Hidden Order and Dynamics of SUSY VBS models

Keisuke Totsuka Yukawa Institute for Theoretical Physics, Kyoto University

Kazuki Hasebe Takuma National College of Technology

29/10/2010 “New development of numerical simulations in low-dimensional quantum systems” @ YITP

Refs: D. Arovas, KH, X-L.Qi and S-C. Zhang, Phys.Rev.B 79, 224404 (2009) KH and KT, preprint (2010) and in preparation

Outline

IntroductionValence-Bond Solid (VBS) model(s)

definition, matrix product & analogiesHidden (string) order

Supersymmetric VBS stateHoles in VBS statesAnalogies (FQHE, BCS) & hidden orderLow-lying excitation (SMA)Some generalizations

Summary

Introduction

VBS model ... a paradigmatic model for spin-gap ststems in 1D

generalization to other groups: SU(n), SO(n), ...

What about adding “holes” to VBS model ?✓a single spin-0 hole ... an exact eigenstate

for a t-J-like model (S=1 VBS background)

✓a single spin-1/2 hole ... a model for Y2BaNiO5

Model with many holes??Hidden order (robust against hole) ? Excitation?Possible future generalizations??

Zhang-Arovas ’89

Penc-Shiba ’95

Tu-Zhang-Xiang ’08, etc.

Valence-bond solid (VBS) states

Valence-bond solid modelsClass of models for which short-range valence-bondcrystals are (rigorous) ground states

Majumdar-Ghosh model (S=1/2 J1-J2 chain with J2/J1=1/2)

2D Shastry-Sutherland model, and many more...

Soluble models which realize Haldane’s scenario:HS=1 =

�

i

�Si·Si+1 +

13

(Si·Si+1)2 +

23

�

HS=2 =�

i

�Si·Si+1 +

29

(Si·Si+1)2 +

163

(Si·Si+1)3 +

107

�

HS=3 =�

i

�Si·Si+1 +

59358

(Si·Si+1)2 +

191611

(Si·Si+1)3 +

13222

(Si·Si+1)4 +

396179

�

Affleck et al ’88, Arovas et al. ’88

Majumdar-Ghosh state: a “Kekule” structure

Valence-bond solid models(Many) important features:✓short-range spin correlation:

✓ “Haldane gap” to triplon excitation

✓edge states & (exact) degeneracy in a finite open system

✓Hidden order

Arovas et al. ’88, Freitag-Müller-Hartmann ’91

�Szi Sz

j � =

(S+1)2

3

�− S

S+2

�|j−i|for i �= j

S(S+1)3 for i = j

Spin Gap (SMA) Approximation

1 2027 → 7.41× 10−1

2 15 → 2.00× 10−1

3 2646265 → 4.21× 10−2

4 71590318 → 7.92× 10−3

5 884634711 → 1.39× 10−3

6 2826251203274644 → 2.35× 10−4

1 2 3 4 5 6 7 8

10 4

0.001

0.01

0.1

1

spin

gap

KT-Suzuki ’95

Supersymmetric VBS states

A quick look at supersymmetry... Schwinger boson construction of SU(2)

Generators (“spin”):commutation relations:

Schwinger-boson representation:

Label for irreps. = “boson number” = spin “S”

{Sx , Sy , Sz}

[Sa , Sb] = i �abc Sc

a†a + b†b = 2S

S+ = a†b , S− = b†a , Sz = (a†a− b†b)/2 (a†a + b†b = 2S)

|S;m� =1�

(S + m)!(S −m)!(a†)S+m(b†)S−m|0�

A quick look at supersymmetry ...Schwinger operator construction of UOSp(1|2)

5 Generators & 3 Schwinger op.✓ Generators-(i) (“bosonic” = spin):✓ Generators-(ii) (“fermionic”):

commutation relations:

Label of irreps.

two SU(2) irreps:

{Sx , Sy , Sz}

K1=12(x−1fa† + xf†b) , K2=

12(x−1fb† − xf†a)

