Bose-Hubbard realization of fracton defects

Bose-Hubbard models are simple paradigmatic lattice models used to study dynamics and phases of quantum bosonic matter. We combine the extended Bose-Hubbard model in the hard-core regime with ring-exchange hoppings. By investigating the symmetries and low-energy properties of the Hamiltonian we argue that the model hosts fractonic defect excitations. We back up our claims with exact numerical simulations of defect dynamics exhibiting mobility constraints. Moreover, we confirm the robustness of our results against fracton symmetry breaking perturbations. Finally, we argue that this model can be experimentally realized in recently proposed quantum simulator platforms with big time crystals, thus paving a way for the controlled study of many-body dynamics with mobility constraints.

Figure 1

An illustration of all possible hopping and interaction processes in the model on a square 2D lattice: standard nearest-neighbor hopping $t_1$, ring-exchange terms $t_2$ and the nearest-neighbor interaction $V$. The ring-exchange interaction we are considering consists of a simultaneous tunneling of two particles to the second neighbors along $1\times 2$ and $2\times 1$ plaquettes. Site labels A and B denote two bipartite sublattices of a square lattice. Throughout the article we are considering strongly repulsive regime at half filling where, in the ground state, all the particles fill completely only one sublattice realizing the checkerboard charge density wave (CDW) order. The focus of our analysis is put on the lowest lying excited energy bands consisting of fracton excitations.

Figure 2

Schematic view of the CDW dislocation movement due to ring-exchange interactions. White circles show particles missing from the checkerboard pattern and red circles show particles present in a sublattice, which is empty in the ground state. A red cross indicates a prohibited hopping. When the system is in the ground state (GS) there are no dislocations and no hopping can take place. For nonlocal dislocations, hopping is still not possible, and hence the motion of such excitations is zero-dimensional (0D). For a single local dislocation one has one-dimensional (1D) motion and for a double local dislocation one has two-dimensional (2D) motion.

Figure 3

Exact time propagation of elementary dislocations. While panels [(a)–(c)] show a propagation of a single local dislocation, panels [(d)–(f)] illustrate a motion of a bound double dislocation. Each row is associated with time point $\tau =0$, $\tau =3.5/t_2$, and $\tau =7/t_2$. The panels confirm the analytical arguments of the main text that a single local dislocation can propagate only in one dimension but two bound dislocation state propagates freely in two dimension. The movement of the later is twice slower compared to a single dislocation state. The lattice size is 48 x 48 sites, each site is represented by a single pixel. Positive probability (blue) shows particle density in empty lattice, as compared to the CDW ground state. Negative probability (red) is used to show hole density, as compared to the CDW ground state. The color scale is cut at low values to be consistent along all images in a column. Model parameters are $t_1=0$, $t_2=0.01V$.

Figure 4

Stability of the fractonic behavior of a single initial dislocation as seen in the top-left panel of Fig. 3. The color scale represents root sum of squared differences of initial and final marginal distributions. The plotted probability is the value achieved during time evolution on $20\times 20$ lattice for time $\tau =2/t_2$. $V=1$ is assumed.

Figure 5

Exact time propagation for a single dislocation at time $\tau =4/t_2$ for $t_2=1V$ and (a) $t_1=0$, for (b) $t_1=0.2V$ and for (c) $t_2=0.1V$ and $t_1=0.2V$. The lattice size is 20 x 20 sites, each site is represented by a single pixel.

Figure 6

Exact time evolution of few fractons for $t_1=0$. Columns present snapshot of the evolution for times $\tau =0,1,7$. [(a)–(c)] The evolution of a double defect for $t_2=0.5V$. Here the value of $t_2$ is much larger than in panels [(d)–(f)] in Fig 6, and because of this, one-dimensional defects, which have a larger interaction energy, appear out of the squared shape because of their larger group velocity. [(d)–(f)] The evolution of two orthogonal separated single defects for $t_2=0.5V$. We see that these defects move straight through each other and no two-dimensional mobility arises, but due to their interaction the time evolution is asymmetric as the propagation is delayed after the collision. [(g)–(i)] The scattering of a lineon on a zero-dimensional immobile fracton.

Figure 7

Schematic of the experimental protocol. Panels [(a)–(c)] illustrate subsequent building blocks of the protocol, which is summarized on panel (d). Note that for illustrative purposes each panel presents a relatively small $5\times 5$ lattice, but experimentally relevant sizes can go up to $100\times 100$ sites (cf. Ref. [90] for accessible lattice sizes in 1D experiments). Panel (a) shows a 2D time crystal with $2\times 1$ and $1\times 2$ ring-exchange interactions. The axes ${\theta }_x$ and ${\theta }_y$ represent directions of the flat torus shown as black rectangle, with parts of it repeated to illustrate the boundary conditions. Such a lattice is achieved using drives: ${\text{drive}}_x\left(\tau \right)={\text{drive}}_y\left(\tau \right)=cos\left(5\omega \tau \right)$. Note that in the first step only a subset of the ring-exchange interactions is realized and only a few of them are shown for illustrative purposes. Panel (b) shows the adiabatic movement step, which is introduced to restore the full translational symmetry of the effective Hamiltonian. This is achieved by introducing a slow adiabatic shift in one of the drives, e.g., ${\text{drive}}_x\left(\tau \right)=cos[5\omega \tau +{\theta }_0\left(\tau \right)]$. This drive change is stopped after the lattice is moved by a single position to the right ${\theta }_0\left({\tau }_{end}\right)=2\pi$, which also means that the $x$ drive made one more oscillation then the $y$ drive. After this step new set of ring-exchange interactions is realized [panel (c)]. After 5 such steps, we return to original positions. Time averaged Bose-Hubbard Hamiltonian realizes exactly the model (1) described in this article.