Computational Approaches to Quantum Many-Body Problems ISSP , U of Tokyo, July 22, 2019 Monte Carlo Renormalization Flows with Dangerously Irrelevant Operators Anders W Sandvik Boston University and Institute of Physics, Chinese Academy of Sciences, Beijing Hui Shao (Beijing Normal University) Wenan Guo (Beijing Normal University) arXiv:1905.13640
Motivations I. To better understand emergent U(1) symmetry in quantum magnets Antiferromagnet 4 − fold degenerate VBS VBS orderparameter distribution P ( D x , D y ) L = 12 L = 24 max L = 64 0 Emergent U(1) symmetry close to deconfined quantum-critical point - a new length scale 휉 ’ > 휉 - role of 휉 ’ in finite-size scaling (Shao, Guo, Sandvik, Science 2016) II. Better general understanding of dangerously irrelevant perturbations - in classical and quantum systems - how to best analyze them in Monte Carlo simulations?
q-fold clock perturbation of the 3D XY model X X H = − J cos( Θ i − Θ j ) − h cos q Θ i h ij i i q = 6 Dangerously irrelevant perturbation - irrelevant at T c , relevant for T<T c ξ 0 ∝ ( g − g c ) � ν 0 - correlation length and emergent U(1) length ξ ∝ ( g − g c ) − ν ν 0 > ν Jose, Kadanoff, Kirkpatrick, Nelson, PRB 1977 Clock models Okubo et al, PRB 2015 Fixed points: P = paramagnet X = 3D XY critical point Y = XY symmetry breaking Q = Z q symmetry breaking Cross-over from XY ordering to Z q ordering at length scale 흃 ’ q RG flows can be observed in MC simulations “phenomenological renormalization”
MC simulations of classical 3D clock model X X H = − J cos( Θ i − Θ j ) − h cos q Θ i (soft clock model) h ij i i X H = − J cos( Θ i − Θ j ) q clock angles (hard clock model) h ij i q = 6 Standard order parameter (m x , m y ) N N m y = 1 m x = 1 X X → global angle θ sin( Θ i ) cos( Θ i ) N N i =1 i =1 Probability distribution P(m x , m y ) shows cross-over from U(1) to Z q for T<T c Can be quantified with “angular order parameter”: Z 2 π φ q = d θ cos( q θ ) P ( θ ) 0 흋 q > 0 only if q-fold anisotropy Finite-size scaling of 흋 q can be used to extract length scale 휉 ’ > 휉 and associated scaling dimension y q Lou, Balents, Sandvik, PRL 2007
Angular order parameter 흋 q reflects the dangerously irrelevant field Relevant field accessed through the Binder cumulant: U m = 2 � h m 4 i h m 2 i 2 MC RG flows in the plane (U m , 휑 q ) Okubo et al. (PRB 2015) 1 1 (a) 0.9 3D, q=6 Z 6 L = 2 , 3 , . . . 0.8 0 . 8 0 . 06 0.7 T = 1 . 0 T = T c 0.6 0 . 04 0 . 6 T = 3 . 0 0.5 φ 6 0 . 02 0.4 L=32 0 . 4 0.3 L=48 0 L=64 0.2 0 . 8 0 . 9 1 L=96 0 . 2 L=128 0.1 G XY NG 0 0 -0.1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 -4 -2 0 2 4 6 8 10 -10 U Entire RG flow can be explained by The exponent 휈 ’ can be directly phenomenological scaling function with two relevant arguments: extracted from 휑 q when it is large � q = L y q Φ ( tL 1 / ν , tL 1 / ν 0 - follows from scaling function q )
Conventional finite-size scaling (general) A ∝ t κ t ∝ T − T c Quantity with thermodynamic-limit critical form A ( t, L ) = L − κ / ν f ( tL 1 / ν , λ 1 L − ω 1 , λ 2 L − ω 2 , . . . ) Finite-size scaling form: Take infinite-size limit: A ( t, L → ∞ ) ∝ L − κ / ν ( tL 1 / ν ) κ + . . . ∝ t κ + . . . Example: 2D Ising model, squared order parameter h m 2 i / | t | 2 β ( T < T c ) β = 1 / 8 T > T c ν = 1 h m 2 i / L − 2 2 L − 2 β / ν f ( tL 1 / ν ) ∝ L − 2 T c /J = √ ln( 2 + 1) ∝ | t | 2 β L − 2 β / ν ( tL 1 / ν ) x ∝ L − 2 ≈ 2 . 269 x = 2( β − ν ) = − 7 / 4
Relevant and irrelevant perturbations i m i = hM ( ≡ hNm = hL d m ) H = H 0 + h P RG description of effects of hM at a critical point. Free energy density: f s ( t, h, L ) = L � d F s ( tL 1 / ν , hL y ). � h s y � d h s f s / hL y � d Taylor expand at t=0: � to leading order, p y = d � ∆ → h e f s = h h m i / hL � ∆ ; From Hamiltonian: Irrelevant perturbation if y<0 . How about dangerously irrelevant? L increases i ale ξ 0 / t � ν 0 h ξ / | t | � ν . (let t = T c − T now) Proposal ξ , ξ 0 < L ) to describe all scaling regimes ( L < ξ , ξ 0 , ξ < L < ξ 0 , f s ( t, h, L ) = L � d F s ( tL 1 / ν , tL 1 / ν 0 , hL y , λ L � ω ) , Two relevant fields tuned by same parameter - first introduced for deconfined criticality (Shao, Guo, Sandvik, Science 2016) - Here: detailed tests and new insights for classical clock models Relationship between 휈 ’ and y?
