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Long-range hopping models with off-diagonal disorder and the concept of random-singlet state

Róbert Juhász

Published Oct 5, 2021• 0 citations
OpenAlex

Abstract

We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with power-law decaying random hopping amplitudes, and address the question how far an analogue of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\alpha>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with the decay exponent $\alpha$ of the hopping amplitudes. The large $\alpha$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\alpha\le 2$, while for $\alpha>2$ it is possibly valid. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Besides the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\alpha$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless it becomes more and more accurate for increasing $\alpha$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.

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Testing the validity of random-singlet state for long-range hopping models through the scaling of entanglement entropy

Róbert Juhász

We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|{t}_{l}|\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\alpha}}$, and address the question how far an analog of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\ensuremath{\alpha}>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with $\ensuremath{\alpha}$. The large-$\ensuremath{\alpha}$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\ensuremath{\alpha}\ensuremath{\le}2$, while for $\ensuremath{\alpha}>2$ it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Aside from the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\ensuremath{\alpha}$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless, it becomes more and more accurate for increasing $\ensuremath{\alpha}$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.

arXiv:2110.01985•Physical review. B./Physical review. B•Jan 31, 2022•4 citations

Entanglement properties of disordered quantum spin chains with long-range antiferromagnetic interactions

Y. Mohdeb, Javad Vahedi, N. Moure +5 more

Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge $c=\mathrm{ln}(2)$, independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size $l$, the EE fits a critical behavior with a central charge close to $c=1$, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins $N$, we find deviations which are rather consistent with the strong disorder fixed point central charge $c=\mathrm{ln}(2)$. Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.

arXiv:2009.11286•Physical review. B./Physical review. B•Dec 1, 2020•17 citations

Strong-disorder renormalization group method for bond-disordered antiferromagnetic quantum spin chains with long-range interactions: Ground-state properties

Stefan Kettemann

We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings $J$ smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths $l$ is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length $n$ increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.

arXiv:2501.07298•Physical review. B./Physical review. B•Nov 10, 2025•1 citation

Beyond universal behavior in the one-dimensional chain with random nearest-neighbor hopping

Akshay Krishna, R. N. Bhatt

We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\ensuremath{\rho}(E)\ensuremath{\sim}1/|E{ln}^{3}|E||$ and of the localization length $\ensuremath{\xi}(E)\ensuremath{\sim}|ln|E||$, near the band center $E=0$. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at $E=0$ shows a universal, subexponential decay at large distances $\ensuremath{\sim}exp[\ensuremath{-}\sqrt{r/{r}_{0}}]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\ensuremath{\sim}1/[t{ln}^{\ensuremath{\lambda}+1}(1/t)]$, characterized by a single, continuously tunable parameter $\ensuremath{\lambda}>0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\ensuremath{\lambda}>2$, it no longer holds in the interval $0<\ensuremath{\lambda}<2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter $\ensuremath{\lambda}$; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below $\ensuremath{\lambda}=1$. For $\ensuremath{\lambda}<2$, the fall-off of the $E=0$ state at large distances also deviates from the universal result and is of the form $\ensuremath{\sim}exp[\ensuremath{-}{(r/{r}_{0})}^{1/\ensuremath{\lambda}}]$, which decays faster than an exponential for $\ensuremath{\lambda}<1$.

arXiv:2004.00064•Physical review. B./Physical review. B•Jun 15, 2020•10 citations

Quantum entanglement in random and inhomogeneous spin chains

Ángel Ramírez

Entanglement can be quantified in bipartite systems using the entanglement entropy which is, in many cases, bounded by an area law. Nonetheless, some critical 1D systems violate this area law and present logarithmic corrections which are parametrized by the central charge of the associated conformal field theory (CFT). Furthermore, other violations to the area law may appear in random systems and inhomogeneous systems. In the first part of the work we focus on the properties of entanglement in random hopping models, focusing on the similarities between the CFT predictions for the clean case and the strong disorder renormalization group (SDRG) predictions. We use a combination of methods: exact diagonalization, a transformation of the SDRG and a new tool based on the study of random permutations. All techniques coincide in providing a compelling image, based on a bond-picture. The second part focus on the engineering of an inhomogeneous fermionic system, in order to obtain a maximal growth of the block entanglement entropy in the ground state. We show that the deformation parameter allows us to interpolate between an uniform model and the strong disorder limit, which is a maximally entangled state called the rainbow state. This state can be seen as a thermo field state. The last part focus on time-evolution of some valence bond states after a global quench to a homogeneous Hamiltonian in 1D. We focused on the evolution of the entanglement in the rainbow and dimer states. We proposed a generalization of the quasiparticle picture, which we have called the ballistic picture.

