Transcript Topological characterization of quantum phase transitions

Jordan-Wigner Transformation and
Topological characterization of quantum
phase transitions in the Kitaev model
Guang-Ming Zhang (Tsinghua Univ)
Xiaoyong Feng (ITP, CAS)
T. Xiang (ITP, CAS)
Cond-mat/0610626
Outline



Brief introduction to the Kitaev model
Jordan-Wigner transformation and a novel
Majorana fermion representation of spins
Topological characterization of quantum
phase transitions in the Kitaev model
Kitaev Model
H  J1
x x

 n m  J 2
x link
y y

 n  m  J3
y link
z z

 n m
z link
Ground state can be rigorously solved
A. Kitaev, Ann Phys 321, 2 (2006)
4 Majorana Fermion Representation of Pauli Matrices
  ib c j
x
j
x
j
cj, bjx, bjy, bjz are Majorana
fermion operators
  ib c j
a , a   2
  ib c j
a 1
y
j
z
j
y
j
z
j
i
j
ij
2
i
Physical spin: 2 degrees of freedom per spin
Each Majorana fermion has 21/2 degree of freedom
4 Majorana fermions have totally 4 degrees of freedom
4 Majorana Fermion Representation of Kitaev Model
H  i  J ujk c j ck
  x, y , z
jk

 
u jk  ib j bk
  ib c j
x
j
x
j
u 

jk
y
  ib c j
y
j
y
j
  ib c j
z
j
z
j
2
1
Good quantum number
x
z
x
y
2D Ground State Phase Diagram
The ground state is in a zero-flux phase (highly degenerate,
ujk = 1), the Hamiltonian can be rigorously diagonalized
non-Abelian anyons in
this phase can be used
as elementary “qubits”
to build up fault-tolerant
or topological quantum
computer
4 Majorana Fermion Representation: constraint
   i
x
j
y
j
z
j
  ib c j
x
j
x
j
  ib c j
y
j
y
j
  ib c j
z
j
z
j
Dj  b b b cj  1
x
j
P
j
y
j
z
j
1 Dj
2
 phys  P 
Eigen-function
in the extended
Hilbert space
3 Majorana Fermion Representation of Pauli Matrices
Dj  b b b c  1
x y z
j j j j
  ib c j
  ib c j
  ib c j
  ib c j
  ib c j
  ib b
x
j
y
j
z
j
x
j
y
j
z
j
x
j
y
j
z
j
x
j
y
j
y
j
x
j
Totally 23/2 degrees of freedom,
still has a hidden 21/2 redundant
degree of freedom
Kitaev Model on a Brick-Wall Lattice
H
 J 
1
i  j even
 ix1, j  J 2 iy1, j iy, j  J3 iz, j iz, j 1 
x
i, j
x
y
z
y
x
z
x
y
y
x
y x
y x
z
z
z
y
x y
x y
y
z
x
x
z
Brick-Wall Lattice
H  J1
honeycomb Lattice
x x

 n m  J 2
x link
y y

 n  m  J3
y link
z z

 n m
z link
Jordan-Wigner Transformation


i, j

z
i, j

i, j
 2a e
i

k  j ,l
al,k al ,k 
 li
al, j al , j


i, j i, j
 2a a  1
Represent spin
operators by
spinless fermion
operators
Along Each Horizontal Chain
x
y
x
y
H    J1 2xi 1 2xi  J 2 2yi 2yi 1 
i
  J1  a2i 1  a2i 1  a2i  a2i   J 2  a2i  a2i  a2i 1  a2i 1 
i
Two Majorana Fermion Representation

i

i

i

i
ci  i(a  ai ), di  a  ai
i  odd
di  i(a  ai ), ci  a  ai
i  even
H    J1 2xi 1 2xi  J 2 2yi 2yi 1 
i
 i   J1c2i 1c2i  J 2c2i c2i 1 
i
Onle ci-type Majorana fermion operators appear!
Two Majorana Fermion Representation

ij

ij
i  j  odd

ij

ij
i  j  even
cij  i(a  aij ) dij  a  aij
dij  i (a  aij ) cij  a  aij
ci and di are Majorana fermion operators
A conjugate pair of fermion operators is
represented by two Majorana fermion operators
No redundant degrees of freedom!
Vertical Bond
  ici di
z
i
 iz zj   ici di   ic j d j 
No Phase String
2 Majorana Representation of Kitaev Model
H

