self-dual topological tensor-network state Guang-Ming Zhang ( ) - PowerPoint PPT Presentation

Gapless Coulomb state emerging from a self-dual topological tensor-network state Guang-Ming Zhang ( 张广铭 ) Department of Physics, Tsinghua University, Beijing, China Reference: Guo-Yi Zhu and G. M. Zhang, PRL 122, 176401 (2019); arXiv:1901.10184.

Brief history of Z 2 spin liquids Anderson (1973 & 1987): RVB — > frustrated quantum magnet, high Tc • Kivelson-Rokhsar-Sethena (1987): deconfined soliton in short-range RVB • Wen (1991) parton mean-field + gauge fluctuation: Z 2 deconfined phase • • Kitaev (1997, 2006): exactly solved toric code model & anyon models Moessner & Sondhi (2001) numerics: quantum dimer on triangular lattice • … • arXiv: 9707021

What kind of phase transition occurs out of Z 2 topological phase along the electric-magnetic self-dual path?

An alternative path: wave-function approach • AKLT VBS state for the Haldane phase (1987) • Laughlin wave-function for FQHE (1983) plasma analogy: quantum — > classical Ground state: minimal coding of all topological data

Previous numerical calculation: still unsolved Haegeman, et.al., PRX (2015) Quantum fidelity indicates a continuous phase transition along the e-m dual line? But the Landau theory says no. Haegeman et.al. Nature Comm. (2015) 1D quantum transfer operator spectrum reveals the Higgs (confining) transition.

along the self-dual line

Z 2 orbifold CFT 𝑇 = 𝑕 4𝜌 ∫ |𝛼𝜚| 2 𝑒 2 𝑦 2 𝑕 = 2 𝑆 2 = 𝑆 𝑒𝑣𝑏𝑚 2 SU(2) 1 , KT, q=4 Potts

Generic quantum-classical operators correspondence @𝜄 = 𝜌 4

Quasi-long range order parameter: Correlation function:

Scaling dimensions of generic anyon correlators across the KT transition point ∼ 4/𝑆 2 1 𝑆 2 Τ by exact diagonalizing the quantum transfer operator of Ly=10 cylinder @𝜄 = 𝜌 4

Finite-size spectrum of the anyon sectors at two end points

Numerical verification of the phase transitions away from the e-m duality by measuring quantum fidelity metric

Wave-function path V.S. Hamiltonian path ？ ? Tupitsyn, Kitaev, Prokof’ev and Stamp (10) Zhu, Zhang (19) Along the self-dual line, increasing the bond-dimension of the double-layer tensor network, we expect that: Gapless Coulomb state gets confined — > 1st order line ? quantum KT — > 3D XY ? 2D Ising universality into 3D Ising universality in the Higgs/confining transition

Flow from anisotropic to Lorentz invariant QCP Isakov, Fendley, Ludwig, Trebst & Troyer (2011) Castelnovo, S. Trebst, and M. Troyer, Topological Order and Quantum Criticality (2010)

Wave-function path V.S. Hamiltonian path tuning wave- tuning Hamiltonian: function: Tagliacozzo, Celi & Lewenstein PRX (14) Tupitsyn, Kitaev, GYZ & GMZ (19) Prokof'ev & Stamp (10)

Summary & Outlook • exactly elucidate a novel quantum phase transition along e-m-self-dual path out of Z 2 topological phase • a potential new route to tackle the long standing puzzle • Generalization to Z n or non-abelian topological phase? • Generalization to phase transition between gapped and gapless spin liquid?

How to characterize the quantum phase transitions out of a non-abelian topologically ordered state?