•Jan 1, 2015•0 citations

Properties of the random-singlet phase: From the disordered Heisenberg chain to an amorphous valence-bond solid

Yu-Rong Shu, Dao‐Xin Yao, Chih-Wei Ke +2 more

We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study $S=1/2$ spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

arXiv:1603.04362•Physical review. B./Physical review. B•Nov 28, 2016•28 citations
We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with power-law decaying random hopping amplitudes, and address the question how far an analogue of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For α>1\alpha>1α>1, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with the decay exponent α\alphaα of the hopping amplitudes. The large α\alphaα limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for α≤2\alpha\le 2α≤2, while for α>2\alpha>2α>2 it is possibly valid. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Besides the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small α\alphaα. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless it becomes more and more accurate for increasing α\alphaα. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.
We consider a sublattice-symmetric free-fermion model on a one-dimensional lattice with random hopping amplitudes decaying with the distance as $|{t}_{l}|\ensuremath{\sim}{l}^{\ensuremath{-}\ensuremath{\alpha}}$, and address the question how far an analog of the random-singlet state (RSS) conceived originally for describing the ground state of certain random spin chains is valid for this model. For this purpose, we study the effective central charge characterizing the logarithmic divergence of the entanglement entropy (EE) and the prefactor of the distribution of distances between localization centers on the two sublattices, which must fulfill a consistency relation for a RSS. For $\ensuremath{\alpha}>1$, we find by exact diagonalization an overall logarithmic divergence of the entanglement entropy with an effective central charge varying with $\ensuremath{\alpha}$. The large-$\ensuremath{\alpha}$ limit of the effective central charge is found to be different from that of the nearest-neighbor hopping model. The consistency relation of RSS is violated for $\ensuremath{\alpha}\ensuremath{\le}2$, while for $\ensuremath{\alpha}>2$ it is possibly valid, but this conclusion is hampered by a crossover induced by the short-range fixed point. The EE is also calculated by the strong-disorder renormalization group (SDRG) method numerically. Aside from the traditional scheme, we construct and apply a more efficient minimal SDRG scheme having a linear (nearest-neighbor) structure, which turns out to be an accurate approximation of the full SDRG scheme for not too small $\ensuremath{\alpha}$. The SDRG method is found to provide systematically lower effective central charges than exact diagonalization does, nevertheless, it becomes more and more accurate for increasing $\ensuremath{\alpha}$. Furthermore, as opposed to nearest-neighbor models, it indicates a weak dependence on the disorder distribution.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size lll, the EE fits a critical behavior with a central charge close to c=1c=1c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins NNN, we find deviations which are rather consistent with the strong disorder fixed point central charge c=ln(2)c=\mathrm{ln}(2)c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.
We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings JJJ smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths lll is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length nnn increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
We study the one-dimensional nearest-neighbor tight-binding model of electrons with independently distributed random hopping and no on-site potential (i.e., off-diagonal disorder with particle-hole symmetry, leading to sublattice symmetry, for each realization). For nonsingular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\ensuremath{\rho}(E)\ensuremath{\sim}1/|E{ln}^{3}|E||$ and of the localization length $\ensuremath{\xi}(E)\ensuremath{\sim}|ln|E||$, near the band center E=0E=0E=0. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain.) Simultaneously, the state at E=0E=0E=0 shows a universal, subexponential decay at large distances $\ensuremath{\sim}exp[\ensuremath{-}\sqrt{r/{r}_{0}}]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small ttt is of the form $\ensuremath{\sim}1/[t{ln}^{\ensuremath{\lambda}+1}(1/t)]$, characterized by a single, continuously tunable parameter $\ensuremath{\lambda}>0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\ensuremath{\lambda}>2$, it no longer holds in the interval $0<\ensuremath{\lambda}<2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the nonuniversal parameter $\ensuremath{\lambda}$; simultaneously, the localization length shows a less divergent form at low energies and ceases to diverge below $\ensuremath{\lambda}=1$. For $\ensuremath{\lambda}<2$, the fall-off of the E=0E=0E=0 state at large distances also deviates from the universal result and is of the form $\ensuremath{\sim}exp[\ensuremath{-}{(r/{r}_{0})}^{1/\ensuremath{\lambda}}]$, which decays faster than an exponential for $\ensuremath{\lambda}<1$.
Entanglement can be quantified in bipartite systems using the entanglement entropy which is, in many cases, bounded by an area law. Nonetheless, some critical 1D systems violate this area law and present logarithmic corrections which are parametrized by the central charge of the associated conformal field theory (CFT). Furthermore, other violations to the area law may appear in random systems and inhomogeneous systems. In the first part of the work we focus on the properties of entanglement in random hopping models, focusing on the similarities between the CFT predictions for the clean case and the strong disorder renormalization group (SDRG) predictions. We use a combination of methods: exact diagonalization, a transformation of the SDRG and a new tool based on the study of random permutations. All techniques coincide in providing a compelling image, based on a bond-picture. The second part focus on the engineering of an inhomogeneous fermionic system, in order to obtain a maximal growth of the block entanglement entropy in the ground state. We show that the deformation parameter allows us to interpolate between an uniform model and the strong disorder limit, which is a maximally entangled state called the rainbow state. This state can be seen as a thermo field state. The last part focus on time-evolution of some valence bond states after a global quench to a homogeneous Hamiltonian in 1D. We focused on the evolution of the entanglement in the rainbow and dimer states. We proposed a generalization of the quasiparticle picture, which we have called the ballistic picture.
We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S=1/2S=1/2S=1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that cannot be generated by the SDRG scheme. In the model with multispin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.