Critical behavior of the thermal phase transition of U(1) lattice gauge systems

Abstract

We model the phase transition of a superconductor as a U(1) lattice gauge system, and determine its critical behavior. For this, we perform Monte Carlo simulations, treating the order parameter field and the gauge field on equal footing, without additional approximations. As the defining correlation function, we determine the order parameter correlation function including a gauge string, thus achieving a gauge-invariant characterization of the long-range behavior explicitly. We obtain a critical exponent $β$ that is consistent with the exponent of the U(1) transition of neutral bosons, i.e. of Bose-Einstein condensation. We determine the critical behavior of the heat capacity, which displays a temperature depends consistent with an XY transition. These results clarify the universality class of the phase transition of this system.

Figures (11)

• Figure 1: Sketch of a lattice U(1) gauge theory that we use as a model for the superconducting state. The amplitude $\abs{\psi}$ and the phase $\phi$ of the complex order parameters $\psi$ are defined on the lattice sites, which are represented by black spheres in the sketch. The components $a_j$ of the vector potential, in unitless form, are defined on the links connecting neighboring sites along the directions $j \in {x, y, z}$, depicted as brown cubes. The magnetic $B_i$-fields, which are derived from the surrounding values of the vector potential $a_j$, are defined on the plaquettes of the lattice and are shown exemplary for one $B_x$-value in blue in the sketch. The tunneling strength between the lattice sites is denoted by $t$. For the Wilson line $W_{\boldsymbol{r_1},\boldsymbol{r_2}}$ all vector potential components $a_i$ on the connecting line between $\boldsymbol{r_1}$ and $\boldsymbol{r_2}$ are accumulated (shown in purple).
• Figure 2: Snapshots of the vorticity $\nu$ of the system on a lattice of $4^3$ grid points for three typical configurations with increasing temperature. In (a) at $\Tilde{T}=0.0253$ first vortex rings form. (b) At $\Tilde{T}=0.0272$ more complex vortex clusters emerge. In (c) at $\Tilde{T}=0.0290$$\approx \Tilde{T_c}$ the number of vortices further increases and double vortices form.
• Figure 3: Single particle correlation function $\mathcal{C}(s)$ with Wilson line as a function of the distant in grid points $s$. For $T<T_c$ the correlation function converges to a nonzero Cooper pair density $n_0$ ($\Tilde{T}=0.0253$ and $\Tilde{T}=0.0282$ in the plot). For $T>T_c$ ($\Tilde{T}=0.0308$ in plot) only short-range correlations are present and $n_0\rightarrow0$. The recovery of the correlation function for large distances $s$ reflects the periodic boundary conditions combined with the gauge invariance.
• Figure 4: (a) The Cooper pair density $n_0$ as a function of the unitless temperature $\Tilde{T}$ for the superconducting case, i.e. in the U(1) gauge theory (SC, blue) and the condensate density of a Bose-Einstein condensate without gauge field (BEC, green) for comparison for one of the evaluated temperature intervals. Near the phase transition finite size effects causes a smearing of the phase transition, these transparent points are not used for the fit. Through the non-transparent data points a fit of the order parameter and the critical temperature is shown (dashed lines). (b) Logarithmic plot of the Cooper pair density $n_0$ as a function of reduced temperature $\tau=(T_c-T)/T_c$ for the same fitting interval as above. The critical exponent extracted from all evaluated temperature intervals for the superconducting case with gauge string, $\beta_{\text{U(1) gauge}}=0.344 \pm 0.014$, is consistent with the BEC case and the expected value for the U(1) universality class.
• Figure 5: Heat capacity $C$ in units of $k_B$ as a function of the unitless temperature $\Tilde{T}$ for different lattice sizes with fits for the superconducting case with gauge field (SC, data=circles, fit=dotted lines) and the gauge-free BEC case (BEC, data=squares, fit=dashed lines). In addition plotted in grey is the extrapolation to infinite lattice sizes. Shown in (a) is the heat capacity for the parameter set 1 of \ref{['tab:parameter']} with large fluctuations. In (b) the heat capacity of the parameter set 2 with only small fluctuations in the particle density per site is shown, proving that the regime of small density fluctuations does not change the behavior. The heat capacity of both parameter sets have a clear XY-transition.