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Detection of symmetry protected topological phases in 1D Frank Pollmann Max-Planck-Institut fr komplexer Systeme, Dresden, Germany FP. and A. M. Turner, arxiv:1204.0704 Florence, May. 29, 2012 Detection of symmetry protected topological

slide-1
SLIDE 1

Frank Pollmann

Max-Planck-Institut für komplexer Systeme, Dresden, Germany

Florence, May. 29, 2012

  • FP. and A. M. Turner, arxiv:1204.0704

Detection of symmetry protected topological phases in 1D

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SLIDE 2

Overview

  • Introduction: Symmetry protected topological phases
  • Non-local order parameters
  • Summary

Florence, May. 29, 2012

Detection of symmetry protected topological phases in 1D

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SLIDE 3
  • Quantum phases: Two gapped quantum states belong

to the same phase if they are adiabatically connected

  • Phases in condensed matter are usually understood using

local order parameters (“symmetry breaking”)

  • Magnets: spin rotation

and TR symmetry broken Magnetization as order parameter

  • Topological phases not characterized by any

symmetry breaking

  • We introduce non-local order parameter for

symmetry protected topological phases in 1D

Symmetry protected topological phases

slide-4
SLIDE 4
  • Example: Spin-1 chain

(time reversal, inversion, symmetry, ...)

  • Hidden symmetry breaking [Kennedy-Tasaki ’92]
  • String order parameter [den Nijs ‘89]

H = P

j

Sj · Sj+1 + D P

j(Sz j )2

[Haldane ‘83]

|Sz = ±1, |Sz = 0

...

...

Z2 × Z2 Z2 × Z2

Symmetry protected topological phases

D Phase transition no symmetry broken no symmetry broken |0i|0i|0i|0i AKLT

[Affleck ‘87]

“Haldane Phase” “Large D Phase”

slide-5
SLIDE 5
  • Spin-1 chain with less symmetries [Gu et al. ‘09]

➡no symmetry ➡Haldane phase still well defined

Which symmetries are required? How to detect “topological” phases?

➡Idea: Use entanglement and matrix-

product states (capturing non-local properties) H = J P

j

Sj · Sj+1 + D P

j(Sz j )2 + Bx

P

j Sx j

Z2 × Z2

Symmetry protected topological phases

     

 





  

slide-6
SLIDE 6

Schmidt decomposition (SVD )

  • Decompose a state into a

superposition of product states:

  • Schmidt states: , Schmidt values:
  • are eigenstates of the reduced density matrix

A B |ψi with C = UDV †

Symmetry protected topological phases

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SLIDE 7
  • Example: Spin-1 Heisenberg chain
  • Schmidt values decay rapidly in ground states of

gapped, local Hamiltonians (area law! [Hastings et al. ’07]): Matrix-Product representation

20 40 60 80 100 10

−10

10

−5

10 λ2

α

α

H = P

j ~

Sj · ~ Sj+1

...

A B

...

|ψ0 =

NA

X

i=1 NB

X

j=1

Cij|iA|jB = X

γ

λγ|φγA|φγB

γ

γ X

γ

λ2

γ = 1

!

Symmetry protected topological phases

slide-8
SLIDE 8
  • Matrix product state (MPS) representation
  • Matrices not uniquely defined: Canonical Form

is directly related to the Schmidt decomposition: Aj = ΓjΛ

           

1 . . . χ 1 . . . d

[Vidal ’02]

ψ...,j1,j2,... =

Symmetry protected topological phases

|Ψi = X

j1,...,jL

BT Aj1 . . . AjLB | {z }

ψj1,...,jL

|j1, . . . , jLi

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SLIDE 9
  • Matrices are directly related to the Schmidt decomposition
  • Left/right transfer matrices T have largest eigenvalue one

with the identity as corresponding eigenstate

               

Symmetry protected topological phases

...

A B

...

