Topological Scars - pks.mpg.de · Scar characteristics Non-integrable model [Turner et. al, Nature...
Transcript of Topological Scars - pks.mpg.de · Scar characteristics Non-integrable model [Turner et. al, Nature...
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Topological Scars
Titus Neupert Dresden, January 25, 2019
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Collaboration
Seulgi Ok
2. Non-topological scar state in 1D
Content
4. Topological scars in 1D, 2D, 3D
arxiv:1901.01260
1. Scar states and ETH violation
3. General construction
Kenny Choo Claudio Castelnovo Claudio Chamon Christopher Mudry
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ScarsRydberg atom experiments: Emergent oscillations in many-body dynamics after sudden quench
[Lukin group, Nature 551, 579-584 (2017)]Theoretical explanation: special “scar” states in the middle of the spectrum
[Turner et. al, Nature Physics (2018)]
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Scar characteristics
Non-integrable model
[Turner et. al, Nature Physics (2018)]
Low (sub-volume law) entanglement entropy states
No disorder (distinct from many-body localization)
“Few” scar states distributed over spectrum
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Analytical scars[S. Moudgalya, et al., PRB 2018, 2X]
Infinite series of low-entropy states in AKLT chain
MPS based construction
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Eigenstate thermalization hypothesis
Classical thermalization: equivalence of time average and ensemble average
Quantum thermalization: Assume system behaves thermally
Unitary evolution cannot construct thermal state
Thermal behavior in quantum systems occurs on the level of individual eigenstates
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Eigenstate thermalization hypothesis… and how it fails in many-body quantum systems
entanglemententropy
ener
gy
ETH obeying generic non-integrable system
nearby states have same expectation values of any observable
strong ETH
system with scar states
entanglemententropy
ener
gy
weak ETH
many-body localized
entanglemententropy
ener
gy
no ETH
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What we do
Recipe for analytical construction of scar states
In systems of any dimension (1D, 2D, 3D)
In systems with topological character (SPT/topological order), inherited by the scar states
Area law entanglement scar states
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2. Non-topological scar state in 1D
Content
4. Topological scars in 1D, 2D, 3D
1. Scar states and ETH violation
3. General construction
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Simple (non-topological) example
1D spin-1/2 chain
Transverse field Ising model+ NNN coupling+ sign-alternating coupling
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Simple (non-topological) exampleProof of non-integrability: energy level statistics
in one momentum and inversion symmetry sector
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Simple (non-topological) exampleWhy the scar state?
H̃(�) :=X
i
Qi(�)
Qi(�) := e��(Zi�1Zi+ZiZi+1) �Xi
Consider alternative Hamiltonian
Each term is positive semidefinite
Qi(�)2 = 2 cosh (� (Zi�1Zi + ZiZi+1)) Qi(�)
State annihilated by all terms= ground state area law entanglement
|scar(�)i := exp
0
@�
2
X
j
ZjZj+1
1
AO
i
|+ixi
H(�) :=X
i
↵iQi(�) ↵i := ↵+ (�1)iExact eigenstate of in the middle of the spectrum
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2. Non-topological scar state in 1D
Content
4. Topological scars in 1D, 2D, 3D
1. Scar states and ETH violation
3. General construction
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Recipe
As A2s = 1 [As, As0 ] = 0
Operators constituting a commuting projector model
[M̄s, As] = 0
{Ms, As} = 0Such that
M =X
i
Oi Ms =X
i2s
Oi
M̄s =X
i 62s
Oi[Oi, Oj ] = 0
Extra set of operators we will use to deform the model
Commuting projector model
Scar model (nonintegrable)
Ground state properties Scar state properties
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RecipeFs = e�/2M (1�As)e
��/2M
= 1� e�MsAs
= e�Ms(e��Ms �As) =: e�MsQs
Deformation:
Qs = Q†s is local operator!
Q2s =(e��Ms �As)(e
��Ms �As)
=e�2�Ms � (e��Ms + e��Ms)As + 1
=2 cosh(�Ms)Qs
Positive semidefinite
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Recipe
| 0i =Y
s0
(1 +As0)|⌦i
(1�As)| 0i =(1�As)Y
s0
(1 +As0)|⌦i
=(1�As)(1 +As)Y
s0 6=s
(1 +As0)|⌦i
=0.
