Multicritical point, conformal field theory andduality
Kiyohide NOMURA
Kyushu Universiry, Dept. Physics

2020/Sep/30

Seminer@Tokyo Univerisity, Dept. Phys, (online)

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Collaborators

1. A. Kitazawa
2. S. Moriya (Doctor@Kyushu Univ.)
3. T. Tonegawa (Kobe Univ., Osaka Prefecture Univ.)
4. K. Okamoto (Univ. of Hyogo)
5. T. Sakai (Univ. of Hyogo, Spring 8)

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Introduction

Critical phenomena are one of the important subjects in condensed
matter physics. Many developments about critical phenomena, such as
renormalization group, numerical methods etc. have been done. But,
when the model has a multicritical point, the scaling behaviors become
diﬃcult due to the interference of multiple critical lines. So, conventional
numerical methods are not useful near a multicritical point.
We have studied several multicritical phenomena combining with theconformal field theory (CFT) and numerical methods (level spectroscopyetc). And we discuss the relation with the duality, such as theKramers-Wannier duality and the Ashkin-Teller self-duality.

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Introduction

Topics about multicritical phenomena:
1. bifurcation at c=3/2 CFT to the two c=1 CFT lines(Takhtajan-Babujian)
2. bifurcation of c=1 CFT to the two c=1/2 CFT lines (Ashkin-Teller)
3. crossing of c=1 CFT (TLL) with c=1/2 CFT (Schulz multicriticalpoint)

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S=1 BLBQ chain with bond-alternation
A. Kitazawa and K. N. : Phys. Rev. B 59,pp.11358 (1999)

1D S=1 Bilinear-Biquadratic model with bond-alternation
Ĥ =L()∑(1 − (−1)j δ) cos θŜj Ŝj+1 + sin θ(Ŝj Ŝj+1 )2(1)j=1

1. θ = −π/4 and δ = 0: Takhtajan-Babujian (TB) point
Bethe Ansatz solvable, massless, c=3/2 Conformal field theory(CFT) or SU(2) level 2 Wess-Zumino-Witten (WZW) model
2. θ < −π/4 and δ = 0:
dimer order, two-fold degenerate
3. θ > −π/4:
there are two c=1 CFT critical lines

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S=1 BLBQ chain with bond-alternation
Phase diagram of 1D S=1 BLBQ with bond-alternation

Figure: Phase diagram

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S=1 BLBQ chain with bond-alternation

Figure: energies of TBC

Twisted boundary condition(TBC)
yxzŜL+j= −Ŝjx , ŜL+j= −Ŝjy , ŜL+j= Ŝjz .

(2)

A. Kitazawa: J. Phys. A 30, L285 (1997)
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S=1 BLBQ chain with bond-alternation

Figure: TBC crossing lines at various sizes

relevant scaling dimensions: x = 3/8 and x = 1
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S=1 BLBQ chain with bond-alternation

Figure: Eﬀective central charge

Eg (L)
πvc= eg −L
6L2(3)
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Ashkin-Teller and S=1/2 bond-alternating XXZ chain
M. den Nijs, M. Kohmoto and L. P. Kadanoﬀ: Phys. Rev. B 24 pp. 5229 (1981)

2D classical Ashkin-Teller model∑
H=(K1 Si Sj + K2 Ti Tj + K3 Si Sj Ti Tj ),(Si = ±1, Ti = ±1) (4)i,j→ 1D quantum Ashkin-Teller model (transfer matrix of the 2D classicalAshkin-Teller model)
Ĥ =LL∑∑zzzz(σ̂jz σ̂j+1+ τ̂jz τ̂j+1+ λσ̂jz σ̂j+1τ̂jz τ̂j+1)−(σ̂jx + τ̂jx + λσ̂jx τ̂jx )jj(5)(σ̂, τ̂ : Pauli matrices)→ S=1/2 bond-alternating chain
Ĥ =L()∑yxz(1 − (−1)j δ) Ŝjx Ŝj+1+ Ŝjy Ŝj+1+ ∆Ŝjz Ŝj+1(6)j=1
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Ashkin-Teller and S=1/2 bond-alternating XXZ chain

S=1/2 bond-alternating chain
Ĥ =L()∑yxz(1 − (−1)j δ) Ŝjx Ŝj+1+ Ŝjy Ŝj+1+ ∆Ŝjz Ŝj+1
j=1▶

Berezinskii-Kosterlitz-Thouless (BKT) transition on δ = 0 line
▶ scaling dimensions at δ = 0, ∆ = 1 point: x = 2 and x = 1/2▶

numerically diﬃcult to caluculate multicritical lines.

