Entanglement Behavior of 2D Quantum Models

Entanglement Behavior of
2D Quantum Models
Shu Tanaka (YITP, Kyoto University)
Collaborators:
Hosho Katsura (Univ. of Tokyo, Japan)
Anatol N. Kirillov (RIMS, Kyoto Univ., Japan)
Vladimir E. Korepin (YITP, Stony Brook, USA)
Naoki Kawashima (ISSP, Univ. of Tokyo, Japan)
Lou Jie (Fudan Univ., China)
Ryo Tamura (NIMS, Japan)
VBS on symmetric graphs, J. Phys. A, 43, 255303 (2010)
“VBS/CFT correspondence”, Phys. Rev. B, 84, 245128 (2011)
Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012)
Nested entanglement entropy, Interdisciplinary Information Sciences, 19, 101 (2013)
審査希望分野
1.
申請研究課題
氏名
姓
論文での2.
3.

Digest
Entanglement properties of
2D quantum systems
Physical properties of
1D quantum systems
VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder
Volume exclusion effect
VBS state on 2D lattice Quantum lattice gas on ladder
Total system Entanglement
Hamiltonian
Square lattice 1D AF Heisenberg
Hexagonal lattice 1D F Heisenberg
Hamiltonian
Square ladder 2D Ising
Triangle ladder 2D 3-state Potts

Introduction
- Entanglement
- Motivation
- Preliminaries

Introduction
EE is a measure to quantify entanglement.
Total
system Subsystem
A
Subsystem
B
|
Divide
Schmidt decomposition

|! =

!|[A]
! |[B]
!
[A]
HA, [B]
{|[A]
HB
}, {|[B]
}
!A = TrB|!| =
: Orthonormal basis

2
|#[A]
!#[A]
|
Reduced density matrix
Normalized GS
von Neumann entanglement entropy
S = Tr!A ln !A =

2
ln 2

then move to the analysis and discussion of the results, a
summary of which is provided by Fig. 1. The XXZ model,
Eq. (2), can be analyzed by using the Bethe ansatz [15].
We have numerically determined the ground state j!gi of
HXXZ for a chain of up to N % 20 spins, from which SL
can be computed. We recall that in the XXZ model, and
due to level crossing, the nonanalyticity of the ground-state
Introduction
Entanglement properties in 1D quantum systems!!
1D gapped systems: EE converges to some value.
1D critical systems: EE diverges logarithmically with L.
energy characterizing a phase transition already
occurs for a finite chain. Correspondingly, already for a
chain of N % 20 spins it is possible to observe a distinct,
characteristic behavior of SL depending on whether the
values ##; $ in Eq. (2) belong or not to a critical regime.
coefficient is related to the central charge.
XY(a = , = 0)
XY (a =1, = 1)
XXZ( =1, = 0)
XXZ( =2.5, = 0)
XY(a = 1.1, = 1)
10 20 30 40
NUMBER OF SITES − L −
2.5
2
1.5
1
ENTROPY − S −
FIG. 1. Noncritical entanglement is characterized by a satu-ration
of SL as a function of the block size L: noncritical Ising
i !x
i+1 + !y
i )
Entanglement properties in 2D quantum systems??
chain (empty squares), HXY#a % 1:1; ( % 1$; noncritical XXZ
chain (filled squares), HXXZ## % 2:5; % 0$. Instead, the en-tanglement
of a block with a chain in a critical model displays
a logarithmic divergence for large L: SL ( log2#L$=6 (stars) for
the critical Ising chain, HXY#a % 1; ( % 1$; SL ( log2#L$=3
HXXZ =

i
(!x
i !y
i+1 + !z
i !z
i+1 !z
G. Vidal et al. PRL 90, 227902 (2003)
XXZ model under magnetic !eld
XY model under magnetic !eld
L A B

Preliminaries: re!ection symmetric case
Pre-Schmidt decomposition Re#ection symmetry
Subsystem
A
Subsystem
B
|! =

|[A]
! {|[A]
! |[B]
}, {|[B]
}
Linearly independent
(but not orthonormal)
! |[A]
(M[A])! := ![A]
! |[B]
, (M[B])! := ![B]

Overlap matrix
Useful fact
If M[A] = M[B] = M and M
is real symmetric matrix,
S =

p ln p, p = d2

where are the eigenvalues of .
d2

d M
J. Phys. A, 43, 255303 (2010)
Re!ection symmetry M[A] = M[B] = M

VBS (Valence-Bond-Solid) state
Valence bond = Singlet pair
AKLT (Affleck-Kennedy-Lieb-Tasaki) model
I. Affleck, T. Kennedy, E. Lieb, and H. Tasaki, PRL 59, 799 (1987).
H =
!
i

