Itinerant electrons in the Coulomb phase

We study the interplay between magnetic frustration and itinerant electrons. For example, how does the coupling to mobile charges modify the properties of a spin liquid, and does the underlying frustration favor insulating or conducting states? Supported by Monte Carlo simulations, our goal is in particular to provide an analytical picture of the mechanisms involved. The models under consideration exhibit Coulomb phases in two and three dimensions, where the itinerant electrons are coupled to the localized spins via double exchange interactions. Because of the Hund coupling, magnetic loops naturally emerge from the Coulomb phase and serve as conducting channels for the mobile electrons, leading to doping-dependent rearrangements of the loop ensemble in order to minimize the electronic kinetic energy. At low electron density $\rho$, the double exchange coupling mainly tends to segment the very long loops winding around the system into smaller ones while it gradually lifts the extensive degeneracy of the Coulomb phase with increasing $\rho$. For higher doping, the results are strongly lattice dependent, displaying loop crystals with a given loop length for some specific values of $\rho$. By varying $\rho$, they can melt into different mixtures of these loop crystals, recovering extensive degeneracy in the process. Finally, we contrast this to the qualitatively different behavior of analogous models on kagome or triangular lattices.

Figure 1

In the Coulomb phase, each frustrated unit (crossed squares in 2D checkerboard, left, and tetrahedra in 3D pyrochlore, right) possesses two up and two down spins, respectively colored in dark blue and light red. Those are the so-called ice rules or divergence-free conditions. Connecting spins of the same color forms a network of loops, as illustrated for the checkerboard. This model is equivalent to the nearest-neighbor spin ice model.

Figure 2

Single-particle energy levels of itinerant electrons confined to loops of length $\ell =8$ (left) and $\ell =12$ (right), due to the hopping term in Hamiltonian (1). There are nondegenerate levels at the highest and lowest energies, $\pm 2t$, independent of loop length; the other levels are doubly degenerate.

Figure 3

Histogram of distribution of the number of loops $N_{\ell }$ per configuration in two dimensions (left) and three dimensions (right), in the absence of electrons. The distribution is scaled according to the Gaussian expression of Eq. (2) collapsing all system sizes onto the same curve ($L=100$ to 600 in two dimensions and $L=4$–60 in three dimensions). In two dimensions, the two Gaussians have the same width and correspond to even and odd $N_{\ell }$ (upper and lower curve). In three dimensions, finite-size corrections are visible for $L=4$ (red crosses).

Figure 4

Mean value of the loop length $\langle \ell \rangle$ as a function of doping. Left: checkerboard, $L=10$ (red dots) and 60 (blue triangles). Right: pyrochlore, $L=6$ (red dots) and 20 (blue triangles). The dashed straight lines are predictions from Eq. (7), without any fitting parameters. For both panels, the blue triangles data points are very close to thermodynamic limit.

Figure 5

Loop length distribution $P_{3D}(\ell ,\rho )$ in three dimensions for different values of doping ($\rho =\{0,0.00075,0.0015,0.002\}$) and system size $L=10$. The distribution is normalized such that $\int \ell P_{3D}(\ell ,\rho )d\ell =16L^3=N$. The dashed line indicates the power-law fit ${\ell }^{-1}$ at $\rho =0$ and the arrow shows the shifting of the distribution for increasing $\rho$. The exponents of the two power-law regions (before and after $\ell \approx L^2=100$) do not vary, but the peak for long winding loops $\ell \sim 8L^3=N/2$ gets smaller with increasing $\rho$.

Figure 6

Main: Loop length distribution $P_{2D}(\ell ,\rho )$ in two dimensions for different values of doping ($\rho =\{0,0.0025,0.0056,0.0087\}$) and system size $L=40$. The distribution is normalized such that $\int \ell P_{2D}(\ell ,\rho )d\ell =4L^2=N$. The dashed line indicates power law $L^2/{\ell }^{15/7}$ and the arrow shows the shifting of the distribution for increasing $\rho$. On this scale, it is not obvious whether the exponent of the power law varies or not. Inset: The power-law exponent defined locally on $\ell$, ${\tau }_{\mathrm{local}}$ for $\ell =8\left(\blacksquare \right),16\left(▴\right),32(•)$ for $L=40$ (similar behavior obtained for $L=20$ and 60).

