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

Tensor network approach to ℤ 𝟑 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 ℤ 2 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 ℤ 2 SPT states  The ℤ 2 topological states can be obtained by gauging the ℤ 2 𝑡 𝛽 = 𝜀 𝑡 𝛽 𝑚 𝜀 𝑡 𝛽 𝑣 𝜀 𝑚𝑣 SPT state. M. Levin and Z.-C. Gu, PRB, (2012).  Two phases of the ℤ 2 SPT states: a trivial phase and a non- trivial phase . X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, PRB, (2013). 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 ሚ 𝐵 𝑚𝑚 ′ 𝑠𝑠 ′ 𝑣𝑣 ′ 𝑒𝑒 ′ =  CZX model for ℤ 2 SPT states can be expressed as TNSs. The 𝑡 𝛾 𝜀 𝑚 ′ 𝑒 𝑡 𝛿 𝑡 𝜀 𝜀 𝑠 ′ 𝑒 ′ 𝑡 𝛽 𝜀 𝑣 ′ 𝑠 𝐵 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 𝜀 𝑚𝑣 four ℤ 2 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: 𝐵 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 = 1 for all 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 .  The local tensor for non-trivial SPT state: 𝐵 0011 = 𝐵 0110 = −1, othewise 𝐵 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 = 1 .  The wavefunction can be written as tTr ሚ 𝐵 ⊗ ⋯ ሚ 𝜔 = ෍ 𝐵 |𝑡 1 𝑡 2 ⋯ 〉 , {𝑡 𝛽 }

MPO of trivial SPT: MPOs for ℤ 2 SPT states 4 ⊗2 𝑉 𝑦 = ෑ 𝑌 𝑗  Acting the local symmetry ς site 𝑌 ⊗4 on the physical 𝑗=1 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: 𝐵 0011 = 𝐵 0110 = 𝜇, 𝐵 1100 = 𝐵 1001 = 𝜇 , MPO of nontrivial SPT: 4 4 𝐵 𝑡 𝛽 𝑡 𝛾 𝑡 𝛿 𝑡 𝜀 = 1 , otherwise. ⊗2 ෑ 𝑉 𝑑𝑨𝑦 = ෑ 𝑌 𝑗 𝐷𝑎 𝑗,𝑗+1 When 𝜇 = 1 , it is the fixed point of trivial state, and 𝜇 = 𝑗=1 𝑗=1 − 1 it represents the nontrivial state. C.-Y. Huang and T.-C. Wei, PRB, (2016). 1 0 0 0 𝑌 = 0 1 0 1 0 0 0 , 𝐷𝑎 = 0 0 1 0 1 0 0 0 −1

Tensor network states for ℤ 2 topological states 𝒠: 𝒝: 𝑕𝑏𝑣𝑕𝑗𝑜𝑕 ℤ 2 topological states:  ℤ 2 SPT states Introducing the ℤ 2 domain walls. 1. ‘Integrating out’ the ℤ 2 spins. M. Levin and Z.-C. Gu, PRB, (2012). 2. 𝑕𝑏𝑣𝑕𝑗𝑜𝑕 Toric code/ double semion. Trivial/ Nontrivial SPT 3. 𝑗 𝑓  Tensor 𝒠 𝑚𝑚 ′ 𝑠𝑠 ′ = 𝜀 𝑚+𝑚 ′ −𝑗 𝑓 𝜀 𝑚𝑠 𝜀 𝑚 ′ 𝑠 ′ detects the domian walls between Ψ 𝜇 : adjacent plaquettes.  Tensor 𝒝 𝑚𝑚 ′ 𝑠𝑠 ′ 𝑣𝑣 ′ 𝑒𝑒 ′ = 𝐵 𝑚𝑣 ′ 𝑠 ′ 𝑒 𝜀 𝑣𝑚 𝜀 𝑣 ′ 𝑠 𝜀 𝑠 ′ 𝑒 ′ 𝜀 𝑚 ′ 𝑒 : no physical degrees of freedom because the ℤ 2 spins are integrated out.  The tensor network wavefnctions can be expressed as Ψ 𝜇 = ෍ 𝑢𝑈𝑠(𝒝 ⊗ 𝒠 ⋯ 𝒝 ⊗ 𝒠) |𝑗 1 𝑗 2 ⋯ 〉 {𝑗 𝑓 }  𝜇 = 1 : fixed point of toric code (TC) model.  𝜇 = −1 : fixed point of double semion (DS) model.

