Positive specific heat critical exponent in three-dimensional three-state random bond Potts model (α >0f for a disordered d dsystem in 3D) ZHONG Fan School of Physics and Engineering Sun Yat-sen University Guangzhou 510275 stszf@mail.sysu.edu.cn International Workshop on Critical Behavior in Lattice Models, April 1-4, 2013 Beijing Normal University, Beijing
Outline Introduction Model Finite-time scaling Method and results Summary
Introduction Continous phase transitions Pure state Harris criterion (Harris 74): whether uncorrelated quenched randomness coupled to local energy density is relevant α >0relevant relevant,α α <0irrelevant irrelevant. Dirty state α < 0? α < 0 (Ma 76, RG) α > 0 (Kinzel & Domany 81, Andelman & Berker 84) crossover exponent of the perturbation associated with the randomness near the pure decoupled fixed point: φ =2y y - d = (2 - dv)/v ) = α/v. At the random fixed point, the replicas are coupled, not applicable.
ν 2/d (Chayes et al. 86) α <0(If an appropriately defined FSS correlation length diverges at a nontrivial value of the disorder with an exponent ν ) ν < 2/d (Pazmandi et al. 97) α > 0 (For systems that t lack self-averaging, intrinsicν finite size ν ) ν 2/d, α < 0 (Aharony98) (to avoid the RG iterations flowing into unphysical regime) The sign of α is not yet conclusive.
First-order phase transitions Quenched disorder coupled to local energy density D 2: continous (rigorous (Aizenman & Wehr, Hui & Berker 89], true for QPTs [Greenblatt et al 09)] D = 3: strong disorder continuous (Uzelac et al 95, Cardy &Jacobsen97] Less well studied. Whether the continuous transition derived from rounding of afirst-order phase transition is different from that from a pure continuous one?
Violation of hyperscaling law α = 2 ( d θ) ν Randomness is relevant at zero-temperature fixed point (Grinstein 76, Fisher 86) Tricritical point of the large-q qrbpmcanbe mapped to RFIM (Cardy & Jacobsen 97)
Model Random-bond Potts model = H Kijδ σσ i j < i, j> σ = 1, 2,3. i K ij three states K, p = 05 0.5 = rk 0, p= 0.5 Simple cubic lattice with periodic boundary conditions
Phase diagram (Yin, Zheng, & Trimper 2005)
Summary of previous results Studied: Site dilutions and bond dilutions, three states and four states, random bond three states and large states (large q) Obtained exponents varied slightly and depending on disorder strength ν > 2/d (FSS, STCD after correction) α >0(Mercaldo et al 06) but claimed to be far away from asymptotic region
Finite-time scaling Finite-size it i scaling Theory Finite size! Numerical simulation L > ξ L < ξ Experiment Bulk behavior Finite-size scaling
Critical Slowing down Diverging orrelation time: t eq ~ ξ z Equilibrium: eq prerequisite for accurate measurement Dynamics: t eq ~ L z dynamic finite-size scaling, good for asymptotic region of long time, large size, and small τ = T-T c But: the simulation time is limited! Analogous situations ξ large but L < ξ FSS t eq large but simulation time limited FTS?
Finite-size scaling Pictorial representation lnm M L β / v FSS L 1 L 2 L L 3 lnm lnll M t β / vz Finite-time scaling? Initial slip 0 t mic 1 1 L 4 ξ ~ τ -ν t eq ~ ξ z R 1 R 1 2 R 1 3 R 1 4 R 5 t eq STCD lnh C H C / vr R βδ / H lnt FTS lnr
Main idea: Need a controllable external time scale: affects evolution, but is still able to probe intrinsic scaling Aim: The system does not evolve by itself to equilibrium, which takes a lot of time, but follows the new controllable finite time scale.
How to realize FTS? Apply an external field H M t H t f t Ht t t H / z 1/ z / z (, τ, ) = β ν ( τ ν, βδ ν ) / eq ~ ν z z z τ ~ ξ, ex ~ ν βδ The external time t ex depends on H. A static H just puts the system off its criticality and a varying H varies t ex too. How to have a controllable t ex? Let H vary linearly with time: H = Rt M t R t f t Rt / 1/ / 1 (,, ) β ν z ( ν z z, βδ ν + τ = τ ) t R ν βδ ν ~ z/( z) ex + just depends on R R introduces a new time scale. Varying R then produces a controllable t ex
Another derivation Scale transformation M t H b M tb b Hb / 1/ / (,, ) β ν z τ = (, τ ν, βδ ν ) valid for time-dependent H r H A new rate exponent: H = Rt R ' = H ' ' ' r H = R t = b z Rt = = b R βδ / ν βδ / ν b H b Rt r = z+ βδ / ν H reflecting the combined effects of H and t. relating static and dynamic exps.
