Random SU(2)-symmetric spin-$S$ chains

We study the low-energy physics of a broad class of time-reversal invariant and SU(2)-symmetric one-dimensional spin-$S$ systems in the presence of quenched disorder via a strong-disorder renormalization-group technique. We show that, in general, there is an antiferromagnetic phase with an emergent $\mathrm{SU}(2S+1)$ symmetry. The ground state of this phase is a random singlet state in which the singlets are formed by pairs of spins. For integer spins, there is an additional antiferromagnetic phase which does not exhibit any emergent symmetry (except for $S=1)$. The corresponding ground state is a random singlet one but the singlets are formed mostly by trios of spins. In each case the corresponding low-energy dynamics is activated, i.e., with a formally infinite dynamical exponent, and related to distinct infinite-randomness fixed points. The phase diagram has two other phases with ferromagnetic tendencies: a disordered ferromagnetic phase and a large spin phase in which the effective disorder is asymptotically finite. In the latter case, the dynamical scaling is governed by a conventional power law with a finite dynamical exponent.

Figure 1

Schematic decimation procedure. The decimated spins $S_2$ and $S_3$ are either replaced by an effective spin $\tilde{S}$ (“1st order” case) or removed from the system (“2nd order” case), depending on whether the local ground state of $H_{2,3}$ has a degeneracy of $2\tilde{S}+1$ or is a nondegenerate singlet, respectively. The set of renormalized couplings $\tilde{\mathbf{K}}=\left(K^{\left(1\right)},\cdots ,K^{\left(J_{\max }\right)}\right)$ is given by Eqs. (18) and (25), respectively.

Figure 2

A schematic depiction of two possible random singlet states. (a) A pairwise random singlet state in which each spin forms a singlet state with another spin indicated by the connecting line. (b) A triplewise random singlet state in which spin trios (connected by the lines) form a singlet state. Occasionally, a group of six (or other multiple of 3) spins can also from a singlet state. Most of the (pairs or trios of) singlets are formed by nearby spins, but there are also singlets formed by spins that are arbitrarily far apart from each other.

Figure 3

Schematic drawing of the hyperoctant formed by all the fully stable AF fixed points of pairwise singlet nature [black circles on the ${\left(-1\right)}^{J+1}{K}^{\left(J\right)}$ semiaxes]. Semistable AF fixed points exist on the hyperplanes formed by two (red triangles) or more (not shown) coordinate axis directions. There is also a totally unstable fixed point (green square) somewhere in the middle of the hyperoctant. This totally unstable fixed point is exactly $\mathrm{SU}(2S+1)$ symmetric, while the other ones have emergent $\mathrm{SU}(2S+1)$ symmetry.

Figure 4

Schematic phase diagram of the disordered spin-$\frac{3}{2}$ chain on and around the AF octant $K^{\left(1\right)}\times -K^{\left(2\right)}\times K^{\left(3\right)}$. The thick dots on the surface of the sphere represent initial conditions of the numerical RG flow. The dots are colored according to their fate in the flow and, therefore, map out the distinct basins of attraction. The beige dots flow to the $K^{\left(1\right)}>0$ (Heisenberg) fixed point (beige circle), the brown ones to the $K^{\left(2\right)}<0$ fixed point (brown circle), while the white ones have only $K^{\left(3\right)}>0$ couplings at the fixed-point Hamiltonian (white circle on the north pole). The pink fixed point in the middle of the octant is totally unstable and has an exact SU(4) symmetry. The other semistable fixed points are represented as black circles. The ground state of the local Hamiltonians $[H_{23}$ in Eq. (13)] is indicated by the background color; see Table 2 for the color scheme (the FM $\tilde{S}=3$ orange region is not visible from this viewing angle). The red and green regions (not marked by any dot for clarity) correspond to the LSP fixed point. On the red line, the system has an exact SO(5) symmetry.

Figure 5

Schematic flow diagram of the average angular variables in the AF octant of random spin-$\frac{3}{2}$ chains (based on the phase diagram of Fig. 4). See text for details about the fixed points ${\mathbf{s}}_i^{*}$.