S
i · S
i+1 +
1
3
#
S
i · Si+1
$2
%
Ground state: VBS state
(S = 1)
Valence bond
S = 1
(projection)
- Exact unique ground state; S=1 VBS state
- Rigorous proof of the “Haldane gap”
- AFM correlation decays fast exponentially

VBS state = Singlet-covering state
2D square lattice 2D hexagonal lattice
MBQC using VBS state
T-C. Wei, I. Affleck, and R. Raussendorf, Phys. Rev. Lett.106, 070501 (2011).
A. Miyake, Ann. Phys. 326, 1656 (2011).

Schwinger boson representation
| ! = a†|vac, | # = b†|vac
Valence bond solid (VBS) state
!
|VBS! =
!k,l

a†kb†l
− b†ka†l
#
|vac!
n(b)
k = b†kbk
a†kak + b†kbk = 2Sk
n(a)
k = a†kak
0 1 2 3 4
4
3
2
1
0
S=0 1/2 1 3/2 2

Re#ection symmetry
Subsystem
A
Subsystem
B
Subsystem A Subsystem B Subsystem A Subsystem B

Subsystem A Subsystem B
|VBS! =
!
!k,l

a†kb†l
− b†ka†l
#
|vac!
=
$
{}
! # |[B]
|[A]
!
{} =
!
1, · · · , |A|

Auxiliary spin: i = ±1/2
#bonds on edge: |A|
- Local gauge transformation
- Re#ection symmetry
Overlap matrix
M{!},{} : 2|A| 2|A| matrix
Each element can be obtained by Monte Carlo calculation!!
SU(N) case can be also calculated. Phys. Rev. B, 84, 245128 (2011)
cf. H. Katsura, arXiv:1407.4262

Entanglement properties
- Entanglement entropy
- Entanglement spectrum
- Nested entanglement entropy

Entanglement properties of 2D VBS states
Lx
Ly
PBC
OBC

Entanglement entropy of 2D VBS states
cf. Entanglement entropy of 1D VBS states
|VBS! =
N!
i=0

a†i
b†i
+1 − b†i
a†i
+1
#S
|vac!
Subsystem A S=8
Subsystem B
S=6
S=4
S=3
S=2
S=1
H. Katsura, T. Hirano, and Y. Hatsugai, PRB 76, 012401 (2007).
S = ln (# Edge states)

Entanglement entropy of 2D VBS states
0
Lx
Ly
PBC or OBC
OBC
S
|A|
= ln2
1D = 0
square hexagonal
square hexagonal #bonds on edge: |A|

Entanglement spectra of 2D VBS states
Reduced density matrix
Entanglement Hamiltonian
LOU, TANAKA, KATSURA, AND KAWASHIMA FIG. 3. (Color online) Entanglement spectra of the (left) square
and (right) hexagonal VBS states with cylindrical geometry (PBC).
In both cases, Ly = 16 and Lx = 5, in which case results have
the FM spectrum left panel can spectrum in the the excitations with spin-wave spectrum to the translational theorem applies are exact eigenstates C. In order to further the holographic Heisenberg chain, nested entanglement properties. One of the ground-state structure in the a subtle point A = eHE (HE = ln A)
des Cloizeaux-
Pearson mode
Hexagonal
(Lx=5, Ly=32)
Spin wave
Square
(Lx=5, Ly=16)
H. Li and F. D. M. Haldane, Phys. Rev. Lett. 101, 010504 (2008).
!A =

!
e|[A]
! ![A]
! |
1D antiferro
Heisenberg
1D ferro
Heisenberg
cf. J. I. Cirac, D. Poilbranc, N. Schuch, and F. Verstraete, Phys. Rev. B 83, 245134 (2011).

