Tricritical points in field theory and statistical mechanics: from - PowerPoint PPT Presentation

Tricritical points in field theory and statistical mechanics: from Potts models to finite density QCD Claudio Bonati 1 1 Istituto Nazionale di Fisica Nucleare, Pisa “New Frontiers in Lattice Gauge Theory”, Florence, 29/08/2012

Outline Tricritical points: general results 3D three states Potts model in external field (2+1)D three states Potts (3+1)D U(1) gauge theory Imaginary chemical potential QCD Conclusions

Tricritical point ◮ The point in which a line of first order transition becomes a line of second order transition. To have a line of 2 nd order transitions we must have two relevant variables and a marginal one. At the tricritical point the third variable becomes relevant. ◮ The point at which three-phase coexistence terminates in an extended parameter space. H L + L − S + H S − S − 1 st H † L τ L τ ( T t , H t ) T S 0 L λ 2 st L λ T N T T

Tricritical points in Landau theory In the simplest case in which the order parameter is a scalar the free energy near a phase transition (small | η | ) can be parametrized as 2( ∇ η ) 2 + µ 2 F = 1 2 η 2 + λη 4 + κη 6 where κ > 0 to ensure stability. ◮ λ > 0, µ 2 = 0 second order phase transition ◮ λ < 0, µ 2 = λ 2 2 κ first order phase transition ◮ λ = 0, µ 2 = 0 tricritical point The upper critical dimension for tricritical points is 3 and the classical critical indices are (up to logarithmic corrections) α = 1 β = 1 ν = 1 γ = 1 δ = 4 η = 0 2 4 2 see e.g. Landau & Lifshitz “Statistical Physics” § 150

Scaling near a tricritical point A tricritical point is an isolated point on a line of first/second order transitions. Where can we see tricritical scaling? L In a finite system of size L we have 3D Ising first order scaling scaling tricritical scaling in a neighborhood of h tric , with the size of the neighborhood going to zero as L → ∞ . It can be shown that tricritical scaling L c ∝ | h − h tric | − 1 in the simplest case. h h tric C. B., M. D’Elia Phys. Rev. D 82 , 114515 (2010).

The 3D three state Potts model in external field The energy is � � H = − β δ ( s i , s j ) − h δ ( s i , s h ) � i , j � i where s i ∈ N , 1 ≤ s i ≤ 3 and s h is the external field direction. For h = 0 the transition is first order. First order transitions are stable ⇒ h ≈ 0 is first order too. For h → + ∞ all spins are completely polarized along s h , no residual symmetry. A critical endpoint is expected for h > 0. F. Karsch, S. Stickan Phys. Lett. B 488 , 319 (2000). For h → −∞ no spin is directed along s h and the system becomes a 3D Ising model. A tricritical point is expected for h < 0. C. B., M. D’Elia Phys. Rev. D 82 , 114515 (2010).

How to search for tricritical points Possible strategies: ◮ estimate the discontinuities of the first order side and look for the point where the discontinuities vanish ◮ use RG invariant observables and look for crossing Observables for the different approaches: ◮ susceptibilities (energy, order parameter), Binder-Challa-Landau cumulant of energy ◮ correlation length, Binder cumulant of the order parameter ν γ α γ/ν α/ν 3 D Ising 0.6301(4) 1 . 2372(5) 0.110(1) ∼ 1 . 963 ∼ 0 . 175 Tricritical 1/2 1 1/2 2 1 1 st Order 1/3 1 1 3 3

Vanishing of the gaps 0.0015 2 ∆ 0.015 B 0.001 0.01 2 B ∆ 0.0005 0.005 0 0 -0.005 -0.004 -0.003 -0.002 -0.001 h � 2 χ max ∼ const + ∆ 2 B = 2 3 − B 4 | min = 1 � ∆ E 4 L 3 3 E

Crossing of the Binder cumulant 2 2 L=40 L=50 L=60 1.8 1.8 L=70 L=80 U 4 at transition 1.6 1.6 1.4 1.4 1.2 1.2 -0.005 -0.005 -0.004 -0.004 -0.003 -0.003 -0.002 -0.002 h U 4 = � ( δ M ) 4 � � ( δ M ) 2 � 2

(2+1)D three states Potts No external field but one dimension is compactified and the lattice extent along this dimension is N t . N t = + ∞ is the 3 D model, first order transition. As far as the correlation length at the transition is � N t we expect first order. N t = 1 is the 2 D model, second order transition. We expect a change in the order of the transition by varying the value of N t . P. de Forcrand, M. Fromm thesis

(2+1)D three states Potts C. B., M. D’Elia in preparation 1.5 1.25 2 1 ∆ 300xB 0.75 0.5 0.25 0 18 19 20 21 22 23 24 25 26 27 28 N t ν γ α γ/ν α/ν 2 D Z 3 5/6 13/9 1/3 26/15 6/15 Z 3 Tricritical 7/12 19/18 5/6 38/21 10/7 1 st Order 1/2 1 1 2 2

(3+1)D U (1) gauge theory N t = ∞ is 4 D U (1) gauge theory, whose transition is first order and we expect first order also for large N t . As far as the correlation length at the transition is � N t we expect first order. N t = 1 is 3 D U (1), whose transition is second order. We expect a change in the order of the transition by varying the value of N t . ν γ α γ/ν α/ν 3 D XY 0.67155(27) 1.3177(5) -0.0146(8) ∼ 1.962 ∼ -0.022 Tricritical 1/2 1 1/2 2 1 1 st Order 1/3 1 1 3 3

