Random quantum magnets in d 2 dimensions: Critical behavior and - PowerPoint PPT Presentation

Random quantum magnets in d ≥ 2 dimensions: Critical behavior and entanglement entropy Ferenc Igl´ oi (Budapest) in collaboration with Istv´ an Kov´ acs Phys. Rev. B 83 , 174207 (2011) J. Phys.: Condens. Matter 23 , 404204 (2011) EPL 97 , 67009 (2012) Phys. Rev. B 87 , 024204 (2013) Statistical Physics of Quantum Matter: Taipei, July 28-31, 2013 1

Introduction • Quantum phase transitions – quantum spin glasses: LiHo x Y 1 − x F 4 – takes place at T = 0 • Paradigmatic model: random transverse – due to quantum flactuation Ising model (RTIM) – by varying a quantum control pa- – in 1d exact results, also through rameter strong disorder RG method • Examples ∗ infinite disorder scaling at the critical point – rare-earth magnetic insulators ∗ dynamical (Griffiths-McCoy) sin- – heavy-fermion compounds gularities outside the critical – high-temperature superconductors point – two-dimensional electron gases – in 2d numerical implementation of the SDRG method – quantum magnets: LiHoF 4 ∗ infinite disorder scaling • Disorder often play an important rˇ ole ∗ but contradictionary results – Anderson localization through the quantum cavity ap- proach – many-body localization 2

Aim of the present talk • Improve the numerical algorithm of the • Study the boundary critical behavior at SDRG method surfaces, corners, edges • Study the critical behavior of the RTIM for d > 2 • Study the entanglement entropy and its • Study Erd˝ os-R´ enyi random graphs ( d → ∞ ) singularity at the critical point 3

Random transverse Ising model H = − ∑ � i , j � J ij σ z i σ z j − ∑ i h i σ x i . • box-h disorder – p ( J ) = Θ ( J ) Θ ( 1 − J ) ( Θ ( x ) : Heaviside • J i j couplings step-function) • independent random numbers from the – q ( h ) = 1 Θ ( h ) Θ ( h b − h ) distribution p ( J ) h b • h i transverse fields • independent random numbers from the • fix-h disorder distribution q ( h ) – p ( J ) = Θ ( J ) Θ ( 1 − J ) – q ( h ) = δ ( h − h f ) . Quantum control parameter: θ = log ( h b ) or θ = log ( h f ) . 4

Strong disorder RG approach (Ma, Dasgupta, Hu 1979, D.S. Fisher 1992, F.I. & Monthus 2005) • sort the couplings and transverse fields, Ω = max ( J i , h i ) • eliminate the largest parameter - reduce the number of spins by one • generate new effective parameters between the remaining spins – i and j form a ferromagnetic cluster - aggregation Ω = J i j in an effective field: � h i j = h i h j J ij having a moment: � µ i j = µ i + µ j – site i is decimated out - annihilation Ω = h i J jk = J ij J ik new effective couplings between sites j and k : � h i • repeate the transformation • at the fixed point Ω is reduced to Ω ∗ = 0 . • final result: set of connected clusters with different masses, µ , decimated at different energies, Ω . 5

Exact results in 1D Infinite disorder fixed point (IDFP) moment: µ ∼ [ ln ( Ω 0 / Ω )] φ ∼ L d f • Critical point: log ( J ) = log ( h ) √ √ 5 + 1 x = 3 − 5 – distribution of the effective parame- d f = φψ = d − x , φ = , 2 4 ters is logarithmically broad – asymptotically exact decimation steps • In the Griffiths phase: θ − θ c > 0 – strongly anisotropic scaling – non-singular static behaviour: ξ < ∞ ln Ω L ∼ L ψ , ψ = 1 / 2 – singular dynamical behaviour: Ω ∼ L − z dynamical exponent: z = z ( θ ) – large effective spin clusters χ ( T ) ∼ T − 1 + d / z , C v ( T ) ∼ T d / z size: ξ ∼ | θ − θ c | − ν ν = 2 6

Numerical implementation of the RG procedure for D > 1 • differences with the 1D procedure • problems with the na ¨ ıve implementation – change in the topology – h -decimations induce several new bonds – application of the maximum rule (valid at IDFP) – the lattice transforms to a fully con- nected cluster – slow algorithm: for N sites works in Ω = J ij Ω = h i a O ( N 3 ) time J ad a d j i i • improved algorithm J ij h i – concept of local maxima - which can b c be decimated independently a ˜ a J d – concept of optimal RG trajectory - J ′ along which the time is minimal i J ′ h ′ – filtering out irrelevant bonds - get- ting rid of latent couplings c b J ′ = J ai J bi /h i J ′ = max( J ai , J aj ) – improved algorithm works in h ′ = h i h j /J ij ˜ J = max( J ad , J ai J di /h i ) O ( N log N ) time 7

Bulk critical behavior 10 0 4D 10 -1 • Finite-size critical points - θ c ( S , L ) 10 -1 10 -2 ~ p 10 -3 – two-copies of the same sample ( S 10 -2 10 -4 and S ′ ) are coupled together p -0.2 0 0.2 10 -3 y L=8 12 16 10 -4 24 32 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 S S’ θ c • Finite-size scaling – shift of the mean: � � – continuously increase θ and monitor � θ c − θ c ( L ) � ∼ L − 1 / ν s the clusters, which are built of iden- tical sites in the copies – width of the distribution: – at θ c ( S , L ) the last correlated cluster ∆ θ c ( L ) ∼ L − 1 / ν w disappears, thus for θ > θ c ( S , L ) we are in the paramagnetic phase – numerical estimates: ν s = ν w • Distribution of pseudocritical points like in a conventional random fixed 10 0 10 -1 point 10 -2 10 -1 ~ p 10 -3 10 -2 10 -4 p 8 10 -3 -0.4 -0.2 0 0.2 0.4 y L=16 24 10 -4 32 48 64 96 -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 θ c

