Frustration-induced emergent Hilbert space fragmentation

Although most quantum systems thermalize locally on short timescales independent of initial conditions, recent developments have shown this is not always the case. Lattice geometry and quantum mechanics can conspire to produce constrained quantum dynamics and associated glassy behavior, a phenomenon that falls outside the rubric of the eigenstate thermalization hypothesis. Constraints “fragment” the many-body Hilbert space due to which some states remain insulated from others and the system fails to attain thermal equilibrium. Such fragmentation is a hallmark of geometrically frustrated magnets with low-energy “icelike manifolds” exhibiting a broad range of relaxation times for different initial states. Focusing on the highly frustrated kagome lattice, we demonstrate these phenomena in the Balents-Fisher-Girvin Hamiltonian (easy-axis regime), and a three-coloring model (easy-plane regime), both with constrained Hilbert spaces. We study their level statistics and relaxation dynamics to develop a coherent picture of fragmentation in various limits of the XXZ model on the kagome lattice.

Figure 1

Structure of Balents-Fisher-Girvin Hamiltonian. (a)  Balents-Fisher-Girvin Hamiltonian $H_{\mathrm{BFG}}$ contains first-, second-, and third-nearest-neighbor XXZ interactions, and the ice Hamiltonian $H_{\bowtie }$ in Eq. (2) acts on bowtie motifs. The filled and empty circles respectively represent up and down spins in an example ice configuration (three ups and three downs for every hexagonal plaquette). (b) Connectivity graph of $H_{\bowtie }$ on the ice manifold of a 12-site lattice. A vertex represents an ice configuration, a basis state for the ice manifold, and an edge represents a nonzero matrix element of the Hamiltonian between two basis states. The three large connected components each form a topological sector, and the 16 isolated ice configurations are the $2\times 2$ “triangular pinwheel” configurations [shown in panel (e)] related by lattice symmetry. (c), (d) Example ice configurations on the 12-site lattice from topological sectors $(w_1,w_2)=(1,1)$ and $(1,-1)$ respectively. $w_i$ are the spin parities along the two lattice directions marked by the green dashed lines in (c). (e) $2\times 2$ triangular pinwheel-ordered state.

Figure 2

Level statistics of the BFG Hamiltonian. (a) Level statistics of disordered ice Hamiltonian for a 30-site lattice, where $J_i^{\bowtie }$ is site dependent, and chosen from a normal distribution centered at zero $[J_i^{\bowtie }\sim \mathcal{N}(0,1)]$. There are 16568 configurations in the ice manifold, partitioned into four topological sectors of the same size. The teal (labeled “ice-all”) and pink [labeled “ice-(1,1)”] histograms respectively represent the probability density functions $P\left(\tilde{r}\right)$ of the whole ice manifold and the topological sector $(w_1,w_2)=(1,1)$, both within the spin-flip-even sector. (b) Density of states of disordered BFG Hamiltonian for an 18-site lattice, where the XY interaction is bond dependent and chosen from a uniform box distribution $J_{ij}^{\perp }\sim U(-J^{\perp },J^{\perp })$, while the Ising interaction is uniformly fixed to be $J_{ij}^z=1$. The inset shows the “ice gap” ${\mathrm{\Delta }}_{\mathrm{ice}}\equiv min{E}_{N_{\mathrm{ice}}+1}-max{E}_{N_{\mathrm{ice}}}$ (averaged over disorder configurations) as a function of $J$, which closes at $J^{\perp }=J_c^{\perp }\approx 0.17$ (see Appendix pp3, Fig. 9). (c), (d) Low energy level statistics of disordered BFG Hamiltonian $H_{\mathrm{BFG}}$ with nearest-neighbor XY interaction only, for a 24-site lattice in the spin-flip-even symmetry sector. $J^{\perp }=$ (c) 0.01, (d) 0.2.