i  j  even
 i
x
x
y
y
z
z
J



J



J


 1 i, j i1, j 2 i1, j i, j 3 i, j i, j 1 
 J c
i  j  even
c
1 i , j i 1, j
 J 2ci 1, j ci , j  J 3 Di , j ci , j ci , j 1 
Di, j  idi , j di , j 1
good quantum numbers
Ground state is in a zero-flux
phase Di,j = D0,j
Phase Diagram
Single chain
x
0
1
J1/J2
Critical point
Quasiparticle excitation:
 k ,   J12  J 22  2 J1 J 2 cos k
Ground state energy E0 

k
k ,
y
x
y
Phase Diagram
J3=1
Critical lines
= J 1 – J2
Two-leg ladder
Multi-Chain System
Chain number = 2 M
Thick Solid Lines:
Critical lines
How to characterize
these quantum
phase transitions?
J3=1
Classifications of continuous phase transitions
Conventional: Landau-type
• Symmetry breaking
• Local order parameters
Topological:
• Both phases are gapped
• No symmetry breaking
• No local order parameters
QPT: Single Chain
H    J1 2xi 1 2xi  J 2 2yi 2yi 1 
i
x
Duality Transformation
 xj   xj1 xj
,  jy 
2N
y

k
kj
j
 jy   jy  jy1 ,  xj   kx
k 1
H    J1 2xi 2 2xi  J 2 2yi 
i
y
x
y
Non-local String Order Parameter
 x  lim  1x 2x   2xn ~ lim  0x 2xn
n 
n 
 1  J / J 2 1/ 4
  2 1  

~

0


,
J1  J 2
,
J1  J 2
Another String Order Parameter
 y  lim  2y 3y   2yn 1
n 
 1  J / J 2 1/ 4
  1 2  

~

0


,
J1  J 2
,
J1  J 2
Two-leg ladder
J3 = 1
= J1 – J2
Phase I: J1 > J2 + J3
H    J1 2xi 1 2xi  J 2 2yi 2yi 3  J3 2zi 2zi 1 
i
In the dual space:
H    J1 2xi 2 2xi  J 2Wi 2yi 2 2yi  J3 2zi 
i
Wi  
 xj
  xj1 xj
,
 zj
 
x
z
x
2i 3 2i 1 2i 1
2N
  kz
kj
j
 zj   zj  zj1 ,  xj   kx
k 1
W1 = -1 in the ground state
String Order Parameters
H    J1 2xi 2 2xi  J 2Wi 2yi 2 2yi  J3 2zi 
i

 x  lim  1x  2x     2xn
n 
 y  lim  2y  3y     2yn 1
n 
2 J / J



1 J 3 / J  2
1 J  / J 
~

0




0

~
2 J / J



1 J 3 / J  2
1 J  / J 


1
4

1
4
,
J   J3
,
J   J3
,
J   J 3
,
J   J 3
QPT: multi chains
Chain number = 2 M
QPT in a multi-chain system
4-chain ladder M = 2
H  i   J1c2i 1, c2i ,  J 2c2i , c2i 3, 1  J 3 (1)i c2i , c2i 1, 
2N M
i 1  1
Fourier Transformation
2N M

H  i   J1c2n1, c2n,  J 2c2n, c2n3, 1  J 3 (1) n c2n, c2n1,
n 1 1
1
iqri
ci , 
e
ci ,q

M q
2 m
q
,
m  0,1,..., M  1
M
H   Hq
q
H q  i   J1c2i 1, q c2i ,q  J 2eiq c2i , q c2i 3,q  J 3 (1)i c2i , q c2i 1,q 
i

q=0
Hq0  i  J1c2i 1,0c2i ,0  J 2c2i,0c2i 3,0  J3 (1)i c2i,0c2i 1,0 
i
ci,0 is still a Majorana fermion
operator
Hq=0 is exactly same as the
Hamiltonian of a two-leg
ladder
String Order Parameter
 x ,0  lim(1)i c1,0 c2,0  c2i ,0
i 
 0


 0

 y ,0  lim(1)i c2,0 c3,0  c2i 1,0
i 
 0


 0

J   J3
J   J3
J    J3
J    J3
q=
Hq  i  J1c2i 1,0c2i,0  J 2c2i,0c2i 3,0  J3 (1)i c2i,0c2i 1,0 
i
ci, is also a Majorana
fermion operator
Hq= is also the same as the
Hamiltonian of a two-leg
ladder, only J2 changes sign
String Order Parameter
 x ,  lim (1) n c1, c2,    c2n,
n
 y ,  lim (1) n c2, c3,    c2n1,
n
 0


 0

 0


 0

J   J 3, J1  J 2
else
J   J 3, J1  J 2
else
Summary
• Kitaev model = free Majorana fermion model with local
Ising field without redundant degrees of freedom
• Topological quantum phase transitions can be
characterized by non-local string order parameters
• In the dual space, these string order parameters
become local
• The low-energy critical modes are Majorana fermions,
not Goldstone bosons