philipp schmoll & romÁn orÚs johannes … · exact tensor network states for the kitaev...

EXACT TENSOR NETWORK STATES FOR THE KITAEV HONEYCOMB MODEL

PHILIPP SCHMOLL & ROMÁN ORÚS JOHANNES GUTENBERG UNIVERSITY MAINZ

ENTANGLEMENT IN STRONGLY CORRELATED SYSTEMS BENASQUE, 17. FEBRUARY 2017

▸ Quantum state of many-body system with N particles

▸ Decompose wave function using the entanglement structure

▸ Kitaev honeycomb model obeys area-law in all phases

GENERAL INTRODUCTION

TENSOR NETWORKS & QUANTUM MANY-BODY ENTANGLEMENT

| i =X

i1,i2,...,iN

ci1,i2,...,iN |i1, i2, . . . , iNi

2

Orús, Annals of Physics 349 (2014) Yao & Qi, PRL 105 080501 (2010)

MPS

PEPS

arbitrary TN

i1i2 i3 i4

. . .iN

ci1,i2,...,iN

Fannes, Nachtergaele and WernerComm. Math. Phys. Volume 144, Number 3

Verstraete and CiracarXiv:cond-mat/0407066

GENERAL INTRODUCTION

FERMIONIC TENSOR NETWORKS

▸ Fundamental difference

▸ Fermionization rules for tensor networks

Spins Fermionscicj = + cjci cicj = - cjci

TP

P P

P

P

=T

Ti1,i2,...,iM = 0 if P(i1)P(i2) . . .P(iM) 6= 1

i1 i2

i2 i1

X

i1 i2

j1 j2

Xi1,i2,j1,j2 = �i1j2�i2j1S (P(i1)P(i2))

S (P(i1)P(i2)) =

�-1 if P(i1) = P(i2) = -1+1 otherwise

3

Corboz, Orús, Bauer, Vidal, Phys. Rev. B 81, 165104 (2010)

THE KITAEV HONEYCOMB MODEL


▸ Model of spin-1/2 on the sites of a 2d honeycomb lattice

▸ Fully anisotropic coupling along x, y and z bonds

▸ HamiltonianH = - J

x

X

x-links

�x

i

�x

j

- Jy

X

y-links

�y

i

�y

j

- Jz

X

z-links

�z

i

�z

j

Kitaev, Annals of Physics 321 (2006)

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THE SOLUTION

▸ Mapping to a free fermion model in various steps

▸ Jordan-Wigner transformation

▸ Introduction of Majorana fermions

▸ Recombination of Majorana fermions

▸ Fourier transformation

▸ Bogoliubov transformation

Feng, Zhang & Xiang, Phys. Rev. Lett. 98, 087204 (2007) Chen & Nussinov, Journal of Physics A 41 075001 (2008)

5


PHASE DIAGRAM

▸ Phase diagram from dispersion relations

▸ Gapless B phase for

Jz

= 1

Jx

= 1 Jy

= 1

Az

Ax

Ay

B

|Jx

| 6 |Jy

|+ |Jz

|

|Jy

| 6 |Jz

|+ |Jx

|

|Jz

| 6 |Jx

|+ |Jy

|

6

Kitaev, Annals of Physics 321 (2006)

CONSTRUCTION OF TN EIGENSTATES

GENERAL PROCEDURE

▸ Start with eigenstates of diagonalHamiltonian

▸ Ground state is , Bogoliubov vacuum

▸ Procedure to build TN

▸ Reverse steps in the solution of the model

▸ Find respective tensor network representation

▸ Transform quantum state step-by-step

�-⇡

2 ,-⇡2

�

�-⇡

2 ,+⇡2

��+⇡

2 ,-⇡2

�

�+⇡

2 ,+⇡2

�

| i = |0i⌦N

7

INTERMEDIATE TENSOR NETWORK


▸ Bogoliubov + Fourier transformation

▸ Fourier transformation as 2d Spectral TN plus additional twiddle factors

▸ Eigenstates of translational invariantHamiltonian in real space

▸ Model on the 2d square lattice

�-⇡

2 ,-⇡2

�

�-⇡

2 ,+⇡2

��+⇡

2 ,-⇡2

�

�+⇡

2 ,+⇡2

�

F̂2

F̂2

F̂2

F̂2

!04

!14

!14

!24

Bk

Bk

Ferris, PRL 113 010401 (2014)

8

MAJORANA FERMIONS


▸ Reintroduce conserved quantities

▸ Recombine Majorana fermions from FT and CQ

▸ Need to be treated as non-Abelian anyons

▸ Exchange leads to braiding of anyons

▸ No occupation number for Majorana fermions

▸ How to treat them in the language of TNs?

