Entanglement entropy in random quantum spin-$S$ chains

We discuss the scaling of entanglement entropy in the random singlet phase (RSP) of disordered quantum magnetic chains of general spin $S$. Through an analysis of the general structure of the RSP, we show that the entanglement entropy scales logarithmically with the size of a block, and we provide a closed expression for this scaling. This result is applicable for arbitrary quantum spin chains in the RSP, being dependent only on the magnitude $S$ of the spin. Remarkably, the logarithmic scaling holds for the disordered chain even if the pure chain with no disorder does not exhibit conformal invariance, as is the case for Heisenberg integer-spin chains. Our conclusions are supported by explicit evaluations of the entanglement entropy for random spin-1 and spin-$3∕2$ chains using an asymptotically exact real-space renormalization group approach.

Figure 1

Schematic picture of the RSP. Spin singlets are composed at arbitrary distances.

Figure 2

Modified MDH renormalization procedure for spin-$S$ chains.

Figure 3

(Color online) Logarithmic scaling of entanglement entropy for the spin-1 and spin-$3∕2$ REHACs in the RSP. Blocks of spins up to $10000$ sites are considered in a chain initially with $200000$ sites. The logarithmic fitting shown is the best curve fitting obtained numerically.

Figure 4

(Color online) Logarithmic scaling of entanglement entropy for the random biquadratic spin-1 chain and the spin-$1∕2$ REHAC in the RSP. Blocks of spins up to $10000$ sites are considered in a chain initially with $200000$ sites. The logarithmic fitting shown is the best curve fitting obtained numerically.