Effective models of doped quantum ladders of non-Abelian anyons

Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t\text{−}J$ models of $\mathrm{SU}\left(2\right)$ spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by $\mathrm{SU}{\left(2\right)}_3$ theories, which are quantum deformations of the $\mathrm{SU}\left(2\right)$ algebra. Inspired by the physics of $\mathrm{SU}\left(2\right)$ spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t\text{−}J$ models. Here, we aim to explore the combined effects of extended dimensionality and doping by studying ladders composed of coupled chains of interacting itinerant Fibonacci anyons. We show analytically that in the limit of strong rung couplings these models can be mapped onto effective 1D models. These effective models can either be gapped models of hole pairs, or gapless models described by $t\text{−}J$ (or modified $t\text{−}J\text{−}V$) chains of Fibonacci anyons, whose spectrum exhibits a fractionalization into charge and anyon degrees of freedom. The charge degrees of freedom are described by the hardcore boson spectra while the anyon sector is given by a chain of localized interacting anyons. By using exact diagonalizations for two-leg and three-leg ladders, we show that indeed the doped ladders show exactly the same behavior as that of $t\text{−}J$ chains. In the strong ferromagnetic rung limit, we can obtain a new model that hosts two different kinds of Fibonacci particles, which we denote as the heavy $\tau$'s and light $\tau$'s. These two particle types carry the same (non-Abelian) topological charge but different (Abelian) electric charges. Once again, we map the two-dimensional ladder onto an effective chain carrying these heavy and light $\tau$'s. We perform a finite size scaling analysis to show the appearance of gapless modes for certain anyon densities, whereas a topological gapped phase is suggested for another density regime.

Figure 1

(a) Illustration of the standard fusion tree with site labels $Y_i$ (that can be either $\tau$ or $\mathbf{1}$) and bond labels $x_i$. (b) The flat version of the fusion tree. (c) A basis change to a different fusion tree using an $F$ move.

Figure 2

Convention for the right handed braid to exchange two particles.

Figure 3

Schematic representation of a braid on the fusion tree.

Figure 4

(a) Nearest-neighbor magnetic interaction of amplitude $J$. (b) The kinetic hopping (of amplitude $t$) of an anyon to its nearest-neighbor vacant site. The blue circles represent $\tau$ anyons, while the white circles denote vacant sites or holes.

Figure 5

Two-leg ladder: (a) interactions along the leg and rung directions. (b) The zigzag fusion path. The couplings have been indicated.

Figure 6

Evaluation of Hamiltonian terms on a two-leg ladder along the leg direction that involve the braid operation. The blue circles represent $\tau$ particles white the white circles are for vacant sites. The green ellipses denote the particles that are interacting along the leg direction. (a) The magnetic term (b) the kinetic term.

Figure 7

Schematic energy spectra in the presence of a parabolic Coulombic charging energy. One can tune the chemical potential and the repulsion between the particles in order to gap out the higher energy sectors. This allows restricting the calculations to a low-energy subspace.

Figure 8

All possible fusion outcomes for a rung of a two-leg ladder. The blue circles represent $\tau$'s, while white circles represent vacant sites. The $\mathrm{U}\left(1\right)$ and topological charges for the different configurations are denoted in the parenthesis. The superscript $U\left(L\right)$ refers to the $\tau$ lying on the upper (lower) leg.

Figure 9

The various fusion outcomes possible for a doped three-leg ladder. The blue circles represent $\tau$'s while white circles represent vacant sites. The first labels in the parentheses signify the $\mathrm{U}\left(1\right)$ charge, corresponding to the number of $\tau$'s present on each rung and the second refers to their fusion outcome. The superscripts $U,M,L$ refer to the positions of the $\tau$ on the different legs (upper, middle, lower) of the ladder.

Figure 10

Ground-state phase diagram for (a) two-leg ladder and (b) three-leg ladder in the isolated rung limit (with or without charging energy). The lowest energy sectors for isolated rungs with different anyon numbers (labelled by $N_{\mathrm{rung}}$) on ladders of itinerant Fibonacci anyons are indicated. The notations are the same as in Figs. 8 and 9.

