Figure 1

(Color online) Possible magnetic textures. (a) A single spin-spiral. (b) A superposition of four spirals. The longitudinal magnetic component has been suppressed for clarity.Reuse & Permissions

Figure 2

(Color online) Cartoon for the formation of metamagnetism. At the metamagnetic transition, the majority band increases its filling through the van Hove singularity. The presence of this singularity leads to a reduced cost in single-particle energy allowing the gain in interaction energy to win out.Reuse & Permissions

Figure 3

(Color online) The phase diagram for the Stoner model as a function of $\mu$, $h$, and $T$ for the next-nearest-neighbor tight-binding dispersion. The line of second-order transitions is given by ${\partial }_m^{2}F=0$, the line of metamagnetic critical end points by ${\partial }_m^{3}F={\partial }_m^{2}F=0$ and the tricritical point by ${\partial }_m^{4}F={\partial }_m^{2}F=0$. The shaded area on the $\mu ,h$ plane is a schematic representation of the region of this phase diagram, which is observed in ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$.Reuse & Permissions

Figure 4

(Color online) The phase diagram for a general Landau expansion. The inset shows how this phase diagram maps onto the microscopic parameters of a theory. This should be compared with the calculated phase diagram for the Stoner model, Fig. 3.Reuse & Permissions

Figure 5

(Color online) The transverse susceptibility and associated features in the Fermi surface and DoS of the next-nearest-neighbor tight-binding model discussed in the text. The first column shows transverse susceptibility, ${\Pi }_{\perp }\left(\mathbf{q}\right)$ evaluated for two situations, (a) $t=0.8$, $\mu =-0.6$, and (b) $t=0.2$, $\mu =-0.37$, chosen to illustrate various possible wave vectors of distortion. In both cases $\overline{M}$ is chosen such that the spin-splitting places one species’ Fermi surface at the van Hove singularity, in (a) this is the minority Fermi surface and in (b) the majority. The susceptibility shows peaks at both large and small $\mathbf{q}$. In the second column, the spin-up and -down Fermi surfaces are shown for each case. Displacements of these Fermi surfaces corresponding to the wave vectors of the peaks in the susceptibility are indicated. These displacements show the origin of the peaks. Wave vectors $A$, $B$, and $C$ are all nesting vectors, however, $D$ is not, it corresponds to a different situation described below. Note that in (a), we have indicated the equivalent displacement of the hole rather than electron pocket for ease of visualization. The third column shows the Fermi surfaces after hybridization for the low $\mathbf{q}$ cases, these are the Fermi surfaces in the spiral state. The wave vector $D$ corresponds to opening the neck of the Fermi surface across the van Hove singularity. The black dotted line shows the Fermi surface before hybridization. The rightmost column shows the densities of states for the two spin-species before (black dotted line) and after (solid lines) the spiral state forms. In (a), a new peak has appeared below the Fermi level. In (b), the peaks in the densities of states have split and moved below the Fermi level. The energetic origin of the nesting $\left(A,B,C\right)$ and non-nesting ($D$) vectors is therefore similar, being associated with the appearance of new peaks in the DoS below the Fermi surface.Reuse & Permissions

Figure 6

(Color online) (a) Energy dispersions for the spin-up [red (dark gray, lower plot)] and spin-down [blue (light gray, upper plot)] Fermi surfaces shifted by $\pm \mathbf{q}/2$. (b) Energy dispersions including a transverse magnetization. This hybridizes the bands, causing anticrossing and the formation of new saddle points.Reuse & Permissions

Figure 7

(Color online) (a) The line of metamagnetic critical end points projected onto the $\mu$, $T$ plane. (b) $K_{\perp }$ evaluated along the line of metamagnetic critical end points. As $\mu$ gets further from the van Hove singularity, this becomes negative. Plots evaluated for the dispersion discussed in the text with $t=0.2$ and $g=1.7$.Reuse & Permissions

Figure 8

(Color online) Free energy with $\mathbf{q}$-dependent region. Shown in black is the free energy when the magnetization is homogeneous. The green (gray) line is the free energy when the magnetization is modulated. The dotted regions show where this is unphysical, either increasing the free energy or corresponding to an imaginary magnetization. In the solid region, the magnetization is real and lowers the free energy. Also shown are the transverse magnetization and wave vector in this region.Reuse & Permissions