H = Another derivation Scale transformation Rt M t H b M tb b Hb H R ' = b R (, τ, ) = / ( z 1/, τ ν, βδ / ) r -- Choose R and t as independent variables: M t R b M tb b Rb M t R R f tr R + ( r = z+ βδ / ν) β / ν z 1/ ν r (, τ, ) = (, τ, H ) β / / 1/ (,, ) r Hν H H 1( z r, r ν τ = τ ) ~ z/ rh ~ z/( z) t R R ν βδ ν ex H -- Choose R and H as independent variables: M H R b M b Hb Rb r H M(, H, R) R f ( R, HR ) / 1/ / ( τ,, ) = β ν ( τ ν, βδ ν, ) β / r Hν 1/ r Hν βδ / r Hν τ = 2 τ
Comparisons of FTS with FSS and short-time critical dynamics FTS: H = Rt, r = z+ βδ / ν H M H R b M Hb b Rb r H / / 1/ (, τ, ) = β ν ( βδ ν, τ ν, ) / / 1/ 1 β ν βδ ν ν FSS: M( H, τ, L) = b M( Hb, τb, L b) β / 1/ / FTS: (,, ) r Hν H H ( r ν βδ M H R R f R, HR r ν τ = τ ) FSS: FTS: STD: 2 β / ν 1/ ν βδ / ν M( τ, H, L) = L f( τl, HL ) Mt R t m t Rt (, τ, ) = β / νz 1/ νz ( τ, r H / z ) β νz νz (2) 2 / 1/ M (, t τ) = t f( t τ) Similar in forms, apparent differences due to dimensions. But lying behind is a controllable external time scale
Temperature driving β ν M ( t, τ ) = b M ( tb τ = T T = Rt, τb / z 1/ ν r R' = b R r z 1/ τ ' = R't ' = b Rt = b ν τ r = z+ 1/ ν M R b M b Rb c β / ν 1/ ν r ( τ, ) = ( τ, ) M R R f R β / r ν (, ) ( 1/ rν τ = τ ) χ ( τ, R ) R f ( τr ) γ / rν 1/ rν = 1 C( τ, R) = R f ( τr ) α / r ν 1/ r ν 2 )
Finite-time and finite-size scaling M R L R f R L R β / rν 1/ rν 1 1/ r ( τ,, ) = ( τ, ) An effective length scale: R -1/r : effective FSS DRIVING SIMULATIONS: Effective finite scaling of whatever effect one needs/wants to consider Crossovers Renormalization-group theory
Corrections to scaling FTS: STCD: FSS: M R Y R f R YR β / rν 1/ rν ω/ r ( τ,, ) = ( τ, ) M t Y t f t Yt β / νz 1/ νz ω/ z ( τ,, ) = ( τ, ) M L Y L f L YL β / ν 1/ ν ω ( τ,, ) = ( τ, ) r = z +1/ν Small correction-to-scaling exponent ω/r Good news: corrections o to scaling can be ignored Bad news: correction-to-scaling exponent is hard to be estimated
Method and results Monte Carlo renormalization group Two lattice matching: r, ν z ( r = z+ 1/ ν ) Best data collapses: β, β γ, γ α M R R f R β / r ν (, ) ( 1/ rν τ = τ ) χ ( τ, R ) R f ( τr ) γ / rν 1/ rν = 1 C( τ, R) = R f ( τr ) α / r ν 1/ r ν 2
MCRG Two lattice matching β / ν z 1/ ν r M (, t τ, R) = b M( tb, τb, Rb ) z 1/ ν ( t t )' = ( t t ) b, τ ' = τb, R' = Rb c c ' ' nn, Lb p nn, L ps s G ( T, R') G ( T, R ) Large lattice: Lb Small lattice: L = nearest correlation function r Invariance under coarse graining: z = r 1/ ν χτ (, R) = R f ( τr ) γ / rν 1/ r ν 1 An independent check of 1/rv A method to determine T c
Critical temperatures r 0 T c T c (FTS) (STCD) 2.5 0.3166(4) 0.31626(4) 10 01025(4) 0.1025(4) 010265(5) 0.10265(5) Effective interaction approximation (Turban, 1980)
5.075 0.450 0.448 2.853 4.918 0.464 0.438 2.765 4.282 0.530 0.440 2.396 4.259 0.537 0.438 2.387
