Tensor network approach to topological quantum phase transitions - - PowerPoint PPT Presentation

Tensor network approach to ℤ𝟑 topological quantum phase transitions and their criticalities

Wen-Tao Xu Department of Physics, Tsinghua University, Beijing, China

Reference: W.-T. Xu and G.-M. Zhang, PRB. 98, 165115 (2018).

CAQMP, ISSP, Kashiwa, Japan July 29, 2019

Outline

 Introduction & Motivation  Tensor network wavefunctions for ℤ2 topological states  Transfer operator and phase diagram  Mapping the wavefunction norm to the classical statictics model  Conformal quantum criticality

Introduction & Motivation

 Topological order in 2D bosonic systems (gapped systems): well-established;

Levin & Wen (2005)

 All fixed point wavefunctions for non-chiral topological states have tensor network

state(TNS) representations.

Gu, Levin, Swingle, Wen (2009); Buerschaper, Aguado, Vidal (2009).

 In the tensor network formalism, matrix product operators(MPOs) encode the

topological order.

Williamson, Bultinck, Marien, Sahinoglu, Haegeman, Verstraete (2016).

 Topological phase transitions (gapless systems): not well-understood.  The topological phase transitions: properly deforming the fixed point TNSs.

Tensor network wavefunctions for ℤ2 SPT states

 The ℤ2 topological states can be obtained by gauging the ℤ2

SPT state. M. Levin and Z.-C. Gu, PRB, (2012).

 Two phases of the ℤ2 SPT states: a trivial phase and a non-

trivial phase. X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, PRB, (2013).

 CZX model for ℤ2 SPT states can be expressed as TNSs. The

four ℤ2 spins in a plaquette form GHZ state 0000 + |1111〉.

• X. Chen, Z.-X. Liu and X.-G. Wen, PRB, (2011).

 The local tensor for trivial SPT state:

𝐵𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀 = 1 for all 𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀.

 The local tensor for non-trivial SPT state:

𝐵0011 = 𝐵0110 = −1, othewise 𝐵𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀= 1.

 The wavefunction can be written as

𝜔 = ෍

{𝑡𝛽}

tTr ሚ 𝐵 ⊗ ⋯ ሚ 𝐵 |𝑡1𝑡2 ⋯ 〉 ,

ሚ 𝐵𝑚𝑚′𝑠𝑠′𝑣𝑣′𝑒𝑒′

𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀

= 𝐵𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀𝜀𝑚𝑣

𝑡𝛽𝜀𝑣′𝑠 𝑡𝛾 𝜀𝑚′𝑒 𝑡𝜀 𝜀𝑠′𝑒′ 𝑡𝛿

𝜀𝑚𝑣

𝑡𝛽 = 𝜀𝑡𝛽𝑚𝜀𝑡𝛽𝑣

MPOs for ℤ2 SPT states

 Acting the local symmetry ςsite 𝑌⊗4 on the physical

degrees of is equvialent to acting the MPO on virtual degrees of freedom.

• D. J. Williamson, N. Bultinck, M. Marien, M. B. Sahinoglu, J. Haegeman, and F.

Verstraete, PRB, (2016).

 We can parametrize the tensors such that the fixed point

tensors of two phases can be connected: 𝐵0011 = 𝐵0110 = 𝜇, 𝐵1100 = 𝐵1001 = 𝜇 , 𝐵𝑡𝛽𝑡𝛾𝑡𝛿𝑡𝜀 = 1, otherwise. When 𝜇 = 1, it is the fixed point of trivial state, and 𝜇 = − 1 it represents the nontrivial state.

C.-Y. Huang and T.-C. Wei, PRB, (2016).

𝑌 = 0 1 1 0 , 𝐷𝑎 = 1 1 1 −1 MPO of trivial SPT: 𝑉𝑦 = ෑ

𝑗=1 4

𝑌𝑗

⊗2

MPO of nontrivial SPT: 𝑉𝑑𝑨𝑦 = ෑ

𝑗=1 4

𝑌𝑗

⊗2 ෑ 𝑗=1 4

𝐷𝑎𝑗,𝑗+1

Tensor network states for ℤ2 topological states

 ℤ2 SPT states

𝑕𝑏𝑣𝑕𝑗𝑜𝑕 ℤ2 topological states:

1.