Figure 11

Phase diagrams of the two-leg ladder in the strong rung coupling limit. (a) is without a rung charging term and (b) with a large rung charging term $V_{\mathrm{rep}}$. Here, the radius denotes the density of anyons. Depending on filling and coupling several phases can be distinguished: a totally gaped phase (T), effective golden chain models (G), effective $t\text{−}J$ chains (C), paired phases (P), and a phase with two different types of $\tau$ anyons (D). The legend indicates which rung states are relevant in the various phases. See the text for details.

Figure 12

Phase diagrams of the three-leg ladder in the strong rung coupling limit. (a) is without a rung charging term and (b) with a large rung charging term $V_{\mathrm{rep}}$. Here, the radius denotes the density of anyons. Depending on filling and coupling several phases can be distinguished: a totally gaped phase (T), effective golden chain models (G), effective $t\text{−}J$ chains (C), paired phases (P), phase separated phases (PS), and a phase with two different types of $\tau$ anyons (D). The legend indicates which rung states are relevant in the various phases. See the text for details.

Figure 13

Charge spectrum at $J_{\mathrm{rung}}=t_{\mathrm{rung}}=1000,t_{\mathrm{leg}}=1,J_{\mathrm{leg}}=0,V_{\mathrm{rep}}=\infty$. The solid lines denote the HCB spectrum (with an external flux) given by Eq. (32), different colors corresponding to the different charge branches [labelled by $p$ in Eq. (32)]. The black circles denote the spectrum of a $2\times 8$ ladder with $\rho =3/4$. The blue crosses correspond to the effective chain spectrum for $L=8,\tilde{\rho }=1/2$ [see Eq. (33)].

Figure 14

Energy splittings of each of the eight $E\le 0$ energy levels at $K=0$ of the $2\times 8$ ladder with $\rho =3/4$ (symbol/line colors here match the colors of the parabolic branches of Fig. 13) as a function of $t_{\mathrm{leg}}/t_{\mathrm{rung}}$. The values of the leg couplings are $t_{\mathrm{leg}}=1,J_{\mathrm{leg}}=0,V_{\mathrm{rep}}=\infty$ and $t_{\mathrm{rung}}(=J_{\mathrm{rung}})$ takes different values from 100 to 10 000. The fits correspond to the expected $t_{\mathrm{rung}}^{-1}$ behavior.

Figure 15

Splitting of the degenerate levels on switching on different values of $J_{\mathrm{leg}}$ on a $2\times 8$ ladder with $\rho =3/4$ and $J_{\mathrm{rung}}=t_{\mathrm{rung}}=1000,t_{\mathrm{leg}}=1,V_{\mathrm{rep}}=\infty$. The parabolas show the continuous HCB spectrum relevant for $J_{\mathrm{leg}}=0$ (see Fig. 13) and the red crosses represent the ladder spectrum at (a) $J_{\mathrm{leg}}=0.1$ and (b) 1.

Figure 16

Comparison of the energy difference spectra, after subtracting the charge contribution to the energy of each state, of a $2\times 8$ ladder, $\rho =3/4$ with that of the effective $t\text{−}J$ chain $L=8,\tilde{\rho }=1/2$. The couplings on the ladder are $J_{\mathrm{rung}}=t_{\mathrm{rung}}=1000,t_{\mathrm{leg}}=1,V_{\mathrm{rep}}=\infty$ and (a) $J_{\mathrm{leg}}=0.1$ and (b) $J_{\mathrm{leg}}=1$.

