Kosterlitz-Thouless signatures in the low-temperature phase of layered three-dimensional systems

We study the quasi-two-dimensional quantum O(2) model, a quantum generalization of the Lawrence-Doniach model, within the nonperturbative renormalization-group approach and propose a generic phase diagram for layered three-dimensional systems with an O(2)-symmetric order parameter. Below the transition temperature we identify a wide region of the phase diagram where the renormalization-group flow is quasi-two-dimensional for length scales smaller than a Josephson length $l_J$, leading to signatures of Kosterlitz-Thouless physics in the temperature dependence of physical observables. In particular the order parameter varies as a power law of the interplane coupling with an exponent which depends on the anomalous dimension (itself related to the stiffness) of the strictly two-dimensional low-temperature Kosterlitz-Thouless phase.

Figure 1

Schematic phase diagram of a layered 3D system with an O(2)-symmetric order parameter as a function of temperature and interplane coupling $J_{\perp }$. The behavior of the correlation length $\xi$ in the parallel direction near the transition has been studied in previous works and is not discussed in the paper. Our main focus is the existence of a wide region (region A, shown in gray) in the low-temperature phase where signatures of KT physics are observable. Region A is separated from region B, which behaves as a standard anisotropic 3D phase, by a crossover line $T\simeq J_{\perp }$ where $J_{\perp }$ is the interplane coupling.

Figure 2

Transition temperature vs $J_{\perp }$.

Figure 3

Anomalous dimensions ${\eta }_{\parallel ,k}$ and ${\eta }_{\perp ,k}$ vs $ln(\mathrm{\Lambda }/k)$ at the transition temperature $T_c\simeq 1.5T_{\mathrm{KT}}$ for $J_{\perp }=T_{\mathrm{KT}}/4$. The two vertical lines correspond to the thermal scale $k_T=1/l_T$ and Josephson scale $k_J=1/l_J$.

Figure 4

Anomalous dimensions ${\eta }_{\parallel ,k}$ and ${\eta }_{\perp ,k}$ vs $ln(\mathrm{\Lambda }/k)$ for $T=T_{\mathrm{KT}}/4$ and $J_{\perp }=0, T_{\mathrm{KT}}/40, 2.5T_{\mathrm{KT}}$ corresponding to the points $A_0, A_1$, and $B$ loosely shown in Fig. 1. The vertical lines correspond to the thermal scale $k_T [ln\left(\mathrm{\Lambda }{l}_T\right)\simeq 5.1]$ and to the scale $k_v$ associated with the size $l_v$ of the largest vortex-vortex bound pairs. The maximum of ${\eta }_{\perp ,k}$ corresponds to the Josephson scale $l_J [ln\left(\mathrm{\Lambda }{l}_J\right)\simeq 9.2$ for $A_1$ and $ln\left(\mathrm{\Lambda }{l}_J\right)\simeq 4.6$ for $B]$.

Figure 5

Same as Fig. 4 but for $J_{\perp }=T_{\mathrm{KT}}/40$ and $T=T_{\mathrm{KT}}/4, T_{\mathrm{KT}}/2$, and $1.05T_{\mathrm{KT}}$ (points $A_1, A_2$, and $C$ loosely shown in Fig. 1). The vertical line corresponds to the Josephson scale $[ln\left(\mathrm{\Lambda }{l}_J\right)\simeq 9.2]$.

Figure 6

Order parameter ${\rho }_0$ vs $J_{\perp }$ for various temperatures. The (black) symbols correspond to the power-law dependence (14).

Figure 7

Temperature dependence of the order parameter for various values of $J_{\perp }$. The (black) dashed lines show the expected low-temperature result (21).

Figure 8

Crossover line between regions A and B (see Fig. 1) obtained by comparing the NPRG result for ${\rho }_0(J_{\perp },T)$ to the small-$J_{\perp }$ expression (14) or the low-temperature limit (21). The dashed line corresponds to $T=J_{\perp }$.

Figure 9

Temperature dependence of the ratio $2{\rho }_0/{\rho }_s=1/Z_{\parallel }(T,J_{\perp })$ between the square of the order parameter and the in-plane stiffness for various values of $J_{\perp }$.