Figure 7

Loop length $\ell \left(\rho \right)$ minimizing the energy (11) for a system filled with loops of length $\ell$ only, assuming no lattice constraints other than ${\ell }_{\min }=4$ (left panel, checkerboard) or 6 (right panel, pyrochlore). $\rho$ takes all rational values $p/q$, with $q\in \{0,1,2,...,1000\}$ and $p\in \{0,1,2,...,q\}$, and we consider all loop lengths from ${\ell }_{\min }$ to ${\ell }_{\max }=100$ (blue squares) and 1000 (red dots): plotting two different values of ${\ell }_{\max }$ provides a graphical way to visualize the two values of $\rho$ where the most favorable loop length is infinite, namely $\rho =1/3$ and $2/5$ (and their symmetric images with respect to $\rho =1/2$). If several loop lengths give the same energy, we plot the smallest one.

Figure 8

Minimum energy $E\left(\rho \right)$ corresponding to the value of $\ell \left(\rho \right)$ plotted in Fig. 7, for a system filled with loops of length $\ell$ only, assuming no lattice constraints other than ${\ell }_{\min }=4$ (left, checkerboard) or 6 (right, pyrochlore). $\rho$ takes all rational values $p/q$, with $q\in \{0,1,2,...,1000\}$ and $p\in \{0,1,2,...,q\}$. We consider all loop lengths from ${\ell }_{\min }$ to ${\ell }_{\max }=1000$. We chose arbitrarily $t=1/2$. A Maxwell construction can be visualized from this figure. The red dots are a set of points, such that the dashed lines connecting them are always below the red curve $E\left(\rho \right)$. Hence a mixture of two phases corresponding to two consecutive red dots $({\ell }_1,{\rho }_1)$ and $({\ell }_2,{\rho }_2)$ has a lower energy than a single phase with a unique density of electrons $\rho$ on a unique type of loop length $\ell$. For increasing values of $\rho$, the red dots correspond to the most favored loop length $\ell =\{4,8,6,8,4\}$ (left panel) and $\ell =\{6,12,10,8,6,8,10,12,6\}$ (right panel), as can be read from Fig. 7.

Figure 9

Zero-temperature “phase diagram” of loop length configuration as a function of electron density $\rho$, obtained without accounting for lattice constraints other than ${\ell }_{\min }=4\left(6\right)$ for checkerboard (pyrochlore). A phase is defined by the length of its constituting loops and the density of electrons on them. The values of $\rho$ and $\ell \left(\rho \right)$ for each phase are given below and above the lines.

Figure 10

Schematic representation of the checkerboard lattice with 16 sites and different loop coverings. Left: 4 loops of length 4, corresponding to the ground state at $\rho =1/4$ with 1 electron per loop. Center: 2 loops of length 8, corresponding to the ground state at $\rho =3/8$ with 3 electrons per loop. Between 1/4 and 3/8, a mixture of these two configurations will occur, their respective ratio being set by the total number of electrons $N_e=\rho N$. This arrangement of two loops of length 8 appears as a “defect” in a crystal of loops of length 4. Right: a red loop of length 6 spanned by another blue loop; the shortest way to close this blue loop requires 10 sites. It is thus impossible to have a mixture of loops of length 6 and 8 only.

Figure 11

Average number of filaments, i.e., those segments of loops spanning the entire system from one border to the opposite one, as a function of system size $L$. We use two definitions of filaments; they can either cross the orthogonal borders thanks to the periodic boundary conditions $(•)$ or not $\left(\blacksquare \right)$. The cartoon on the right shows such a boundary-crossing filament, which would be excluded in the $\blacksquare$ data. Dashed lines are guides to the eye for the linear behavior with $L$. Both $x$ and $y$ axes are on a logarithmic scale.

Figure 12

Average number of filaments in two dimensions (left) and three dimensions (right), corresponding, respectively, to systems of sizes $L=10,20,40$ and $L=6,12,18$ (red squares, green dots, blue triangles) as a function of doping $\rho$.

Figure 13

Configurations on the triangular (top) and kagome (bottom) lattices respecting the antiferromagnetic frustrated constraints. (a) and (c) are random configurations, while (b) is the conducting hopping network minimizing the kinetic energy on the triangular lattice, and (d) is the insulating state maximizing the number of closed loops on kagome. Panel (d) also illustrates the arrangement of loops described in Sec. 4c2 in the two-dimensional kagome layers, which are part of the three-dimensional pyrochlore lattice.

Figure 14

On the left, the same loop configuration as in Fig. 1. The thick dashed green line represents a possible worm, as built by our algorithm; flipping the spins along this worm does not break the ice rules (see on the right) and allows us to visit the configurational space of the Coulomb phase, ensuring ergodicity.