Ground state of commuting projector model
| �i = e��/2M | 0i
Fs| �i =e��/2M (1�As)e�/2Me��/2M
Y
s0
(1 +As0)|⌦i
=0
) Qs| �i = 0
Deformed state is annihilated by all Qs
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Recipe
| �i = e��/2M | 0i is ground state of H =X
s
Qs
H0 =
X
s
↵sQsAnd excited eigenstate of if coefficients of different sign
No need for deformed Hamiltonians to be integrablesince may belong to different s
[Qs, Qs0 ] 6= 0Oi
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2. Non-topological scar state in 1D
Content
4. Topological scars in 1D, 2D, 3D
1. Scar states and ETH violation
3. General construction
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SPT exampleHSPT =
X
i
Zi�1XiZi+1Nontrivial gapped paramagnet
Z1, X1Z2, Y1Z2Pauli algebra of chain end operators
• commute with Hamiltonian• double degeneracy of each state (including the ground state)• anticommute with protecting symmetries (cannot be added as
perturbations)
Y
i2even
Xi
Y
i2odd
Xi T = KY
i
Xi
Z2 ⇥ Z2 ZT2
or
spin-1/2 chain
Protect topological degeneracy at each end, stable to symmetry-preserving deformations (unitary spectral evolution)
Z2 ⇥ Z2
classification
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SPT exampleHSPT =
X
i
Zi�1XiZi+1Deform SPT paramagnet
H = H1 +H2 Ha =X
i2SLa
↵a,iQa,i
↵a,i = ↵+ (�1)(i�a)/2 Qa,i = e��a(Xi�1+Xi+1) � Zi�1XiZi+1
Has scar state:
|scari = exp
�1
2
X
i2SL1
Xi�1
!exp
�2
2
X
i2SL2
Xi�1
!|+, · · · ,+i
But: integral of motion H = H1 +H2
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SPT example
Unitary transformation:
Ha =X
i2SLa
↵a,iQa,i
Qa,i = e��a(Xi�1+Xi+1) � Zi�1XiZi+1
Q̃a,i = e��a(Zi�2Xi�1Zi+ZiXi+1Zi+2) �Xi
+ +
For diagonalization of alone, fix X state of every other spin (integral of motion)
H = H1 +H2
Qi(�) := e��(Zi�1Zi+ZiZi+1) �Xi
Same as trivial scar Hamiltonian:- non-integrable (as argued before)- scar state is not ground state of integral of motion
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Toric code example
As
Bp
Bp =Y
i2p
Zi
As =Y
i2s
Xi
H = H1 +H2
H1 = �X
s
As
H2 = �X
p
Bp 4-fold degenerate ground state due to Wilson loops (and every excited state)
deformation
H1 =X
s
"exp
��1
X
i2 s\P1
Zi
!�As
#
H2 =X
p
2
4exp
0
@��2
X
i2 p\P2
Xi
1
A�Bp
3
5
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H = H1 +H2
Choice of paths:
- non-intersecting - cover all sites- connected
Guarantees no extra integrals of motion
P1
P2
Consider only : Z eigenvalues for spins not on path are integrals of motionIn each sector, reduces to 1D scar paramagnet
H = H1 +H2H = H1 +H2
H2 =X
p
↵p
2
4exp
0
@��2
X
i2 p\P2
Xi
1
A�Bp
3
5
H1 =X
s
↵s
"exp
��1
X
i2 s\P1
Zi
!�As
#
Non-integrable, supports scar state in the middle of spectrum+ 4-fold “topological” degeneracy of scar state
Toric code example
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Let’s go 3D
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Fracton topological orderTopological order beyond the gauge theory paradigm
- 3D (or higher D) - (partially) immobile excitations- exponential in L (ground) state degeneracy: great topological
quantum memory
Motivation: T-stable topological order (did not work)
[C. Castelnovo, C. Chamon, and D. Sherrington, PRB 81, 184303 (2010)]
First model
Haah code
[J. Haah, PRA 83, 042330 (2011)]
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X-cube model
As =Y
i2s
Xi
spin 1/2 on cubic lattice bonds
three types of stars
Bc =Y
i2c
Zi
cubes
H = �X
c
Bc �X
s
As
degeneracy on LxLxL system with pbc: 6L-3
eigenvalues of independent stars/cubes
Excitations:corners of membrane
operators, mobile only in pairs1D confined
particles
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X-cube modelDeformed version
H1 =X
s
↵s
"exp
��1
X
i2 s\P1
Zi
!�As
#
H2 =X
c
↵c
"exp
��2
X
i2 c\P2
Xi
!�Bc
#
H = H1 +H2
- alternating sign coefficients- same conditions on paths as in 2D
Consider only : Z eigenvalues for spins not on path are integrals of motionIn each sector, reduces to 1D scar paramagnet
H = H1 +H2H = H1 +H2
AGAIN:
Non-integrable, supports scar state in the middle of spectrum
|scar; ⇣i = exp
�1
2
X
i2P1
Zi
!exp
�2
2
X
i2P2
Xi
!|+, · · · ,+; ⇣i
+ exp(L) “topological” degeneracy of scar state
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Conclusions- new models with scar
states in all dimensions [only single scar state]
- strong numerical evidence for their non-integrability
- topological features of the scar states, in particular exp(L) degeneracy for fracton models in 3D
Open questions- Is there any sense of stability of the “topological” scar state degeneracies?
- Stability of scar states in general?
- Observables/physical consequences of single scar states?arxiv:1901.01260
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Open questions
2D model:phase transition (in ground state)
E
�
H1 =X
s
↵s
"exp
��1
X
i2 s\P1
Zi
!�As
#
topologically degenerate ground states
symmetry breaking ground states
related to Ising model phase transition
Turn into scar state: phase transition in excited states?
E
�
?