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arXiv:2003.04472 · Physical review. B./Physical review. B

Entanglement and correlations in a one-dimensional quantum spin- 12 chain with anisotropic power-law long-range interactions

Jie Ren, Wen‐Long You, Xiaoqun Wang

Published Mar 9, 2020• 23 citations
arXivHTMLPDF

Abstract

The correlations, entanglement entropy, and fidelity susceptibility are calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law long-range interactions by employing the density matrix renormalization group method. In particular, this long-range interaction is assigned to ferromagnetic for transversal components, while it can be either ferro- or antiferromagnetic for the longitudinal spin component. Two ground-state phase diagrams are established versus the anisotropy of the interactions, which not only changes the phase boundaries of the counterparts with short-range interactions, but also leads to the emergence of exotic phases. We found that the long-range interactions of the $z$ component results in a Wigner crystal phase, whereas the transversal one may break a continuous symmetry, resulting in a continuous symmetry-breaking phase.

DOI
10.1103/physrevb.101.094410

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The correlations, entanglement entropy, and fidelity susceptibility are calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law long-range interactions by employing the density matrix renormalization group method. In particular, this long-range interaction is assigned to ferromagnetic for transversal components, while it can be either ferro- or antiferromagnetic for the longitudinal spin component. Two ground-state phase diagrams are established versus the anisotropy of the interactions, which not only changes the phase boundaries of the counterparts with short-range interactions, but also leads to the emergence of exotic phases. We found that the long-range interactions of the zzz component results in a Wigner crystal phase, whereas the transversal one may break a continuous symmetry, resulting in a continuous symmetry-breaking phase.
We investigate the ground-state properties of the spin-1/21/21/2 XXZ model with power-law-decaying ($1/{r}^{\ensuremath{\alpha}}$) interactions, which describe spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hard-core bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation, and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent $\ensuremath{\alpha}$ and number of dimensions ddd: (1) a short-range regime for $\ensuremath{\alpha}>d+{\ensuremath{\sigma}}_{c}$ (where ${\ensuremath{\sigma}}_{c}=1$ in the gapped N\'eel antiferromagnetic phase exhibited by the XXZ model, and ${\ensuremath{\sigma}}_{c}=2$ in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions ($\ensuremath{\alpha}=\ensuremath{\infty}$); (2) a long-range regime $\ensuremath{\alpha}<d$, sharing the same properties as those of the infinite-range interactions ($\ensuremath{\alpha}=0$) in the thermodynamic limit; and (3) a most intriguing medium-range regime for $d<\ensuremath{\alpha}<d+{\ensuremath{\sigma}}_{c}$, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation $\ensuremath{\omega}\ensuremath{\approx}{\mathrm{\ensuremath{\Delta}}}_{g}+c{k}^{z}$ in the gapped phase, and $\ensuremath{\omega}\ensuremath{\sim}{k}^{z}$ in the gapless phase, exhibiting a continuously varying dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}d)/{\ensuremath{\sigma}}_{c}$. In the gapless phase of the model the zzz exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal subdominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.
We study the phase diagram of a spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power law $\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}1)$ accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, the effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to a quantitative change of the phase boundaries and reduces the order of the phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-DDD phase occurs for slowly decaying antiferromagnetic interactions.
We study the fidelity susceptibility in an antiferromagnetic spin-1 XXZ chain numerically. By using the density-matrix renormalization group method, the effects of the alternating single-site anisotropy D on fidelity susceptibility are investigated. Its relation with the quantum phase transition is analyzed. It is found that the quantum phase transition from the Haldane spin liquid to periodic Néel spin solid can be well characterized by the fidelity. Finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. The results are confirmed by the second derivative of the ground-state energy. We also study the relationship between the entanglement entropy, the Schmidt gap and quantum phase transitions. Conclusions drawn from these quantum information observables agree well with each other.
We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.
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 $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.