[Sa , Sb]=i �abcSc , [Sa , Kµ] =12(σa)νµKν

{Kµ , Kν} =12(iσyσa)µνSa

S = L (nf = 0) , S = L−1/2 (nf = 1)

na + nb + nf = 2L (∈ Z) spin-1 state

1-hole (S=1/2) state

L=1

(a† , b† , f†)

SUSY VBS state... construction

prepare two (super)spin-1/2s on each site:

“Physical” state @site = superspin-1Schwinger-op rep. :“spin-1” or “1-hole(S=1/2)”

Form “entangled pairs”:

SUSY-VBS state:

a†jaj + b†jbj + f†j fj = 2L = 2

(a†jb†j+1−b†ja

†j+1−r f†

j f†j+1)

Def |sVBS� =�

j

(a†jb†j+1 − b†ja

†j+1 − r f†

j f†j+1)|0� (r ≡ x2 ∈ R)

spin-↑ spin-↓ hole

☞ super-qubit... L=1/2(spin-↑, spin-↓, hole) ⇔ (a† , b† , f†)

SUSY VBS state... Physical picture

Schwinger-op. rep. :

expansion w.r.t. “r”:lowest-order in “r”... S=1 VBS state

highest-order in “r”... S=1/2 Majumdar-Ghosh

r→0: VBS, r→∞: MG

a†jaj + b†jbj + f†j fj = 2L = 2

Def |sVBS� =�

j

(a†jb†j+1 − b†ja

†j+1 − r f†

j f†j+1)|0� (r ≡ x2 ∈ R)

spin-↑ spin-↓ hole

SUSY VBS state ... Parent Hamiltonian

usual VBS (S=1):valence-bond = “SU(2) singlet”:addition of angular-momenta:

Projection operator:

SUSY VBS:valence-bond = “UOSp(1|2) singlet”:addition of angular momenta:

Projection operator:

(a†jb†j+1−b†ja

†j+1)

1⊗ 1 � 0⊕ 1⊕ 2

(a†jb†j+1−b†ja

†j+1−rf†

j f†j+1)

(L = 1)⊗ (L = 1) � (0)⊕ (1/2)⊕ (1)⊕ (3/2)⊕ (2)

V3/2(P†3/2P3/2) + V2(P †

2 P2) (V3/2, V2 > 0)

hlocal = V2 P2 =V2

2

�Si·Si+1 +

13

(Si·Si+1)2 +

23

�

cf) Klein model ’82

Analogy to FQHE

✓coherent state w.f.

✓projection to J≧S+1 state

VBS models

HVBS =2S�

J=S+1

VJPJ(Sj ·Sj+1)

✓spin-hole coherent state w.f.✓projection to Ltot ≧ L+1/2

state

SUSY VBS models

✓Laughlin w.f. on sphere

✓two-particle int:

FQHE (Laughlin w.f.)

�

i<j

(uivj − viuj)m

H =�

i<j

�

J>2L−m

VJ PJ(i, j)

✓SUSY Laughlin w.f.

✓projection to J>2L-m state ... exact G.S.

SUSY FQHE (Laughlin w.f.)

�

j

(ujvj+1 − vjuj+1)S

Arovas et al ’88

Haldane ’83

Matrix-product representation... SUSY VBS states

Schwinger-op. rep. :Matrix-product representation:

size = number of (L/R) edge states (up, down & 1-hole...3)KEY: “valence-bond insertion” = “matrix multiplication”

gj =

−a†jb

†j −(a†j)

