Quantum spin ladders of non-Abelian anyons

Quantum ladder models, consisting of coupled chains, form intriguing systems bridging one and two dimensions and have been well studied in the context of quantum magnets and fermionic systems. Here we consider ladder systems made of more exotic quantum mechanical degrees of freedom, so-called non-Abelian anyons, which can be thought of as certain quantum deformations of ordinary SU(2) spins. Such non-Abelian anyons occur as quasiparticle excitations in topological quantum fluids, including $p_x+i{p}_y$ superconductors, certain fractional quantum Hall states, and rotating Bose-Einstein condensates. Here we use a combination of exact diagonalization and conformal field theory to determine the phase diagrams of ladders with up to four chains. We discuss how phenomena familiar from ordinary SU(2) spin ladders are generalized in their anyonic counterparts, such as gapless and gapped phases, odd and even effects with the ladder width, and elementary “magnon” excitations. Other features are entirely due to the topological nature of the anyonic degrees of freedom. In general, two-dimensional systems of interacting localized non-Abelian anyons are anyonic generalizations of two-dimensional quantum magnets.

Figure 1

Three coupled chains of interacting anyons (indicated by the filled circles). The interaction along ($J_{\mathrm{leg}}$) and perpendicular ($J_{\mathrm{rung}}$) to the chains are indicated by the ellipses.

Figure 2

Representation of the parameter space of the model on a circle. The couplings $J_{\mathrm{rung}}$ and $J_{\mathrm{leg}}$ can be either positive (AFM) or negative (FM). The different phases in each of the quadrants labeled by $J_{\mathrm{rung}}$–$J_{\mathrm{leg}}$ are given in Table . The (gray) squares at $\pi /4$ correspond to isotropic couplings $|J_{\mathrm{rung}}/J_{\mathrm{leg}}|=1$. The (orange) stars mark the parameters in the vicinity of the strong-coupling limit $\left|J_{\mathrm{rung}}\right|\gg \left|J_{\mathrm{leg}}\right|$ used in this work, namely, $\theta =\{3\pi /7,4\pi /7,10\pi /7,11\pi /7\}$ labeled from 1 to 4, respectively.

Figure 3

(a) The fusion path connecting $N$ Fibonacci [su(2)${}_3$] anyons. Basis states correspond to all admissible labelings of the edges $x_i$ with anyon charges $0$ and $1/2$ that satisfy the fusion rules. (b) A nearest-neighbor coupling in a chain of anyons can be calculated by using the $F$ matrix to transform to a unique basis in which the fusion product of the two anyons is one of the variables. (c) Longer-range interactions first need to be mapped to nearest-neighbor interaction by braiding anyons. (d) A Dehn twist, giving an additional phase factor is needed if the interaction winds around the torus. For su(2)${}_3$ the phase is ${\psi }_x=0$ for $x=0$ and ${\psi }_x=-4\pi /5$ for $x=1/2$.

Figure 4

Sketch of a typical ladder system of extent $4\times L$ ($L=6$), where the dots indicate the location of $\tau$ anyons. The fusion path $\mathcal{C}$ (dotted line) with labelings $x_i$ (or $y_i$) is used define an (arbitrary) ordering of the sites, which is used in the definition of the basis states. The exchange couplings $J_{\mathrm{leg}}$ and $J_{\mathrm{rung}}$ are indicated.

Figure 5

A ladder system as in Fig. 4, but with periodic boundary conditions, where an additional rung-coupling term connects the two out legs (vertical red segments).

Figure 6

Dispersion of a generalized “magnon” excitation along the ladder direction for two-leg and four-leg ladders and various couplings. The magnon excitation is created by flipping a local label $x_i$ from $1$ to $\tau$, which can then propagate down the ladder. Data is shown for $2\times 20$ (and $2\times 21$) ladders (open symbols) as well as $4\times 8$ ladder (closed symbols) and coupling parameters $\theta =\pi /4$ ($J_{\mathrm{leg}}=J_{\mathrm{rung}}=\sqrt{1/2}$) and $\theta =3\pi /4$ ($J_{\mathrm{leg}}=-J_{\mathrm{rung}}=-\sqrt{1/2}$). The solid lines are a guide to the eye, obtained from Fourier series fit to the data. The arrows indicate extrapolations to the thermodynamic limit as shown in Fig. 8.

