Berezinskii-Kosterlitz-Thouless Criticality in the q-state Clock - PowerPoint PPT Presentation

Berezinskii-Kosterlitz-Thouless Criticality in the q-state Clock Model Tao Xiang txiang@iphy.ac.cn Institute of Physics Chinese Academy of Sciences

Outline ✓ Brief introduction to the tensor-network renormalization group methods ✓ Critical properties of the q-state clock model

Road Map of Renormalization Group Numerical Renormalization Group Tensor-network RG Kondo impurity 2D Kadanoff Wilson 0D White 1982 DMRG 1D Phase transition and Critical phenomena Quantum field theory Stueckelberg Gell-Mann Low QED 1965 EW 1999 QCD 2004 1950 1970 1990 2010 year

Basic Idea of Renormalization Group Basic Idea of Renormalization Group 𝑶 𝒖𝒑𝒖𝒃𝒎 𝑶≪𝑶 𝒖𝒑𝒖𝒃𝒎  = ෍ 𝒍  | ۧ 𝒃 𝒍 | ۧ 𝒃 𝒍 | ۧ ෍ 𝒍 𝒍=𝟐 𝒍=𝟐 To find a small but optimized basis set to represent accurately a quantum state Scale transformation: refine the wavefunction by local RG transformations

Is Quantum State Renormalizable? 𝑂 total = 2 𝑀 2 𝑶 ≪ 𝑶 𝐮𝐩𝐮𝐛𝐦  = | ۧ 𝒃 𝒍 | ۧ ෍ 𝒍 B 𝒍=𝟐 Area Law of Entanglement entropy L A  𝐦𝐩𝐡 𝑶 S  𝑴 𝑶 ~ 𝟑 𝑴 << 𝟑 𝑴 𝟑 = N total L

How to Determine the Optimized Basis States? Pump-Probe Use a sub-system as a pump to probe the other System Environment part of the system block block Importance is measured by the entanglement or Tr e  −  = H reduced density matrix sys env reduced density matrix

Tensor-Network State Faithful representation of partition functions of classical/quantum models Variational wavefunctions of ground states of quantum lattice models

Example: Tensor-network representation of the Clock Model 𝜄 𝑗 = 2𝜌𝑜 𝐼 = − ෍ cos 𝜄 𝑗 − 𝜄 𝑟 (𝑜 = 0, … , 𝑟 − 1) 𝑘 ⟨𝑗𝑘ۧ q-state clock model = discretized XY-model

Example: Tensor-network representation of the Clock Model 𝐽 𝑛 𝑓 𝑗𝑛𝜄 /𝑟 𝑊 𝜄,𝑛 = 𝜄 𝑗 𝑛 𝜄 𝑘 𝑓 𝛾 cos(𝜄 𝑗 −𝜄 𝑘 ) = 𝑊 ∗ 𝑊 𝑟 𝑓 −𝑗𝑛𝜄 𝑜 𝑓 𝛾 cos 𝜄 𝑜 𝐽 𝑛 = ෍ Fourier transformation 𝑜=1 𝑘 𝑊 ∗ 𝑙 ∝ 𝐽 𝑗 𝐽 𝑘 𝐽 𝑙 𝐽 𝑚 𝜀 mod 𝑗+𝑘−𝑙−𝑚,𝑟 = 𝑗 𝑊 𝜄 𝑗 𝑚

Tensor-network representation in the dual lattice 𝜐 𝜏 1 = 𝜄 1 − 𝜄 4 𝜐 𝑗𝑘𝑙𝑚 = 𝜇 𝑗 𝜇 𝑘 𝜇 𝑙 𝜇 𝑚 𝜀 mod 𝑗+𝑘−𝑙−𝑚,𝑟 𝜏 2 = 𝜄 2 − 𝜄 1 𝜏 3 = 𝜄 3 − 𝜄 2 𝜏 4 = 𝜄 4 − 𝜄 3 𝜇 𝑛 = 𝑓 𝛾 cos 𝜄 𝑛

