Disorder-induced rounding of the phase transition. in the large-q-state Potts model. F. Iglói SZFKI - Budapest

Transcription

1 Disorder-induced rounding of the phase transition in the large-q-state Potts model M.T. Mercaldo J-C. Anglès d Auriac Università di Salerno CNRS - Grenoble F. Iglói SZFKI - Budapest

2 Motivations 2. CRITICAL PROPERITES OF DISORDERED SYSTEMS EFFECTS OF DISORDER ON FIRST ORDER PHASE TRANSITIONS RANDOM BOND POTTS MODEL (RBPM), ESPECIALLY IN THE LARGE-q-LIMIT, IS A PERFECT GROUND TO ANALIZE THIS ASPECT

3 Motivations 2. CRITICAL PROPERITES OF DISORDERED SYSTEMS EFFECTS OF DISORDER ON FIRST ORDER PHASE TRANSITIONS RANDOM BOND POTTS MODEL (RBPM), ESPECIALLY IN THE LARGE-q-LIMIT, IS A PERFECT GROUND TO ANALIZE THIS ASPECT Outline POTTS MODEL IN THE RANDOM CLUSTER REPRESENTATION INTRODUCING DISORDER RESULTS AND PERSPECTIVES

4 q-state Potts model 3. Z {σ} q βh({σ}) H = i,j J ij δ(σ i, σ j ) σ i = 0, 1,, q 1 J ij FM random couplings Random cluster representation Z = G E q c(g) ij G ν ij ν ij = e βj ij 1

5 4 q=

6 5 c(g)= = 22 x x x x x x x x x x

7 c(g) βj Π (e e 1) e x x x x x x x x x x

8 . Properties of homogeneous Potts model 7

9 . Properties of homogeneous Potts model q < q c 2 nd order PT q > q c 1 st order PT 7

10 . Properties of homogeneous Potts model 7 q < q c 2 nd order PT q > q c 1 st order PT in 2D: q c = 4 (exact result) in the q limit the PT is strongly 1 st order

11 Properties of homogeneous Potts model 7. q < q c 2 nd order PT q > q c 1 st order PT in 2D: q c = 4 (exact result) in the q limit the PT is strongly 1 st order Systems with Disorder Continuous PT HARRIS criterion α P > 0 disorder is relevant α P < 0 disorder is irrelevant 1 st order PT criterion it is only known that DISORDER will SOFTEN the transition

12 Properties of homogeneous Potts model 7. q < q c 2 nd order PT q > q c 1 st order PT in 2D: q c = 4 (exact result) in the q limit the PT is strongly 1 st order Systems with Disorder Continuous PT HARRIS criterion α P > 0 disorder is relevant α P < 0 disorder is irrelevant 1 st order PT criterion it is only known that DISORDER will SOFTEN the transition Relevant disorder Conventional Random FP (different values of critical exponents) Infinite Randomness FP (IRFP) (disorder grows without limits)

13 q limit 8. T T = T ln q f(t ) f(t ) ln q Z = q c(g) [ q βj ij 1 ] G E ij G q Z = G E q φ(g) φ(g) = c(g) + β ij G J ij

14 q limit 8. T T = T ln q f(t ) f(t ) ln q Z = q c(g) [ q βj ij 1 ] G E ij G q Z = G E q φ(g) φ(g) = c(g) + β ij G J ij Z = n 0 q φ (1 +...) where φ = max G φ(g) and φ = βnf

15 9. All information about the is contained in the OPTIMAL SET G

16 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ

17 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ free energy, internal energy, specif ic heat,...

18 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ free energy, internal energy, specif ic heat,... MAGNETIZATION AND CORRELATION FUNCTIONS ARE OBTAINED FROM THE GEOMETRICAL STRUCTURE OF G

19 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ free energy, internal energy, specif ic heat,... MAGNETIZATION AND CORRELATION FUNCTIONS ARE OBTAINED FROM THE GEOMETRICAL STRUCTURE OF G C(r), average correlation function, is related to the distribution of clusters

20 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ free energy, internal energy, specif ic heat,... MAGNETIZATION AND CORRELATION FUNCTIONS ARE OBTAINED FROM THE GEOMETRICAL STRUCTURE OF G C(r), average correlation function, is related to the distribution of clusters m, magnetization, is the fraction of sites in the infinite cluster

21 9. All information about the is contained in the OPTIMAL SET G THERMAL PROPERTIES ARE CALCULATED FROM φ free energy, internal energy, specif ic heat,... MAGNETIZATION AND CORRELATION FUNCTIONS ARE OBTAINED FROM THE GEOMETRICAL STRUCTURE OF G C(r), average correlation function, is related to the distribution of clusters m, magnetization, is the fraction of sites in the infinite cluster ξ, correlation lenght, is the average size of the clusters

22 maximize φ 10.

23 maximize φ 10. One has to f ind the max over the 2 E possible conf iguration!