(3+1)D U (1) gauge theory, previously confusing results A high precision study of (3+1)D U (1) gauge theory was performed in 2006 for temporal extent N t = 4 , 5 , 6 and spatial N s � 18 with the result that N τ =N s “The exponents are 80 N τ =6 N τ =5 consistent with 3 D N τ =4 40 Gaussian values, but C max not with either first 20 order transitions or the 10 universality class of the 3 D XY model.” 5 8 16 N s B. A. Berg, A. Bazavov Phys. Rev. D 74 , 094502 (2006). First order behaviour was observed before on a lattice 6 × 48 3 but with very low statistics. M. Vettorazzo, P. de Forcrand Phys. Lett. B 604 , 82 (2004).

Explanation of the previous results For N t = 6 the transition is in fact first order, but very large lattices have to be used in order to see first order scaling since we are near the tricritical point. b = 0 . 2778(82) 1.75 c = 0 . 000123(3) 1.5 first order / tricritical 1.25 40 lattice 6 × L 3 lattice 6 × L 3 1 30 0.75 Specific heat fit with a+bx+cx 3 20 0.5 0.25 10 0 10 20 30 40 50 60 L 0 20 30 40 50 60 L

The location of the tricritical point 0.0005 0.0004 0.0003 B 0.0002 0.0001 0 4 5 6 7 8 9 N t C. B., M. D’Elia in preparation.

The Roberge-Weiss transition endpoint We consider QCD at finite density with imaginary quark chemical potential (no sign problem) 1 3 µ B = µ q ≡ i µ I θ = µ I / T It can be shown that Z ( T , µ I ) is a periodic function of θ with period 2 π/ 3 (3 = N c ) and that at θ = (2 k + 1) π/ N c an exact Z 2 symmetry is present. At low temperature this Z 2 symmetry is realized ` a la Wigner, while in the high temperature region it is spontaneously broken. It was proposed that the structure of phase diagram of zero density QCD is determined by the RW endpoints. M. D’Elia, F. Sanfilippo Phys. Rev. D 80 , 111501(R) (2009). P. de Forcrand, O. Philipsen Phys. Rev. Lett. 105 , 152001 (2010).

The order of the RW endpoint for N f = 2 The transition is definitely first order for low and large quark masses but gets weaker at intermediate masses. Since there is a change of symmetry there must be a transition for all mass values. Are first orders becoming weaker or are they turning second orders? 0.006 0.01 2 /4 2 /4 ∆ ∆ 1.2e-05 0.005 B B 0.00025 0.008 0.004 0.0002 0.006 8e-06 0.003 0.00015 0.004 0.0001 0.002 4e-06 0.002 0.001 5e-05 0 0 0 0 0.0 0.5 1.0 1.5 0 0.01 0.02 0.03 0.04 0.05 a m 1 / (a m) Two tricritical points, one at low and one at high mass. C. B., G. Cossu, M. D’Elia, F. Sanfilippo Phys. Rev. D 83 , 054505 (2011).

The “accepted” QCD phase diagram ∞ O (4)? 1 st T Z 2 crossover Z 2 m s 1 st m u = m d ∞ 0 The first order regions shrink as a chemical potential is turned on. P. de Forcrand, O. Philipsen Nucl. Phys. B 673 , 170 (2003). The following results are obtained by using simple staggered fermions with N t = 4.

The extended “accepted” QCD phase diagram We will look for this blue line ( µ /T) 2 ( µ /T) 2 ( µ /T) 2 m s m s m u,d m u,d ❶ ❶ 0 0 0 m u,d m u,d ∞ ∞ ∞ ❂ ❂ ❷ ❷ ❷ ❷ -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 -( π /3) 2 ❸ ❸ ❸ ❸

Surprise! (mu tric (m=0)/T) 2 = 0.55 0 mc1 mc2 m q = 0 . 0025 -0.2 -0.4 First order Crossover (mu/T) 2 -0.6 m q = 0 . 00375 -0.8 -1 ❷ 0 0.05 0.1 0.15 0.2 0.25 2/5 m q C. B, P. de Forcrand, M. D’Elia, O. Philipsen, F. Sanfilippo work in progress

Not really a surprise . . . previous studies supporting this picture M. D’Elia, A. Di Giacomo, C. Pica Phys.Rev. D 72 114510 (2005) It is shown that O (4) critical indices are not compatible with the chiral transition for N f = 2 G. Cossu, M. D’Elia, A. Di Giacomo, C. Pica arXiv:0706.4470 It is shown that first order critical indices are compatible with the chiral transition for N f = 2

“Personal” QCD phase diagram ∞ In this region it can be { 1 st misleading to look at ¯ ψψ : we are near the Z 2 crossover tricritical line and Z 2 γ � m s O (4) ∼ 1 . 977 � ν � γ � 1 st tric = 2 � ν � m u = m d ∞ 0

Conclusions ◮ tricritical points appear in very different physical systems the existence of tricritical points can explain some puzzling ◮ results the zero density QCD phase diagram can still have some ◮ surprise