Confirmation of consistency r 0 1/rv 1/rv T (from T p ) (from MCRG) 2.5 0.447(5) 0.451(7) 5 0.428(6) 0.431(6) 7.5 0.435(12) 0.426(6) 10 0.421(7) 0.419(6) 9.749(5) 15 0.413(8) 0.408(7) 20 0.416(12) 0.406(9) 25 0.390(12) 0.394(8) 30 0.361(16) 0.353(9) 10 0.375(13) 0.339(13) 9.71 T c
Effects of r 0 A fixed probability distribution of the random bonds v z 1/rv R = 5*10-5 1 2 3 4 1 2 3 4 1 2 3 4 10-7.5 0.617 06 0.615 06 0.669 0.636 2.592 2.601 2.385 2.511 0.385 0385 0.385 0385 0.385 0385 0.385 0385 10-10 0.504 0.505 0.549 0.535 2.685 2.679 2.456 2.527 0.425 0.425 0.426 0.425 10-20 0.324 0.327 0.349 0.341 2.512 2.473 2.225 2.312 0.551 0.553 0.563 0.560 1/rv depends on R Only comparison with identical r 0 is reasonable square: 10-20 circle: 10-10 triangle: 10-7.5
No effect of lattice size
Exponets obtained ν <2/d α >0 The exponents vary with r 0 ; but in some range, the variations are slight.
Verification of the exponents by data collapses (a) r 0 = 2.5, (b) 7.5, (c) 10, (d) 20, and (e) 30
The exponents are correct. Corrections to scaling are negligible. Lines are fits according to C(T c ) R α/rν Within some ranges, C(T c )=c 1 c 2 R α/rν, (a negative α) but C curves collapse badly. The same method yielded α < 0 correctly for 3D random-bond Ising model.
Test of scaling laws Tricritical fixed point Crossover Random fixed point Crossover Percolation fixed point? combined with the plateaus in the variations
Critical exponents of the random fixed point (averages over r 0 = 10 to 20) ν = 0.568(22), β = 0.32(4), α = 0.29(6), γ = 1.15(7), z = 2.58(6), 1/ r ν = 0.407(11), r = 4.34(9), β / ν = 0.56(8), γ / ν = 2.03(16).
Comparison r 0 =10 β/vz (d-2β/v)/z z β/v v FTS 0.216(13) 0.764(56) 2.51(4) 0.54(6) 0.554(9) STCD 0.221(1) 0761(14) 0.761(14) 249(5) 2.49(5) 248(5) 2.48(5) 0548(13) 0.548(13) 0.584(18) 0.704(35) no corrn no corrn no corrn corrected no corrn corrected Without corrections, β/vz and (d-2β/v)/z z = 2.49(5) and β/v = 0.550(11) 1/vz = 0.573(16) 20% bigger v = 0.584(18) ~ 0.554(9) Fitting RG r = z +1/ν 1/rv 1/rv=1-z/r r v z C α α = 2-dv
Summary A possible case of α > 0 and ν < 2/d If confirmed What is the mechanism? First-order phase transition of the pure system or/and nonperturbative phenomena FTS can probe the intrinsic i i ν Through FTS, a lot of exponents can be estimated independently and scaling laws tested Scaling laws and asymptotic exponents (One or two scaling laws validated but not asymptotic) Critical exponents of the RBPM in 3D
Acknowledgements Former students Xiong Wanjie Fan Shuangli CNSF
Relevant publications Xiong, W. Zhong, F. & Fan, S. Comput. Phys. Commun. 183, 1161 (2012). Gong, S.; Zhong, F.; Huang, X. & Fan, S. New J. Phys. 12, 043036 (2010). Zhong, F. in Applications of Monte Carlo Method in Science and Engineering. g Chap. 18, p. 469, Intech 2011. Available at http://www.intechopen.com/articles/show/title/finite-time-scalingand-its-applications-to-continuous-phase-transitions
Thank you for your attention!