24 maximize φ 10. One has to f ind the max over the 2 E possible conf iguration! φ is a supermodular function φ(a) + φ(b) φ(a B) + φ(a B) A, B E

25 maximize φ 10. One has to f ind the max over the 2 E possible conf iguration! φ is a supermodular function φ(a) + φ(b) φ(a B) + φ(a B) A, B E theorem of discrete math a combinatorial optimization method to maximize it in polynomial time

26 maximize φ 10. One has to f ind the max over the 2 E possible conf iguration! φ is a supermodular function φ(a) + φ(b) φ(a B) + φ(a B) A, B E theorem of discrete math a combinatorial optimization method to maximize it in polynomial time for φ(g) of the Potts model a specif ic algorithm has been formulated Angles d Auriac et al.jpa35, 973 (2002)

27 maximize φ 10. One has to f ind the max over the 2 E possible conf iguration! φ is a supermodular function φ(a) + φ(b) φ(a B) + φ(a B) A, B E theorem of discrete math a combinatorial optimization method to maximize it in polynomial time for φ(g) of the Potts model a specif ic algorithm has been formulated Angles d Auriac et al.jpa35, 973 (2002) d = 2 L =

28 11 M.T. Mercaldo, J-C. Angle s d Auriac, F. Iglo i

29 12 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.200

30 13 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.1

31 14 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.0

32 15 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.050

33 1 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.042

34 17 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.033

35 18 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.025

36 19 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.01

37 20 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.008

38 21 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 1.000

39 22 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.992

40 23 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.983

41 24 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.97

42 25 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.941

43 2 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.7

44 27 P (J) = 1 ( 2 δ J 1 ) + 12 ( δ J 5 ) T = 0.500

45 28.

46 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT

47 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT IN 3D WEAK DISORDER DOES NOT DISTROY PHASE COEXISTENCE i.e. disorder has to be strong enough to soften the PT into 2 nd order PT

48 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT IN 3D WEAK DISORDER DOES NOT DISTROY PHASE COEXISTENCE i.e. disorder has to be strong enough to soften the PT into 2 nd order PT

49 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT IN 3D WEAK DISORDER DOES NOT DISTROY PHASE COEXISTENCE i.e. disorder has to be strong enough to soften the PT into 2 nd order PT δ 1 2D 2nd order above breaking lenght ordered phase 2nd order below breaking lenght 1 T/d

50 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT IN 3D WEAK DISORDER DOES NOT DISTROY PHASE COEXISTENCE i.e. disorder has to be strong enough to soften the PT into 2 nd order PT δ 1 2D δ 1 3D 2nd order above breaking lenght 2nd order ordered phase 2nd order below breaking lenght ordered phase 1st order above breaking lenght 1st order below breaking lenght 1 T/d 1 T/d

51 28. IN 2D DISORDER DESTROY PHASE COEXISTENCE it softens the 1 st order PT into a 2 nd order PT IN 3D WEAK DISORDER DOES NOT DISTROY PHASE COEXISTENCE i.e. disorder has to be strong enough to soften the PT into 2 nd order PT δ 1 2D δ 1 3D 2nd order above breaking lenght 2nd order ordered phase 2nd order below breaking lenght ordered phase 1st order above breaking lenght 1st order below breaking lenght 1 T/d 1 T/d In a f inite size system weak disorder fluctuation could not be suff icient to break phase coexistence

52 29 Through extreme value statistics one can estimates the breaking length scale L exp[(1/δ) 2 ] [the f inite length scale L at which breaking of phase coexistence take place] strenght of disorder δ = /J

53 29 Through extreme value statistics one can estimates the breaking length scale L exp[(1/δ) 2 ] [the f inite length scale L at which breaking of phase coexistence take place] strenght of disorder δ = /J

54 29 Through extreme value statistics one can estimates the breaking length scale L exp[(1/δ) 2 ] [the f inite length scale L at which breaking of phase coexistence take place] L bimodal distribution L (J/ ) 2 J/ strenght of disorder δ = /J

55 30. Free Energy: F = c(g )T ij G J ij Internal Energy: E = ij G J ij For a given sample E is a piecewise costant function of temperature it shows discontinuities The average over disorder generally smears out discontinuities The behavior of averaged quantities is different for the discrete and the continuous distributions E bimodal distribution Energy per bond E /m L=128 L=4 L=32 Continuous distribution L=1 0.2 L=32 L=4 L=128 L= β β

56 31 a At the critical point the largest cluster of G is a fractal and its mass M L d f d f = d β ν d f = According to scaling theory, cumulative distribution of the mass of the cluster R(M, L) = M τ R(M/L d f ) R 1.8 Square L=25 Square L=128 Square L=4 Square L=32 Hexagonal L=128 Bimodal distribution a) L=32 L=4 L=128 L=25 m(r) m s 3 28 m τ P(m) d f m s 3 b) L R L m/l d f

57 Conclusions 32. RESULTS IN 2D α = 0, β = 3 5, ν = 1 as for the RTIM IRFP! 4 We can argue that the RTIM is the Hamiltonian version of the 2D Ref.: Mercaldo, Anglès d Auriac, Iglói, PRE 9, 041xx (2004);. Anglès d Auriac, Iglói PRL 90, (2003)

58 Conclusions 32. RESULTS IN 2D α = 0, β = 3 5, ν = 1 as for the RTIM IRFP! 4 We can argue that the RTIM is the Hamiltonian version of the 2D Ref.: Mercaldo, Anglès d Auriac, Iglói, PRE 9, 041xx (2004);. Anglès d Auriac, Iglói PRL 90, (2003) QUESTIONS IN 3D is the transition line T c /d = 1 for δ 1? is δ = 1/2 the tricritical point? does the critical line depend on the disorder distribution? Ref.: Anglès d Auriac, Iglói, Mercaldo work in progress

59 Conclusions 32. RESULTS IN 2D α = 0, β = 3 5, ν = 1 as for the RTIM IRFP! 4 We can argue that the RTIM is the Hamiltonian version of the 2D Ref.: Mercaldo, Anglès d Auriac, Iglói, PRE 9, 041xx (2004);. Anglès d Auriac, Iglói PRL 90, (2003) QUESTIONS IN 3D is the transition line T c /d = 1 for δ 1? is δ = 1/2 the tricritical point? does the critical line depend on the disorder distribution? Ref.: Anglès d Auriac, Iglói, Mercaldo work in progress RELATED PROBLEM Critical Properties of Quantum Potts model Ref.: Mercaldo, De Cesare, work in progress