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S=1/2 bond-alternating XXZ chain
S. Moriya and K.N.: J.P.S.J. Vol. 89,093001 (2020)

Phase diagram of S=1/2 bond-alternating chain
Ĥ =L()∑yxz(1 − (−1)j δ) Ŝjx Ŝj+1+ Ŝjy Ŝj+1+ ∆Ŝjz Ŝj+1j=112 / 36

S=1/2 bond-alternating XXZ chainĤ = βL/2 (∑y yx xzz
Ŝ2jŜ2j+1 + Ŝ2jŜ2j+1 + ∆Ŝ2jŜ2j+1)

j=1+L/2 (∑yyxxzzŜ2j−1Ŝ2j+ Ŝ2j−1Ŝ2j+ ∆Ŝ2j−1Ŝ2j)(7)

j=1(β = (1 − δ)/(1 + δ))In ∆ → ∞, β → 0, ∆β = O(1) limit, the main contribution of thezzterm.
Hamiltonian is the ∆Ŝ2j−1Ŝ2j| ↑2j−1 , ↓2j ⟩ = | ↑′j ⟩| ↓2j−1 , ↑2j ⟩ = | ↓′j ⟩(8)

Pertubative Hamiltonian:Ĥ1 = β∆L/2∑zzŜ2j+Ŝ2j+1j=1∑yyxx
Ŝ2j−1+ Ŝ2j−1Ŝ2jŜ2j)j=1

L/2= β∆L/2 (∑)
1 ∑( +−−+Ŝ2j−1 Ŝ2j+ Ŝ2j−1Ŝ2j2L/2zz
Ŝ2jŜ2j+1+(9)13 / 36

S=1/2 bond-alternating XXZ chain

1 ′|↑ ⟩2 j+11z
Ŝ2j+1| ↓′j+1 ⟩ = − | ↓′j+1 ⟩21 ′z′Ŝ2j | ↑j ⟩ = | ↑j ⟩21z′
Ŝ2j | ↓j ⟩ = − | ↓′j ⟩2zŜ2j+1| ↑′j+1 ⟩ =(10)and−+Ŝ2j| ↓′j+1 ⟩ = | ↑′j+1 ⟩Ŝ2j−1+−Ŝ2j−1Ŝ2j| ↑′j+1 ⟩ = | ↓′j+1 ⟩(11)

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S=1/2 bond-alternating XXZ chain
In summary, the eﬀective Hamiltonian:Ĥ ′ =L/2 (∑′z−β∆Ŝj′z Ŝj+1+ Ŝj′x)
(12)

j=1

By operating exp(iπIsing (TFI) model:′∑Ĥ = β∆Ŝj′z ), this is equivalent with the Transverse Field

L/2 (∑j=1)′z−Ŝj′z Ŝj+1− γ Ŝj′x ,(γ ≡1)
β∆(13)(TFI model comes from the transfer matrix of the classical 2D Isingmodel).
Note that γ = 1 is a critical point with 2D Ising type.

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TFI model, Kramers-Wannier duality and boundary
condition∑L/2−1Ĥ ′ = −j=1′z′z
Ŝj′z Ŝj+1− g ŜL/2Ŝ1′z − γL/2∑Ŝj′x ,(14)j=1(g = 1:periodic boundary condition (PBC), g = −1:antiperiodic boundarycondition (ABC)).
From duality (using Jordan-Wigner type transformation), one obtain
E0 (L, g = 1, Uπy = −1) = E0 (L, g = −1, Uπy = 1) + 2(γ − 1)

(15)

Thus one can determine the critical point with the crossing
E0 (L, g = 1, Uπy = −1) = E0 (L, g = −1, Uπy = 1)(16)16 / 36

1D BA XXZ model and BCz-axis twisted boundary condition (zTBC):yxzŜL+j= −Ŝjx , ŜL+j= −Ŝjy , ŜL+j= Ŝjz .

(17)

y-axis BC (yTBC):
yxz
ŜL+j= −Ŝjx , ŜL+j= Ŝjy , ŜL+j= −Ŝjz .

(18)

1. PBC,zTBC of the 1D BA XXZ ↔ g = 1 of the 1D TFI
2. yTBC of the 1D BA XXZ ↔ g = −1 of the 1D TFIThus, in 1D BA XXZ,
E0P BC (M = 0, Uπy = −1) = E0yT BC (M = even, Uπy = 1)(19)

E0zT BC (M = 0, Uπy = −1) = E0yT BC (M = even, Uπy = 1)

(20)

or

17 / 361D BA XXZ model and BC

On δ = 0 we can show
E0zT BC (L, M = 0, Uπy = −1) = E0zT BC (L, M = 0, Uπy = 1)

(21)

Thus, at δ = 0 and ∆ = 1 (isotropic) point,E0zT BC (M = 0, Uπy = −1) = E0yT BC (M = even, Uπy = 1)(22)