Figure 7

Finite-size extrapolations of the energy gaps of three-leg ladder systems for various couplings $\theta$, with ferromagnetic rung and antiferromagnetic leg couplings in the (a) $K=0$ and (b) $K=\pi$ momentum sectors. Extrapolations to the thermodynamic limit are obtained by fitting the numerical data to the form $\Delta (L)\simeq \Delta (\infty )+\frac{C}{L}\exp (-L/\xi )$, where $\xi$ is a correlation length. In the $K=0$ sector, the first excitation energy (open symbols) extrapolates to zero (indicating that the GS is twofold degenerate in the thermodynamic limit), while the extrapolation of the second excitation energy (filled symbols) indicated a finite gap.

Figure 8

Extrapolated values $\Delta (L\to \infty )$ of the gaps at the two crystal momenta $K=0$ and $K=K_0$ corresponding to the zero-energy modes of the single chains. $\Delta (\infty )$ is plotted as a function of coupling parameter $\theta$, i.e., $\theta \in [0,\pi ]$ for two- and four-leg ladders with AFM rung coupling (a) and $\theta \in [-\pi ,0]$ for a three-leg ladder with FM rung coupling (b). For AFM (FM) leg coupling, $K_0=\pi$ ($K_0=2\pi /3$) and clusters up to $2\times 20$ ($2\times 21$), $3\times 12$ ($3\times 12$), and $4\times 8$ have been used. $\Delta (\infty )$ is deduced by fitting the data as $\Delta (L)\simeq \Delta (\infty )+\frac{C}{L}\exp (-L/\xi )$, where $\xi$ is a correlation length.

Figure 9

Finite-size scaling of $3\times L$ (three-leg) ladders of sizes up to $L=12$, with AFM rung coupling and with both AFM (top) and FM (bottom) leg couplings. The lowest eigenenergies per rung (symbols) are plotted vs $1/L^2$. The right-hand and left-hand panels correspond to two different crystal momenta, the ones characterizing the zero-energy modes of a single chain. The $\theta$ values (indicated on the plot) correspond to the strong rung couplings $|J_{\mathrm{rung}}/J_{\mathrm{leg}}|~4.38$ labeled as “1” and “2” in Fig. 2. CFT scalings are shown: (i) A linear fit of the GS energy (labeled by “0”) accurately provides the overall energy scale (i.e., the velocity); and (ii) all other straight lines are expected CFT scalings using the conformal weights $2h_n$ and the central charge $c$ indicated on the plots [see Eq. (12) in the text].

Figure 10

Finite-size scaling of $2\times L$ (two-leg) ladders with FM rung coupling, with both FM (top) and AFM (bottom) leg couplings and with up to $2\times 21$ and $2\times 20$ sites, respectively. The $\theta$ values (indicated on the plot) correspond to the strong rung couplings $|J_{\mathrm{rung}}/J_{\mathrm{leg}}|~4.38$ labeled as “3” and “4” in Fig. 2. Same notations and same analysis as in Fig. 9.

Figure 11

Phase diagrams of the two-leg and four-leg ladders (a) and of the three-leg ladders (b) vs the couplings $J_{\mathrm{leg}}=\cos \theta$ and $J_{\mathrm{rung}}=\sin \theta$. The central charge of the gapless phases is indicated.

Figure 12

Finite-size scaling of the lowest eigenenergies (normalized per rung) of a two-leg ladder with $J_{\mathrm{rung}}=0$ and $J_{\mathrm{leg}}=\pm 1$ (decoupled chains) vs $1/L^2$. The right-hand and left-hand panels correspond to two different crystal momenta, the ones characterizing the zero-energy modes of a single chain. Same procedure and notations as Figs. 9 and 10.

