Anyonic quantum spin chains: Spin-1 generalizations and topological stability

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)${}_k$ anyons, well-known deformations of SU(2), in which only the first $k+1$ angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin $S=1$ chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter $k$ when considering su(2)${}_k$ anyonic theories with $k\ge 5$, as well as for the special case of the su(2)${}_4$ theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-$\frac{1}{2}$ chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

Figure 1

The anyonic spin-1 chain.

Figure 2

The basis transformation for the anyonic spin-1 chain.

Figure 3

Various stages in the calculation of the topological symmetry operator.

Figure 4

Relation between the flux $i$ through the chain and the additional spin-$l$ anyon.

Figure 5

Phase diagrams of the ordinary SU(2) spin-1 chain in a projector representation (9) with $J_1=-\sin \left({\theta }_{2,1}\right)$ and $J_2=\cos \left({\theta }_{2,1}\right)$.

Figure 6

Phase diagrams of the anyonic su(2)${}_k$ spin-1 chain with odd $k$ in a projector representation (9), where $J_1=-\sin \left({\theta }_{2,1}\right)$ and $J_2=\cos \left({\theta }_{2,1}\right)$. With increasing (odd) index $k\ge 5$ the phase boundaries move as indicated by the arrows.

Figure 7

The su(2)${}_5$ spin-1 chain. The energy spectra for the various phases of the phase diagram are shown in the top left panel. For the critical phases/points the energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles indicate the descendant fields. The energies predicted by CFT are given in green (red) for primary (descendant) fields. The topological symmetry sector is indicated by the violet index. Data shown are for system sizes $L=18$ and $L=15$, respectively.

Figure 8

The su(2)${}_7$ spin-1 chain. The energy spectra for the various phases of the phase diagram are shown in the top left panel. For the critical phases/points the energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields, red circles indicate the descendant fields. The energies predicted by CFT are given in green (red) for primary (descendant) fields. The topological symmetry sector is indicated by the violet index. Data shown are for system sizes $L=16$ and $L=14$, respectively.

Figure 9

The AKLT construction of the valence-bond-solid state on a finite chain of spin-1 degrees of freedom. Each solid circle represents a spin-$\frac{1}{2}$ variable, each dotted ellipse corresponds to a spin-1 particle, and and each line connecting two spin-$\frac{1}{2}$ variables symbolizes a singlet bond.

Figure 10

The eigenenergies $\Delta E_i(x_0,x_{L+1}):=E_i(x_0,x_{L+1})-E_0(x_0,x_{L+1})$ ($i\ge 1$) of the su(2)${}_5$ anyonic spin-1 chain with fixed boundaries $x_0$ and $x_{L+1}$ as a function of $1/L$ at ${\theta }_{2,1}=-0.01\pi$. The legend at the lower left side indicates the values of $x_0$ and $x_{L+1}$. The energy $E_0(x_0,x_{L+1})$ is the lowest energy and not necessarily a “ground-state energy.” For $x_0=x_{L+1}=1$ and for $x_0=x_{L+1}=2$, there are two almost degenerate zero-energy states, and $\Delta E_1(x_0,x_{L+1})$ corresponds to the finite-size splitting of the two ground states that decay exponentially with system size (see the inset).

Figure 11

Phase diagram of the even-$k$ anyonic su(2)${}_k$ spin-1 chain in a projector representation (9), where $J_1=-\sin \left({\theta }_{2,1}\right)$ and $J_2=\cos \left({\theta }_{2,1}\right)$. The locations of the phase boundaries correspond to the case $k=6$. Some of the phase boundaries move with increasing (even) $k$; the arrows indicate the direction of the change.

Figure 12

Phase diagrams of the su(2)${}_4$ spin-1 chain in the integer sector and half-integer sector and different projector representations. The colored circles indicate special points in the $c=1$ gapless phase that can be matched to the labeled CFTs.

Figure 13

The su(2)${}_4$ chain: IS. Energy spectra at various in the gapless phases of the phase diagram in Fig. 12a. The energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles indicate the descendant fields. The topological symmetry sector is indicated by the violet index. Data shown are for system size $L=20$.

Figure 14

The su(2)${}_4$ chain: HIS. Energy spectra at various points in the gapless phases of the phase diagram displayed in Fig. 12c. The energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles indicate the descendant fields. The topological symmetry sector is indicated by the violet index. Data shown are for system size $L=20$.

Figure 15

The ground-state energy per site (top panel) and its first and second derivative (middle and bottom panels, respectively) of the IS su(2)${}_4$ chain. Data shown are for system size $L=18$.

Figure 16

The ground-state energy per site of the IS su(2)${}_4$ chain for system sizes $L=8,...,20$, in steps of two.

Figure 17

Numerical estimate (system size $L=20$) of parameter $p$ from the eigenenergies that are associated with operators with scaling dimensions $1/2p$ (red squares) and $4/2p$ (blue dots) in the IS gapless phase with twist operators at $k=0$ and $k=\pi$. The parameter $p$ is about 1 at ${\theta }_{2,0}=0$ and grows to about 9 at ${\theta }_{2,0}=\pi /2$. For angles ${\theta }_{2,0}>\pi /2$, the estimates of $p$ obtained from the two operators start to deviate. The black dots correspond the scaling dimension of the fields with dimensions $h=1/8$ and $h=9/8$ multiplied by eight, as obtained from exact diagonalization. The shaded region indicates the range of ${\theta }_{2,0}$ for which the latter dimension lies between $8.9<8h<9.1$.

Figure 18

Numerical estimate (system size $L=20$) of parameter $p$ from the eigenenergies that are associated with operators with scaling dimensions $1/2p$ (red squares) and $4/2p$ (blue dots) in the HIS gapless phase. The black dots correspond to the scaling dimension of the fields with dimensions $h=1/8$ and $h=9/8$ multiplied by eight, as obtained from exact diagonalization. The shaded region indicates the range of ${\theta }_{2,0}$ for which the latter dimension lies between $8.9<8h<9.1$.

Figure 19

The su(2)${}_2$ spin-$\frac{1}{2}$ chain. Energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles specify the descendant fields. The topological symmetry sector is indicated by the violet index. Data shown are for system size $L=40$.

Figure 20

The su(2)${}_4$ spin-$\frac{1}{2}$ chain. Energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles specify the descendant fields. The topological symmetry sector is indicated by the violet index. Data shown are for system size $L=28$ and $L=24$.

Figure 21

The su(2)${}_5$ spin-$\frac{1}{2}$ chain. Energy spectra have been rescaled to match the CFT prediction given in Eq. (15). Green squares indicate the location of the primary fields; red circles indicate the descendant fields. The topological symmetry sector is indicated by the violet index. Data shown are for system size $L=26$ and $L=25$, respectively.

Figure 22

Phase diagrams of the various spin-1 models considered in the paper: (a) the bilinear-biquadratic spin-1 Heisenberg chain, (b) the generic even $k\ge 6$ anyonic spin-1 chain, (c) the generic odd $k\ge 5$ anyonic spin-1 chain, (d) the special case $k=4$.

Figure 23

Basis (fusion diagram) of a chain of spin-$j$ anyons [$j=1/2$ in the case of the su(2)${}_k$ spin-$\frac{1}{2}$ chain, discussed in Sec. 6, and $j=1$ in the case of the su(2)${}_k$ spin-1 chain, discussed in Sec. 3].

Figure 24

The associativity of the fusion rules.

Figure 25

The relation between the flux of anyon spin $i$ through a loop of anyon spin $l$ to the case without the anyon loop $l$.

Figure 26

Basis transformation used to obtain the fusion product of two neighboring spin-$j$ anyons.