A
( )
x
A
φ
( )
x
A
φ′
τ
x
B
( )
x
B
φ
( )
x
B
φ
A Time evolution (transfer matrix)
sin , cos ϕ ρ ϕ ρ τ = = x
Evolution generated by
ˆ ϕ ∂ ∂ = i K
SLIDE 24 ( ) ( ) ( )
( )
2 S , , ,
II I 2
4 g ~ 2 1
π θ φ ρ
π θ
α
= ∫ = ′
+
∂ × ⋅∂ + ∂ −
∫
Surf Surf y x
n n n i n dxdy A A
e y x n D y x y x n
Higher dimensions: essentially the same procedures 2+1d Entanglement spectrum (restrict to S=even) S=2×odd: ES=dispersion of 1+1d NLσ at θ=π (massless) S=2×even: ES=dispersion of 1+1d NLσ at θ=0 (massive) Strange correlator: ET 2pt correlator of above 1+1d action S=2×odd: field correlator of 1+1d NLσ at θ=π (power law)
∴
S=2×even: field correlator of 1+1d NLσ at θ=π (SRO)
∴
- cf. Lou et al PRB 84 (2011) 245128
SLIDE 25
Conclusions The Haldane semiclassical mapping contains information on entanglement properties of ground state. (c.f. speculation to the contrary: McGreevy’s lecture notes.) Both the strange correlator and the entanglement spectrum inherit properties of sigma models w. topological terms in one dimension lower. There are many further problems of interest: SU(n) generalizations, detailed comparision w. Chen et al’s approach, etc.
SLIDE 26 Protecting symmetry of GS Dual theory （field theory of AF order parameter →field theory of space-time vortex condensate）
( ) [ ]
( )
( )
− + ∂ = ∫ S z g dx d x S dual
eff
π ϕ ϕ π τ τ ϕ
µ
cos 2 8 ,
2 2
Turn on a staggered magnetic field // z-axis (induces staggered magnetization δm while depleting in-plane OP) z:vortex fugacity
( ) ( )
) ( cos 2 cos 2 m S z S z δ π ϕ π ϕ − − → −
Connects odd& even S without closing gap. Suggests that odd S: SPT protected by link-centered inversion symmetry.
different phases. shortened in-plane AF OP Consistent w.work by Pollman et al 2010.
SLIDE 27
Well-known (textbook) feature of effective field theory for 2d AFs : smooth configs.→no topological terms （absence of Berry phase effects）. However… Once we admit singular config.(space-time monopole=hedgehog) Berry Phase terms (→S-dependent quantum effects) will govern GS. The Berry phase effects agree precisely with VBS picture. Gapped/spatially uniform GS → Berry phase argument/VBS picture implies restriction to S=2, 4, 6. .. Haldane conjecture for 2d AF (1988) S=2 S=4 and and so on. Like
SLIDE 28
The essence of what will follow Consider how one can “gap out” edge states via singlet bond formation among edge spins: S=4
SLIDE 29 The essence of what will follow Consider how one can “gap out” edge states via singlet bond formation among edge spins (two classes): S=4 (&8,12,…) Need to break translational symmetry→cannot gap-out if this symmetry is imposed
Possible to gap-out without breaking translational symmetry. We can expect that the edge state (and hence the topological order
- f the bulk) is/is not protected by symmetry in the former/the latter.
S=2 (&6, 10, …) symmetry protection no symmetry protection
SLIDE 30 GS wave functional Again assume pbc and strong coupling limit (g→∞)
( )
[ ]
( )
( ) ( )
( )
∫ ∫
∈ ∂ × ∂ ⋅ ≡ = ∝ ∫ = Ψ
− ∂ ∂ − pbc y x S i a r d d S i y x n y x n GS
n n n dxdy e e y x n D y x n
xy xy i
Z
4 1 Q 1
, ,
xy Q 2 S Q 2 4 , ,
2
π τ
π ε τ
λ ν µ µνλ
S=2, 6, 10 .. : GS sensitive to Skyrmion number Qxy (topological) S=4, 8, 12 .. : GS insensitive to Skyrmion Qxy (trivial) Two classes, in accord with VBS picture.
SLIDE 31 With application to SPT detection in mind, modify to tractable setup: easy plane case (assumed: Haldane gap phase persists)
( )
, sin , cos φ φ ≡ n Planar config. Effective action: Obviously,
( ) ( )
{ }
2 1
2 2
∫
∂ + ∂ = ⇒ φ φ τ
τ σ x XY NL
dx d g S S
Naively,
Q ≡ ≡
x
S
τ θ
Need to redo derivation to address 2nd point correctly.
SLIDE 32 [ ] [ ]
V
Q S π φ φ i S S
XY eff
+ =
Effective action Sensible? Convert to dual (vortex) language: Variant of sine-Gordon action
( )
( )
ϕ π ϕ π
µ
cos S cos 2 8
2 2
z g Ldual + ∂ =
z: fugacity Agrees with Affleck’s meron action. S=half odd integer: KT transition into massive phase cannot occur. S=integer: vortex condensation possible (=Haldane gap phase)
SLIDE 33 top kinetic eff
S S S + =
Proposed action (ST, Pujol, AT, PRB 2016) for d=1, 2, 3
, Q 3 S
m
π i Stop =
NLσ total derivative
3+1d: “Haldane gap phase”(=AKLT-like state) on cubic lattice
) 2 (
4
SU N i N g ∈ ⋅ + = σ
O(4)
( )
) )( )( ( tr 32 1 Q
1 1 1 3 2 m
Z ∈ ∂ ∂ ∂ ∂ =
− − −
∫
g g g g g g r d d
ρ ν µ λ λµνρ
ε τ π
O(4) monopole Berry phase term
General form:
