Fractons from a liquid of singlet pairs

Fracton phases of matter feature a variety of exciting phenomena stemming from the restricted mobility of their quasiparticles. Here we consider a model of interacting electrons in one dimension that describes hopping of spin-singlet pairs and obeys both charge and dipole conservation laws. The model contains Bethe ansatz integrable sectors which allow us to solve the ground state and calculate the exact spin and single-electron excitation gaps. We observe hallmarks of fractonic behavior, including localization of single-electron excitations and propensity to clustering. Our results demonstrate the important role of the dipole moment conservation law in a simple model of spin-1/2 fermions.

Figure 1

Correlated hopping processes that enable the motion of a spin-singlet pair. The two processes amount to second-neighbor hopping, with or without spin flip, conditioned to the presence of an electron with opposite spin in the intermediate site.

Figure 2

Three distinct types of particles in the two-electron sector: (a) itinerant singlet pair, (b) two isolated electrons, and (c) localized triplet pair.

Figure 3

Ground-state energy density as a function of the electronic filling $n$.

Figure 4

One-particle (${\mathrm{\Delta }}_{1p}$), two-particle (${\mathrm{\Delta }}_{2p}$), and spin (${\mathrm{\Delta }}_s$) excitation gaps as a function of the electronic density $n$.

Figure 5

Spin-triplet-pair excitation in the singlet-pair liquid. The localized triplet acts as a hard wall for itinerant singlet pairs.

Figure 6

Scattering state with a right-moving singlet pair. From top to bottom, the electron jumps two sites to the left when the singlet pair is transmitted.

Figure 7

Ground-state energy in the (1,1,0) subspace as a function of the dipole moment $D$. The numerical data was obtained for an open chain with $L=81$ sites.

Figure 8

Single-electron probability density in the (1,1,0) space at various values of $D$. The numerical data was obtained for a wire with $L=31$ sites.

Figure 9

Schematic representation of an open chain with two electrons and one singlet pair. The first and second electrons can occupy the positions $d_1\pm 1$ and $d_2\pm 1$, respectively, depending on the position of the singlet pair.

Figure 10

Ground-state energy as a function of the relative distance $r=d_2-d_1$ between the two electrons in the presence of an itinerant singlet pair. The center of mass coordinate is fixed at the center of the chain, $d_1+d_2=L+1$. Here we set $L=81$.

Figure 11

Constrained dynamics of a single electron immersed in the singlet-pair liquid. Here we show a schematic picture of a state in the (4,1,0) subspace. The electron can only hop between the red sites inside the shaded area.

Figure 12

Sketch of the adiabatic approximation scheme. At each position of the single electron, the chain is subdivided into two smaller chains. Our ansatz is then constructed by taking the ground-state wave function of the singlet-pair liquid in each box. The effective length and the number of singlet pairs for each box change when a singlet pair tunnels across the single electron.

Figure 13

Single-electron probability density calculated in the adiabatic approximation for a system with $M=6$ singlet pairs in a chain with $L=31$ sites. The different curves correspond to different values of the dipole moment $D$.

Figure 14

Phase diagram for the model with an external magnetic field. Above the critical point, $h>h_c$, an island of polarized electrons emerges in the ground state.

Figure 15

Critical repulsion $V_c$ as a function of electronic filling $n$. For $n>2/3$, the nearest-neighbor interaction does not modify the one-particle excitation gap.

Figure 16

Two-particle continuum and bound state bands in the spectrum of two-body problem including single-particle hopping $t$ as a perturbation. Here we set $t=0.1$ in units of the pair hopping amplitude.