Fibonacci quantum net wave function ： P. P. Fendley ey, , Annals of P Physics, s, 322, 3113 (2008). Lattice duality transformation: 𝐺 = 〈෠ 𝜐|𝜐〉 = 1 1 𝜚 1|1〉 〈 Ƹ 𝜐|1〉 〈෠ 𝜚 1|𝜐 〉 〈 Ƹ 𝜚 −1 The wave function is self-dual: 𝐸 𝜚 2 |𝐸〉 𝜚 −𝑚 𝐸 𝜓 ෡ Ψ = ෍ 𝐸 1 + ℎ𝜏 𝑨 + ෠ 𝜏 𝑨 |Ψ〉 Ψ ℎ, ෠ ℎ = ෑ ℎ ො Deformed Fibonacci net wave function: 𝑓𝑒𝑕𝑓

• The low temperature of Q-state Potts model: 𝑎 𝑞𝑝𝑢𝑢𝑡 (𝐿, 𝑅) = σ 𝑂 𝑓 −𝐿𝑚 𝑂 𝜓 ෡ 𝑂 (𝑅) . 𝑚 𝑂 : total length of stings (domian walls) in 𝑂; 𝐿: inversed temperature. Mapping to a two coupled 𝜚 2 -Potts model: • 𝑂 𝜚 2 𝜓 ෢ 𝑂 ′ 𝜚 2 ෑ 𝑎 = Ψ ℎ, ෠ ℎ Ψ ℎ, ෠ ℎ = ෍ 𝜓 ෡ 𝑋 𝑜 𝑓 𝑜 𝑓 ′ 𝑂,𝑂 ′ 𝑓𝑒𝑕𝑓 (𝑄 2 ) 11 (𝑄 2 ) 1𝜐 / 𝜚 𝑓 −𝐿 2 1 and the matrix 𝑋 = ∝ 𝑓 −2𝐿 2 −𝐿 4 . 𝑓 −𝐿 2 (𝑄 2 ) 𝜐1 / (𝑄 2 ) 𝜐𝜐 /𝜚 𝜚 Some special So l tr transit ition lin lines in in th the parameter space • On the boundary of parameter space ℎ 2 + ෠ ℎ 2 − 2 2𝜚 − 3 ℎ෠ ℎ = 1 , the W matrix is reduced to 𝑓 −𝐿 2 1 𝑋 = 𝑓 −2𝐿 2 , 𝑓 −𝐿 2 then we have 𝐿 4 = 0 , and two Potts models are decoupled. • When 𝑓 −𝐿 2 1 𝑋 = 𝑓 −𝐿 2 , 𝑓 −𝐿 2 the coupled model acquires additional symmetry and two Potts models strongly coupled and become a single 𝜚 4 -statePotts model.

Gu, Levin, Swingle and Wen, 2009, Tensor network representation Buerschaper, Aguado and Vidal, 2009 • Tensor network states for Fibonacci string-net 3𝑢 𝑂 introducing virtual DOF 𝑂 𝜚 2 |𝑂〉 4 𝜓 ෡ Ψ = ෍ 𝜚 𝑂 𝑗𝑘𝑙 Ψ = ෍ tTr ⊗ vertex 𝑈 ⋯ 𝑗𝑘𝑙 ⋯ , 𝛽𝛾𝛿 𝑗𝑘𝑙⋯ 𝑗𝑘𝑙 is uniquely determined by the Fibonacci topological order. the tensor 𝑈 𝛽𝛾𝛿 • Notice the similarity between two wavefunctions, we can derive the tensor network states for the self-dual Fibonacci wavefunction: 𝑂 𝜚 2 𝑂 ֜ 𝜚 −𝑚 𝑂 𝜓 ෡ Ψ = ෍ 𝑂 Ψ = σ 𝑗𝑘𝑙𝑚⋯ tTr ⊗ vertex 𝜚 − 𝑗+𝑘+𝑙+𝑚 ⋯ 𝑗𝑘𝑙𝑚 ⋯ ， 𝑈 4 𝑡𝑟 𝑈 is the 𝑈𝜚 − 3𝑢𝑂 where ෨ 4 .

Phase diagram C=27/20 C=14/15 There is a conformal quantum tri-critical point C with a fractional supersymmetry, which is described by a coset CFT theory with Z 3 parafermions.

Acknowledgements: Guo-Yi Zhu and Wen-Tao Xu at Tsinghua University . Thanks for Your Attention!