hidden order and dynamics of susy vbs models...hidden order and dynamics of susy vbs models keisuke...

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)