Scaling at T c (XY critical point) 1 Z 6 φ q = L y q Φ ( tL 1 / ν , tL 1 / ν 0 , hL y q , λ L − ω ) 0 . 8 0 . 06 T = 1 . 0 ∝ L y q (1 + hL y q + λ L − ω + . . . ) T = T c 0 . 04 0 . 6 T = 3 . 0 φ 6 0 . 02 0 . 4 0 0 . 8 0 . 9 1 0 . 2 G XY NG -2 10 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 U φ q Scaling dimensions for q = 4 , 5 , 6 q=4, h=1 q 4 5 6 -4 q=5, hard 10 y q q=5, h=5 Ref. [8] -0.2 -1.5 -3.0 q=5, h=2 Ref. [11] -0.114 -1.16 -2.29 q=6, hard Refs. [10, 14] -0.108(6) -1.25 -2.5 This work -0.114(2) -1.27(1) -2.55(6) 10 100 L [8] Oshikawa, PRB 2000 Normally the scaling dimension is [10] Okubo et al., PRB 2015 extracted from the corresponding [11] Leonard & Delamotte, PRL 2015 [14] Hasenbusch & Vicari, PRB 2011 correlation function in the h=0 model - noisy because fast decay
φ q = L y q Φ ( tL 1 / ν , tL 1 / ν 0 , hL y q , λ L − ω ) 1 Distance to the XY critical point Z 6 0 . 8 0 . 06 T = 1 . 0 � q / L y q (1 + tL 1 / ν ) T = T c 0 . 04 0 . 6 T = 3 . 0 φ 6 ) = U XY + tL 1 / ν + L � ω , U ( tL 1 / ν , L − ω ) 0 . 02 0 . 4 0 q 0 . 8 0 . 9 1 Distance: d 1 = φ 2 q + ( U − U XY ) 2 0 . 2 G XY NG q q ( tL 1 / ν + L � ω ) 2 + L 2 y q (1 + tL 1 / ν ) 2 . 0 d 1 / ! ⌧ | y 6 | , d 1 is dom 0 0 . 2 0 . 4 0 . 6 0 . 8 1 U ; d 1 / tL 1 / ν + L � ω . Since ! ⌧ | y 6 | → / Predicted minimum distance and corresponding system size 1 1 / ν + ω = t 0 . 39(2) , ω 1 / ν + ω = t � 0 . 412(4) , L 1 / t � D 1 / t L , e ⌫ = 0 . 6717(1) Known exponent: 5 15 4 e ! = 0 . 94(3), “Effective 휔 ”: 4 2 2 3 10 D 1 × 10 d 1 × 10 L 1 3 Fits to data give exponents: 2 2 0 . 372(1) [ D 1 ] , − 0 . 404(4) [ L 1 ] 5 (a) (b) 10 20 0.001 0.01 (no scaling corrections included) t L
φ q = L y q Φ ( tL 1 / ν , tL 1 / ν 0 , hL y q , λ L − ω ) 1 Z 6 Minimum value of 흋 q 0 . 8 0 . 06 T = 1 . 0 tL 1 / ν , tL 1 / ν 0 ⌧ 1 , tL 1 / ν 0 ⌧ tL 1 / ν T = T c 0 . 04 0 . 6 T = 3 . 0 φ 6 0 . 02 0 . 4 � q / L y q (1 + tL 1 / ν ) 0 0 . 8 0 . 9 1 0 . 2 G XY NG Minimize as function of L at fixed t: 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 D 2 ∝ t − y 6 ν = t 1 . 71(4) , L 2 ∝ t − ν = t 0 . 6717(1) U Used exponents: e ⌫ = 0 . 6717(1) y 6 = − 2 . 55(6) 6 8 16 4 3 3 14 From MC fits D 2 × 10 d 2 × 10 L 2 4 2 12 1 . 88(2) [ D 2 ] , − 0 . 60(3) [ L 2 ] 10 (a) (b) 0 10 20 0.05 0.1 t L