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S=1/2 bond-alternating XXZ chain

Figure: energies of each BC

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S=1/2 bond-alternating XXZ chain

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S=1/2 bond-alternating XXZ chain

Figure: Comparison between yTBC-PBC and yTBC-zTBC

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S=1 XXZ with single ion anisotropy
S=1 XXZ + single ion anisotropy chain∑()yz
x+ ∆Sjz Sj+1+ D(Sjz )2H=Sjx Sj+1+ Sjy Sj+1(23)jW. Chen, K. Hida and C. Sanctuary. : Phys. Rev. B 67,pp.104401 (2003)
22 / 36Bosonization by Schulz(1986)
H.J.Schulz: Phys. Rev. B Vol. 34 (1986) pp.6372

1. correlation functions in the XY1 phase−⟨Sj+ Sj+r⟩ = C⊥ exp(iπr)|r|−η
−⟨(Sj+ )2 (Sj+r)2 ⟩ = C⊥2 |r|−η2z⟨Sjz Sj+r⟩ = Cz |r|−2 + Dz exp(iπr) exp(−|r|/ξ)

(24)
(25)
(26)

(η2 = 4η and 0 < η ≤ 1/4.)
2. correlation functions in the XY2/nTLL phase−⟨Sj+ Sj+r⟩ = C⊥ exp(iπr) exp(−|r|/ξ ′ )−⟨(Sj+ )2 (Sj+r)2 ⟩ = C⊥2 |r|−η2z⟨Sjz Sj+r⟩ = Cz |r|−2 + Dz exp(iπr)|r|−ηz

(27)
(28)
(29)

(ηz = 1/η2 and 0 < η2 ≤ 1.)

23 / 36Bosonization by Schulz(1986)

1. XY1-Haldane phase boundary: BKT transition
2. Haldane-Neel phase boundary: 2D Ising type Universality class
3. XY2/nTLL-Neel phase boundary: BKT transition
4. Haldane, Neel, XY1 and XY2/nTLL phases should cross at one point“Schulz multicritical point”

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Hidden SU(2) symmetry in S=1/2 XY spin ladder system
and S=1 XY spin chain
A. Kitazawa, K. Hijii and KN: J. Phys. A, Vol. 36 (2003) L351▶

S=1 XY + single ion anisotropy chain∑∑yx
H=JSjx Sj+1+ Sjy Sj+1+D(Sjz )2j▶

(30)jS=1/2 XY quantum spin ladder system∑yyyyxxxx(S1,jS1,j+1+ S1,jS1,j+1+ S2,jS2,j+1+ S2,jS2,j+1)
H = Jlegj∑yyyyxxxx+ Jdia
(S1,jS2,j+1+ S1,jS2,j+1+ S2,jS1,j+1+ S2,jS1,j+1)
j+ Jrung,xy∑∑yyxxzz(S1,jS2,j+ S1,jS2,j) + Jrung,zS1,jS2,j(31)jjThere is a hidden SU(2) symmetry in S=1/2 spin ladder system and S=1
XY chain.
↔ BKT transition (in the case of massless)
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Hidden SU(2) symmetry▶

s̃±j ≡▶

1 ± 2 z1(S ) , s̃j ≡ Sjz
2 j2(32)

Commutation relations±[s̃zj , s̃±k ] = δj,k s̃k(33)−z[s̃+j , s̃k ] = 2δj,k s̃k(34)(∵ [(Sj+ )2 , (Sj− )2 ] = −8(Sjz )3 + 4(2S 2 + 2S − 1)Sjz and(Sjz )3 = Sjz (f orS = 1) )▶

zs̃±j , s̃j satisfy an SU(2) algebra.

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Hidden SU(2) symmetry∑Although the operator j s̃zj commutates with the S=1 XY∑Hamiltonian, the operators j s̃±
j do not commutate with Hamiltonian.
▶

Thus, we do a following nonlocal transformations±j =1 ± 2
1(S ) Uj , szj = Sjz (= s̃zj )22(35)

whereU1 = 1,Uj =

j−1
∏
l=1

(1 − 2(Slz )2)= exp(iπj−1∑Slz )(j > 1) (36)l=1z
The operators s±
j , sj satisfy an SU(2) algebra.
∑L∑L±zz▶ One can prove that s± ≡T
j=1 sj commutate
j=1 sj and sT ≡with S=1 XY chain with open boundary condition.
▶

27 / 36Hidden SU(2) symmetry▶

But S=1 XY model with the periodic boundary condition (PBC)
does not commutate.
▶ Combining the twisted boundary condition (TBC), one can prove
the hidden SU(2) symmetry in the S=1 XY chain.
J ∑ + −+(Sj Sj+1 + Sj− Sj+1)
2 j
L−1

H=+J′ + −ππ(SL S1 exp(∓i STz ) + SL− S1+ exp(±i STz ))222(37)▶

The energy for magnetization M = 4n(n: integer) under PBC isdegenerate with that for magnetization M = 4n + 2 under TBC.
▶ One can also discuss the space inversion and the wave number.