[φα]j1,j2... =

           

α | {z }

T Σ

(αα0);(ββ0)

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SLIDE 10
  • Transformation of an MPS under symmetry operations

...wave function only changes by a phase

  • Time reversal ( ) and inversion ( )
  • Matrices are projective representations which tell

us about topological phases

       

 

           

Γj → Γ∗

j

Γj → ΓT

j

Σ Σ

UΣ

[FP et al. ’10, Chen et al ’11]

, [UΣ, Λ] = 0

[Perez-Garcia ’07]

Symmetry protected topological phases

























 

         

                               

               

slide-11
SLIDE 11

Use projective representations to classify phases!

  • Ground state is invariant under a symmetry group

with elements

  • Projective representation of the symmetry group

G g1, g2, . . . , gn gjgk = gl : UgjUgk = eiφjkUgl Ugj |ψ0

Symmetry protected topological phases

slide-12
SLIDE 12

Use projective representations to classify phases!

  • Ground state is invariant under a symmetry group

with elements

  • Projective representation of the symmetry group
  • Phase ambiguities classify the phases

(Schur classes)

➡Complete classification of topological

phases in 1D G g1, g2, . . . , gn gjgk = gl : UgjUgk = eiφjkUgl Ugj |ψ0

[FP , A. Turner, E. Berg, M. Oshikawa ’10, Chen et al ’11]

Symmetry protected topological phases

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SLIDE 13
  • Which symmetries stabilize topological phases?
  • Example : Rotation about single axis

➡Redefining removes the phase

  • Example : Phase for pairs

➡Phases cannot be gauged

away: topological phases

Zn

Rn = 1 ⇒ U n

R = eiφ1

Z2 × Z2 φ = 0, π ˜ UR = e−iφ/nUR RxRy = RyRx ⇒ URxURy = eiφxyURyURx

Symmetry protected topological phases

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SLIDE 14

Symmetry protected topological phases

  • Example S=1 AKLT state

➡

  • Matrix-product state representation
  • Rotations represented by Pauli matrices and

thus

  • Inversion symmetry with :
  • Time reversal with :

|ψ = Z2 × Z2 Γi = σi, i = x, y, z UIU ∗

I = −1

UT RU ∗

T R = −1

UI = σy UT R = σy

[Affleck ‘87]

URxURy = −URyURx

           

slide-15
SLIDE 15

Symmetry protected topological phases

  • Framework to classify topological phases in 1D by looking

at the “entanglement states” / MPS

  • “Topological” phase in a S=1 chain protected by
  • Inversion symmetry
  • Time reversal symmetry
  • Symmetry protected topological phases exist
  • nly in the presence of certain symmetries

(not topologically ordered!) Z2 × Z2

FP , E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010)

slide-16
SLIDE 16

Symmetry protected topological phases

  • How can we detect which phase a given

state belongs to?

  • We discuss two ways to detect

topological phases: (1)Directly extract the projective representations from a matrix-product state representation (very useful for iTEBD / iDMRG ) (2)Non-local order parameters for inversion, and time reversal symmetry and a generalized string-order for internal symmetries

[Vidal ’07] [McCulloch ’08]

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SLIDE 17

| {z }

T ΣX=X

| {z }

T 1=1

  • Get from the dominant eigenvector
  • f the generalized transfermatrix ( )
  • Overlap with transformed Schmidt states

Non-local order parameter (1)

                             

 

             

⇔

Σ

                             

 

             

UΣ X UΣ = X†

                             

 

             

T Σ

(αα0);(ββ0) =

X

j,j0

Σjj0 ˜ Γj0,αβΓ∗

j,α0β0ΛβΛβ0

UΣ = X†

           

slide-18
SLIDE 18

     

 





  

  • S=1 chain
  • stabilizes Haldane phase if

and H = P

j ~

Sj · ~ Sj+1 + D P

j(Sz j )2 + B P j Sx

iMPS obtained using the iTEBD / iDRMG method OZ2×Z2 = ⇢ if symmetry broken

1 χtr

  • UxUzU †

xU † z

  • if symmetry not broken .

B = 0 Z2 × Z2

Non-local order parameter (1)

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SLIDE 19

     

 





  

  • S=1 chain
  • Inversion symmetry stabilizes Haldane phase if

and H = P

j ~

Sj · ~ Sj+1 + D P

j(Sz j )2 + B P j Sx

OI = ⇢ if symmetry broken

1 χtr (UIU ∗ I)

if symmetry not broken .