Tensor-network Methods for Quantum 1D/Classical 2D Systems Thoroughly developed, most accurate quantum many-body computational methods 1. Ground state ✓ Density-matrix renormalization group (DMRG, White 1992) ✓ Simple update, time evolving block decimation (TEBD, Vidal 2004) ✓ Variational minimization of MPS (FBC, PBC) 2. Thermodynamics ✓ Transfer-matrix renormalization group (TMRG, Nishino coworkers/classical 2D 1995, Xiang coworkers/quantum 1D 1996) ✓ Corner transfer matrix renormalization (Nishino et al 1996) ✓ Coarse-graining tensor renormalization (TRG, SRG, HOTRG, HOSRG, TNR, loop-TNR) ✓ Ancilla purification approach (Verstraete et al 2004)

Tensor-network Methods for Quantum 1D/Classical 2D Systems Thoroughly developed, most accurate quantum many-body computational methods 3. Dynamic correlation functions ✓ Lanczos DMRG ✓ Lanczos MPS ✓ Chebyshev MPS ✓ Correction vector method 4. Time-dependent problem ✓ Pace-keeping DMRG ✓ TEBD ✓ Adaptive time-dependent DMRG ✓ Folded transfer matrix method 5. Excitation spectra ✓ MPS ansatz of single-mode approximation

Evolution of Coarse-Graining Tensor-Network Renormalization ✓ Tensor renormalization group (TRG, Levin, Nave, 2007) ✓ Second renormalization group (SRG, Xie et al 2009) ✓ TRG with HOSVD (HOTRG, HOSRG Xie et al 2012) ✓ Tensor network renormalization (TNR, Evenbly, Vidal 2015) ✓ Loop TNR (Yang et al 2016 ) • TNR and loop TNR are more accurate at the critical points • HOTRG and HOSRG can be applied to 2D quantum and 3D classical models

Tensor-network Methods for Quantum 2D/Classical 3D Systems Still under development, already applied to quantum spin/interacting electron models 1. Ground state: based on the PEPS/PESS ansatz 2. Thermodynamics: coarse-graining tensor renormalization 3. Excitations: single-mode approximation Projected Entangled Pair State (PEPS) 𝒏 Physical state 𝑈 𝑦𝑦 ′ 𝑧𝑧 ′ [𝑛 ] = 𝒚 𝒚′ D y' Verstraete & Cirac, cond-mat/0407066 Virtual state

Ground state: Problems need be solved 1. Determination of PEPS/PESS wave function 2. Evaluation of expectation values (high cost) ⟨𝛺 ෠ 𝑃 𝛺ۧ and ⟨𝛺|𝛺ۧ are each a 2D tensor-network

Determination of PEPS/PESS Wave Function ➢ Simple update Jiang, Weng, Xiang, PRL 101, 090603 (2008) Fast and can access large D tensors ➢ Full update Jordan et al PRL 101, 250602 (2008) more accurate than simply update cost high ➢ Variational minimization with automatic differentiation Liao, Liu, Wang, Xiang, PRX 9 , 031041 (2019) most accurate and reliable method cost high

Automatic Differentiation (AD) ➢ a cute technique which computes exact derivatives, whose errors are limits only floating point error ➢ a powerful tool successfully used in deep learning Computation Graph Chain rule of differentiation

TMRG: Fixed Point MPS Method Fixed point MPS equation: Fixed gauge by left and right canonicalization 𝑈 𝑂 𝛺 𝑈|𝛺 = Tr 𝑈 𝑂 = 𝛺 𝛺 𝑂 → ∞

TMRG: Fixed Point MPS Method To determine the local tensor, one needs to solve the following equations:

Outline ✓ Brief introduction to tensor-network renormalization group methods ✓ Critical properties of the q-state clock model

q-state Clock Model 𝜄 𝑗 = 2𝜌𝑜 𝐼 = − ෍ cos 𝜄 𝑗 − 𝜄 𝑟 (𝑜 = 0, … , 𝑟 − 1) 𝑘 ⟨𝑗𝑘ۧ Understanding the nature of topological phase transition without symmetry breaking dislocation 2D melting: I. Halperin and D. R. Nelson, PRL. 41, 121 (1978); Phys. Rev. 8 19, 2457 (1979).

