Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

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26 universal aspects of confinement in 2 + 1 dimensions 3.3 universality If the Z N breaking deconfinement transition of an SU(N) gauge theory is first order, a dimensionally reduced spin system can only provide an effective description. The correlation length ξ is finite, so the effect of microscopic physics shows up at some scale. If both systems have a second order transition, however, then the correlation length ξ diverges at criticality. Svetitsky and Yaffe conjectured that the phase transitions should then have the same universal properties [38]. This is explained by the renormalization group (RG). 4 As the correlation length diverges, the system becomes self-similar at all scales. The microscopic features of the system are encoded in irrelevant operators that vanish at criticality, leaving behind the few relevant operators that are responsible for macroscopic physics. These are the operators that are shared by theories in the same universality class. As scale invariance emerges at criticality, models in the same universality class may be thought of as different realizations of the same theory, with a dictionary relating their operators. In the vicinity of criticality, thermodynamic quantities exhibit scaling laws. For a finite temperature transition, the correlation diverges like, ξ = ξ 0 ±|t| −ν + · · · , t = T T c − 1 → 0 ± . (3.28) Other thermodynamic quantities follow suit and diverge with their own critical exponents. For instance, the specific heat C and susceptibility χ scale with exponents, C ∼ |t| −α , χ ∼ |t| −γ , (3.29) while the order parameter, which is the magnetization |M| in a spin system, scales above T c like, |M| ∼ (−t) β , t = T T c − 1 → 0 + . (3.30) The renormalization group predicts that these exponents should be the same for all models in the same universality class, and, moreover, not all independent. Several scaling, and for d < 4, ‘hyperscaling’ relations reduce the number of critical exponents needed to specify a universality class. The amplitudes, on the other hand, depend on microscopic physics. Nevertheless, they are not all independent. RG predicts the universality of several amplitude ratios [52]. Universality is an extremely powerful tool, as it allows one to use the simplest model in a universality class to make quantitative statements about all of its universal partners. Unfortunately, the finite temperature transition of SU(3) in 3 + 1 d falls just short of universality. It is first order, but only weakly so. Reducing the number of colors or dimensions leads theories with second order transitions. SU(2) in 3 + 1 d is in the universality class of the 3d Ising model, while SU(2) and SU(3) in 2 + 1 d are in the universality classes of the corresponding 2d N-state Potts models. Each of these spin systems possess an exact dual. The 3d Ising model is dual to Z 2 gauge theory in 2 + 1 d, while the 2d N-state Potts models are all self-dual. 4 See [52] for a thorough review.
3.3 universality 27 These dualities are tied to the global symmetries and topology of the theories and emerge at criticality for all models in the same universality class. In what follows we focus on gauge theories in 2 + 1 d, which allows us to explore universality for N = 2, 3 and the borderline case N = 4. 3.3.1 Universality of interface free energies Figure 3.5: Center vortices between Z N center sectors in SU(N) gauge theories (left) correspond to spin interfaces (right). The analogy of spins and Polyakov loops becomes exact for systems in the same universality class in the sense that their correlators map to those of same scale invariant theory at criticality. There is then an equivalence between Z N interfaces. Spin ensembles with cyclically shifted boundary conditions, map to gauge ensembles with temporal twist, s(⃗x + Lî) = e i2πn/N s(⃗x), (3.31) P(⃗x + Lî) = e i2πn/N P(⃗x), (3.32) and we can relate the respective free energies F I of spin interfaces and center vortices F k , F I (⃗ k) = − ln Z ( ⃗ k) , and, (3.33) Z (0) F k (⃗ k) = − ln Z k (⃗ k) Z k (0) , (3.34) where the partition functions with cyclically shifted spins are denoted by Z ( ⃗ k) . At criticality, these free energies converge to the same universal value, which depends on the aspect ratio of the lattice but not on the specific model. 5 Real systems are necessarily finite, so the divergence of the correlation length ξ is tempered. In the vicinity of a phase transition the relevant scale is set by the ratio of the box size and the correlation length, assuming that the correlation length is already much larger than the lattice spacing. Generalized couplings with fixed points such as the interface free energies become functions of L/ξ ∼ L 1/ν t [52]. This means that the behavior of vortices in the gauge theory is universal with spin interfaces not only at T c , but also in a window around criticality. Once the physical 5 A argument for the universality of such partition function ratios is given in Ref. [53].
Page 1 and 2: T O P O L O G Y, S Y M M E T RY, A
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Page 10 and 11: 2 introduction While waiting on exp
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4.5 intuitive picture 77 This illus
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4.6 algebraic formulation 79 It fol
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4.6 algebraic formulation 81 Figure
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4.6 algebraic formulation 83 Applyi
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4.7 the punchline 85 this restricti
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88 fractional electric charge and q
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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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T W I S T AWAY F R O M C R I T I C
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twist away from criticality 123 0.1
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130 more on c-periodic boundary con
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L O W- E N E R G Y R E P R E S E N
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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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ibliography 147 [120] Y. Aoki, G. E
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ibliography 149 [151] M. P. Nightin
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C U R R I C U L U M Name: Samuel Ry
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