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Universality class of nTLL phasecentral charge c=1 conformal field theory(CFT)

Figure: eﬀective central charge

Region ∆AF ≤ 0: c = 1Region ∆AF > 0: c < 1 and c decreases as size LFig. 9 means that the region ∆AF ≤ 0 belongs to the c=1 CFTuniversality.
▶
▶

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Universality class of nTLL phase
Numerical results for scaling dimensions.
1. ∆E(M = 1, q = π): massive
2. ∆E(M = 2, q = 0), ∆E(M = 4, q = 0): massless
3. ∆E(M = 4, q = 0)/∆E(M = 2, q = 0) = 4,4. ∆E(M = 0, q = π): massless
5. Scaling dimensions are consistent with TLL model (one parameterscaling or TL parameter K).
√∆E0 (L; T BC; m = 0, P = −1)
K=(38)∆E0 (L; P BC; m = 2, P = 1)
6. TL-parameter K is consitent with the perturbative mapping (S=1/2XXZ).
∑()yxz
H=Sjx Sj+1+ Sjy Sj+1+ ∆Sjz Sj+1,(39)jK=πarccos(−∆)(40)

They are consistent with XY2 phase by Schulz (1986).
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Universality class of nTLL phase
1. Exponential quasi-degeneracy between PBC and TBC energies.
E0 (L; T BC; m = 0, P = 1) − E0 (L; P BC; m = 0, P = 1)
= C1 exp(−L/ξ1 )(41)

E0 (L; T BC; m = 2, P = 1) − E0 (L; P BC; m = 2, P = 1)
= C2 exp(−L/ξ2 )(42)
E0 (L; T BC; m = 0, P = −1) − E0 (L; P BC; m = 0, P = −1)= C3 exp(−L/ξ3 )(43)

1.1 numerical calculation.
1.2 perturbative mapping for S=1/2 XXZ chain.

2. The above bebavior is completely diﬀerent from∆(L; T BC) − ∆(L; P BC) ∝ 1/L in S=1/2 XXZ chain and S=1
XY1 phases.

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Universality class of nTLL phase:PBC-TBC

Figure: quasi degeneracy between PBC and TBC (nTLL phase),
ΓF = 0.5, ∆AF = −0.05

Exponential quasi-degenaracy (Fig. 10semi-log)
Correlation lengths are ξ1 = 1.08, ξ2 = 1.13 ξ3 = 1.12, thus
ξ1 ≈ ξ2 ≈ ξ3 .
Coeﬃcients are C1 ≈ C2 ≈ −C3 > 0.
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Universality class of nTLL phase:PBC-TBC
JF=-1.0 JAF=0.1 ∆F=1.0 ΓAF=1.0 ΓF=0.5 ∆AF=0.5-8log(∆+)
log(∆2)log(∆−)-10-12-14-16-18-2068101214LFigure: quasi degeneracy between PBC and TBC (Stripe Neel phase),
ΓF = 0.5, ∆AF = 0.5

Exponential quasi-degenaracy (Fig. 11semi-log)
Correlation lengths are ξ1 = 0.89, ξ2 = 0.94 ξ3 = 0.91, thus
ξ1 ≈ ξ2 ≈ ξ3 .
Coeﬃcients are C1 ≈ C2 ≈ −C3 > 0.
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Universality class of nTLL phase:PBC-TBC (perturbation)

Phase-factor between PBC-TBC: (−1)2 = 134 / 36

Summary
1. Takhtajan-Babujian
2. Ashkin-Teller multicritical point
2.1 yTBC-zTBC method
2.2 Now checking universality class etc.

3. Schulz multicritical point
3.1 Both XY1-Haldane and nTLL(XY2)-Neel phase boundaries can be
determined from the level crossing
M = 0, P = −1 in TBC and M = ±2, P = 0 in PBC
and they are BKT transition.
One can also apply numerically these level cross for the S=1/2 spinladder system without rung inversion symmetry.
3.2 Universality class of the nTLL phase▶
▶

c=1 CFTquasi-degeneracy between TBC and PBC.

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Future problem▶

XY1-nTLL phase boundary
level cross between▶
▶

M = 0, q = π in PBC and M = ±1, q = π in PBC
M = 1 and M = 2 in PBC(tentative)▶

Universality class of the Schulz multicritical point.
c=3/2 CFT? not the SU(2) level 2 WZW.

▶

General spin S?

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