Relationship between ힶ q ’ and y q � q = L y q Φ ( tL 1 / ν , tL 1 / ν 0 q ) tL 1 / ν 0 still small ( ξ ⌧ L ⌧ ξ 0 ) Consider the case: tL 1 / ν large , Scaling then requires the form � q = L y q ( tL 1 / ν ) a g ( tL 1 / ν 0 q ) , where the exponent a depends on the physics of the clock model - how does 흋 q depend on L at fixed t when 흋 q is still small (g ~ 1) - it should be some power of L; 흋 q ~ L p φ q = L p t ν ( p � y q ) g ( tL 1 / ν 0 q ) this form should apply also when 흋 q → 1 - scaling then demands g(x)=x b for some exponent b � � - all L and t dependence must go away → ν 0 q = ν (1 � y q /p ) = ν (1 + | y q | /p ) , Lou, Balents, Sandvik (PRL 2007): p=3 Okubo et al (PRB 2015), Leonard & Delmotte (PRL 2015): p=2
Determination of the exponent ힶ q ’ 1 φ q = L p t ν ( p � y q ) g ( tL 1 / ν 0 Z 6 q ) 0 . 8 0 . 06 T = 1 . 0 T = T c 0 . 04 0 . 6 Can write g() asymptotically as T = 3 . 0 φ 6 0 . 02 0 . 4 g ! ( tL 1 / ν 0 q ) b [1 � k ( tL 1 / ν 0 q )], 0 0 . 8 0 . 9 1 0 . 2 G XY NG which gives 흋 q in this regime: 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 φ q ! 1 � k ( tL 1 / ν 0 U q ) Okubo et al. (PRB 2015) 1 where k() must be dimensionless (a) 0.9 3D, q=6 0.8 Determine 휈 ’ by data collapse 0.7 - or cross of 흋 q (L) with constant 0.6 0.5 d ν 0 6 = 1 . 52(4). Our fit: 0.4 φ 6 =0.6 L=32 60 φ 6 =0.55 L=48 0.3 atisfied if p = φ 6 =0.5 L=64 y 6 = − 2 . 55(6) e ⌫ = 0 . 6717(1) 0.6 0.2 50 L=96 φ 6 L c L=128 0.1 ν 0 ν = 1 + | y q | 40 0 q 0.4 -0.1 (a) (b) p 30 -4 -2 0 2 4 6 8 10 -10 20 40 60 0.25 0.3 0.35 is satisfied with p=2 t L
Near the Nambu-Goldstone point 1 φ q = L p t ν ( p � y q ) g ( tL 1 / ν 0 q ) Z 6 0 . 8 0 . 06 T = 1 . 0 Now g()~1 again T = T c 0 . 04 0 . 6 T = 3 . 0 φ 6 0 . 02 Distance to the NG point: 0 . 4 - Need cumulant: U ( tL 1 / ν ) → 1 0 0 . 8 0 . 9 1 0 . 2 G XY NG 1 � U / ( tL 1 / ν ) � r , 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 What is the exponent r? U - surprisingly, it was not known! - we find r=1.52(2) for 3D XY model (a) q q t 2 r ( R � 1) + t 4( ν 0 q � R ν ) , L 3 / t � ν R , -2 D 3 / 10 1-U where R = ( r +2 ν 0 q ) / ( r +2 ν ) L=16 L=32 3 18 0 . 9(1) -4 L=64 10 18 Using our q=6 exponents: L=128 16 3 2 D 3 × 10 d 3 × 10 L=256 e D 3 / t 0 . 9(1) and L 3 / t � 1 . 07(3) . L 3 16 2 14 -1 0 1 2 10 10 10 10 Data fit gives: 1/ υ 14 12 tL (a) (b) 1 10 20 0.15 0.2 1 . 19(3) [ D 3 ] , − 1 . 14(2) [ L 3 ] t L
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