Random Coulomb antiferromagnets: From diluted spin liquids to Euclidean random matrices

We study a disordered classical Heisenberg magnet with uniformly antiferromagnetic interactions which are frustrated on account of their long-range Coulomb form, i.e., $J\left(r\right)\sim -Alnr$ in $d=2$ and $J\left(r\right)\sim A/r$ in $d=3$. This arises naturally as the $T\to 0$ limit of the emergent interactions between vacancy-induced degrees of freedom in a class of diluted Coulomb spin liquids (including the classical Heisenberg antiferromagnets in checkerboard, SCGO, and pyrochlore lattices) and presents a novel variant of a disordered long-range spin Hamiltonian. Using detailed analytical and numerical studies we establish that this model exhibits a very broad paramagnetic regime that extends to very large values of $A$ in both $d=2$ and $d=3$. In $d=2$, using the lattice-Green-function-based finite-size regularization of the Coulomb potential (which corresponds naturally to the underlying low-temperature limit of the emergent interactions between orphans), we find evidence that freezing into a glassy state occurs only in the limit of strong coupling, $A=\infty$, while no such transition seems to exist in $d=3$. We also demonstrate the presence and importance of screening for such a magnet. We analyze the spectrum of the Euclidean random matrices describing a Gaussian version of this problem and identify a corresponding quantum mechanical scattering problem.

Figure 1

Illustration of the orphan spin arising from the introduction of nonmagnetic impurities [(red) circles] on the checkerboard lattice. Its effective moment is half that of a free spin.

Figure 2

$J(x,y)$ used in the simulations in $d=2$.

Figure 3

Spin glass susceptibility (top) and correlation length (bottom) for the LGF interaction. Several system sizes are indicated by different colors. Circles represent MC simulations (error bars are of the order of the circle size), and lines are from the LM approach—correlations are stronger for Heisenberg spins than the ”soft” LM spins throughout. Insets: Scaling collapse for LM for $1/A_c=0$.

Figure 4

Scaling of the number of zero eigenvalues ($m_0$) of matrix $B$ defined in the text and of the spin glass susceptibility (insets) with the number of particles for the LGF (top) and log (bottom) interactions.

Figure 5

Spin glass susceptibility (top) and correlation length (bottom) for the log interaction, as computed in the MC simulations (points) or with the LM approach (lines). Insets: The corresponding scaling collapses.

Figure 6

Disorder-averaged pair correlations with a spin at the origin as a function of the relative coordinates. The center of each circle indicates the position of the spin, and its radius gives the magnitude, with red (black) denoting positive (negative) correlations. The large central (red) circle thus reflects $\langle {\vec{S}}_i^2\rangle =1$. Top: Result for the LGF with $A=100$. Bottom: Result for the log with $A=20$. MC data are in agreement with LM data (not shown).

Figure 7

Spin-glass susceptibility (top) and correlation length (bottom) computed from the LM approach (lines) or measured in MC simulations (points), for the LGF interaction on a cubic lattice.

Figure 8

Scaling of the number of vanishing eigenvalues of matrix $B$ defined in the text and of the spin glass susceptibility (inset) with the number of particles for the LGF in three dimensions at $1/A=0$.

Figure 9

Ground-state eigenvector showing a trimer for a particular disorder realization using the LGF as the interaction on a lattice of size $L=32$ with $N=102$ particles. The components of the eigenvector are proportional to the radii of the circles, which are centered on the corresponding spin position. A red (black) circle indicates a positive (negative) sign.

Figure 10

Spectral density (top) and average $Y$ (Eq. (37), bottom) for a fraction of $x=0.125$ occupied sites in the lattice, using $J_{ij}$ as defined in (16), the LGF interaction. The inset shows the fluctuations of $Y$.

Figure 11

Spectral density (top) and average $Y$ (bottom) for a fraction of $x=2^{-13}$ occupied sites in the lattice, using $J_{ij}$ as defined in (16), the LGF interaction. Inset: Fluctuations of $Y$.

Figure 12

Uniform susceptibility as computed from MC simulations (points) compared to a weak-coupling expansion (WCE) averaged over 200 disorder realizations (dashed lines). The Curie-Weiss constant increases approximately linearly with the number of particles (inset), yielding a vanishing radius of convergence of the weak-coupling expansion already at leading order.

Figure 13

Correlation function exhibiting screening: numerical results (for a single disorder realization with $N=200$ points on a square of unit size) compared to the analytical form predicted from the chain diagrams.

Figure 14

Left: The sign function in the first quadrant for a lattice of side $L=100$ used to resum the correlations. Right: Scaling of the new susceptibility proposed to describe the ordering occurring with the log interaction.

Figure 15

Numerically obtained Fourier transform of the log (red symbols) and LGF (blue symbols) for a lattice of size $L=100$. Circles connected by lines indicate the edge $k_y=0$, while squares indicate the diagonal $k_x=k_y$. The index $n$ labeling the $x$ axis indicates the index of the wave vector: $k_x=\frac{2\pi }{L}n$. Inset: The region near the global minimum in more detail.

Figure 16

The average in each binning block of the spin glass susceptibilities plotted against the logarithm (base 2) of the size of the corresponding binning block. Data shown here correspond to MC simulation of the LGF in a cubic lattice at $A=200$.