Quantum vs thermal fluctuations in phase transitions of two-dimensional superconductors

TL;DR

The paper addresses how quantum and thermal phase fluctuations suppress superconducting order in two dimensions by studying the 2D quantum XY model with phase fluctuations controlled by $U$ using path-integral quantum Monte Carlo (PIQMC).A mapped 3D classical action with anisotropic couplings is simulated to obtain a temperature–interaction phase diagram displaying superconducting, metallic, and insulating phases, along with a quantum critical point at $T=0$ governed by 3D‑XY universality.Finite-temperature transitions follow BKT scaling with a universal stiffness jump, while the $T=0$ transition exhibits 3D‑XY critical behavior; quantum fluctuations shift the phase boundaries and induce a finite-frequency conductivity signature in both phases.The study connects thermodynamic signatures to transport, showing Halperin–Nelson resistance scaling for $U<U_c$ and a metal–insulator crossover for $U>U_c$, and it reveals finite-frequency features in the conductivity attributable to quantum fluctuations, with implications for oxide interfaces and related 2D superconductors.

Abstract

We investigate the impact of quantum and thermal phase fluctuations on the suppression of superconducting order in two-dimensional systems. Within the two-dimensional quantum XY model in the phase representation, where on-site interaction terms govern quantum phase fluctuations, we perform extensive path-integral quantum Monte Carlo simulations. The resulting temperature-interaction phase diagram establishes the presence of a well-defined critical line ending at a quantum critical point at vanishing temperature with no indication of reentrant behavior. We further demonstrate that the resistance above the critical line reproduces the two expected different critical behaviors. For stronger interactions, above the quantum critical point, the system exhibits a crossover to an insulating regime at low temperatures. Finally, Monte Carlo calculations of current-current correlation functions enable us to extract the frequency-dependent conductivity in both superconducting and normal regimes, revealing a finite-frequency response that we attribute to quantum phase fluctuations.

Figures (14)

• Figure 1: Phase diagram in the $T-U$ plane. SC denotes the superconducting, M the metallic , and I the insulating phase. The temperature $T$ and the interaction energy $U$ quantifies the strength of thermal and quantum fluctuations, respectively, while $t$ is the Josephson coupling energy, used as reference (see \ref{['sec:model']}). The diagram reports the phase-transition points extracted via finite-size scaling (\ref{['sec:finite-size-scaling']}) of the Monte Carlo simulation data (\ref{['sec:methods']}). Green squares mark the BKT critical temperatures $T_c=T_c^{BKT}$ obtained at fixed $U$, whereas black triangles indicate the BKT critical couplings $U_c^{BKT}$ determined at fixed temperature (\ref{['sec:finite-size-scaling']}). The orange circle marks the quantum critical value $U_c=U_c^{3D}(T=0)$, which belongs to the 3D-XY universality class. The dotted curve emanating from the quantum critical is a fit of the points near the $T=0$ quantum phase transition, obtained using the 3D-XY power-law indicated in the figure (\ref{['sec:3d-xy']}). The purple crosses indicate the pseudo 3D-XY phase transition (\ref{['sec:fake_3d']}). The blue diamonds denote the resistance minimum that signals the M–I crossover from metallic to insulating behaviour (the dashed line is only a guide for the eye) (\ref{['sec:corr-len']}).
• Figure 2: Stiffness $\Upsilon$ as a function of the temperature $T$ for the size $L=48$. (a) Results are shown for the interaction strengths $U/t=0.1,2,4$ at the discretization number $N_\tau$=1000. The dotted line indicates the NK curve following the criterion given in Eq. (\ref{['eq:stiff-scal']}), which is typically used to mark the finite temperature transition points. (b) Results are displayed for discretization numbers $N_\tau=50,100,500,1000$ at the interaction energy $U/t=4$.
• Figure 3: Finite size-scaling for different values of $U$, $T$, and $N_\tau$: (a) $U/t=0.1$, $N_\tau=100$; (b) $U/t=4$, $N_\tau=1000$; (c) $T/t=0.5$, $N_\tau=1000$; (d) $T/t=0.05$, $N_\tau=5000$. The BKT scaling is valid for both directions, $T$ and $U$, of the phase diagram reported in Fig \ref{['fig:diagram']}.
• Figure 4: 3D-XY scaling of the stiffness $\Upsilon$ vs interaction energy $U$ with $N_\tau=150\beta$, with $\beta=L$ and $U_\text{c}/t=5.63(2)$. In the inset, the data collapse with $\nu=0.671$ is showed.
• Figure 5: Variation of the 3D-XY fixed point as a function of $\Delta\tau$. In the limit $\Delta\tau\rightarrow0$, the system converges to the critical value $U_c(0)/t=5.72(1)$. This value is obtained by the data fit $U_c(\Delta\tau)/t=U_c(0)/t+a\Delta\tau$.