Average block entanglement in the 1D XX-model with uncorrelated random couplings is known to grow as the logarithm of the block size, in similarity to conformal systems. In this work we study random spin chains whose couplings present long range correlations, generated as gaussian fields with a power-law spectral function. Ground states are always planar valence bond states, and their statistical ensembles are characterized in terms of their block entropy and their bond-length distribution, which follow power-laws. We conjecture the existence of a critical value for the spectral exponent, below which the system behavior is identical to the case of uncorrelated couplings. Above that critical value, the entanglement entropy violates the area law and grows as a power law of the block size, with an exponent which increases from zero to one. Interestingly, we show that XXZ models with positive anisotropy present the opposite behavior, and strong correlations in the couplings lead to lower entropies. Similar planar bond structures are also found in statistical models of RNA folding and kinetic roughening, and we trace an analogy between them and quantum valence bond states. Using an inverse renormalization procedure we determine the optimal spin-chain couplings which give rise to a given planar bond structure, and study the statistical properties of the couplings whose bond structures mimic those found in RNA folding.
Average block entanglement in the 1D XX-model with uncorrelated random couplings is known to grow as the logarithm of the block size, in similarity to conformal systems. In this work we study random spin chains whose couplings present long range correlations, generated as gaussian fields with a power-law spectral function. Ground states are always planar valence bond states, and their statistical ensembles are characterized in terms of their block entropy and their bond-length distribution, which follow power-laws. We conjecture the existence of a critical value for the spectral exponent, below which the system behavior is identical to the case of uncorrelated couplings. Above that critical value, the entanglement entropy violates the area law and grows as a power law of the block size, with an exponent which increases from zero to one. Interestingly, we show that XXZ models with positive anisotropy present the opposite behavior, and strong correlations in the couplings lead to lower entropies. Similar planar bond structures are also found in statistical models of RNA folding and kinetic roughening, and we trace an analogy between them and quantum valence bond states. Using an inverse renormalization procedure we determine the optimal spin-chain couplings which give rise to a given planar bond structure, and study the statistical properties of the couplings whose bond structures mimic those found in RNA folding.
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.
The ground state of the uniform antiferromagnetic spin-1/2 Heisenberg chain can be viewed as a strongly fluctuating liquid of valence bonds, while in disordered chains these bonds lock into random singlet states on long-length scales. We show that this phenomenon can be studied numerically, even in the case of weak disorder, by calculating the mean value of the number of valence bonds leaving a block of L contiguous spins (the valence-bond entanglement entropy) as well as the fluctuations in this number. These fluctuations show a clear crossover from a small L regime, in which they behave similar to those of the uniform model, to a large L regime, in which they saturate in a way consistent with the formation of a random singlet state on long-length scales. A scaling analysis of these fluctuations is used to study the dependence on disorder strength of the length scale characterizing the crossover between these two regimes. Results are obtained for a class of models that include, in addition to the spin-1/2 Heisenberg chain, the uniform and disordered critical 1D transverse-field Ising model and chains of interacting non-Abelian anyons.
We study the R\'enyi entanglement entropy and the Shannon mutual information for a class of spin-1/2 quantum critical XXZ chains with random coupling constants which are partially correlated. In the XX case, distinctly from the usual uncorrelated random case where the system is governed by an infinite-disorder fixed point, the correlated-disorder chain is governed by finite-disorder fixed points. Surprisingly, we verify that, although the system is not conformally invariant, the leading behavior of the R\'enyi entanglement entropies are similar to those of the clean (no randomness) conformally invariant system. In addition, we compute the Shannon mutual information among subsystems of our correlated-disorder quantum chain and verify the same leading behavior as the n=2 R\'enyi entanglement entropy. This result extends a recent conjecture concerning the same universal behavior of these quantities for conformally invariant quantum chains. For the generic spin-1/2 quantum critical XXZ case, the true asymptotic regime is identical to that in the uncorrelated disorder case. However, these finite-disorder fixed points govern the low-energy physics up to a very long crossover length scale, and the same results as in the XX case apply. Our results are based on exact numerical calculations and on a numerical strong-disorder renormalization group.
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), 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=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.
Using strong-disorder renormalization group, numerical exact diagonalization, and quantum Monte Carlo methods, we revisit the random antiferromagnetic XXZ spin-1∕2 chain focusing on the long-length and ground-state behavior of the average time-independent spin-spin correlation function $C(l)=\ensuremath{\upsilon}{l}^{\ensuremath{-}\ensuremath{\eta}}$. In addition to the well-known universal (disorder-independent) power-law exponent $\ensuremath{\eta}=2$, we find interesting universal features displayed by the prefactor $\ensuremath{\upsilon}={\ensuremath{\upsilon}}_{o}∕3$, if l is odd, and $\ensuremath{\upsilon}={\ensuremath{\upsilon}}_{e}∕3$, otherwise. Although ${\ensuremath{\upsilon}}_{o}$ and ${\ensuremath{\upsilon}}_{e}$ are nonuniversal (disorder dependent) and distinct in magnitude, the combination ${\ensuremath{\upsilon}}_{o}+{\ensuremath{\upsilon}}_{e}=\ensuremath{-}1∕4$ is universal if C is computed along the symmetric (longitudinal) axis. The origin of the nonuniversalities of the prefactors is discussed in the renormalization-group framework where a solvable toy model is considered. Moreover, we relate the average correlation function with the average entanglement entropy, whose amplitude has been recently shown to be universal. The nonuniversalities of the prefactors are shown to contribute only to surface terms of the entropy. Finally, we discuss the experimental relevance of our results by computing the structure factor whose scaling properties, interestingly, depend on the correlation prefactors.
This work provides a full description of the critical behavior of the XX spin-1/2 chain under correlated quenched disorder.Previous investigations have shown that the introduction of correlation between couplings in the random XX model gives rise to a novel critical behavior, where the infinite-randomness critical point of the uncorrelated case is replaced by a family of finite-disorder critical points that depends on the disorder strength.Here it is shown that most of the critical exponents of the XX model with correlated randomness are equal to clean (without disorder) chain values and do not depend on disorder strength, except the critical dynamical exponent and the anomalous dimension.The former increases monotonically with disorder strength, whereas the results obtained for the latter are unreliable.Furthermore, the scaling relations between the critical exponents were also tested and it was found that those involving the system dimensionality, namely the hyperscaling and Fisher's scaling relations, are not respected.Measurements of the Rényi entanglement entropy of the system at criticality have also been performed, and it is shown that the scaling behavior of the correlated-disorder case is similar to the theoretical prediction for the clean chain, displaying the same finite-size correction and a disorder-dependent effective central charge in the leading term of the scaling.Further corrections to the scaling of the entanglement entropy were also investigated, but the results are inconclusive.The model was studied via exact numerical diagonalization of the corresponding Hamiltonian.