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

Scaling of entanglement entropy at quantum critical points in random spin chains

Prashant Kumar, R. N. Bhatt

We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ${c}_{\mathrm{eff}}=1.17(4)$ at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.We study the scaling properties of entanglement entropy (EE) near the quantum critical points in interacting random antiferromagnetic (AFM) spin chains. Using density-matrix renormalization group, we compute the half-chain EE near the topological phase transition between the Haldane and random singlet phases in a disordered spin-1 chain. It is found to diverge logarithmically in system size with an effective central charge ${c}_{\mathrm{eff}}=1.17(4)$ at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.

arXiv:2307.00062Physical review. B./Physical review. BDec 15, 20232 citations

Entanglement entropy in ordered and quantum critical systems

Wenxin Ding

Jan 1, 20120 citations

Entanglement in quantum-critical spin systems

Tommaso Roscilde, Paola Verrucchi, Andrea Fubini +2 more

arXiv:quant-ph/0412056Springer eBooksMay 29, 20061 citation

Entanglement spectrum of random-singlet quantum critical points

Maurizio Fagotti, Pasquale Calabrese, Joel E. Moore

The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{\underline{}}{\mathrm{Tr}{\mathrm{\ensuremath{\rho}}}_{\mathrm{A}}^{\mathrm{\ensuremath{\alpha}}}}$ of the reduced density matrix ${\ensuremath{\rho}}_{A}$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random $\mathit{XX}$ model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the $\mathit{XX}$ case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $\stackrel{\underline{}}{\mathrm{Tr}{\mathrm{\ensuremath{\rho}}}_{\mathrm{A}}^{\mathrm{\ensuremath{\alpha}}}}$ of the reduced density matrix ${\ensuremath{\rho}}_{A}$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random $\mathit{XX}$ model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the $\mathit{XX}$ case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.

arXiv:1009.1614Physical Review BJan 31, 201178 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.

{c}_{\mathrm{eff}}=1.17(4)

at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent $\ensuremath{\nu}=2.28(5)$. Our unbiased calculation establishes that the QCP is in the universality class of the infinite-randomness fixed point predicted by previous studies based on the strong disorder renormalization group technique. However, in the disordered spin-1/2 Majumdar-Ghosh chain, where a valence bond solid phase is unstable to disorder, the crossover length exponent obtained from a scaling analysis of EE disagrees with the expectation based on the Imry-Ma argument. We provide a possible explanation.

\mathit{XX}

model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the

\mathit{XX}

case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.