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56 universal aspects of confinement in 2 + 1 dimensions Figure 3.22: N t = 4 pseudo-critical couplings obtained from the pairwise intersection of vortex free energies (red), and intersection with the universal value for interface free energies in the 4-state Potts universality class (blue). Clear first order transitions were reported in Ref. [41] for N t = 1, 2, for which the couplings are still near to the bulk crossover. Finite size scaling consistent with second order Potts model was found on their available N t = 3, 4 volumes, however. More recent evidence [39, 40] suggests that the transition is in fact weakly first order. We will check for evidence of this in center vortex ensembles. 3.8.1 Scaling of vortex free energies at criticality To obtain an estimate of the critical lattice coupling β c that assumes a second order transition, but is independent from 4-state Potts model scaling, we have extracted it from pairwise intersections of center vortex free energies F k (1, 0) on lattices with N s ratios of 2 : 1. We obtain the estimate β c = 26.283(9) for N t = 4. It may be compared with the result obtained from intersecting vortex free energies with the exact universal value for interfaces in the 4-state Potts model given in [57]. This yields the consistent value β c = 26.294(2). Both extrapolations are plotted in Fig. 3.22. They are likewise consistent with β c = 26.228(75) from [39], but deviate with significance from β c = 26.251(16) in [40], in which first order scaling was assumed. Under the assumption of second order scaling, the critical exponent ν may be extracted from the slopes of F k (β) at β c , exactly as for SU(2) and SU(3). Fitting to an assumed Ns 1/ν dependence in a one-sigma interval around β c yields ν = 0.59(5) for our N t = 4 volumes. This favors the lower bound ν = 2/3 for the 4-state Potts model along the critical line of symmetric Ashkin-Teller models, which is consistent with the slight underestimations of ν for SU(2) and SU(3). Indeed, center vortex free energies on finite 4 × Ns 2 lattices with up to N s = 80 scale well around criticality under the assumption of 4-state Potts scaling, ν = 2/3 (Fig. 3.23).
3.8 su(4) in 2 + 1 d 57 14 12 10 32x32x4 40x40x4 56x56x4 80x80x4 universal value 10 data (N t = 4) slope 3/2 Fk(x) 8 6 slope d 4 1 2 0 0 0.5 1 1.5 2 2.5 x = ±N s(±t) 2/3 16 20 24 28 32 40 48 56 80 N s Figure 3.23: (left) Scaling of vortex free energies under the assumption of Potts model exponents. (right) Slopes of vortex free energies at β c compared to Potts model scaling ν = 2/3. 3.8.2 Distinguishing first from second order scaling Finite size scaling laws for a second order transition follow from the idea that, as the correlation length ξ diverges at criticality, it is eventually limited by the finite size L of the system and the ratio ξ/L determines the behavior of thermodynamic quantities. If ξ is large but finite at criticality, as for a weakly first order transition, then this ratio becomes effectively infinite for small and intermediate volumes. From this point of view, it is not surprising that we observe second order scaling in intermediate volumes for SU(4) in 2 + 1 d, if the transition is weakly first order. The evidence in Refs. [39, 40] that best supports a first order transition stems from the behavior of the Polyakov loop near criticality. If the SU(4) phase transition is first order, then the system may tunnel between the disordered, confined phase and the four deconfined, ordered center secters that all coexist at T c . This is revealed by peaks in the probability distribution for the Polyakov loop P for each sector. A probability density plot of | 1 V ∑ ⃗x P(⃗x)| (3.101) exhibits tell-tale peaks near zero for the disordered phase and at 1/4 for the Z 4 centor sectors, since the trace for the Polyakov loop is normalized by the number of colors N c = 4. This was observed for various volumes in [39, 40]. There is a slight subtlety. A double peak structure may also be observed on finite volumes near a second order transition, since the volume average of the Polyakov may pass through zero when the system tunnels between deconfined center sectors. We demonstrate this for SU(3) in 2 + 1 d on a 4 × 40 2 lattice in Fig. 3.24 (left). In this case, examination of the Monte Carlo history in Fig. 3.24 (right) indicates that the deconfined sectors do not coexist with a confined phase in which the the volume averaged Polyakov loop is disordered. SU(4) Monte Carlo histories were presented in Ref. [40] for N t = 3, which revealed evidence for phase coexistence near T c and strengthened the case for a first order transition.
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T O P O L O G Y, S Y M M E T RY, A
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To Angela.
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Z U S A M M E N FA S S U N G Wir un
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viii contents 4.6 Algebraic formula
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2 introduction While waiting on exp
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C O N C L U D I N G R E M A R K S 6
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twist away from criticality 123 0.1
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B I B L I O G R A P H Y [1] S. Glas
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ibliography 141 [30] L. von Smekal,
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ibliography 143 [60] J. Engels, E.
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ibliography 145 [89] A. S. Kronfeld
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C U R R I C U L U M Name: Samuel Ry
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