Entanglement entropy of the random $s=1$ Heisenberg chain

Random spin chains at quantum critical points exhibit an entanglement entropy between a segment of length $L$ and the rest of the chain that scales as ${\log }_{2}L$ with a universal coefficient. Since for pure quantum critical spin chains this coefficient is fixed by the central charge of the associated conformal field theory, the universal coefficient in the random case can be understood as an effective central charge. In this paper we calculate the entanglement entropy and effective central charge of the spin-1 random Heisenberg model in its random-singlet phase and also at the critical point at which the Haldane phase breaks down. The latter is the first entanglement calculation for an infinite-randomness fixed point that is not in the random-singlet universality class. Our results are consistent with a $c$-theorem for flow between infinite-randomness fixed points. The formalism we use can be generally applied to calculation of quantities that depend on the RG history in $s⩾1$ random Heisenberg chains.

Figure 1

(Color online) Phase diagram of the spin-1 random Heisenberg model. At $r=0$ the chain is in the gapped Haldane phase and its ground state resembles a valence-bond solid (VBS). As randomness is increased, the gap is destroyed at $r_G$, but the VBS structure survives up to the critical point, $r=r_c$. At $r>r_c$ the chain is in the spin-1 random-singlet phase. At the critical point, a different infinite randomness fixed point obtains, which has $\psi =1∕2$, and $\chi =2$. We concentrate on the entanglement entropy of this point.

Figure 2

(a) At very low disorder, the ground state of the spin-1 Heisenberg chain is well described as a valence-bond solid. Each spin-1 site is described by two spin-$1∕2$ parts (black dots) that are symmetrized. Each site forms a spin-$1∕2$ singlet to its right and to its left. (b) As disorder grows, defects appear in the VBS structure, and the gap is suppressed. (c) At very high disorder, a phase transition occurs to the spin-1 random singlet (RS) phase.

Figure 3

(a) Two domains are possible in the spin-$1∕2$ Heisenberg model, (1,0) and (0,1). A domain wall between them gives rise to a spin-$1∕2$ effective site. (b) In the case of a spin-1 chain, there are three possible domains: (1,1), (2,0), and (0,2), domain walls between them are effectively spin-$1∕2$ and spin-1 sites, respectively.

Figure 4

(Color online) The possible outcomes of a spin-1 site when our partition is located inside the site. When the bond to the right (left) of the site gets decimated, it either forms a complete singlet—in case the domain to the left of the bond is the same as that to its right—or makes the spin join a (1,1) domain, in which case the bond strength has probability distribution $S_0\left(\beta \right)$. These two possibilities occur with the same probability, $1∕2$, due to the domain-wall permutation symmetry. The (1,1) domain can be decimated by forming another singlet (eventuality $e_0$), by forming a triplet (eventuality $s_0$) or by joining on its left (right) another (1,1) domain through a Ma-Dasgupta decimation (eventuality $f_0$). It could also form another bond to its left, giving rise to the next line of diagrams, and the eventualities $S_1\left(\beta \right),e_1,s_1,f_1$. The subscript indicates the number of bonds to the left of the partition that are involved in the (1,1) domain. Note that the $e_n$ and $s_n$ eventualities occur on a partition-bond decimation $(\beta =0)$, when $D_L=D_R$ and $D_L\ne D_R$, respectively, and thus with the same probability due to domain-wall permutation symmetry ($D_{R∕L}$ are the domains to right/left of the bond).

Figure 5

(Color online) The possible outcomes of a partition-bond of domain (2,0). When the partition-bond gets decimated, and it is between two identical domains, $D_L=D_R$, then the Ma-Dasgupta rule is applied, Depending on whether $D_L=D_R=(1,1)$ or not, eventualities $p_{-1},p_{-1}^{\prime }$ are obtained. If $D_L\ne D_R$, when the partition-bond gets decimated it joins a (1,1) domain, in which case the bond strength has probability distribution $S_0\left(\beta \right)$. The (1,1) domain can be decimated by forming another singlet (eventuality $e_0$), by forming a triplet (eventuality $s_0$) or by joining on its left (right) another (1,1) domain through a Ma-Dasgupta decimation (eventuality $f_0$). It could also form another bond to its left, giving rise to the next line of diagrams, and the eventualities $S_1\left(\beta \right),e_1,s_1,f_1$. The subscript indicates the number of bonds to the left of the partition that are involved in the (1,1) domain. Note that the $e_n$ and $s_n$ eventualities occur when $D_L=D_R$ and $D_L\ne D_R$, respectively, and thus with the same probability due to domain-wall permutation symmetry ($D_{R∕L}$ are the domains to right/left of the bond).

Figure 6

(Color online) The possible outcomes of a partition-bond of domain (1,1). When the partition-bond gets decimated, if it is between two identical domains, $D_L=D_R$, then the Ma-Dasgupta rule is applied, otherwise, it forms a partition-spin-1.

Figure 7

(Color online) Starting with a partition-spin-1, the various possibilities are illustrated on the same chart as Fig. 4. $\sigma$ is the effective weight of the partition-spin-1 that we begin with. Note that we omit here the future reduced entanglement in the case of the resulting partition-spin-1.

Figure 8

(Color online) Reduced entanglement starting with a (2,0) partition-bond. The various possibilities are illustrated on the same chart as Fig. 5. Note that we omit here the future reduced entanglement in the case of the resulting partition-spin-1.

Figure 9

(Color online) Reduced entanglement starting with a (1,1) partition-bond. The various possibilities are illustrated on the same chart as Fig. 6. Note that we omit here the future reduced entanglement in the case of the resulting partition-spin-1.

Figure 10

(Color online) Numerical calculation of the reduced entropy $S_{\mathrm{red}}$ in the $s=1$ chain for intervals of various sizes $N$ within a chain of length ${10}^4$ sites. The fit is to $S_{\mathrm{red}}=S_0+(c∕3){\log }_{2}N$. The statistical standard deviation for each point shown is less than 5%.

Figure 11

(Color online) (a) A typical $e$-eventuality configuration. (b) Using the chain-rule singlet row reduction, we can reduce a chain of $n+1$ spin-1 sites with two dangling spin-$1∕2$’s, to just two exact spin-$1∕2$’s, which are related to the dangling edge spin-$1∕2$’s as in Eq. (105). (c) Application of the chain rule to the configuration in (a).

Figure 12

(Color online) (a) A typical $f$-eventuality following a FM decimation. The entanglement entropy across the partition site can be calculated exactly if we neglect correlations between the two spin-$1∕2$ components of the partition site, a state which is denoted by $s_{\infty }$. (b) A typical $e$-eventuality configuration after a FM decimation. (c) Using the row-reduction rule we reduce the configuration from (a) to one of only six spin-$1∕2$’s. Neglecting internal correlations in the partition site, we can calculate this entanglement entropy exactly.