B 6= 0

Non-local order parameter (1)

iMPS obtained using the iTEBD / iDRMG method

slide-20
SLIDE 20
  • What if we do not have access to the transfermatrix

(i.e., Monte Carlo or experiments)?

  • Inversion symmetry: Inverting part of the wave function

    

hΨ|I1,2n|Ψi =

Non-local order parameter (2)

slide-21
SLIDE 21
  • What if we do not have access to the transfermatrix

(i.e., Monte Carlo or experiments)?

  • Inversion symmetry: Inverting part of the wave function

                                               

hΨ|I1,2n|Ψi =

Non-local order parameter (2)

slide-22
SLIDE 22
  • What if we do not have access to the transfermatrix

(i.e., Monte Carlo or experiments)?

  • Inversion symmetry: Inverting part of the wave function

                                                 

 

hΨ|I1,2n|Ψi =

Non-local order parameter (2)

slide-23
SLIDE 23
  • What if we do not have access to the transfermatrix

(i.e., Monte Carlo or experiments)?

  • Inversion symmetry: Inverting part of the wave function

    

 

 

 

   

hΨ|I1,2n|Ψi =

Non-local order parameter (2)

slide-24
SLIDE 24
  • What if we do not have access to the transfermatrix

(i.e., Monte Carlo or experiments)?

  • Inversion symmetry: Inverting part of the wave function
  • Distinguishes the

Haldane phase from the trivial phase in presence of inversion symmetry

                                               

SI =

Non-local order parameter (2)

     

 





  

slide-25
SLIDE 25
  • Time reversal symmetry: two copies of the wave function

with swapping operators [Isakov et al ’11] and

Non-local order parameter (2)

ST R =

                                    



 

                                







                                  

          





  

  

    

 

               

   

 

                                    



 

                                







                                  

          





  

  

    

 

               

   

 

Σ = e−iπSy

       

 

hR1n|Swapn+1,2n|ψ2i hψ2|R1ni

slide-26
SLIDE 26
  • Time reversal symmetry: two copies of the wave function

with swapping operators [Isakov et al ’11] and

Non-local order parameter (2)

ST R =

                                    



 

                                







                                  

          





  

  

    

 

               

   

 

Σ = e−iπSy

slide-27
SLIDE 27
  • General internal symmetries symmetry: multiple

copies of the wave function

  • Example:

Non-local order parameter (2)

SG = Tr(UaUbU −1

a U −1 b

)

























 

         

                               

               

[See also: Haegemann et al.: arXiv:1201.4174]

slide-28
SLIDE 28

                

  • If phases are topological, where is the torus??
  • Express wave function as

partition function on the half plane:

  • Sandwich of a string
  • perator:

Non-local order parameter (2)

slide-29
SLIDE 29

       

  • If phases are topological, where is the torus??
  • Sandwich the symmetry string operators a / b, then deform

them and finally glue them together

Non-local order parameter (2)

slide-30
SLIDE 30

            

  • If phases are topological, where is the torus??
  • Sandwich the symmetry string operators a / b, then deform

them and finally glue them together

Non-local order parameter (2)

slide-31
SLIDE 31
  • If phases are topological, where is the torus??
  • Sandwich the symmetry string operators a / b, then deform

them and finally glue them together

            

SG = Tr(UaUbU −1

a U −1 b

)

Non-local order parameter (2)

























 

         

                               

               

slide-32
SLIDE 32
  • If phases are topological, where is the torus??
  • Sandwich the symmetry string operators a / b, then deform

them and finally glue them together

            

SG = Tr(UaUbU −1

a U −1 b

)

                

=

                      

Non-local order parameter (2)

slide-33
SLIDE 33
  • Derivation of non-local order parameters which

can be used to detect/distinguish all symmetry protected topological phases in 1D

  • Can be obtained directly from a generalized transfermatrix
  • Expressions which can be evaluated

using any numerical methods, e.g., Quantum Monte Carlo

  • Measuring string order experimentally: High-resolution

imaging of low-dimensional quantum gases

[M. Endres et al ’11]

Summary

  • FP. and A. M. Turner, arxiv:1204.0704