Introducing the ℤ2 domain walls.

2.

‘Integrating out’ the ℤ2 spins. M. Levin and Z.-C. Gu, PRB, (2012).

3.

Trivial/ Nontrivial SPT

𝑕𝑏𝑣𝑕𝑗𝑜𝑕Toric code/ double semion.

 Tensor 𝒠𝑚𝑚′𝑠𝑠′

𝑗𝑓

= 𝜀𝑚+𝑚′−𝑗𝑓𝜀𝑚𝑠𝜀𝑚′𝑠′ detects the domian walls between adjacent plaquettes.

 Tensor 𝒝𝑚𝑚′𝑠𝑠′𝑣𝑣′𝑒𝑒′ = 𝐵𝑚𝑣′𝑠′𝑒𝜀𝑣𝑚𝜀𝑣′𝑠𝜀𝑠′𝑒′𝜀𝑚′𝑒: no physical

degrees of freedom because the ℤ2 spins are integrated out.

 The tensor network wavefnctions can be expressed as

Ψ 𝜇 = ෍

{𝑗𝑓}

𝑢𝑈𝑠(𝒝 ⊗ 𝒠 ⋯ 𝒝 ⊗ 𝒠) |𝑗1𝑗2 ⋯ 〉

 𝜇 = 1: fixed point of toric code (TC) model.  𝜇 = −1: fixed point of double semion (DS) model.

𝒠: 𝒝: Ψ 𝜇 :

MPOs for ℤ2 topological states

 The MPOs for ℤ2 topological states are the same

as those for SPT states.

 The local tensors are invariant under the MPOs

without needs for symmetry actions on physical indices.

• D. J. Williamson, N. Bultinck, M. Marien, M. B. Sahinoglu, J.

Haegeman, and F. Verstraete, PRB, (2016).

Toric code : Double semion :

𝑌 = 0 1 1 0 ,

𝐷𝑎 = 1 1 1 −1

MPOs for ℤ2 topological states

 The ground state of topological state on the torus is degenerate, other ground state is

• btained by inserting MPOs. N. Schuch, I. Cirac, D. Pérez-García, Annals of Physics, (2010).

Toric code: |Ψ𝑦〉 Double semion: |Ψ𝑑𝑨𝑦〉

𝑉𝑦: 𝑉𝑑𝑨𝑦:

Transfer operator

 Transfer operator 𝕌 is defined via the tensor network norm.

Ψ Ψ = 𝑈𝑠 𝕌𝑂𝑦 .

 The correlation length:

𝜊 = −1/ln(𝑒2/𝑒1); 𝑒1: the dominant eigenvalue; 𝑒2: the second dominant eigenvalue

 The wavefunction norm

Ψ𝜚 Ψ𝜚 = 𝑈𝑠 (𝕌𝜚

𝜚)𝑂𝑦 ,

𝕌𝜚

𝜚 denotes the transfer operator with MPO insertion, 𝜚 is ‘𝑦’

for 𝜇>0 and ‘𝑑𝑨𝑦’ for 𝜇<0.

 The transfer operator inherits the MPO symmetry, it has the

ℤ2 ⊗ ℤ2 symmetry: 𝐽 ⊗ 𝐽, 𝐽 ⊗ 𝑉𝜚, 𝑉𝜚 ⊗ 𝐽, 𝑉𝜚 ⊗ 𝑉𝜚 .

• N. Schuch, D. Poilblanc, J. I. Cirac, and D. Pérez-García, PRL, 111, (2013).

𝕌𝑦

𝑦:

𝕌𝑑𝑨𝑦

𝑑𝑨𝑦 :

𝕌

Ψ Ψ :

Phase diagram ℤ2 topological TNS

 The correlation length 𝜊 from several dominant

eigenvalues of the complete transfer operator 𝕌 ⊕ 𝕌𝜚

𝜚

(𝕌: transfer operator without MPO insertion).

 The phase diagram is obatined from 𝜊: 1.

For 𝜇 > 1.73: symmetry breaking (SB) phase;

2.

For −1.73 < 𝜇 < 0: DS phase;

3.

For 0 < 𝜇 < 1.73: TC phase.