Figure 3

Three-coloring states and Kempe loops. (a), (b), (c) Three-coloring states of a kagome lattice. (a) and (b) are respectively referred to as $q=0$ and $\sqrt{3}\times \sqrt{3}$ states. The orange lines example red-blue Kempe loops: In the $q=0$ state, Kempe loops are all global, while in the $\sqrt{3}\times \sqrt{3}$ state, they are all local. (d) Number of connected components in Kempe connectivity graph with loops of length 6 ($N_{k6}$) vs number of sites ($N_s$), for various system geometries. The orange dashed line is an exponential fit $N_{k6}\sim W^{N_s}$ to the largest number of components for every system size (marked by a red cross). For comparison, we present an exponential with the $W$ for the scaling of the number of colorings [68] (shown as the green dashed-dotted line), normalized to match the data point at 108 sites. (e) Scaling analyses for the exponential fits. The $y$ axis represents the parameter $W$ from fitting to data points in the range $60\le N_s\le N_s^{\max }$.

Figure 4

Kempe connectivity of three-coloring manifold. (a) Kempe connectivity graph of the three-coloring manifold of a 36-site lattice (168 colorings). A vertex represents a three-coloring, and an edge represents a Kempe move that connects two colorings. The color of a vertex represents the component it belongs to, and the color of an edge represents the length of the move. Two special configurations are highlighted by blue and red circles, $q=0$ and $\sqrt{3}\times \sqrt{3}$, respectively shown in Figs. 5 and 5. The graph clusters into two connected components (labeled A and B) when loops of all lengths are allowed. (b) The hierarchical component structure of the graph in (a). A leaf node (circle) represents a three-coloring, and an internal node (rectangle) represents a connected component when Kempe moves of a certain length or shorter (represented by the gray horizontal lines) are allowed. Vertical lines mark component-subcomponent relationship; a child node is a subcomponent of its parent node. The number shown in an internal node is the number of colorings in the connected component. The colors of vertices and edges match those of panel (a). (c) Kempe connectivity graph of the three-coloring manifold on a 81-site lattice (45 184 colorings). Length dependent structures are presented in the Supplemental Material [69], Video 1.

Figure 5

Dynamics of three-coloring states. (a) Loschmidt echoes $F\left(t\right)\equiv {\left|\langle \psi \left(0\right)|\psi \left(t\right)\rangle \right|}^2$ for length-6 Kempe model on a 81-site kagome lattice. The blue and red curves respectively represent Loschmidt echoes of $q=0$ state and $\sqrt{3}\times \sqrt{3}$ state. The black dotted curves are Loschmidt echos of other coloring states. (b) Degree histogram of the Kempe connectivity graphs with loops of length 6, i.e., distribution of numbers of Kempe loops $d$ for all three-colorings. Inset: $F^{″}\left(t\right)$ vs number of Kempe loops, confirming the relationship $F\left(t\right)\approx 1-d{{\kappa }_6}^2{t}^2$. (c) Loschmidt echoes for XXZ model $H_{\text{XXZ}}^{\text{n.n.}}$ at $\mathrm{\Delta }{J}^z\equiv J^z+1/2=0.01$, 0.02, 0.03, with 18 sites.

Figure 6

Geometry of a finite size kagome lattice with shape $(2,-2)\times (2,4)$. The purple arrows on the right mark the lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ of the Bravais lattice (represented by green pluses). The blue arrows are the lattice vectors of the superlattice (represented by red crosses): ${\mathbf{b}}_1=2{\mathbf{a}}_1-2{\mathbf{a}}_2$ and ${\mathbf{b}}_2=2{\mathbf{a}}_1+4{\mathbf{a}}_2$.

Figure 7

(a) Connectivity graph of $H_{\bowtie }$ on Balents-Fisher-Girvin ice manifold on a 36-site kagome lattice with 107 176 ice configurations. Each vertex represents an ice configuration, and an edge represents a nonzero tunneling element between two ice configurations that the edge connects. The ice manifold fragments into four topological sectors, shown as four connected components in the graph. The color of a vertex indicates the connected component it belongs to. (b)–(d) Representative configurations in each of the four topological sectors (chosen arbitrarily), for $(2,-2)\times (2,4)$ cluster. Opaque black circles (open circles for down spins, and filled circles for up spins) mark the sites within the cluster, and gray circles represent translated image sites under periodic boundary condition. Corresponding values of topological invariant $(w_1,w_2)$ are (b) $(1,-1)$, (c) (1,1), (d) $(-1,1)$, and (e) $(-1,-1)$.