Berezinskii-Kosterlitz-Thouless Transition XY-model 𝐼 = − ෍ cos(𝜄 𝑗 − 𝜄 𝑘 ) ⟨𝑗𝑘ۧ Berezinkii Thouless Kosterlitz T BKT phase: critical T c paramagnetic

Effective Low Energy Theory XY-model Sine-Gordon Model: 𝐼 = − ෍ cos(𝜄 𝑗 − 𝜄 𝑘 ) 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = ⟨𝑗𝑘ۧ Berezinkii Thouless Kosterlitz T BKT phase: critical T c paramagnetic

Scaling Dimension Δ Sine-Gordon Model: Δ cos( 2𝜒) = 𝐿 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = Δ < 2 relevant 𝐿 > 4 free boson K < 4 non-critical 𝛦 = 2 marginal 𝛦 > 2 irrelevant K = 4 T BKT phase: critical T c paramagnetic

Central Charge c Sine-Gordon Model: Δ cos( 2𝜒) = 𝐿 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = Δ < 2 relevant 𝐿 > 4 free boson K < 4 non-critical 𝛦 = 2 marginal 𝑑 = 1 𝑑 = 0 𝛦 > 2 irrelevant K = 4 T BKT phase: critical T c paramagnetic

q-state Clock Model: Large q Limit + 𝑕 2 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2𝜌 ∫ 𝑒 2 𝑠 cos(𝑟 2 𝜄 ） 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = XY Model Clock Model 𝜖 𝑦 𝜒 = −𝜖 𝑧 𝐿𝜄 𝜖 𝑧 𝜒 = 𝜖 𝑦 (𝐿𝜄) 𝜾 is dual to 𝝌 ∶ P. B. Wiegmann, J. Phys. C 11, 1583(1978)

q-state Clock Model: Large q Limit + 𝑕 2 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2𝜌 ∫ 𝑒 2 𝑠 cos(𝑟 2 𝜄 ） 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = Scaling dimension Δ cos( 2𝜒) = 𝐿 Δ < 2 relevant 2 𝛦 = 2 marginal Δ cos(𝑟 2𝜄) = 𝑟 2 𝛦 > 2 irrelevant 2𝐿

q-state Clock Model: Large q Limit + 𝑕 2 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2𝜌 ∫ 𝑒 2 𝑠 cos(𝑟 2 𝜄 ） 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = Δ cos( 2𝜒) > 2 Δ cos( 2𝜒) > 2 Δ cos( 2𝜒) < 2 Δ cos(𝑟 2𝜄) < 2 Δ cos(𝑟 2𝜄) > 2 Δ cos(𝑟 2𝜄) > 2 𝑑 = 0 𝑑 = ? 𝑑 = 0 Δ cos(𝑟 2𝜄) = 2 Δ cos( 2𝜒) = 2 T T c1 T c2 Ferromagnetic BKT phase: critical paramagnetic J. V. Jose, et al, PRB 16,1217(1977)

q-state Clock Model: Self-dual Point + 𝑕 2 2𝜌𝐿 ∫ 𝑒 2 𝑠 ∇𝜒 2 + 𝑕 1 1 2𝜌 ∫ 𝑒 2 𝑠 cos(𝑟 2 𝜄 ） 2𝜌 ∫ 𝑒 2 𝑠 cos( 2 𝜒) 𝑇 = When K = q, g 1 = g 2 , the model is invariant under dual transformation 𝜒 ↔ 𝑟𝜄 At the self-dual point 2𝜒 = Δ 𝑑𝑝𝑡(𝑟 2𝜄) = 𝑟 Δ cos 2 → 𝐿 𝑡𝑒 = 𝑟 The self-dual point is a critical point for 𝒓 ≤ 𝟓 The self-dual point is not a critical point when 𝒓 > 𝟓

q-state Clock Model: Small q Limit 𝐔 𝐝 q c 2𝐦𝐨 −𝟐 (𝟐 + 2 1/2 Ising, Majorana fermion 𝟑) (𝟒/𝟑)𝐦𝐨 −𝟐 (𝟐 + 3 4/5 Z 3 Parafermion 𝟒) 𝐦𝐨 −𝟐 (𝟐 + 4 1 Two copies of Ising 𝟑) Self-dual Point T T c1 Ferromagnetic BKT phase: critical T c T c2 paramagnetic