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 ${\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}L$ 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 $s\ensuremath{\geqslant}1$ random Heisenberg chains.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 ${\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}L$ 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 $s\ensuremath{\geqslant}1$ random Heisenberg chains.

arXiv:cond-mat/0703038Physical Review BJul 13, 200747 citations

Entanglement entropy, conformal invariance, and the critical behavior of the anisotropic spin- s Heisenberg chains: DMRG study

J. C. Xavier

Using the density-matrix renormalization group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to $s=9/2$ . We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly $c$ , the sound velocity ${v}_{s}$ , the anomalous dimension ${x}_{bulk}$ , and the surface exponent ${x}_{s}$ of the anisotropic spin- $s$ Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entropy $S(L,{l}_{A},s)$ of the spin- $s$ Heisenberg chains satisfies the relation $S(L,{l}_{A},s)\ensuremath{-}S(L,{l}_{A},s\ensuremath{-}1)=1/(2s+1)$ for $s>3/2$ in the thermodynamic limit.Using the density-matrix renormalization group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to $s=9/2$. We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly $c$, the sound velocity ${v}_{s}$, the anomalous dimension ${x}_{bulk}$, and the surface exponent ${x}_{s}$ of the anisotropic spin-$s$ Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entropy $S(L,{l}_{A},s)$ of the spin-$s$ Heisenberg chains satisfies the relation $S(L,{l}_{A},s)\ensuremath{-}S(L,{l}_{A},s\ensuremath{-}1)=1/(2s+1)$ for $s>3/2$ in the thermodynamic limit.

arXiv:1002.0531Physical Review BJun 1, 201030 citations

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

{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}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 $s\ensuremath{\geqslant}1$ random Heisenberg chains.

Using the density-matrix renormalization group, we investigate the critical behavior of the anisotropic Heisenberg chains with spins up to

s=9/2

. We show that through the relations arising from the conformal invariance and the DMRG technique it is possible to obtain accurate finite-size estimates of the conformal anomaly

c

, the sound velocity

{v}_{s}

, the anomalous dimension

{x}_{bulk}

, and the surface exponent

{x}_{s}

of the anisotropic spin-

s

Heisenberg chains with relatively good accuracy without fitting parameters. Our results indicate that the entanglement entropy

S(L,{l}_{A},s)

of the spin-

s

Heisenberg chains satisfies the relation $S(L,{l}_{A},s)\ensuremath{-}S(L,{l}_{A},s\ensuremath{-}1)=1/(2s+1)$ for

s>3/2

in the thermodynamic limit.