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 Z2 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: Z2 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

• ut of Z2 topological phase

along the electric-magnetic self-dual path?

An alternative path: wave-function approach

• Laughlin wave-function for FQHE (1983)
• AKLT VBS state for the Haldane phase (1987)

Ground state: minimal coding of all topological data plasma analogy: quantum —> classical

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

Z2 orbifold CFT

𝑕 = 2 𝑆2 = 𝑆𝑒𝑣𝑏𝑚

2

2 𝑇 = 𝑕 4𝜌 ∫ |𝛼𝜚|2𝑒2𝑦

SU(2)1, KT, q=4 Potts

Generic quantum-classical

• perators correspondence

@𝜄 = 𝜌 4

Quasi-long range order parameter: Correlation function:

Scaling dimensions of generic anyon correlators across the KT transition point

@𝜄 = 𝜌 4 Τ 1 𝑆2 ∼ 4/𝑆2

by exact diagonalizing the quantum transfer operator of Ly=10 cylinder

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

Castelnovo, S. Trebst, and M. Troyer, Topological Order and Quantum Criticality (2010) Isakov, Fendley, Ludwig, Trebst & Troyer (2011)

Wave-function path V.S. Hamiltonian path

GYZ & GMZ (19)

tuning wave- function:

Tupitsyn, Kitaev, Prokof'ev & Stamp (10)

tuning Hamiltonian:

Tagliacozzo, Celi & Lewenstein PRX (14)

Summary & Outlook

• exactly elucidate a novel quantum

phase transition along e-m-self-dual path out of Z2 topological phase

• a potential new route to tackle the

long standing puzzle

• Generalization to Zn or non-abelian

topological phase?

• Generalization to phase transition

between gapped and gapless spin liquid?

How to characterize the quantum phase transitions

• ut of a non-abelian topologically ordered state?

Fibonacci quantum net wave function：

Lattice duality transformation: 𝐺 = 〈෠ 1|1〉 〈 Ƹ 𝜐|1〉 〈෠ 1|𝜐 〉 〈 Ƹ 𝜐|𝜐〉 = 1 𝜚 1 𝜚 𝜚 −1 The wave function is self-dual: Ψ = ෍

𝐸

𝜚−𝑚𝐸 𝜓 ෡

𝐸 𝜚2 |𝐸〉

Deformed Fibonacci net wave function:

Ψ ℎ, ෠ ℎ = ෑ

𝑓𝑒𝑕𝑓

1 + ℎ𝜏𝑨 + ෠ ℎ ො 𝜏𝑨 |Ψ〉

P.

• P. Fendley

ey, , Annals of P Physics, s, 322, 3113 (2008).

• The low temperature of Q-state Potts model:

𝑎𝑞𝑝𝑢𝑢𝑡(𝐿, 𝑅) = σ𝑂 𝑓−𝐿𝑚𝑂𝜓 ෡

𝑂(𝑅).

𝑚𝑂: total length of stings (domian walls) in 𝑂; 𝐿: inversed temperature.

• Mapping to a two coupled 𝜚2-Potts model:

𝑎 = Ψ ℎ, ෠ ℎ Ψ ℎ, ෠ ℎ = ෍

𝑂,𝑂′

𝜓 ෡

𝑂 𝜚2 𝜓෢ 𝑂′ 𝜚2 ෑ 𝑓𝑒𝑕𝑓

𝑋𝑜𝑓𝑜𝑓

′

and the matrix 𝑋 = (𝑄2)11 (𝑄2)1𝜐/ 𝜚 (𝑄2)𝜐1/ 𝜚 (𝑄2)𝜐𝜐/𝜚 ∝ 1 𝑓−𝐿2 𝑓−𝐿2 𝑓−2𝐿2−𝐿4 .

• On the boundary of parameter space ℎ2 + ෠

ℎ2 − 2 2𝜚 − 3 ℎ෠ ℎ = 1, the W matrix is reduced to 𝑋 = 1 𝑓−𝐿2 𝑓−𝐿2 𝑓−2𝐿2 , then we have 𝐿4 = 0, and two Potts models are decoupled.

• When

𝑋 = 1 𝑓−𝐿2 𝑓−𝐿2 𝑓−𝐿2 , the coupled model acquires additional symmetry and two Potts models strongly coupled and become a single 𝜚4-statePotts model.

So Some special l tr transit ition lin lines in in th the parameter space

Tensor network representation

• Tensor network states for Fibonacci string-net

Ψ = ෍

𝑂

𝜚

3𝑢𝑂 4 𝜓 ෡ 𝑂 𝜚2 |𝑂〉 introducing virtual DOF

Ψ = ෍

𝑗𝑘𝑙⋯

tTr ⊗vertex 𝑈

𝛽𝛾𝛿 𝑗𝑘𝑙

⋯ 𝑗𝑘𝑙 ⋯ , the tensor 𝑈

𝛽𝛾𝛿 𝑗𝑘𝑙 is uniquely determined by the Fibonacci topological order.

• Notice the similarity between two wavefunctions, we can derive the tensor

network states for the self-dual Fibonacci wavefunction: Ψ = ෍

𝑂

𝜚−𝑚𝑂 𝜓 ෡

𝑂 𝜚2 𝑂 ֜

Ψ = σ𝑗𝑘𝑙𝑚⋯ tTr ⊗vertex 𝜚−𝑗+𝑘+𝑙+𝑚

4

𝑈

𝑡𝑟

⋯ 𝑗𝑘𝑙𝑚 ⋯ ， where ෨ 𝑈is the 𝑈𝜚−3𝑢𝑂

4 .

Gu, Levin, Swingle and Wen, 2009, Buerschaper, Aguado and Vidal, 2009

C=27/20 C=14/15

Phase diagram

There is a conformal quantum tri-critical point C with a fractional supersymmetry, which is described by a coset CFT theory with Z3 parafermions.

Thanks for Your Attention!

Acknowledgements: Guo-Yi Zhu and Wen-Tao Xu at Tsinghua University.