Figure 6

Diagrams representing the two-spin ground multiplet for two spins $3/2$ and the AF RG fixed points of the disordered spin-$\frac{3}{2}$ chain. Panels (a), (b), and (c) represent the $K^{\left(3\right)}=0,K^{\left(2\right)}=0$, and $K^{\left(1\right)}=0$ planes, respectively. The total angular momentum of the ground multiplets can be $\tilde{S}=0,1,2$, or 3, which are identified by the color scheme of Table 2. The small circles represent the AF angular fixed points. The stable AF ones lie on the semiaxes with $K^{\left(1\right)}>0,K^{\left(2\right)}<0$, and $K^{\left(3\right)}>0$. The unstable fixed points can also be found analytically (see the main text for details).

Figure 7

Energy-length relation for couplings on the $K^{\left(3\right)}$ axes, starting with all spins equal to $S=2$. For negative initial values, the numerical value of the tunneling exponent is $\psi \approx 0.38$, while for positive initial values $\psi =0.54$, compatible with the predicted values of $\psi =\frac{1}{3}$ and $\psi =\frac{1}{2}$, respectively. We have decimated chains of size $N_{\mathrm{sites}}=8\times {10}^6$.

Figure 8

Diagrams representing the two-spin ground multiplet for two spins-2 and the AF RG fixed points of the disordered spin-2 chain in the various two-dimensional planes in $\mathbf{K}$ space. The total angular momentum of the ground multiplets vary from $\tilde{S}=0$ to $\tilde{S}=4$ and it is identified by the color scheme of Table 2. The small circles represent the AF angular fixed points. The stable AF ones lie on the semiaxes with $K^{\left(1\right)}>0,K^{\left(2\right)}<0,K^{\left(3\right)}>0$, and $K^{\left(3\right)}<0$. The unstable fixed points can also be found analytically (see the Appendix pp3).

Figure 9

Schematic phase/flow diagram of the disordered spin-2 chain on the unit two-sphere of the hyperspace with $K^{\left(4\right)}=0$ and axes $K^{\left(1\right)}\times K^{\left(2\right)}\times K^{\left(3\right)}$. The regions on the sphere are colored according to the spin $\tilde{S}$ of the ground multiplet of the two-spin Hamiltonian; see Table 2 for the color scheme (the region with $\tilde{S}=4$ is not visible from this viewing angle). The thick dots on the sphere's surface represent initial conditions of the numerical RG flow, keeping always $K_i^{\left(4\right)}=0$. A generic flow starting at a point in the blue region ends up at one of the stable fixed points on the semiaxes $K^{\left(1\right)}>0,K^{\left(2\right)}<0$, and $K^{\left(3\right)}>0$ (dots have the same color as their final stable fixed points, which are represented by large circles on the semiaxes). Also in the blue region, semistable planar fixed points belonging to 2-planes (where only two $K^{\left(J\right)}$ are nonzero) are represented in black. A single planar fixed point (belonging to the hyperplane $K^{\left(4\right)}=0)$ that is fully unstable on this two-sphere is shown in pink. RG flows starting in the green region end up on a fixed point on the $K^{\left(3\right)}$ axis with random signs, corresponding to the triplewise RSP with exponent $\psi =\frac{1}{3}$. The red and orange regions (not marked by any dot for clarity) are attracted to LSP fixed points.

Figure 10

RG step when the spins are equal to $\frac{3}{2}$ and $K^{\left(3\right)}<0$. The two-spin ground-state multiplet is $\tilde{S}=1$ and the first-order perturbation theory yields a vanishing renormalization. Unlike in the cases discussed in Sec. 3a, the second-order step generates tensors of ranks that were not present in the original chain.

Figure 11

Schematic representation of the RG decimations after the decimation procedure shown in Appendix pp6 is implemented and the generation of zero couplings after some RG time. The strongest coupled pair is represented by the red line. The first decimation generates an effective spin $\tilde{S}=1$ (light red), which is coupled to its neighbors via $K^{\left(1,2\right)}$, but not $K^{\left(3\right)}$. The second decimation leads back to an effective spin-$\frac{3}{2}$, with its left neighbor having $K^{\left(3\right)}=0$. The third decimation is a second-order singlet-formation decimation and the final effective coupling of the edge spins is zero. This is so because ${\tilde{K}}^{\left(J\right)}$ involves the product of neighboring couplings of same rank and, whereas the right neighbor has only $K^{\left(3\right)}\ne 0$, the left neighbor has $K^{\left(3\right)}=0$.