Nested entanglement entropy
“Entanglement” ground state := g.s. of H E : |0
HE|0 = Egs|0 A|0 = 0|0
Maximum eigenvalue
Nested reduced density matrix
!() := Tr+1,··· ,L [|0!0|]
HE = ln A
Nested entanglement entropy
S(!, L) = Tr1,··· , [(!) ln(!)]
1D quantum critical system (periodic boundary condition)
P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.
A B

Nested entangElNeTmAeNnGt LeEnMtrEoNpTy SPECTRA OF THE TWO-DIMENSIONAL S(!, L) = Tr1,··· ,[(!) ln(!)]
1.4
1.2
A B
1
0.6
Square 0.8
c=1.01(7) Lx=5, Ly=16
0 2 4 6 8 10 12 14 16
FIG. 4. (Color online) Nested entanglement entropy as a function of the subchain length ! for Lx = 5 and (a) and (b) show results obtained for square VBS states with OBC, respectively. Fits to the CFT predictions, Eqs. (14) S(l,16)
l
(a)
s1=0.77(4)
fitting
0.8
0.7
S(0.6
0.5
0 2 4 6 8 10 l,16)
l
(b)
a=0.393(1)
c1/v=0.093(3)
Lx=5, Ly=fitting
lattice
(PBC)
Square ladder
(OBC)
Central charge: c = 1 1D antiferromagnetic Heisenberg
des Cloizeaux-Pearson mode in ES supports this result.
VBS/CFT correspondence

Rydberg Atom
Rydberg atom (excited state)
Interaction
Ground state Max Planck Institute
H = !

i
x
i +

i
ni + V

i,j
ninj
|ri rj |

Quantum hard-core lattice gas model
i + 1
2
ni = z
Construct a solvable model
Hsol = −z

i!
(+
i + −i )P#i$ +

i!
[(1 − z)ni + z]P#i$
Hsol =

i!
h†i
(z)hi(z), hi(z) := [−i − z(1 − ni)]P#i$
Creation/annihilation Interaction btw particles
chemical potential
1-dim chain
P!i :=

j#Gi
(1 nj)
Transverse Ising model
with constraint
Hamiltonian is positive semi-de!nite. Eigenenergies are non-negative.
Zero-energy state (ground state)
Unique (Perron-Frobenius theorem)
H =
!L
i=1
P

−zx
i + (1 − 3z)ni + zni1ni+1 + z
#
P
|z! =
1 !
(z)

i!
exp(z+
i Pi#) |## · · · #! |!! · · · ! : Vacuum state

GS of the quantum hard-core lattice gas model
!

unnormalized ground state: |!(z) :=
(z)|z =
CS
znC/2|C
: classical con!guration of particle on
C
S
!C|C = C,C (　|C is orthonormal basis)
: set of classical con!gurations with “constraint”
nC : number of particles in the state C
Normalization factor
= Partition function of classical hard-core lattice gas model
z : chemical potential
!(z) = !(z)|(z) =

CS
znC

Periodic boundary condition is imposed in the leg direction.
Square ladder Triangle ladder
Subsystem A
1 2 L
1 2 L
unnormalized ground state:
Subsystem A
1 2 L
1 2 L
d c
a b
= w(a, b, c, d)
1 z1/4 z1/4 z1/4 z1/4 z1/2 z1/2 1 z1/4 z1/4 z1/4 z1/4 z1/2
[T(z)]!, =
L
i=1
z(i+!i)/2(1 !ii)(1 !i!i+1)(1 ii+1) [T(z)]!, =
L
i=1
z(i+!i)/2(1 !ii)(1 !i!i+1)(1 ii+1)(1 i!i+1)
Subsystem B
Subsystem B
|(z)! =
!
!
!

[T(z)],!|!! |!, [T(z)],! :=
L
i=1
w(i, i+1, !i+1, !i)

Periodic boundary condition is imposed in the leg direction.
Subsystem A
1 2 L
1 2 L
unnormalized ground state:
Subsystem A
1 2 L
1 2 L
d c
a b
= w(a, b, c, d)
Subsystem B
Subsystem B
|z! =
1 !
(z)

!

[T(z)],! |!! |!
|(z)! =
!
!
!

[T(z)],!|!! |!, [T(z)],! :=
L
i=1
w(i, i+1, !i+1, !i)
Overlap matrix
M =
1
(z)
[T(z)]TT(z)
Phys. Rev. A, 86, 032326 (2012)

Entanglement entropy

S = Tr [M lnM] =

p ln p p ( = 1, 2, · · · ,NL)
5
a),
5
4
3
2
1
0
0 10 20 30
S
z
(a)
5
4
3
2
1
0
0 5 10 15 20 25
S
L
!=0.2272(3)
S0=-0.036(6)
(c)
2
10
8
6
4
2
0
0 50 100 150
S
z
(b)
10
8
6
4
2
0
0 5 10 15 20 25
S
L
!=0.4001(3)
S0=0.020(5)
(d)
2
# of states