Scaling at the critical point • Cluster structure – distribution function: L ( µ ) = L d f ˜ P ( µ L − d f ) P – power-law tail for large µ L − d f = u P ( u ) ∼ u − τ , with τ = 1 + d ˜ d f 0.5 10 0 0.4 0.3 d f /d 10 -2 0.2 0.1 10 -4 0 10 2 10 3 10 4 10 5 10 6 P N correlation (left) and energy (right) 10 -6 clusters 2D 3D • Correlation clusters → magnetization 10 -8 4D – mass: µ = N # of connected sites 1 10 100 1000 µ – typical mass: µ ∼ L d f 9

Energy clusters → dynamical scaling • energy scale: ε L smallest gap associ- • We use: γ L = log ε L ated with the energy cluster • Critical point • (disordered) Griffiths phase – typical value: γ L ∼ L ψ – typical value: γ L ∼ z log ( L ) [ ε L ∼ L − z ] – distr.: log p ( γ ) ≈ − ( d / z ) γ , ( γ ≫ 1 ) γ = ( γ L − γ 0 ) L − ψ – scaling combination ˜ – scaling comb.: ˜ γ = ( γ L − z ln ( L ) − γ 0 – Infinite disorder scaling 10 -1 3D 3D 4D 10 -1 10 -1 10 -2 10 -2 p ~ p 10 -3 10 -3 10 -4 10 -2 10 -4 -10 -5 0 5 10 p 0 10 20 30 40 50 60 70 0 10 20 30 40 50 γ -z ln(L) γ γ L=24 L=8 10 -1 32 12 10 -3 48 16 10 -2 64 24 L=24 ~ 96 32 p L=48 128 48 10 -3 L=96 10 -4 10 -4 0 5 10 15 20 25 30 γ 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 10 ~ ~ γ γ 10

Erd˝ os-R´ enyi (ER) random graphs D → ∞ • Construction • log-energy scaling – N sites ER 1.5 – kN / 2 edges in random positions 10 -1 1.2 w 0.9 – k > 1 random graph is percolating 0.6 10 -2 0.3 • Distribution of the pseudocritical points p 0 10 1 10 2 10 3 10 4 10 5 10 6 N N=2 12 10 -3 N=2 14 N=2 16 N=2 18 ER 10 -1 N=2 20 10 -1 N=2 22 10 -4 10 -2 0 5 10 15 20 25 30 35 40 ~ p γ 10 -3 10 -2 inset: fixed- h +, box- h ⊡ 10 -4 p 10 -3 -5 0 5 10 15 • logarithmically infinite disorder scaling y N=2 10 N=2 12 N=2 14 10 -4 N=2 16 • width of the distribution: N=2 18 W ≈ W 0 + W 1 log ε N 1 2 3 4 5 6 θ c ε = 1 . 3 ( 2 ) 11

Bulk critical parameters 1D 2D 3D 4D ER L max 2048 128 48 4 . 2 × 10 6 2 . 1 × 10 6 5 . 3 × 10 6 4 . 2 × 10 6 N max θ ( b ) 1 . 6784 ( 1 ) 2 . 5305 ( 10 ) 3 . 110 ( 5 ) 2 . 775 ( 2 ) 0 c θ ( f ) − 1 . − 0 . 17034 ( 2 ) − 0 . 07627 ( 2 ) − 0 . 04698 ( 10 ) − 0 . 093 ( 1 ) c d ν w 2 . 2 . 48 ( 6 ) 2 . 90 ( 15 ) 3 . 3 ( 1 ) 7 . ( 2 ) d ν s 2 . 50 ( 6 ) 2 . 96 ( 5 ) 2 . 96 ( 10 ) 5 . ( 1 ) √ 3 − 5 x / d 0 . 491 ( 8 ) 0 . 613 ( 3 ) 0 . 68 ( 3 ) 0 . 81 ( 2 ) 4 ψ / d 1 / 2 0 . 24 ( 1 ) 0 . 15 ( 2 ) 0 . 11 ( 2 ) 0 . ( log ) . Conclusions at this point 1 • Infinite disorder fixed point at any di- x/d mensions ψ 0.8 1/(d ν s ) 1/(d ν w ) • Strong disorder RG approach is asymp- 0.6 totically exact 0.4 • Spin glass and random ferromagnet are in the same universality class 0.2 12 0 0 0.2 0.4 0.6 0.8 1 1/d

Boundary critical behavior • Different finite geometries – interpolation formula m l = A . slab wedge pyramid L x [ sin ( πλ )] x [ cos ( πλ / 2 )] x α λ = l / L α • results in 3D N N L N L L 1.2 10 -1 L L L 1.1 r 1 0.9 • Different local exponents m 0.8 10 -3 . surface edge corner 0 0.2 0.4 0.6 0.8 1 λ • magnetization profiles 10 -5 fixed – from the fixed boundary: box (Fisher-de Gennes) 0 0.2 0.4 0.6 0.8 1 m l ∼ l − x b , x = x b , l ≪ L λ – from the free part slab geometry, inset: ratio with the m l ′ ∼ ( l ′ ) x α b , l ′ = L α − l + 1 ≪ L α interpolation formula x α b = x α − x , α : s , c , e 13