Shaping Magnetic Order by Local Frustration for Itinerant Fermions on a Graph

Abstract

Kinetic magnetism is an iconic and rare example of collective quantum order that emerges from the interference of paths taken by a hole in a sea of strongly interacting fermions. Here the lattice topology plays a fundamental role, with odd loops frustrating ferromagnetism, as seen in recent experiments. However, the resulting magnetic order on a general graph has remained elusive. Here we systematically establish a general principle: that local frustration centers bind singlets while sharing a delocalized hole. This collective effect -- absent in exchange magnetism -- extends from rectangular grids to random graphs, producing sharp and predictable variation with tunable frustration measures. Our findings demonstrate that one can shape the spin order and tune the net magnetization by embedding kinetic frustration, opening ways of spatially resolved quantum control of many-body systems. We outline a protocol to realize some of the key findings in existing cold-atom setups.

Figures (15)

• Figure 1: (a) Square with tunneling $t= 1-u$ along the sides (black) and $t^{\prime}=u$ on the diagonals (green), containing $N=3$ fermions with total spin $S_{\text{total}} \in \{1/2, 3/2\}$. (b) Magnetization (circles) and reflection parity (triangles) of the ground state as a function of $u$. The spin gap $\Delta$ (minimum energy gap from the ground-state sector) is plotted in the background. The inset shows a "cross-dimer" structure of the ground state in region III, where blue and red ellipses denote a singlet and a delocalized hole, respectively. (c)-(e) Hilbert-space graph representation of the ground state. The vertices correspond to the position of the hole in I and those of the hole and the $\downarrow$ spin (magnon) in II and III. The signs indicate the relative amplitudes and dashed lines show frustrated edges. (f)-(h) Spin correlations showing anti-aligned spins on the vertical bonds in II and on the diagonal bonds in III.
• Figure 2: (a) Ladder with a central frustrated plaquette. (b) Minimum energy sector and ground-state parity as a function of $u = t^{\prime} = 1-t$, with the excitation gap $\Delta$ plotted in the background. (c) Hole density $\langle \hat{n}_h \rangle$ and (d) $\downarrow$-spin density $\langle \hat{n}_{\downarrow}\rangle$, showing that the magnons are strongly bound to the center even when the hole is not. (e) Sketch of the cross-dimer type ground state for large $u$ (see text). The results are for a ladder of length $l=10$; however, they do not change beyond $l \gtrsim 4$.
• Figure 3: (a) Ladder with two frustrated plaquettes at a distance $p$ from the center. (b) Phase diagram for $l=24$, where $N/2 - S_{\rm total} = 2$ (II and V), $4$ (IV), $\sim \text{min}(4p, 2l-4p)$ (III). (c) Hole density and (d) magnon density for $l = 24, p =5$. (e)-(f) Spin correlations for $l=16, p=4$, showing singlet on each diagonal in region IV and two domain walls in region III. Sites are numbered from left to right, with odd integers for the lower leg and even integers for the upper leg, as in Fig. \ref{['Fig1']}(a). The frustrated plaquettes behave collectively in III and IV.
• Figure 4: (a) Reduction of total spin with the number of diagonal bonds in a $4 \times 5$ grid, where the bonds are added randomly with at most one per plaquette. Dots show the average values and gray scale shows the distributions for up to $1000$ graphs. Allowing two diagonals per plaquette produces a similar variation. (b) Distribution of the hole, given by the radii of the black dots, and spin correlation on the bonds, given by the color scale, for a sample graph, showing singlets on the diagonals while the other sites are aligned. Generally, correlations between non-neighboring sites (not shown) are weaker.
• Figure 5: (a) Correlation between $S_{\rm total}$ and the number of loops of a given size $r$ in $20$-site random graphs, where $\bar{k}$ is the average number of neighbors per site. (b) Average correlation between the spin alignment across individual bonds, $\langle \hat{S}_i.\hat{S}_j \rangle$, and the number of loops that the bond is a part of, showing that the alignment is strongly governed by local frustration. (c)-(e) Oscillation of the distribution (gray scale) and average value (yellow dots) of $S_{\rm total}$ with the minimum loop size, $r_{\rm min}$, in a graph. The statistics for each parameter set are obtained from an ensemble of $1000$ nonseparable graphs.