3
Estimation of zc
2
1
(z) :=
1
ln[p(1)(z)/p(2)(z)]
p(1)(z)
p(2)(z)
6
S L
4
2
!=0.4001(3)
S0=0.020(5)
: the largest eigenvalue of M
: the second-largest eigenvalue of M
0
0 5 10 15 20 25
S
L
!=0.2272(3)
S0=-0.036(6)
(c)
2
1
0
0 2 4 6
(z)/L
z
(e)
0
(d)
0 5 10 15 20 25
2
1
0
0 5 10 15
(z)/L
z
(f)
correlation length crosses at
Finite-size scaling for correlation length
FIG. 4: (color online) (a) EE (S) of the state Eq. (8) on square ladder as a function of activity z and (b) the same

Finite-size scaling
(z)/L = f[(z zc)L1/]
2.5
2
1.5
1
0.5
0
-10 0 10
ξ(z)/L
(z-zc)L1/ν
(a)
L= 6
L= 8
L=10
L=12
L=14
L=16
L=18
L=20
L=22
2
1.5
1
0.5
0
2D 3-state Potts
-50 0 50
ξ(z)/L
(z-zc)L1/ν
(b)
L= 6
L= 9
L=12
L=15
L=18
L=21
Finite-size scaling relation:
2D Ising
= 1 = 5/6

Entanglement spectra at z=zc
Eigenvalues of entanglement Hamiltonian at 6
6
4
4
2
0
z = zc
c = 1/2 (2-dim Ising) ! !0 =
0 0.25 0.5 0.75 1
-!0
k/2
2
2
1
2v
L
(hL, + hR,)
: Velocity
: Holomorphic
conformal weight
: Antiholomorphic
conformal weight
v
hL,
hR,
hL, + hR,
Scaling dimension
Triangular ladder Primary !eld
!-!0 k/2
1
0
0 0.25 0.5 0.75 1
-!0
k/2
FIG. 6: (color online) Low-energy spectra of the entangle-ment
1.4
1.4
1.2
1.2
1
s(1
0 0.5 1 1.5 s(0.8
0.8
0.6
0 0.5 1 1.5 l,L)
ln[g(l)]
(a)
c=1/2
Descendant !eld
FIG. 7: (color online) NEE for the triangular ladder (b). eye and indicate the slope of are used for the square and The solid circle indicates NEE are the same as in Fig. 5.
Square ladder 2
0
0 0.25 0.5 0.75 1
!-!0
k/2
0
0 0.25 0.5 0.75 1
0.6
l,L)
ln[g(l)]
(a)
c=1/2
FIG. 7: (color online) NEE for the triangular ladder (b). eye and indicate the slope of are used for the square and The solid circle indicates NEE are the same as in Fig. 5.
c = 4/5 (2-dim 3-state Potts)
M. Henkel “Conformal invariance and
critical phenomena” (Springer)

Nested entanglement entropy at z=zc
7
z = zc
0: Ground state of entanglement Hamiltonian ( )
1.4
1.2
L)
l,1
s(0.8
0.6
0 0.5 1 1.5 2 ln[g(l)]
(a)
c=1/2
1.4
1.2
1
0.8
0.6
0 0.5 1 1.5 2
s(l,L)
ln[g(l)]
(b)
c=4/5
|!nested reduced density matrix: () := Tr+1,··· ,L[|#0!#0|]
nested entanglement entropy: s(!,L) := Tr1,··· ,[(!) ln (!)]
Phys. Rev. B 84, 245128 (2011).
Interdisciplinary Information Sciences, 19, 101 (2013)
s(!, L) = c
3
ln[g(!)] + s1, g(!) = L

sin
!
!
L

Triangle ladder
L=6-24
Square ladder
L=6-24
A B
2D Ising 2D 3-state Potts

Conclusion
Entanglement properties of
2D quantum systems
Physical properties of
1D quantum systems
VBS on square lattice VBS on hexagonal lattice Quantum lattice gas on ladder
Volume exclusion effect
VBS state on 2D lattice Quantum lattice gas on ladder
Hamiltonian
Square lattice 1D AF Heisenberg
Hexagonal lattice 1D F Heisenberg
Hamiltonian
Square ladder 2D Ising
Triangle ladder 2D 3-state Potts

Thank you for your attention!!
VBS on symmetric graphs, J. Phys. A, 43, 255303 (2010)
“VBS/CFT correspondence”, Phys. Rev. B, 84, 245128 (2011)
Quantum hard-square model, Phys. Rev. A, 86, 032326 (2012)
Nested entanglement entropy, Interdisciplinary Information Sciences, 19, 101 (2013)

Entanglement Behavior of 2D Quantum Models

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