2 −√

ra†jf†j

(b†j)2 a†jb

†j

√rb†jf

†j

−√

rf†j b†j −

√rf†

j a†j 0

g†j =

−ajbj (bj)2 −

√rbjfj

−(aj)2 ajbj −√

rajfj

−√

rfjaj√

rfjbj 0

|sVBS� =�

j

(a†jb†j+1−b†ja

†j+1−r f†

j f†j+1)|0�

g1 ⊗ g2 ⊗ · · ·⊗ gL⊗gL+1

|sVBS� =

�g1 ⊗ · · · ⊗ gj ⊗ · · · gL . . . for open chain

STr [g1 ⊗ · · · ⊗ gj ⊗ · · · gL] . . . for periodic chain

Fannes et al. ’92,93

hole↓ ↑

hole

↑↓

⇧Left

⇦Right

Correlation functions (i) ...spin-spin correlation functions

“transfer-matrix” formalism:

spin-spin correlation function:

Correlation length:9 edge states((2L+1)2, in general)

Klümper et al ’92, KT-Suzuki ’94-95, etc..

Arovas et al. ’09

�Sαi Sα

j � =

2(r2+√

8r2+9+3)√8r2+9(

√8r2+9+3) for i = j

13r2+24+(r2+8)√

8r2+9

2√

8r2+9(√

8r2+9+3)

�− 2√

8r2+9+3

�|j−i|for i �= j

“edge states”

0 5 10 15 20

!0.5

0.0

0.5

site-j

hole

�sVBS|sVBS� = GL ,

�sVBS|OAx O

By |sVBS� = Gx−1G(OA)Gy−x−1G(OB)GL−y

Ga,a;b,b ≡ g∗(a, b)g(a, b) , G(OA)a,a;b,b ≡ g∗(a, b)OA g(a, b)

Correlation functions (ii) ...”superconducting” correlation

Possible “hole pairing”...

Need to modify “transfer-matrix” formalism:

Order parameter for “hole pairing”: Arovas et al. ’09

S=1 VBS

S=1/2 MG�∆j� =

�(ajbj+1 − bjaj+1)f†

j f†j+1

�

=144r

�r2 + 3 +

√8r2 + 9

�

√8r2 + 9

�3 +

√8r2 + 9

�2 �5 +

√8r2 + 9

�

(ajbj+1 − bjaj+1)f†j f†

j+1

�sVBS|sVBS� = GL ,

�sVBS|OAx O

By |sVBS� = Gx−1G(OA) �Gy−x−1G(OB)GL−y

�Ga,a;b,b ≡ g∗(a, b)(−1)F g(a, b) , G(OA)a,a;b,b ≡ g∗(a, b)OA g(a, b)

1 2 3 j

Analogy ... BCS wavefunction

BCS wavefunction:

gk=0: (trivial) vacuum0< gk <∞: superconducting gk=∞: filled Fermi sphere (normal state)

SUSY VBS state:

“vacuum” = spin-1 VBS stater=0: pure VBS state0<r<∞: “hole-pair condensate”r=∞: Majumdar-Ghosh state

electron pair

hole pair|sVBS� =

�

j

(a†jb†j+1 − b†ja

†j+1 − r f†

j f†j+1)|0� (r ≡ x2 ∈ R)

|ΨBCS� =�

k

(1 + gkc†k,↑c†−k,↓)|0�

Hidden Order... (usual) VBS state

Order in VBS state:No ordinary (local) magnetic LRO, no dimerization...But there is “hidden” order (“string order”)

How to detect ?Néel order:VBS:

den Nijs-Rommelse ’89, Tasaki ’91Pérez-García et al ’08

Oshikawa ’92, KT-Suzuki ’95

S=2

Szj (−1)nSz

j+n ⇒ ferro

Girvin-Arovas ’90

cf) Gu-Wen ’09

Ozstring = lim

n�∞�sVBS|Sj exp

iπj+n−1�

k=j

Szk

Szj+n|sVBS�

Hidden Order... SUSY VBS state

“string order” in SUSY VBS ?matrix-product rep. ⇒ ”almost” flat surface

string order parameter captures hidden order...

gj =

−a†jb

†j −(a†j)

