Correlations and order parameters in infinite matrix product states - PowerPoint PPT Presentation

Correlations and order parameters in infinite matrix product states Ian McCulloch Jason Pillay University of Queensland School of Mathematics and Physics December 6, 2019 Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 1 / 38

Outline Approaches to detecting quantum criticality 1 Cumulants 2 Examples 3 Ising model More complicated examples 4 string order parameters SPT and time reversal BKT transition Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 2 / 38

Approaches to detecting criticality Bipartite fluctuations Bipartite entropy Entanglement spectrum Bipartite fluctuations in quantum numbers (eg J. Stat. Mech. (2014) P10005 from Karyn le Her’s group; Kjall, Phys. Rev. B 87, 235106 (2013)) Transfer matrix spectrum Eigenvalues λ i λ 0 = 1 by construction (normalization condition) 1 Spectrum of correlation lengths ξ i = − ln λ i Or consider ǫ i = − ln λ i = 1 /ξ i – Behaves like an energy scale Choice of which quantity to use as the scaling variable Scaling with respect to the bond dimension Scaling with respect to the correlation length – diverges at critical point Scaling with respect to δ = ǫ 2 − ǫ 1 (or some other combination of ǫ i ) Order parameter and order parameter fluctuations (higher moments) Lots of history behind finite-size scaling, can be modified for finite-entanglement scaling J. Pillay and IPM, arXiv:1906.03833 (to appear in PRB) Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 3 / 38

Inverse correlation length spectrum m=5 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=6 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=13 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=16 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=20 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=25 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=30 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Inverse correlation length spectrum m=40 0.5 0.4 0.3 1/length 0.2 0.1 0 0.98 0.99 1 1.01 1.02 lambda Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 4 / 38

Higher moments It is straight forward to evaluate a local order parameter, eg � M = M i i The first moment of this operator gives the order parameter, � M � = m 1 ( L ) It is also useful to calculate higher moments, eg � M 2 � = m 2 ( L ) or generally � M k � = m k ( L ) These are polynomial functions in the system size L . Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 5 / 38

Cumulant expansions Express the moments m i in terms of the cumulants per site κ j , m 1 ( L ) = κ 1 L 1 L 2 + κ 2 L κ 2 m 2 ( L ) = 1 L 3 + 3 κ 1 κ 2 L 2 + κ 3 L κ 3 m 3 ( L ) = 1 L 4 + 6 κ 2 1 κ 2 L 3 + ( 3 κ 2 2 + 4 κ 1 κ 3 ) L 2 + κ 4 L κ 4 m 4 ( L ) = κ 1 is the order parameter itself κ 2 is the variance (related to the susceptibility) κ 3 is the skewness κ 4 is the kurtosis The cumulants per site κ k are well-defined for a translationally invariant iMPS The moments (and hence cumulants) can be obtained directly as a polynomial-valued expectation value (arXiv:1008.4667) (I don’t know of a good way to calculate the cumulants per site for a finite system!) Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 6 / 38

The cumulants have power-law scaling at a critical point, κ i ∝ m α i Ising model example Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 7 / 38

Scaling functions For finite systems, scale with respect to system size L Control parameter h ≡ λ − λ c λ c Critical exponents Exponent relation | h | − ν ν ξ ∝ ( − h ) β β � M � ∝ σ 2 = � M 2 � − � M � 2 | h | γ γ ∝ Scaling relation 2 β + γ = ν d Note: no fluctuation-dissipation theorem: γ is not related to dM / dH . Finite-size scaling functions L X ( h L 1 /ν ) ξ = L − β/ν M ( h L 1 /ν ) � M � = L γ/ν G ( h L 1 /ν ) σ 2 = U ( h L 1 /ν ) U L = (Binder ratio) Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 8 / 38

Finite entanglement Effective ’system size’ L → m κ New scaling functions that scale with t m κ/ν . Finite-entanglement scaling functions m κ X ( h m κ/ν ) ξ = m − βκ/ν M ( h m κ/ν ) � M � = m γκ/ν G ( h m κ/ν ) σ 2 = U ( h m κ/ν ) = U L Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 9 / 38

