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 ceff=1.17(4) at the quantum critical point (QCP). Moreover, a scaling analysis of EE yields the correlation length exponent \ensuremathν=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.
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.
The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean and a random part, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion S∼lnln(L), where L is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance l from the contact point as ∼l−κ. For κ<1/2, the entropy scales as S(κ)≃(1−2κ)S(κ=0), while for κ≥1/2, when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.
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.
In the present work, we study the fidelity susceptibility and the entanglement entropy in an antiferromagnetic spin-1 chain with additional next-nearest neighbor interactions and three-site interactions, which are given by H=(J1SiSi+1+ J2SiSi+2)+[J3(SiSi+1)(Si+1Si+2)+ h.c.]. By using the density matrix renormalization group method, the ground-state properties of the system are calculated with very high accuracy. We investigate the effect of the three-site interaction J3 on the fidelity susceptibility numerically, and then analyze its relation with the quantum phase transition (QPT). The fidelity measures the similarity between two states, and the fidelity susceptibility describes the associated changing rate. The QPT is intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a peak of the fidelity susceptibility indicates a QPT and the location of the peak denotes the critical point. For the case of J2=0, a peak of the fidelity susceptibility is found by varying J3, and the height of the peak grows as the system size increases. The location of the peak shifts to a slightly lower J3 up to a particular value as the system size increases. Through a finite size scaling, the critical point J3c=0.111 of the QPT from the Haldane spin liquid to the dimerized phase is identified. We also study the effect of the three-site interaction on the entanglement entropy between the right half part and the rest. It is noted that the peak of the entanglement entropy does not coincide with the critical point. Instead, the critical point is determined by the position at which the first-order derivative of the entanglement entropy takes its minimum, since a second-order QPT is signaled by the first derivative of density matrix element. Moreover, the entanglement entropy disappears when J3=1/6, which corresponds to the size-independent Majumdar-Ghosh point. The positions of quantum critical points extracted from these two quantum information observables agree well with those obtained by the string order parameters, which characterizes the topological order in the Haldane phase. Secondly, we also study the case of J20, and obtain the critical points by both the fidelity susceptibility and the entanglement entropy. Finally we provide a ground-state phase diagram of the system. To sum up, the quantum information observables are effective tools for detecting diverse QPTs in spin-1 models.
Abstract The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean subsystem and a random subsystem, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion S ∼ ln ln ( L ) , where L is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance l from the contact point as ∼ l − κ . For κ < 1 / 2 , the entropy scales as S ( κ ) ≃ ln 2 ( 1 − 2 κ ) 6 ln L , while for κ ⩾ 1 / 2 , when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.
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.