Classical discrete time crystals

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: the discrete time crystal (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator–prey models to Faraday waves and a.c.-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially long timescales. Power-law correlations are present along a first-order line, which terminates at a critical point. We analyse the transition by mapping it to the locked-to-sliding transition of a d.c.-driven charge density wave. Finally, building upon results from the field of probabilistic cellular automata, we conjecture the existence of classical time crystals with true long-range order, where time-translation symmetry is broken out to infinite times

Parton Production Via Vacuum Polarization

We discuss the production mechanism of partons via vacuum polarization during the very early, gluon dominated phase of an ultrarelativistic heavy-ion collision in the framework of the background field method of quantum chromodynamics.Comment: 3 pages, Latex, 3 figures (eps), to be published in JPhysG, SQM2001 proceeding

Renormalization Group Equation and QCD Coupling Constant in the Presence of SU(3) Chromo-Electric Field

We solve renormalization group equation in QCD in the presence of SU(3) constant chromo-electric field E^a with arbitrary color index a=1,2,...8 and find that the QCD coupling constant \alpha_s depends on two independent casimir/gauge invariants C_1=[E^aE^a] and C_2=[d_{abc}E^aE^bE^c]^2 instead of one gauge invariant C_1=[E^aE^a]. The \beta function is derived from the one-loop effective action. This coupling constant may be useful to study hadron formation from color flux tubes/strings at high energy colliders and to study quark-gluon plasma formation at RHIC and LHC.Comment: 13 pages latex, 4 eps figs, Eur. Phys. J.

Magnetic field-tuned Aharonov-Bohm oscillations and evidence for non-Abelian anyons at v=5/2

We show that the resistance of the v=5/2 quantum Hall state, confined to an interferometer, oscillates with magnetic field consistent with an Ising-type non-Abelian state. In three quantum Hall interferometers of different sizes, resistance oscillations at v=7/3 and integer filling factors have the magnetic field period expected if the number of quasiparticles contained within the interferometer changes so as to keep the area and the total charge within the interferometer constant. Under these conditions, an Abelian state such as the (3,3,1) state would show oscillations with the same period as at an integer quantum Hall state. However, in an Ising-type non-Abelian state there would be a rapid oscillation associated with the "even-odd effect" and a slower one associated with the accumulated Abelian phase due to both the Aharonov-Bohm effect and the Abelian part of the quasiparticle braiding statistics. Our measurements at v=5/2 are consistent with the latter.Comment: 10 pages, 8 figures, includes Supplemental Material

Reduction of derived Hochschild functors over commutative algebras and schemes

We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finite type flat maps f: X->Y of noetherian schemes, with f^!(O_Y) in place of D.Comment: 32 pages. Minor changes from previous version. To appear in the Advances in Mathematic