Cumulant exponent relation Following V. Privman and M.E. Fisher, Phys. Rev. B 30, 322 (1984) Singular part of the free energy: � C 1 tL 1 /ν , C 2 hL ( β + γ ) /ν � f ≃ L − d Y Finite-entanglement version: � C 1 tm κ/ν , C 2 hL ( β + γ ) κ/ν � f ≃ m − κ d Y Order parameter: κ 1 = − ∂ f ∂ h = C 2 m − βκ/ν Y ′ � C 1 tm κ/ν , C 2 hL ( β + γ ) κ/ν � Higher cumulants: κ n = − ∂ n f ∂ h n = C n 2 m (( n − 2 ) β +( n − 1 ) γ ) κ/ν Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 10 / 38

Cumulant exponent relation This gives relation for all of the exponents (recall κ n ∼ m α n ) α n = [( n − 2 ) β + ( n − 1 ) γ ] κ ν α 1 = − β α 2 = γ This also works for α 0 = − ( 2 β + γ ) κ/ν = d κ which is the correlation length exponent! Alternative self-contained expression α n = − ( n − 2 ) α 1 + ( n − 1 ) α 2 Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 11 / 38

Ising model example Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 12 / 38

Ising model cumulant scaling functions Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 13 / 38

Ising model correlation length scaling function Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 14 / 38

Binder cumulant scaling For finite systems, the Binder cumulant of the order parameter cancels the leading-order finite size effects U L = 1 − � m 4 � 3 � m 2 � 2 0 . 8 1 L = 50 L = 50 L = 100 L = 100 L = 150 0 . 6 L = 200 L = 150 0 . 75 L = 200 iDMRG |� ∆ N L/ 2 �| finite size scaling U L 0 . 4 0 . 5 0 . 425 0 . 2 0 . 4 0 . 25 0 . 215 0 . 216 0 0 0 . 18 0 . 2 0 . 22 0 . 24 0 . 18 0 . 2 0 . 22 0 . 24 Ω Ω The 2-component Bose-Hubbard model, with a linear coupling between components, has an Ising-like transition from immersible (small Ω ) to miscible (large Ω ). Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 15 / 38

Binder Cumulant for iMPS Naively taking the limit L → ∞ for the Binder cumulant doesn’t produce anything useful: if the order parameter κ 1 � = 0 , U L = 1 − � m 4 � L → 2 3 � m 2 � 2 3 L 2 L 2 + k 4 L if κ 1 = 0 , then m 4 ( L ) = 3 k 2 Hence 2 L 2 + k 4 L U L = 1 − 3 k 2 → 0 3 k 2 2 L 2 Finally, a step function that detects whether the order parameter is non-zero Better approach, in the spirit of finite-entanglement scaling: Evaluate the moment polynomial using L ∝ correlation length Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 16 / 38

Choice of scaling parameter L = s ξ is important! s = 2 (top) and s = s ∗ = 5 . 31 (bottom) chosen by optimisation For Ising model the point of intersection doesn’t depend so much on s , but important in other models! Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 17 / 38

String parameters Order parameters do not have to be local Mott insulator string order parameter O 2 | j − i |→∞ � Π j k = i ( − 1 ) n k � P = lim We can write this as a correlation function of ‘kink operators’, p i = Π k < i ( − 1 ) n k − 1 This turns the string order into a 2-point correlation function: O 2 P = | j − i |→∞ � p i p j � lim Or as an order parameter: ( − 1 ) n − 1 � � I � P = p i MPO: P = 0 I i Then O 2 L 2 � P 2 � 1 p = Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 18 / 38

Example: Kondo lattice J Pillay, IPM, Phys. Rev. B 97, 205133 (2018), arXiv:1906.03833 (PRB) Simplified model of topological Kondo insulators suggested by Piers Coleman’s group M. Dzero, K. Sun, V. Galitski, and P . Coleman, Phys. Rev. Lett. 104, 106408 (2010) Kondo lattice with local + topological couplings, n = 1 particle per site � � � � H ⊥ = J ⊥ S j · � H K = J K S j · � π j s j j j where � π is a vector of non-local spins, π j = 1 � p † � j ,α � σ α,β p j ,β 2 α,β 1 p j ,σ = √ ( c j + 1 ,σ − c j − 1 ,σ ) 2 Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 19 / 38

Topological Kondo chain J ⊥ / J K large: topologically trivial insulator, Kondo singlet formation J K / J ⊥ large: SPT protected by spatial reflection Phase transition between these two limits: charge gap vanishes but spin gap remains finite Ian McCulloch Jason Pillay (UQ) iMPS December 6, 2019 20 / 38