arXiv:1009.1614 · Physical Review B

Entanglement spectrum of random-singlet quantum critical points

Maurizio Fagotti, Pasquale Calabrese, Joel E. Moore

Published Jan 31, 2011• 78 citations

Abstract

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.

DOI: 10.1103/physrevb.83.045110

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arXiv:1009.1614 · Physical Review B

Entanglement spectrum of random-singlet quantum critical points

Maurizio Fagotti, Pasquale Calabrese, Joel E. Moore

Published Jan 31, 2011• 78 citations

arXiv HTML PDF

Abstract

\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.

DOI: 10.1103/physrevb.83.045110

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Schmidt gap in random spin chains

Giacomo Torlai, Kenneth D. McAlpine, Gabriele De Chiara

We numerically investigate the low-lying entanglement spectrum of the ground state of random one-dimensional spin chains obtained after partition of the chain into two equal halves. We consider two paradigmatic models: the spin-1/2 random transverse-field Ising model, solved exactly, and the spin-1 random Heisenberg model, simulated using the density matrix renormalization group. In both cases we analyze the mean Schmidt gap, defined as the difference between the two largest eigenvalues of the reduced density matrix of one of the two partitions, averaged over many disorder realizations. We find that the Schmidt gap detects the critical point very well and scales with universal critical exponents.We numerically investigate the low-lying entanglement spectrum of the ground state of random one-dimensional spin chains obtained after partition of the chain into two equal halves. We consider two paradigmatic models: the spin-1/2 random transverse-field Ising model, solved exactly, and the spin-1 random Heisenberg model, simulated using the density matrix renormalization group. In both cases we analyze the mean Schmidt gap, defined as the difference between the two largest eigenvalues of the reduced density matrix of one of the two partitions, averaged over many disorder realizations. We find that the Schmidt gap detects the critical point very well and scales with universal critical exponents.

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Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.

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\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.

{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.