Dual simulation of the 2d U(1) gauge Higgs model at topological angle $θ= π\,$: Critical endpoint behavior

TL;DR

The paper studies the critical endpoint of the 2d U(1) gauge-Higgs model at topological angle $\theta=\pi$, where charge conjugation is an exact symmetry. By reformulating the lattice theory in a dual Villain representation with a properly quantized $\theta$-term, the authors avoid the complex action problem and preserve $2\pi$ periodicity, enabling sign-free simulations. Finite-size scaling of the topological charge density and its susceptibility reveals a second-order transition at a critical mass $M_c$, with critical exponents matching the 2d Ising universality class. This work validates the Ising nature of the endpoint and demonstrates a practical, sign-problem-free lattice framework for studying topological terms and related phase structure in gauge-Higgs systems, with potential extensions to more scalars and higher dimensions.

Abstract

We simulate the 2d U(1) gauge Higgs model on the lattice with a topological angle $θ$. The corresponding complex action problem is overcome by using a dual representation based on the Villain action appropriately endowed with a $θ$-term. The Villain action is interpreted as a non-compact gauge theory whose center symmetry is gauged and has the advantage that the topological term is correctly quantized so that $2π$ periodicity in $θ$ is intact. Because of this the $θ= π$ theory has an exact $Z_2$ charge-conjugation symmetry $C$, which is spontaneously broken when the mass-squared of the scalars is large and positive. Lowering the mass squared the symmetry becomes restored in a second order phase transition. Simulating the system at $θ= π$ in its dual form we determine the corresponding critical endpoint as a function of the mass parameter. Using a finite size scaling analysis we determine the critical exponents and show that the transition is in the 2d Ising universality class, as expected.

Figures (5)

• Figure 1: The phase diagram of the theory as function of $m^2$ and $\theta$, as suggested in Komargodski:2017dmc.
• Figure 2: The topological charge density $\langle q \rangle$ (top row of plots) and the gauge action density $\langle s_G \rangle$ (bottom) as a function of $\Delta \theta$. We compare different volumes and use $\beta = 3.0, \lambda = 0.5$. For the mass parameter $M = 4 + m^2$ we use $M = 2.0$ (lhs. column of plots) and $M = 3.5$ (rhs.).
• Figure 3: The topological charge density $\langle | q | \rangle$ and the topological susceptibility shifted by a constant $\chi_t+\frac{\beta}{4\pi}$ at $\theta = \pi$ for different volumes. We compare the results for $\beta = 3.0$ (top plots) and $\beta = 5.0$ (bottom) at $\lambda = 0.5$ and plot the observables as a function of our control parameter, i.e., the mass parameter $M = 4 + m^2$. For comparison on the top of the plots we also show the horizontal axis labelled with $m^2$, which at the point of the transition is negative.
• Figure 4: Lhs.: The maxima of the mass-derivative of the Binder cumulant $U$ and the logarithmic derivatives of $\langle |q| \rangle$ and $\langle q^2 \rangle$ plotted as a function of the lattice size $L$. The solid lines show fits with $A \, L^{\frac{1}{\nu}}$ which were used to determine $\nu$. Rhs.: The pseudo-critical values $M_{pc}(L)$ determined as the positions of the maxima of different observables (see the labels in the legend), plotted against $L^{-\nu}$. The solid lines are fits according to (\ref{['equ:estimate_Mc']}) and their values extrapolated to $L = \infty$ were used for the determination of $M_c$. In both plots we annotate the final results for the parameters determined from the fits.
• Figure 5: Lhs.: Scaling of the topological charge $\langle |q| \rangle$. The symbols indicate the values of $\langle |q| \rangle_{(0,L)}$ at the critical mass $M_c$ (i.e., at reduced mass $m_r = 0$) for different lattice sizes. The solid line is the fit to these data according to (\ref{['FSS1']}). Rhs.: Scaling of the topological susceptibility $\chi_t$. The symbols show the values of $\chi_{t\,(0,L)}$ at the critical mass for different lattice sizes. The solid line is again the fit to the finite size scaling formula (\ref{['FSS1']}). In both plots we annotate the final results for the parameters determined from the fits.