Figure 17

Three-leg ladder. The black circles denote the spectrum of a $3\times 6$ ladder with $\rho =5/18$ while the red crosses are for the effective chain with $L=6$ and $\tilde{\rho }=5/6$. The couplings on the ladder are $J_{\mathrm{rung}}=t_{\mathrm{rung}}=1000,t_{\mathrm{leg}}=1,V_{\mathrm{rep}}=\infty$. (a) Charge spectrum at $J_{\mathrm{leg}}=0$. The solid lines denote the HCB spectrum. (b) Energy difference spectrum for $J_{\mathrm{leg}}=0.1$.

Figure 18

Energy splittings of all $E\le 0$ energy levels at $K=0$ of the $3\times 6$ ladder with $\rho =5/6$ as a function of $t_{\mathrm{leg}}/t_{\mathrm{rung}}$. The values of the couplings are $t_{\mathrm{leg}}=1,J_{\mathrm{leg}}=0,V_{\mathrm{rep}}=\infty$, and $t_{\mathrm{rung}}(=J_{\mathrm{rung}})$ takes different values from 100 to 10 000.

Figure 19

Hopping of heavy (or equivalently light) $\tau$'s. (a) Microscopic ladder model: the blue circles denote $\tau$ particles and the white circles are vacant sites on the ladder. (b) The effective chain.

Figure 20

Low-energy spectrum of the two-leg ladder with strong FM rung couplings compared to the 1D heavy and light $\tau$ model. The black circles are for a $2\times 8$ ladder and anyon density $\rho =3/4$, while the red crosses are for the effective chain that it maps to $\left(L=8,\tilde{\rho }=1/2\right)$. The values of the couplings are $t_{\mathrm{leg}}=1,J_{\mathrm{leg}}=0,V_{\mathrm{rep}}=\infty$, and $J_{\mathrm{rung}}=-2t_{\mathrm{rung}}=-2000$.

Figure 21

Energy dispersions of a single heavy (or light) $\tau$ amidst $L-1$ light (or heavy) $\tau$'s for even (circles) and odd (crosses) length chains, with couplings $t=-1$ and $J=0$.

Figure 22

(a) Spectra for the effective chains for heavy and light $\tau \text{"s}$ at $\tilde{\rho }=1/4$ with $t=1,J=0$. Note the momenta have been shifted by $\pi$ for $L$ odd to make all spectra similar and the ground-state energy has been subtracted out. The lowest energy levels at momenta 0, $2\pi /L,\pi -2\pi /L$ and $\pi$ have been tagged as A, B, C, and D, respectively. The second excitations at momenta $\pi$ and $\pi -2\pi /L$ are labeled by E and F, respectively. (b) Finite-size scaling analysis of the A, B, C, D, E, and F energy excitations. Linear (dashed lines) and exponential (full lines) fits are shown around $K=\pi$ and 0, respectively (see text). The scalings of the B, C, and F gaps are also reported in (a).

Figure 23

(a) Spectra of the chains of heavy and light $\tau$'s of different sizes at $\tilde{\rho }=1/2$ and $t=1,J=0$. The ground-state energy has been subtracted out and the momenta are shifted by $\pi$ for odd number of particles of each type so as to get the same zero ground-state momentum in all cases. The lowest energy levels at momenta 0, $2\pi /L,\pi -2\pi /L$ and $\pi$ have been tagged as B, E, G, A, respectively. The second excitations at momenta 0 and $\pi$ are labeled by F and C, respectively. The third excitation at momentum $\pi$ is labeled by D. (b) Finite-size scaling analysis of the energy gaps of (a) vs $1/L$ (see text). The scalings of the E and G gaps are also reported in (a).

Figure 24

Illustration of the Hilbert space for two possible site configurations for a section of the fusion path with two particles and two vacant sites. The state for the site labels is (a) $|{\psi }_{\mathrm{site}}^a\rangle =|0011\rangle$ and (b) $|{\psi }_{\mathrm{site}}^b\rangle =|0101\rangle$. The state for the bond labels is $|{\psi }_{\mathrm{bond}}\rangle =|x_1{x}_2{x}_3{x}_4{x}_5\rangle \in \mathcal{B}\left(\right|{\psi }_{\mathrm{site}}\rangle )$.