Random matrix model for antiferromagnetism and superconductivity on a two-dimensional lattice

We suggest a new mean-field method for studying the thermodynamic competition between magnetic and superconducting phases in a two-dimensional square lattice. A partition function is constructed by writing microscopic interactions that describe the exchange of density and spin fluctuations. A block structure dictated by spin, time-reversal, and bipartite symmetries is imposed on the single-particle Hamiltonian. The detailed dynamics of the interactions are neglected and replaced by a normal distribution of random matrix elements. The resulting partition function can be calculated exactly. The thermodynamic potential has a structure which depends only on the spectrum of quasiparticles propagating in fixed condensation fields, with coupling constants that can be related directly to the variances of the microscopic processes. The resulting phase diagram reveals a fixed number of phase topologies whose realizations depend on a single coupling parameter ratio, $\alpha$. Most phase topologies are realized for a broad range of values of $\alpha$ and can thus be considered robust with respect to moderate variations in the detailed description of the underlying interactions.

Figure 1

Parametrization of the first Brillouin zone. Regions 1 and 2 are related by a momentum shift by $\pm \mathbf{Q}=\pm \left(\pi \hslash /a,\pi \hslash /a\right)$, as are regions 3 and 4. Regions 1 and 3 and regions 2 and 4 are related by momentum reversal.

Figure 2

Phase diagram for $t/t_{\text{TH}}=0.5$ and $\alpha <0.1$. The transition from the AF to the PA is second order at half filling and first order at zero temperature (thick line). These two lines merge at a tricritical point, $t$. Here, temperature is plotted in units of $T_c\equiv T_c\left(\mu =0,t=0\right)$, which is the transition temperature at half filling in the limit $t\to 0$, while chemical potential is plotted in units of ${\sigma }_0\equiv 1/\sqrt{2B}$, which represents the AF field at half filling, zero temperature, and for $t\to 0$.

Figure 3

Phase diagram for $t/t_{\text{TH}}=0.5$ and $\alpha =0.2$. In addition to the phase structure of Fig. 2, there is a SC emerging out of the antiferromagnetic phase via a first-order transition. This phase undergoes a second-order transition to the paramagnetic phase at either higher $\mu$ or higher $T$. Here, $t$ is a tricritical point. The scales $T_c$ and ${\sigma }_0$ are those defined in the caption of Fig. 2.

Figure 4

Zero-temperature auxiliary fields as a function of $\mu$ for the parameters corresponding to the phase diagram of Fig. 3. The scale ${\sigma }_0$ is defined in the caption of Fig. 2.

Figure 5

Phase diagram for $t/t_{\text{TH}}=0.5$ and $\alpha =0.8$. The superconducting phase has grown beyond the tricritical point of Fig. 3. The two second-order lines (thin lines) separating, respectively, the antiferromagnetic and superconducting phases from the paramagnetic phase now meet at a bicritical point, $b$, which is also the end point of a first-order line (thick line) between the AF and SC phases. The scales $T_c$ and ${\sigma }_0$ are those defined in the caption of Fig. 2.

Figure 6

Phase diagram for $t/t_{\text{TH}}=0.7$ and $\alpha =0.05$. The transition from the AF to the PA is second order at half filling (thin line) and first order at zero temperature. These two lines merge at a tricritical point, $t$. The scales $T_c$ and ${\sigma }_0$ are those defined in Fig. 2.

Figure 7

Phase diagram for $t/t_{\text{TH}}=0.7$ and $\alpha =0.2$. Thin lines are second order and the thick line is first order; $t$ is a tricritical point. The scales $T_c$ and ${\sigma }_0$ are those defined in Fig. 2.

Figure 8

Phase diagram for $t/t_{\text{TH}}=0.7$ and $\alpha =0.8$. Thin lines are second order and the thick line is first order. $b$ is a bicritical point. The scales $T_c$ and ${\sigma }_0$ are those defined in Fig. 2.