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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 
Total system Entanglement 
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
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 
Total system Entanglement 
Hamiltonian 
Square ladder 2D Ising 
Triangle ladder 2D 3-state Potts
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 (Valence-Bond-Solid) state 
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).
VBS (Valence-Bond-Solid) state 
VBS state = Singlet-covering state 
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
VBS (Valence-Bond-Solid) state 
Re#ection symmetry 
Subsystem 
A 
Subsystem 
B 
2D square lattice 2D hexagonal lattice 
Subsystem A Subsystem B Subsystem A Subsystem B
VBS (Valence-Bond-Solid) state 
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 
VBS state = Singlet-covering state 
2D square lattice 2D hexagonal lattice 
Subsystem A Subsystem B Subsystem A Subsystem B 
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 
2D square lattice 2D hexagonal lattice 
Subsystem A Subsystem B Subsystem A Subsystem B 
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
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 
Total system Entanglement 
Hamiltonian 
Square ladder 2D Ising 
Triangle ladder 2D 3-state Potts
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 
Total system Entanglement 
Hamiltonian 
Square ladder 2D Ising 
Triangle ladder 2D 3-state Potts
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
GS of the quantum hard-core lattice gas model 
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) 
Square ladder Triangle ladder 
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)
GS of the quantum hard-core lattice gas model 
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) 
Square ladder Triangle ladder 
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 
Square ladder Triangle ladder 
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 
Square ladder Triangle ladder 
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 
Square ladder Triangle ladder 
-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
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 
Total system Entanglement 
Hamiltonian 
Square ladder 2D Ising 
Triangle ladder 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 
Total system Entanglement 
Hamiltonian 
Square lattice 1D AF Heisenberg 
Hexagonal lattice 1D F Heisenberg 
Total system Entanglement 
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)

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Entanglement Behavior of 2D Quantum Models

  • 1. 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.
  • 2. 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 Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 3. Introduction - Entanglement - Motivation - Preliminaries
  • 4. 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
  • 5. 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
  • 6. 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
  • 7. 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 Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 8. 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
  • 9. VBS (Valence-Bond-Solid) state 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).
  • 10. VBS (Valence-Bond-Solid) state VBS state = Singlet-covering state 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
  • 11. VBS (Valence-Bond-Solid) state Re#ection symmetry Subsystem A Subsystem B 2D square lattice 2D hexagonal lattice Subsystem A Subsystem B Subsystem A Subsystem B
  • 12. VBS (Valence-Bond-Solid) state 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
  • 13. Entanglement properties - Entanglement entropy - Entanglement spectrum - Nested entanglement entropy
  • 14. Entanglement properties of 2D VBS states VBS state = Singlet-covering state 2D square lattice 2D hexagonal lattice Subsystem A Subsystem B Subsystem A Subsystem B Lx Ly PBC OBC
  • 15. 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)
  • 16. Entanglement entropy of 2D VBS states 0 2D square lattice 2D hexagonal lattice Subsystem A Subsystem B Subsystem A Subsystem B Lx Ly PBC or OBC OBC S |A| = ln2 1D = 0 square hexagonal square hexagonal #bonds on edge: |A|
  • 17. 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).
  • 18. 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
  • 19. 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
  • 20. 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 Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 21. 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 Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 22. Rydberg Atom Rydberg atom (excited state) Interaction Ground state Max Planck Institute H = ! i x i + i ni + V i,j ninj |ri rj |
  • 23. 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
  • 24. 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
  • 25. GS of the quantum hard-core lattice gas model 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) Square ladder Triangle ladder 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)
  • 26. GS of the quantum hard-core lattice gas model 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) Square ladder Triangle ladder 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)
  • 27. Entanglement entropy S = Tr [M lnM] = p ln p p ( = 1, 2, · · · ,NL) 5 a), 5 4 3 2 1 0 Square ladder Triangle ladder 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
  • 28. 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 Square ladder Triangle ladder 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
  • 29. Finite-size scaling (z)/L = f[(z zc)L1/] 2.5 2 1.5 1 0.5 0 Square ladder Triangle ladder -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
  • 30. 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)
  • 31. 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
  • 32. 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 Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 33. 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 Total system Entanglement Hamiltonian Square lattice 1D AF Heisenberg Hexagonal lattice 1D F Heisenberg Total system Entanglement Hamiltonian Square ladder 2D Ising Triangle ladder 2D 3-state Potts
  • 34. 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)