Urs Gerber

Topological lattice actions

We consider lattice field theories with topological actions, which are invariant against small deformations of the fields. Some of these actions have infinite barriers separating different topological sectors. Topological actions do not have the correct classical continuum limit and they cannot be treated using perturbation theory, but they still yield the correct quantum continuum limit. To show this, we present analytic studies of the 1-d O(2) and O(3) model, as well as Monte Carlo simulations of the 2-d O(3) model using topological lattice actions. Some topological actions obey and others violate a lattice Schwarz inequality between the action and the topological charge Q. Irrespective of this, in the 2-d O(3) model the topological susceptibility

Interpretation of topologically restricted measurements in lattice σ-models

{\chi_t} = {{{\left\langle {{Q^2}} \right\rangle }} \left/ {V} \right.}

CP(N−1) quantum field theories with alkaline-earth atoms in optical lattices

is logarithmically divergent in the continuum limit. Still, at non-zero distance the correlator of the topological charge density has a finite continuum limit which is consistent with analytic predictions. Our study shows explicitly that some classically important features of an action are irrelevant for reaching the correct quantum continuum limit.

Berezinskii–Kosterlitz–Thouless transition with a constraint lattice action

We consider models with topological sectors, and difficulties with their Monte Carlo simulation. In particular we are concerned with the situation where a simulation has an extremely long auto-correlation time with respect to the topological charge. Then reliable numerical measurements are possible only within single topological sectors. The challenge is to assemble such restricted measurements to obtain an approximation for the full-fledged result, which corresponds to the correct sampling over the entire set of configurations. Under certain conditions this is possible, and it provides in addition an estimate for the topological susceptibility χt. Moreover, the evaluation of χt might be feasible even from data in just one topological sector, based on the correlation of the topological charge density. Here we present numerical test results for these techniques in the framework of non-linear σ-models.

The constraint effective potential of the staggered magnetization in an antiferromagnet

We propose a cold atom implementation to attain the continuum limit of (1+1)-d CP(N-1) quantum field theories. These theories share important features with (3+1)-d QCD, such as asymptotic freedom and

Extracting Physics from Topologically Frozen Markov Chains

\theta

Measuring the Topological Susceptibility in a Fixed Sector

vacua. Moreover, their continuum limit can be accessed via the mechanism of dimensional reduction. In our scheme, the CP(N-1) degrees of freedom emerge at low energies from a ladder system of SU(N) quantum spins, where the N spin states are embodied by the nuclear Zeeman states of alkaline-earth atoms, trapped in an optical lattice. Based on Monte Carlo results, we establish that the continuum limit can be demonstrated by an atomic quantum simulation by employing the feature of asymptotic freedom. We discuss a protocol for the adiabatic state preparation of the ground state of the system, the real-time evolution of a false

Constraint effective potential of the magnetization in the quantum XY model

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Topological susceptibility from slabs

-vacuum state after a quench, and we propose experiments to unravel the phase diagram at non-zero density.

Combining ordinary and topological finite volume effects for fixed topology simulations

The 2D XY model exhibits an essential phase transition, which was predicted long ago—by Berezinskii, Kosterlitz and Thouless (BKT)—to be driven by the (un)binding of vortex–anti-vortex pairs. This transition has been confirmed for the standard lattice action, and for actions with distinct couplings, in agreement with universality. Here we study a highly unconventional formulation of this model, which belongs to the class of topological lattice actions: it does not have any couplings at all, but just a constraint for the relative angles between nearest neighbour spins. By means of dynamical boundary conditions we measure the helicity modulus , which shows that this formulation performs a BKT phase transition as well. Its finite size effects are amazingly mild, in contrast to other lattice actions. This provides one of the most precise numerical confirmations ever of a BKT transition in this model. On the other hand, up to the lattice sizes that we explored, there are deviations from the spin wave approximation, for instance for the Binder cumulant U4 and for the leading finite size correction to . Finally we observe that the (un)binding mechanism follows the usual pattern, although free vortices do not require any energy in this formulation. Due to this observation, one should reconsider an aspect of the established picture, which estimates the critical temperature based on this energy requirement.