Topological critical slowing down: Variations on a toy model

Numerical simulations of lattice quantum field theories whose continuum counterparts possess classical solutions with nontrivial topology face a severe critical slowing down as the continuum limit is approached. Standard Monte Carlo algorithms develop a loss of ergodicity, with the system remaining frozen in configurations with fixed topology. We analyze the problem in a simple toy model, consisting of the path integral formulation of a quantum mechanical particle constrained to move on a circumference. More specifically, we implement for this toy model various techniques which have been proposed to solve or alleviate the problem for more complex systems, like non-Abelian gauge theories, and compare them both in the regime of low temperature and in that of very high temperature. Among the various techniques, we propose an alternative algorithm which completely solves the freezing problem, but unfortunately is specifically tailored for this particular model and not easily exportable to more complex systems.

Figure 1

Dependence of the autocorrelation time of the topological susceptibility on the (inverse) lattice spacing for the Metropolis update (with $\mathrm{\Delta }=0.5$ and $a{N}_t=2$). The dashed line is the result of a fit with the function $a_{0}exp\left(a_1{N}_t\right)$, from which the values $a_0=0.074\left(10\right)$ and $a_1=0.0290\left(5\right)$ are obtained.

Figure 2

Examples of configurations corresponding to topological charge $Q=0$ (left), $Q=-1$ (center), $Q=2$ (right) on lattices of temporal extent $N_t=500$ and $T=1/2$. Simulation points are connected by shortest distance lines on the spatial circumference.

Figure 3

An example of the tailor move: a piece of the original path (the dotted-line one) is cut and sewed back after reversing it (dashed-line one). As a consequence, a starting path with $Q=0$ is turned into a path with $Q=-1$ and the same action as before.

Figure 4

Dependence of the autocorrelation time of the topological susceptibility on the (inverse) lattice spacing when using the tailor update.

Figure 5

Fit to Eq. (27) to extract the topological susceptibility from subvolume measurements.

Figure 6

Autocorrelation of ${\chi }_s$, defined in Eq. (27), by using the subvolume measures; the case of $x=0.1$ and 0.3 are shown for $\mathrm{\Omega }=0$. The continuous black line is a fit of the form $\tau \propto N_t^3$ to the $x=0.1$ data, while the dashed red line is a fit to the $x=0.3$ data obtained by using the functional form $\tau \propto N_t^2$.

Figure 7

Values of the topological susceptibility obtained with different procedures when open boundary conditions are used: by computing the susceptibility of the topological charge restricted to the slice, by computing the susceptibility of the topological charge restricted to the slice after cooling, by evaluating the integral (28). The horizontal band represents the value obtained from simulation with periodic boundary conditions.

Figure 8

An example of the behavior of ${\int }_0^t\langle q\left(0\right)q\left(t^{\prime }\right)d{t}^{\prime }$ as a function of $t$ for three values of $\mathrm{\Omega }$, with $t$ in physical units, for $1/T=2$ with open boundary conditions and $N_t=400$.

Figure 9

Integrated autocorrelation time of the topological susceptibility obtained from simulations performed with open boundary conditions (with $a{N}_t=2$). The dashed line is a fit of the $\mathrm{\Omega }=0$ data with the functional form $a_0{N}_t^2$.

Figure 10

Effective integrated autocorrelation time of the topological susceptibility measured using parallel tempering simulations (with $a{N}_t=2$). The dashed line is the result of a fit of the form ${\tau }_{\mathrm{eff}}=a_0{N}_t^{a_1}$ and the parameter $a_1$ turns out to be $a_1\approx 2.2$.

Figure 11

Dependence on $N_{PT}$ of the integrated autocorrelation time $\tau$ as measured in the copies with the smaller value of the lattice spacing in parallel tempering simulations (with $a{N}_t=2$). The scaling Eq. (32) is well satisfied.

Figure 12

Topological susceptibility in the very high temperature case. Just the $\mathrm{\Omega }=0$ data are plotted since the difference with the $\mathrm{\Omega }\ne 0$ data is invisible on this scale. The red line is the theoretical expectation of Eq. (16).

Figure 13

Comparison of the results obtained by using different simulation algorithms in the low temperature phase for $\mathrm{\Omega }=0$. Results obtained by the Tailor update are not shown in order to make the figure more readable.

Figure 14

Comparison of the results obtained by using different simulation algorithms in the low temperature phase for ${\mathrm{\Omega }}^2=10$.