Figure 13

Low-energy excitations of four-leg ladders with open boundary conditions. The GS energy (K=0) is used as energy reference. FM (AFM) rung (leg) coupling are considered, i.e., giving rise to the $c=7/10$ phase of Fig. 11a: Moderately strong rung coupling, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}|~1.73$ ($\theta =5\pi /3$), and strong rung coupling, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}~4.38$ ($\theta =11\pi /7$), are shown. (a), (c) Low-energy spectra of a $4\times 8$ ladder vs momentum along the ladder. (b), (d) Finite-size scalings of the low-energy levels for the two cases shown in (a) and (c). Fits to $c=7/10$ CFT invariant spectra are provided: Expected levels corresponding to primary (secondary) fields of the CFT are shown by red boxes (blue circles). The overall energy scale of the CFT is set by adjusting the position of the $K=0$, $2h=1/5$ state to the corresponding energy level of the $4\times 8$ cluster.

Figure 14

Bond-energy correlations in a $4\times 8$ ladder with open boundary conditions (OBC) along the rungs (i.e., with two inner and two outer eight-site legs) as a function of the distance between two (parallel) bonds along the leg direction or across the ladder. The bonds are oriented simultaneously along the rungs (rung-rung correlator) or along the legs (leg-leg correlator). The disconnected part has been subtracted and the data are normalized with regard to the zero-distance autocorrelation. Data are shown for $J_{\mathrm{rung}}<0$ and $J_{\mathrm{leg}}>0$, i.e., in the $c=7/10$ phase of Fig. 11a. (a) Isotropic couplings, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}=1$ ($\theta =7\pi /4$); (b) strong rung coupling, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}~4.38$ ($\theta =11\pi /7$).

Figure 15

Low-energy excitations of four-leg ladders with periodic boundary conditions. The GS energy (K=0) is used as energy reference. (a), (c) Spectra of a $4\times 8$ ladder for values of $\theta$ corresponding to FM (AFM) rung (leg) coupling. Isotropic couplings, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}=1$ ($\theta =7\pi /4$), and moderately strong rung coupling, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}~1.73$ ($\theta =5\pi /3$), are shown. (b), (d) Finite-size scalings of the low-energy levels for the two cases shown in (a) and (c) revealing threefold degenerate GSs and a finite gap.

Figure 16

Low-energy excitations of $4\times 6$ ladders with FM rung and leg couplings. The GS energy ($K=0$) is used as energy reference. (a) Open boundary condition (in the rung direction). The data are obtained for strong rung coupling, $J_{\mathrm{rung}}/J_{\mathrm{leg}}~4.38$ ($\theta =10\pi /7$). A fit to a $c=4/5$ CFT invariant spectrum is shown: Expected levels corresponding to primary (secondary) fields of the CFT are represented by red boxes (blue circles) and the overall energy scale is set by adjusting the position of the $K=0$, $2h=4/5$ energy level. (b) Periodic boundary condition (in the rung direction). The data are obtained for isotropic couplings, $J_{\mathrm{rung}}/J_{\mathrm{leg}}=1$ ($\theta =5\pi /4$). A large gap is seen above three quasidegenerate levels.

Figure 17

Bond-energy correlations in a $4\times 8$ ladder with periodic boundary conditions (PBC) along the rungs as a function of the distance (in the leg direction) between two (parallel) bonds. The bonds are oriented simultaneously along the rungs (rung-rung correlator) or along the legs (leg-leg correlator). The disconnected part has been subtracted and the data are normalized with regard to the zero-distance autocorrelation. Data are shown for $J_{\mathrm{rung}}<0$ and $J_{\mathrm{leg}}>0$, i.e., in the $c=7/10$ phase of Fig. 11a and for isotropic couplings, $\left|J_{\mathrm{rung}}\right|/J_{\mathrm{leg}}=1$ ($\theta =7\pi /4$).