 Three quantum critical points(QCPs): 𝜇 ≃ ±1.73 and

𝜇 = 0.

 At these QCPs, 𝜊 ∝ 𝑂𝑧, and it diverges in

thermodynamical limit.

Mapping to classical statictics model

 The norms 〈Ψ|Ψ〉 of ground state wavefunctions

𝑟𝑣𝑏𝑜𝑢𝑣𝑛−𝑑𝑚𝑏𝑡𝑡𝑗𝑑𝑏𝑚 𝑛𝑏𝑞𝑞𝑗𝑜𝑕partition functions of 2D

classical statistical models.

• F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I.Cirac, PRL, (2006).

 The ground states are superpoisitions of domain wall

configurations.

 The domain wall configurations consist of eight types of

vertices.

 So the wavefunction norm can be mapped to classical

eight-vertex model.

Mapping to calssical statictics model

 The ground states:

Ψ(𝜇 > 0) = σ{𝑗𝑓} 𝜇𝑜5+𝑜6 |𝑗1𝑗2 ⋯ 〉, toric code, Ψ(𝜇 > 0) = σ 𝑗𝑓 𝜇𝑜5+𝑜6 𝑥 𝑗1𝑗2 ⋯ 𝑗1𝑗2 ⋯ , double semion, 𝑜5 and 𝑜6: totoal number of type-5 and type-6 vertices in configuration |𝑗1𝑗2 ⋯ 〉. where 𝑥 𝑗1𝑗2 ⋯ = ±1 is a sign factor.

 𝑥(𝑗1𝑗2 ⋯ ) is cancelled in the wavefunction norm 𝒶 =

〈Ψ|Ψ〉. So the DS model and TC model will be mapped to the same classical model. 𝐵0011 = 𝐵0110 = 𝜇, 𝐵1100 = 𝐵1001 = 𝜇 ,

Mapping to calssical statictics model

 The partition function is

𝒶 = Ψ Ψ = ෍

𝑗𝑓

𝜇2(𝑜5+𝑜6)

 From Baxter’s exact solution, we know that 𝜇 = ± 3 is

the critical point, which is consistent with our calculation.

 When 𝜇 = 0, the eight-vertex model becomes the six-

vertex model, which is also in the critical region of six- vertex model.

• R. J. Baxter, Exact solved models in statistics mechanics.

Conformal Quantum Criticality

 At all QCPs, we find that 𝜗𝑗 = −ln

𝑒𝑗 𝑒1 ∝ 1 𝑂𝑧 , the

scaling behavior is the same as that of CFT spectrum.

 The critical points of 2D eight-vertex model are

described by CFTs.

 We determine that central charges 𝑫 = 𝟐 for all QCPs.  C= 1 CFTs are free boson CFTs compactified on a

circle(𝑓 and 𝑛 are integers, 𝑆 is the radius)

 The scaling dimensions are

Δ 𝑓, 𝑛 = 𝑓2 𝑆2 + 𝑛2𝑆2 4 ,

Where 𝑓 and 𝑛 are integers. Calabrese-Cardy formula: 𝑇 = 𝐷𝑦 + const, 𝑦 =

1 3 ln[ 𝑂𝑧 𝜌 sin( 𝑜𝜌 𝑂𝑧)].

𝜇 = 0 𝜇 = ± 3

Conformal quantum critical points at 𝜇 = ± 3

 The spectrum of the transfer operator 𝕌 ⊕ 𝕌𝜚

𝜚.

 The two sectors correspond to the untwisted and twisted sectors of 𝑉 1 6

free boson CFT compactified on a ℤ𝟑 orbifold with the radius 𝑆 = 6.

 The transfer operator spectra for 𝜇 = ± 3 are exactly the same.  The phase transitions at 𝜇 = ± 3 should belong to different universality

classes, because toric code and double semion are different phases.

 The drawback of transfer operator spectrum: it can not distinguish the

𝜇 = ± 3 QCPs.

Charge -1 sector: Charge 1 sector:

Conformal quantum critical point at 𝜇 = 0

 At the QCP 𝜇 = 0, the TNSs have both the MPO symmetries 𝑉𝑦 and

𝑉𝑑𝑨𝑦.

 It truns out there are two other transfer operators, 𝕌𝑑𝑨, 𝕌𝑑𝑨𝑦

𝑦 .