Figure 8

Level statistics for disordered $H_{\bowtie }$ on a 30-site lattice of shape $(3,-1)\times (1,3)$. The panels show the probability density distributions $P\left(\tilde{r}\right)$, for (a)–(d) ${\mathbb{Z}}_2$ symmetric sector of connected components 1, 2, 3, and 4, respectively, (e)–(h) ${\mathbb{Z}}_2$ antisymmetric sector of connected components 1, 2, 3, and 4, respectively, (i) ${\mathbb{Z}}_2$ symmetric sector (combining all four connected components), (j) ${\mathbb{Z}}_2$ antisymmetric sector, and (k) the complete ice manifold. The blue and red dashed curves respectively mark the $P\left(\tilde{r}\right)$ for Poisson and GOE distributions. For the GOE level statistics, we use the expression for $P\left(\tilde{r}\right)$ from Ref. [85].

Figure 9

(a) Box plots for the “ice gap” ${\mathrm{\Delta }}_{\text{ice}}$, sampled over disorder configurations $\left\{J_{ij}^{\perp }\right\}$. The box ranges from the lower to upper quartiles, with a line marking the median, and the whiskers showing the range of the data. The number above a box indicates the number of sampled disorder configurations, and the dashed lines are fits to the disorder averaged ${\mathrm{\Delta }}_{\mathrm{ice}}$ with the form ${\mathrm{\Delta }}_{\mathrm{ice}}/J^z=1-J^{\perp }/J_c^{\perp }$. (b) $J_c^{\perp }$ vs number of sites. The horizontal lines mark the standard error for $J_c^{\perp }$.

Figure 10

Hierarchical connectivity structures of the three-coloring manifold with Kempe moves at various loop lengths, for (a) 36-site kagome lattice of shape $(2,-2)\times (2,4)$, and (b) 48-site lattice of shape $(4,0)\times (0,4)$. The leaf nodes at the bottom are three-coloring configurations, and the internal nodes labeled $\ell$ are connected components when loops of length $\ell$ or less are allowed, with increasing value of $\ell$ from bottom to top.

Figure 11

(a) Number of connected components in Kempe connectivity graph with loops of length 6 ($N_{k6}$) vs number of sites ($N_s$), for various system geometries, plotted on a log-log scale. The orange dashed line is a power-law fit $N_{k6}\sim {N_s}^{\alpha }$ to the largest number of components for every system size (marked by a red cross). (b) Scaling analyses for the power-law fits. The $y$ axes represent parameters from fits to data points in the range $60\le N_s\le N_s^{\max }$.

Figure 12

(a) Loschmidt echoes for the length-6 Kempe model on an 81-site kagome lattice [same the data as Fig. 3] plotted as $1-{\left|\langle \psi \left(0\right)|\psi \left(t\right)\rangle \right|}^2$ vs ${\kappa }_{6}t$ in log-log scale. For small $t$ (${\kappa }_{6}t\lesssim 0.1$), the curves follow $1-{\left|\langle \psi \left(0\right)|\psi \left(t\right)\rangle \right|}^2\propto t^2$. (b), (c) Degree histograms of the Kempe connectivity graphs with loops of length 6 for systems of shapes $(3,-3)\times (3,6)$ (81 sites) in (b) and $(6,0)\times (0,6)$ (108 sites) in (c). (d) Average degree vs number of sites for various lattice geometries.

Figure 13

Loschmidt echo plot for $q=0$ and $\sqrt{3}\times \sqrt{3}$ states on the 18-site lattice of shape $(3,0)\times (1,2)$, at $\mathrm{\Delta }{J}^z\equiv J^z+1/2=0.01$.