MPOs for ℤ 2 topological states  The MPOs for ℤ 2 topological states are the same Toric code : 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). 𝑌 = 0 1 0 , Double semion : 1 1 0 0 0 0 1 0 0 𝐷𝑎 = 0 0 1 0 0 0 0 −1

MPOs for ℤ 2 topological states  The ground state of topological state on the torus is degenerate, other ground state is obtained by inserting MPOs. N. Schuch, I. Cirac, D. Pérez-García, Annals of Physics, (2010). Double semion: |Ψ 𝑑𝑨𝑦 〉 Toric code: |Ψ 𝑦 〉 𝑉 𝑑𝑨𝑦 : 𝑉 𝑦 :

Transfer operator  Transfer operator 𝕌 is defined via the tensor network norm. Ψ Ψ = 𝑈𝑠 𝕌 𝑂 𝑦 . Ψ Ψ :  The correlation length: 𝜊 = −1/ln(𝑒 2 /𝑒 1 ) ; 𝑒 1 : the dominant eigenvalue; 𝑒 2 : the second dominant eigenvalue 𝕌  The wavefunction norm 𝜚 ) 𝑂 𝑦 , Ψ 𝜚 Ψ 𝜚 = 𝑈𝑠 (𝕌 𝜚 𝜚 denotes the transfer operator with MPO insertion, 𝜚 is ‘ 𝑦 ’ 𝕌 𝜚 𝑑𝑨𝑦 : 𝑦 : 𝕌 𝑦 𝕌 𝑑𝑨𝑦 for 𝜇 >0 and ‘ 𝑑𝑨𝑦 ’ for 𝜇 <0.  The transfer operator inherits the MPO symmetry, it has the ℤ 2 ⊗ ℤ 2 symmetry: 𝐽 ⊗ 𝐽, 𝐽 ⊗ 𝑉 𝜚 , 𝑉 𝜚 ⊗ 𝐽, 𝑉 𝜚 ⊗ 𝑉 𝜚 . N. Schuch, D. Poilblanc, J. I. Cirac, and D. Pérez-García, PRL, 111 , (2013).

Phase diagram ℤ 2 topological TNS  The correlation length 𝜊 from several dominant 𝜚 eigenvalues of the complete transfer operator 𝕌 ⊕ 𝕌 𝜚 ( 𝕌 : transfer operator without MPO insertion).  The phase diagram is obatined from 𝜊: For 𝜇 > 1.73 : symmetry breaking (SB) phase; 1. For −1.73 < 𝜇 < 0 : DS phase; 2. For 0 < 𝜇 < 1.73 : TC phase. 3.  Three quantum critical points(QCPs): 𝜇 ≃ ±1.73 and 𝜇 = 0 .  At these QCPs, 𝜊 ∝ 𝑂 𝑧 , 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 {𝑗 𝑓 } 𝜇 𝑜 5 +𝑜 6 |𝑗 1 𝑗 2 ⋯ 〉 , toric code, Ψ(𝜇 > 0) = σ 𝑗 𝑓 𝜇 𝑜 5 +𝑜 6 𝑥 𝑗 1 𝑗 2 ⋯ 𝑗 1 𝑗 2 ⋯ , double semion, 𝑜 5 and 𝑜 6 : totoal number of type-5 and type-6 vertices in configuration |𝑗 1 𝑗 2 ⋯ 〉 . where 𝑥 𝑗 1 𝑗 2 ⋯ = ±1 is a sign factor. 𝐵 0011 = 𝐵 0110 = 𝜇,  𝑥(𝑗 1 𝑗 2 ⋯ ) is cancelled in the wavefunction norm 𝒶 = 𝐵 1100 = 𝐵 1001 = 𝜇 , 〈Ψ|Ψ〉 . So the DS model and TC model will be mapped to the same classical model.

Mapping to calssical statictics model  The partition function is 𝜇 2(𝑜 5 +𝑜 6 ) 𝒶 = Ψ Ψ = ෍ 𝑗 𝑓  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 𝜇 = ± 3 𝜇 = 0 𝑒 𝑗 1  At all QCPs, we find that 𝜗 𝑗 = −ln 𝑒 1 ∝ 𝑂 𝑧 , 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 Calabrese-Cardy formula: Δ 𝑓, 𝑛 = 𝑓 2 𝑆 2 + 𝑛 2 𝑆 2 , 𝑇 = 𝐷𝑦 + const, 4 𝑂 𝑧 1 𝑜𝜌 𝑦 = 3 ln[ 𝜌 sin( 𝑂 𝑧 )] . Where 𝑓 and 𝑛 are integers.

Charge 1 sector: Conformal quantum critical points at 𝜇 = ± 3 𝜚 .  The spectrum of the transfer operator 𝕌 ⊕ 𝕌 𝜚  The two sectors correspond to the untwisted and twisted sectors of 𝑉 1 6 free boson CFT compactified on a ℤ 𝟑 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. Charge -1 sector:  The drawback of transfer operator spectrum: it can not distinguish the 𝜇 = ± 3 QCPs.