9

MAJORANA FERMIONS


▸ Majorana transformation action on the Fock space

▸ Define (anti-)clockwise swap operators to braid anyons

Leijnse & Flensberg, Semicond. Sci. Technol. 27, 124003 (2012)

Bij = 1/p2 (1+ �i�j) B̄ij = 1/

p2 (1- �i�j)

FT

c c

⌦a ,a†

↵dc

CQ

dd

⌦a ,a†

↵d c

=

B̄34

B̄34

B̄23

FT CQ

⌦a ,a†

↵ ⌦a ,a†

↵

T↵��

↵ �

� �

10

OVERALL TENSOR NETWORK CONSTRUCTION


▸ Some remarks

▸ Fermionic tensor network

▸ Numerical checks up to 32 spins

▸ Modification of boundary termsnecessary

▸ All excited states in the vortex-freesector possible

▸ Reliable scaling to the thermodynamic limit

�-⇡

2 ,-⇡2

�

�-⇡

2 ,+⇡2

�

�+⇡

2 ,-⇡2

�

�+⇡

2 ,+⇡2

�

|0i|0i

|0i|0i

F̂2

F̂2

F̂2F̂2

!04

!14

!14

!24

T

T

T

T

Bk

Bk

11

GROUND STATE FIDELITY

APPLICATIONS 12


APPLICATIONS

▸ Ground state fidelity defined as

▸ Reduces to a simple formF�J̃z, Jz

�= | h

�J̃z�| (Jz)i |2

U

U†

=

W†

W

=

B~k B~k

T T T T

f1 f2 f4 f6f3 f5 f7 f8

!04 !1

4 !14 !2

4

|0i |0i |0i |0i |0i |0i |0i |0i

12


APPLICATIONS



�= | h

�J̃z�| (Jz)i |2

h �J̃z�| (Jz)i =

Q

k

Bk

B̃†k

|0i |0i

h0| h0|

⌘Q

k

Bk |00i

h00| B̃†k

B~k B~k

T T T T

f1 f2 f4 f6f3 f5 f7 f8

!04 !1

4 !14 !2

4

|0i |0i |0i |0i |0i |0i |0i |0i

12


APPLICATIONS



�= | h

�J̃z�| (Jz)i |2

h �J̃z�| (Jz)i =

Q

k

Bk

B̃†k

|0i |0i

h0| h0|

⌘Q

k

Bk |00i

h00| B̃†k

128 spins 2048 spins 32768 spins

12

APPLICATIONS

TWO-POINT CORRELATORS FOR THE GROUND STATE

G. Baskaran, S. Mandal, R. Shankar, Phys. Rev. Lett. 98, 247201 (2007)

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APPLICATIONS



1

2 4

6

3

5 7

8

x

x

y

y

z

z z

z

D�↵i �

�j

E=

�0 if i, j not nearest neighbours

g↵ · �↵� if i, j are nearest neighbours

g↵ 6= 0 only for an ↵-type link

13

APPLICATIONS



1

2 4

6

3

5 7

8

x

x

y

y

z

z z

z

D�↵i �

�j

E=




13

h�

x

2�x

5i =

T T T T

T

†T

†T

†T

†

X X

=

T T T T

T

†T

†T

†T

†

Z X X Z = 0

APPLICATIONS



1

2 4

6

3

5 7

8

x

x

y

y

z

z z

z

D�↵i �

�j

E=




13

h�y2�

y5i =

T T T T

T † T † T † T †

Y Y =

T T T T

T † T † T † T †

Z Y Y Z = 0

APPLICATIONS



1

2 4

6

3

5 7

8

x

x

y

y

z

z z

z

D�↵i �

�j

E=




13

h�z2�

z5i =

T T T T

T † T † T † T †

Z Z =

T T T T

T † T † T † T †

Z Z = Z

h�y2�

y5i =

T T T T

T † T † T † T †

Y Y =

T T T T

T † T † T † T †

Z Y Y Z = 0

CONCLUSION

TN EIGENSTATES OF THE KITAEV HONEYCOMB MODEL

▸ Exact tensor network for the eigenstates in vortex-free sector

▸ Scaling to larger systems for all steps possible

▸ Structure reveals access to physical properties

▸ Tensor network as quantum circuit to generate many-body wave function

▸ TN should provide good initial wave function for numerical simulations

▸ Unclear if contractible to a PEPS with finite bond dimensionP. Schmoll, R. Orús, Phys. Rev. B 95, 045112

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THANK YOU FOR THE ATTENTION!

PHILIPP SCHMOLL & ROMÁN ORÚS JOHANNES GUTENBERG UNIVERSITY MAINZ

ENTANGLEMENT IN STRONGLY CORRELATED SYSTEMS BENASQUE, 17. FEBRUARY 2017

philipp schmoll & romÁn orÚs johannes … · exact tensor network states for the kitaev...

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