Figure 9

(Color online) Phase diagram for the Ginzburg-Landau theory. Green (dark gray) sheets represent first-order transitions in $\varphi$. Blue (pale gray) sheets represent continuous transitions into the inhomogeneous phase. A cut through the phase diagram at constant $K_{\perp }$ and the variation of $\varphi$ and ${\varphi }_{\perp }$ along a path through this cut are shown. This shows both first-order and continuous transitions.Reuse & Permissions

Figure 10

(Color online) Sketch showing how the phase diagram of Fig. 9 fits into the phase diagram for the metamagnetic system. This consists of a rotated version of Fig. 9 placed on the metamagnetic wing of Fig. 4. The light blue (pale gray) region is the inhomogeneous magnetic phase, the dark green (gray) sheet is the $h=0$ transition, the green “wings” are the metamagnetic transition.Reuse & Permissions

Figure 11

(Color online) (a) The experimental phase diagram as inferred from in-plane transport properties. Green (dark gray, below black line) planes correspond to abrupt changes in resistivity as a function of field. Blue (pale gray) shading indicates regions where the in-plane resistivity is anomalously high, becomes highly anisotropic with respect to the in-plane component of the field (Ref. 8) and shows an anomalous temperature dependence: for currents in the direction of maximum resistivity, the resistivity decreases with increasing temperature. The phase diagram obtained from magnetic susceptibility (Ref. 22) shows the same first-order transitions as indicated here in green, but lacks the roof shown in blue. (b) Phase diagram from the Ginzburg-Landau theory rotated into experimental configuration.Reuse & Permissions

Figure 12

(Color online) Angle dependence of simple model. (a) Fermi surface for a minimal model of ${\text{Sr}}_3{\text{Ru}}_2{\text{O}}_7$. Dotted lines show the unhybridized bands. Colored lines indicate the Fermi surface of one spin species for a variety of angles: Red, along $c$ axis. Green, $45°$ from c. Blue, $ab$ plane. (b) DoS for above angles. The peak moves to lower energy as the angle from c increases. (c) An in-plane field component along one axis breaks the symmetry of the dispersion. Inset, the various field angles illustrated in the Fermi surface plots.Reuse & Permissions

Figure 13

(Color online) (a) Phase diagram for a theory with a $U_1{\varphi }^2{\varphi }_{\perp }^2$ cross term. (b) A cut through the phase diagram at $K_{\perp }=-0.5$ with example free energy curves. Note the first-order transition in panel 3 where two minima are degenerate. (We take $L_{\perp }=0.1$)Reuse & Permissions

Figure 14

(Color online) Phase diagram for a theory with $U_1{\varphi }^2{\varphi }_{\perp }^2$ and $V_2{\varphi }^2{\varphi }_{\perp }^4$ cross terms. Cuts show (a) $K_{\perp }=-0.5$, all transitions first order. (b) $K_{\perp }=-0.6$, transitions change from first to second order at a critical point. (c) $K_{\perp }=-0.8$, all transitions second order. (We take $L_{\perp }=0.1$.)Reuse & Permissions

Figure 15

(Color online) (a) Phase diagram for the symmetric theory with particular choices for $K_2$ and $K_3$. Inset shows the tricritical point structure of a conventional Landau theory. (b) A cut through the phase diagram taken at $K_{\perp }=-1$ showing first-order transitions below $R=0.2$ and second-order transitions above. Plots of the effective free energy show (1) a first-order transition, and (2) a second-order transition. The top free energy curve shows how this occurs for the inhomogeneous theory and the bottom for a conventional theory. Black curves are homogeneous terms only and green (gray) curves include inhomogeneous terms. Also shown are magnetization plots as a function of $H$ showing both longitudinal (black) and transverse (red) magnetization. The magnitude of the jump in $\varphi$ at the first-order transition $\left(\Delta \varphi \right)$ is related to the spacing of minima in the free energy. [We take $L_{\perp }=0.1$, $K_2=0.3$, and $K_3=0.2$].Reuse & Permissions

Figure 16

(Color online) Phase diagram for the Ginzburg-Landau theory as in Fig. 9. Green (dark gray) sheets represent first-order transitions in $\varphi$. Blue (pale gray) sheets represent continuous transitions into the inhomogeneous phase. $K_{\perp }$ represents movement along the metamagnetic wing. $H$ moves in a direction perpendicular to the wing. Taking a cut at an angle to the $R$ axis shows how the inhomogeneous region becomes finite. Solid blue (thin) and green (thick) lines show cuts at constant $K_{\perp }$.Reuse & Permissions