Magnetism and $d$-wave superconductivity on the half-filled square lattice with frustration

The role of frustration and interaction strength on the half-filled Hubbard model is studied on the square lattice with nearest- and next-nearest-neighbor hoppings $t$ and $t^{\prime }$ using the variational cluster approximation (VCA). At half-filling, we find two phases with long-range antiferromagnetic (AF) order: the usual Néel phase, stable at small frustration $t^{\prime }∕t$, and the so-called collinear (or superantiferromagnet) phase with ordering wave vector $(\pi ,0)$ or $(0,\pi )$, stable for large frustration. These are separated by a phase with no detectable long-range magnetic order. We also find the $d$-wave superconducting (SC) phase $\left(d_{x^2-y^2}\right)$, which is favored by frustration if it is not too large. Intriguingly, there is a broad region of coexistence where both AF and SC order parameters have nonzero values. In addition, the physics of the metal-insulator transition in the normal state is analyzed. The results obtained with the help of the VCA method are compared with the large-$U$ expansion of the Hubbard model and known results for the frustrated $J_1-J_2$ Heisenberg model. These results are relevant for pressure studies of undoped parents of the high-temperature superconductors: we predict that an insulator to $d$-wave SC transition may appear under pressure.

Figure 1

(Color online) Schematic generalized zero-temperature parameter space for the high-temperature superconductors. Horizontal axis $\delta$ represents doping, vertical axis the $U∕t$ interaction strength, and third direction the $t^{\prime }∕t$ frustration. The parallelepiped indicates the region of parameter space relevant for high-temperature superconductors. In this study, we consider the zero doping $\delta =0$ plane. Experimentally, $U∕t$ can be varied by pressure. The inset shows the definitions of $t$ and $t^{\prime }$ on the square lattice.

Figure 2

The ring-exchange contributions to the large-$U$ expansion of the $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model from Ref. 30. The empty circle in case (c) denotes that the central site does not participate in the plaquette spin-exchange term.

Figure 3

(Color online) Classical phase diagram of the Hubbard model with next-nearest-neighbor hopping $t^{\prime }$ based on the large-$U$ expansion analysis. The usual AF phase (AF1) with the ordering vector $(\pi ,\pi )$ is observed at small ratios of $t^{\prime }∕t$, whereas a so-called superantiferromagnetic phase $(0,\pi )$ (AF2, shaded region) is realized at large frustration $t^{\prime }∕t$. Note that because of the nature of the large-$U$ expansion, this phase diagram is expected to be inaccurate at low $U$ values.

Figure 4

The tilings of the square lattice with the various clusters used in this work containing (a) four sites, (b) six sites, and (c) eight sites. In this example, the gray and white sites are inequivalent since the $(\pi ,\pi )$ AF order is possible. Note that in some cases, such as that of the six-site cluster, a different tiling (d) must be chosen for a $(0,\pi )$ phase to become possible.

Figure 5

(Color online) The magnetic phase diagram of the $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model ignoring the SC solution. There are two magnetic phases: $(\pi ,\pi )$ and $(\pi ,0)$, and the paramagnetic region (shaded area) where both order parameters vanish. The diagram was obtained with the VCA method using an eight-site cluster. The dashed line with filled data points shows the Hartree-Fock prediction from Ref. 41 for the transition between the Néel $(\pi ,\pi )$ and the nonmagnetic phases. The solid line (red) with error bars indicates the transition $U_c\left(t^{\prime }\right)$ between the insulating (Mott-like) phase above and the metallic region below $U_c$ when no magnetic order is allowed.

Figure 6

Fermi surface for $t^{\prime }=0.2t$ (left), $t^{\prime }=0.6t$ (center), and $t^{\prime }=0.8t$ (right). The change in topology (Lifshitz transition) occurs around $t^{\prime }=0.71t$. The best nesting vector $\mathbf{Q}$ is also shown. The Fermi surface for negative $t^{\prime }∕t$ looks like the ones above if one translates the origin to $(\pi ,\pi )$. The phase diagram in other figures depends only on the absolute value of $t^{\prime }∕t$.

Figure 7

(Color online) Finite-size effects on the metal-insulator transition. The crossover interaction strength $U_c$ as a function of frustration $t^{\prime }∕t$ for three clusters studied. Error bars on $U_c$ $(\Delta U_c\approx 0.3t)$ are not shown for clarity.

Figure 8

(Color online) The phase diagram of the $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model obtained with the VCA based on a eight-site cluster. The solid lines denote the phase boundaries between the $(\pi ,\pi )$ and $(\pi ,0)$ antiferromagnetic phases and the $d_{x^2-y^2}$ SC phase. The hatched area on the phase diagram (red and white) denotes the critical region where neither $(\pi ,\pi )$ nor $(\pi ,0)$ order could be found. The coexistence region between AF and SC phases (shaded area, cyan) is contained between two black lines that meet around $t^{\prime }=0.7t$, with the (blue) line in between indicating the area where the free energies of the would-be separate SC and AF phases become equal [cf. the point $U=2.45t$ in Fig. 9b]. Triangles and filled circles denote points on the phase boundary where an order parameter sustains a discontinuity at a first-order phase transition; short dashes inside the coexistence region mark the points where total energies of the AF and SC phases are equal.

Figure 9

(Color online) Details of the AF and SC phase coexistence for a $2\times 3$ cluster at $t^{\prime }=0.2t$. (a) Expectation values of SC ($D$, circles and red line) and AF ($M$, squares and black line) order parameters as a function of increasing interaction $U$; note different scales for the two quantities plotted. Values $U_1$ and $U_2$ denote the positions of the first-order phase transitions into or from the coexistence phase. (b) Blowup of the free energy $F=\Omega +\mu ⟨n⟩$ as a function of $U$ near $U=2.5t$ is shown for all three phases studied: AF phase (blue solid line), SC phase (red dash-dotted line), and the coexistence phase, where both $D$ and $M$ order parameters are nonzero (dashed black line). The latter phase has lower energy than the other two in the whole coexistence region of $2.3<U∕t<3.9$.

Figure 10

(Color online) The expectation values of SC ($D$, circles and red line, left-hand scale) and two types of AF order parameters (shown with squares and black lines), calculated on a $2\times 3$ cluster for (a) $t^{\prime }=0.5t$, the AF $(\pi ,\pi )$ phase order parameter $M$ (right-hand scale), and (b) $t^{\prime }=0.8t$, the $(\pi ,0)$ phase order parameter $Q$ (right-hand scale). Unlike in (a), no coexistence region is seen in (b), where the first-order transition occurs at $U_c=4.7t$.

Figure 11

(Color online) The phase diagram of the $t\text{−}{t}^{\prime }\text{−}U$ Hubbard model obtained with the VCA based on a (a) four-site cluster and a (b) six-site cluster. The hatched area (red and white) denotes the critical region where neither $(\pi ,\pi )$ nor $(\pi ,0)$ order could be found or when the paramagnetic solution had lower energy than any of the AF phases. The coexistence region between SC and AF phases (shaded area, cyan) is shown. The solid lines and as well as open triangles and short dashes on the phase boundaries have the same meaning as in Fig. 8.