2 −√

ra†jf†j

(b†j)2 a†jb

†j

√rb†jf

†j

−√

rf†j b†j −

√rf†

j a†j 0

k�

j=1

gj =

|Sz

tot=0� |Sztot= + 1� |Sz

tot= + 1/2�|Sz

tot=− 1� |Sztot=0� |Sz

tot=− 1/2�|Sz

tot=− 1/2� |Sztot= + 1/2� |Sz

tot=0�

Sztot ≡

k�

j=1

Szj

-1 0

Ozstring = �sVBS|Sj exp

iπj+n−1�

k=j

Szk

Szj+n|sVBS�

S=1 VBS

“S”=1 sVBS

Low-lying excitations ...“spin”

Failure of “Lieb-Schultz-Mattis” like excitation = want for alternative way to physical excitations

triplon = “crackion”

single-mode approximation:

Fáth-Sólyom ’93, KT-Suzuki ’94-95, Rommer-Östlund ‘95,’97

Szj |sVBS� =

12

�−|ψ(0)

j−1�+ |ψ(0)j �

�

ωzSMA(k) = −

12�sVBS| [[H, Sz(k)], Sz(−k)] |sVBS�

�sVBS|Sz(k)Sz(−k)|sVBS�

=�ψ(0)(k)|H|ψ(0)(k)��ψ(0)(k)|ψ(0)(k)�

=12

(1 − cos k)Szz(k)

�ψ(0)x |hx,x+1|ψ

(0)x �

SUSY valence bond

triplet bond

Low-lying excitations ...“hole”

In SUSY, we have two more (fermionic) generators

“charege-crackion” = “spinon-hole pair”

can be treated by MPS, toolack of unitary operationconnecting “spin” and “charge” ➡ different spectra ...

SUSY valence bond

spinon-hole pair

0.0

0.5

1.0

0.0

0.5

1.0

0

1

2

3

K1(j) =12

�1√ra†jfj +

√rf†

j bj

�, K2(j) =

12

�1√rb†jfj −

√rf†

j aj

�

K1,j |sVBS� =12√

r�

|ψ(1/2)j−1 � − |ψ(1/2)

j ��

Generalizations

On Square Lattice...

model-I: 1-species of fermion

model-II: 3-species of fermion

=

|Ψ2D-(I)sVBS � =

�

�i,j�

(a†i b†j−b†ia

†j−r f†

i f†j )|0�

|Ψ2D-(II)sVBS � =

�

�i,j�

(a†i b†j−b†ia

†j−r1 f†

i f†j−r3 g†i g

†j−r3 h†

ih†j)|0�

r1,r2,r3→∞−−−−−−−−→�

all possible dimer coverings

Rokhsar-Kivelson RVB liquid Rokhsar-Kivelson ’88

+ ... +

More fermions...

Case with two species of holes (1D)...

Matrix-product rep.:

Matrix-product formalism:finite string ordersymmetric behavior w.r.t. “r”

|sVBS� =�

j

�a†jb

†j+1−b†ja

†j+1−r(f†

j f†j+1 + h†

jh†j+1)

�|0�

gj =

−a†jb†j −(a†j)

2 −√

ra†jf†j −

√ra†jh

†j

(b†j)2 a†jb

†j

√rb†jf

†j

√rb†jh

†j

−√

rh†jb

†j −

√rh†

ja†j −rh†

jf†j 0

−√

rf†j b†j −

√rf†

j a†j 0 −rf†j h†

j

|0�j

Summary

SU(2) -> UOSp(1|2) generalization of VBS model(s)✓ “spin-L” or “spin-(L-1/2)+one (spin-1/2) hole”

Matrix-product representation in terms of (2L+1)×(2L+1) matrix:

✓ short-range spin-spin cor., finite hole-pairing correlation✓ hidden string order (symmetry-protected phase ?)✓ single-mode approx. ⇒ gapped magnetic excitations

Generalizations:

✓ 2D case ⇒ relation to Rokhsar-Kivelson point...

✓ physics of QDM, topological order ??

(a†jb†j+1−b†ja

†j+1) ⇒ (a†jb

†j+1−b†ja

†j+1−rf†

j f†j+1)