Fractional charge and emergent mass hierarchy in diagonal two-leg $t–J$ cylinders

We define a class of “diagonal” $t–J$ ladders rotated by $\pi /4$ relative to the canonical lattice directions of the square lattice, and study it using density matrix renormalization group. Here, we focus on the two-leg cylinder with a doped hole concentration near $x=\frac{1}{4}$. At exactly $x=\frac{1}{4}$, the system forms a period 4 charge density wave and exhibits spin-charge separation. Slightly away from $\frac{1}{4}$ doping, we observe several topologically distinct types of solitons with well-defined fractionalized quantum numbers. Remarkably, given the absence of any obvious small parameter, the effective masses of the emergent solitons differ by several orders of magnitude.

Figure 1

(a) Diagonal two-leg ladder with cylinder boundary condition (CBC): open in $x$, periodic in $y$; empty circles and dashed lines represent the periodic boundary. (b) The local site-exchange symmetry of two-leg ladder. The dotted circle indicates one rung. Exchanging the two sites on the rung preserves the Hamiltonian.

Figure 2

(a) Density profile of the ground state of a $2\times 123$ diagonal cylinder with $2\times 31$ doped holes. We use an odd length cylinder to minimize the boundary effects. (b) Enlarged view of the red rectangle part in (a). There is a period 4 density pattern $ABA\tilde{B}$. (c) The length dependence of $\delta {\rho }_{A\overline{B}}\equiv \rho \left(A\right)-[\rho \left(B\right)+\rho \left(\tilde{B}\right)]/2$; extrapolated to the limit $L\to \infty$ this difference approaches $3.631\left(3\right)\times {10}^{-2}$. (d) The length dependence of $\delta {\rho }_{\tilde{B}B}\equiv \rho \left(\tilde{B}\right)-\rho \left(B\right)$; extrapolated to the limit $L\to \infty$ this difference approaches $0.279\left(3\right)\times {10}^{-2}$. Here, $\rho$ is the averaged density of one type of site in the bulk. The lattice length varies from $L=67$ to 123.

Figure 3

Schematic illustrations of three topologically distinct domain walls. The first chain is a reference without any domain walls. The three dashed rectangles below enclose the domain walls with different subtended angle $\mathrm{\Delta }{\alpha }_1$. Each domain wall is associated with different solitons $S$.

Figure 4

Density profiles of the $L=123$ lattice with $5\le i\le 118$ (excluding boundary regions). (a) The ground state with $2\times 31$ doped holes and $S_z^{\mathrm{tot}}=1$ supports two neutral solitons. The density and spin are shown in blue and red, respectively. (b) A metastable state with $S_z^{\mathrm{tot}}=0$ and $2\times 31$ doped holes. (c), (d) Two metastable states with $S_z^{\mathrm{tot}}=0$ and $2\times 30$ doped holes. (e), (f) Two metastable states with $S_z^{\mathrm{tot}}=0$ and $2\times 32$ doped holes.

Figure 5

(a) Dependence of the spin gap on $1/L$; the extrapolation $L\to \infty$ yields ${\mathrm{\Delta }}_s=3.0\left(1\right)\times {10}^{-5}$. $L$ varies from 51 to 99. (b) The schematic band dispersion of diagonal two-leg cylinder at $\frac{1}{4}$ doping. The orange line is the unoccupied flat band. The blue one is the dispersive band with two Fermi points (black dots) at $k_F=\pm 3\pi /4$.

Figure 6

Length dependence of inverse of spin-spin correlation length $1/{\xi }_s$ on systems varied from $L=67$ to 195.