From doubled Chern–Simons–Maxwell lattice gauge theory to extensions of the toric code

We regularize compact and non-compact Abelian Chern–Simons–Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group R, each local Hilbert space is analogous to the one of a charged “particle” moving in the link-pair group space R2 in a constant “magnetic” background field. In the compact case, the link-pair group space is a torus U(1)2 threaded by k units of quantized “magnetic” flux, with k being the level of the Chern–Simons theory. The holonomies of the torus U(1)2 give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from U(1) to Z(k). The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern–Simons limit of a large “photon” mass, this results in a Z(k)-symmetric variant of Kitaev’s toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle . Non-Abelian U(k) Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.

Interaction of the Trinuclear Triangular Secondary Building Unit [Cu 3 (μ 3 -OH)(μ-pz) 3 ] 2+ with 4,4′-Bipyridine. Structural Characterizations of New Coordination Polymers and Hexanuclear Cu II Clusters. 2°

By reacting 4,4′-bipyridine (bpy) with selected trinuclear triangular CuII complexes, [Cu3(μ3-OH)(μ-pz)3(RCOO)2(LL′)] [pz = pyrazolate anion; R = CH3, CH3CH2, CH2═CH, CH2═C(CH3); L, L′ = Hpz, H2O, MeOH] in MeOH, the substitution of monotopic ligands by ditopic bpy was observed. Depending on the stoichiometric reaction ratios, different compounds were isolated and structurally characterized. One- and two-dimensional coordination polymers (CPs), as well as two hexanuclear CuII clusters were identified. One of the hexanuclear clusters self-assembles into a supramolecular three-dimensional structure, and its crystal packing shows the presence of two intersecting channels, one of which is almost completely occupied by guest bpy, while in the second one guest water molecules are present. This compound also shows a reversible, thermally induced, single-crystal-to-single-crystal transition.

Synthesis and Structural Characterizations of New Coordination Polymers Generated by the Interaction Between the Trinuclear Triangular SBU [Cu 3 (μ 3 -OH)(μ-pz) 3 ] 2+ and 4,4′-Bipyridine. 3°

The reactions of 4,4′-bipyridine with selected trinuclear triangular copper(II) complexes, [Cu3(μ3-OH)(μ-pz)3(RCOO)2Lx], [pz = pyrazolate anion, R = CH3(CH2)n (2 ≤ n ≤ 5); L = H2O, MeOH, EtOH] yielded a series of 1D coordination polymers (CPs) based on the repetition of [Cu3(μ3-OH)(μ-pz)3] secondary building units joined by bipyridine. The CPs were characterized by conventional analytical methods (elemental analyses, ESI-MS, IR spectra) and single crystal XRD determinations. An unprecedented 1D CP, generated through the bipyridine bridging hexanuclear copper clusters moieties, two 1D CPs presenting structural analogies, and two monodimensional tapes having almost exactly superimposable structures, were obtained. In one case, the crystal packing makes evident the presence of small, not-connected pores, accounting for ca. 6% of free cell volume.

Standard model cross-over on the lattice

With the physical Higgs mass the standard model symmetry restoration phase transition is a smooth cross-over. We study the thermodynamics of the cross-over using numerical lattice Monte Carlo simulations of an effective SU(2)×U(1) gauge+Higgs theory, significantly improving on previously published results. We measure the Higgs field expectation value, thermodynamic quantities like pressure, energy density, speed of sound and heat capacity, and screening masses associated with the Higgs and Z fields. While the cross-over is smooth, it is very well defined with a width of only ∼5  GeV. We measure the cross-over temperature from the maximum of the susceptibility of the Higgs condensate, with the result Tc=159.5±1.5  GeV. Outside of the narrow cross-over region the perturbative results agree well with nonperturbative ones.

Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass

We present results on the nucleon scalar, axial, and tensor charges as well as on the momentum fraction, and the helicity and transversity moments. The pion momentum fraction is also presented. The computation of these key observables is carried out using lattice QCD simulations at a physical value of the pion mass. The evaluation is based on gauge configurations generated with two degenerate sea quarks of twisted mass fermions with a clover term. We investigate excited states contributions with the nucleon quantum numbers by analyzing three sink-source time separations. We find that, for the scalar charge, excited states contribute significantly and to a less degree to the nucleon momentum fraction and helicity moment. Our result for the nucleon axial charge agrees with the experimental value. Furthermore, we predict a value of 1.027(62) in the MS¯¯¯¯¯ scheme at 2 GeV for the isovector nucleon tensor charge directly at the physical point. The pion momentum fraction is found to be ⟨x⟩π±u−d=0.214(15)(+12−9) in the MS¯¯¯¯¯ at 2 GeV.

Comparison of different lattice definitions of the topological charge

We present a comparison of different definitions of the topological charge on the lattice, using a small-volume ensemble with 2 flavours of dynamical twisted mass fermions. The investigated definitions are: index of the overlap Dirac operator, spectral projectors, spectral flow of the HermitianWilson- Dirac operator and field theoretic with different kinds of smoothing of gauge fields (HYP and APE smearings, gradient flow, cooling). We also show some results on the topological susceptibility.

Proof theory for lattice-ordered groups

Proof-theoretic methods are developed and exploited to establish properties of the variety of lattice-ordered groups. In particular, a hypersequent calculus with a cut rule is used to provide an alternative syntactic proof of the generation of the variety by the lattice-ordered group of automorphisms of the real number chain. Completeness is also established for an analytic (cut-free) hypersequent calculus using cut elimination and it is proved that the equational theory of the variety is co-NP complete.