 𝕌 has the symmetry 𝑉𝑦 ⊗ 𝑉𝑦, 𝐽 ⊗ 𝑉𝑑𝑨 and 𝑉𝑦 ⊗ 𝑉𝑑𝑨𝑦.  𝕌𝑑𝑨, 𝕌𝑑𝑨𝑦

𝑦

and 𝕌𝑦

𝑦 have the symmetry 𝐽 ⊗ 𝑉𝑑𝑨, 𝑉𝑦 ⊗ 𝑉𝑑𝑨𝑦 and 𝑉𝑦 ⊗

𝑉𝑦, seperately. 𝕌𝑑𝑨𝑦

𝑦

: 𝕌𝑑𝑨:

Conformal quantum critical point at 𝜇=0

 The spectrum 𝕌of matches the free boson CFT with compactified radius 𝑆 =

8/3.

 The symmetry charges satisfy the relation: 𝑉𝑦 ⊗ 𝑉𝑦 𝐽 ⊗ 𝑉𝑑𝑨 𝑉𝑦 ⊗ 𝑉𝑑𝑨𝑦 −1 𝑓 −1 𝑛 −1 𝑓+𝑛 𝑉𝑦 ⊗ 𝑉𝑦 𝐽 ⊗ 𝑉𝑑𝑨 𝑉𝑦 ⊗ 𝑉𝑑𝑨𝑦

Conformal quantum critical point at 𝜇=0

 Δ also satisfy the formula Δ 𝑓, 𝑛 = 𝑓2/𝑆 + 𝑛2𝑆2/4, where 𝑆 ≃

8/3.

𝕌𝑦

𝑦

𝕌𝑑𝑨 𝕌𝑑𝑨𝑦

𝑦

𝑓 Integers Half-integers Half-integers 𝑛 Half-integers Integers Half-integers Symmetry charges −1 𝑓 −1 𝑛 −𝑗 −1 𝑓+𝑛

𝕌𝑦

𝑦:

𝕌𝑑𝑨: 𝕌𝑑𝑨𝑦

𝑦 :

Understanding the conformal quantum critical points

 It is expected that the QCPs of 2D quantum systems should be described by (2+1)D QFTs.  However, in our cases, the QCPs are described by 2D time-independent CFTs, that is the

equal-time correlation function described to CFT, how to understand it?

 Two scenarios about the topological phase transitions: 1.

Hamiltonian deformation:

①

Described (2+1)D QFT

②

Lorentz invariant with Dynamical critical exponent 𝒜 = 𝟐.

2.

Wavefunction deformation:

①

Rokhsar-Kivelson type conformal QCP, Described 2D CFT

②

Space-time asymmetry with dynamical critical exponent 𝐴 > 𝟐.

③

We believe the conformal QCP will flow to (2+1)D Lorentz invariant QCP under perturbation.

• C. Castelnovo, S. Trebst, and M. Troyer, arXiv,(2009);
• E. Ardonne, P. Fendley, and E. Fradkin, Annals of Physics, (2004).
• S. V. Isakov, P. Fendley, A. W. W. Ludwig, S. Trebst, and M. Troyer, PRB (2011).
• B. Hsu and E. Fradkin, PRB (2013).

Summary

 We construct a one-parameter family of TNS for ℤ2 topological phases.  It incorporates the toric code, double semion and symmetry breaking phases.  By calculating the correlation length and mapping the wavefunction norm to eight-vertex model,

we determine the three QCPs 𝝁 = ± 𝟒 and 𝝁 = 𝟏.

 At 𝝁 = ± 𝟒(QCP between TC/DS and SB), the QCPs are decribed by the 𝑽 𝟐 𝟕 ℤ𝟑 orbifold

CFT (𝑆 = 6).

 At 𝝁 = 𝟏(QCP between TC and DS), the QCP are described by the compactified free boson

CFT with 𝑺 = 𝟗/𝟒, whose scaling dimensions are Δ 𝑓, 𝑛 =

𝑓2 𝑆2 + 𝑛2𝑆2 4 , where 𝑓 and 𝑛 can

be both integers and half-integers.

 The three QCPs are both conformal QCPs.

Acknowledgment