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arXiv:1703.01799 · Physical review. B./Physical review. B

Entanglement and fluctuations in the XXZ model with power-law interactions

Irénée Frérot, Piero Naldesi, Tommaso Roscilde

Published Jun 12, 2017• 77 citations
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Abstract

We investigate the ground-state properties of the spin-$1/2$ 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 $d$: (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 $z$ 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.

DOI
10.1103/physrevb.95.245111

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We investigate the ground-state properties of the spin-1/21/21/2 XXZ model with power-law-decaying (1/r\ensuremathα1/{r}^{\ensuremath{\alpha}}1/r\ensuremathα) 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 \ensuremathα\ensuremath{\alpha}\ensuremathα and number of dimensions ddd: (1) a short-range regime for \ensuremathα>d+\ensuremathσc\ensuremath{\alpha}>d+{\ensuremath{\sigma}}_{c}\ensuremathα>d+\ensuremathσc​ (where \ensuremathσc=1{\ensuremath{\sigma}}_{c}=1\ensuremathσc​=1 in the gapped N'eel antiferromagnetic phase exhibited by the XXZ model, and \ensuremathσc=2{\ensuremath{\sigma}}_{c}=2\ensuremathσc​=2 in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions (\ensuremathα=\ensuremath∞\ensuremath{\alpha}=\ensuremath{\infty}\ensuremathα=\ensuremath∞); (2) a long-range regime \ensuremathα<d\ensuremath{\alpha}<d\ensuremathα<d, sharing the same properties as those of the infinite-range interactions (\ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0) in the thermodynamic limit; and (3) a most intriguing medium-range regime for d<\ensuremathα<d+\ensuremathσcd<\ensuremath{\alpha}<d+{\ensuremath{\sigma}}_{c}d<\ensuremathα<d+\ensuremathσc​, 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 \ensuremathω\ensuremath≈\ensuremathΔg+ckz\ensuremath{\omega}\ensuremath{\approx}{\mathrm{\ensuremath{\Delta}}}_{g}+c{k}^{z}\ensuremathω\ensuremath≈\ensuremathΔg​+ckz in the gapped phase, and \ensuremathω\ensuremath∼kz\ensuremath{\omega}\ensuremath{\sim}{k}^{z}\ensuremathω\ensuremath∼kz in the gapless phase, exhibiting a continuously varying dynamical exponent z=(\ensuremathα\ensuremath−d)/\ensuremathσcz=(\ensuremath{\alpha}\ensuremath{-}d)/{\ensuremath{\sigma}}_{c}z=(\ensuremathα\ensuremath−d)/\ensuremathσc​. 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.
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 study the XXZ square lattice bilayer model with antiferromagnetic nonfrustrating long-range interactions that decay as a power law with the distance. Employing large-scale high-order series expansions with classical Monte Carlo integration (pCUT+MC)(\mathrm{pCUT}+\mathrm{MC})(pCUT+MC) about the limit of isolated Heisenberg dimers in the rung-singlet phase, we investigate the one-triplon dispersion and the corresponding spectral weight along the parameter axes of the long-range decay exponent and the XXZ anisotropy. By tuning the latter, we observe two extended regions of 3D XY and Ising universality as well as 3D Heisenberg critical exponents at the isotropic point. Along the decay exponent axis, we demonstrate mean-field behavior for strong long-range couplings, the aforementioned three universality classes for sufficiently weak interactions, and continuously varying critical exponents in between. Using extrapolations, we are able to determine the one-triplon dispersion in a quantitative fashion up to the quantum-critical breakdown of the rung-singlet phase. This allows us to extract the dynamical critical exponent zzz as a function of the decay exponent, displaying a universal behavior. The detected z<1z<1z<1 for small decay exponents is in agreement with the expected properties of the anomalous Goldstone modes in the ordered phases with broken continuous symmetry.
We study the XXZ square lattice bilayer model with antiferromagnetic non-frustrating long-range interactions that decay as a power law with the distance. Employing large-scale high-order series expansions with classical Monte Carlo integration (pCUT+MC) about the limit of isolated Heisenberg dimers in the rung-singlet phase, we investigate the one-triplon dispersion and the corresponding spectral weight along the parameter axes of the long-range decay exponent and the XXZ anisotropy. By tuning the latter, we observe two extended regions of 3d XY and Ising universality as well as 3d Heisenberg critical exponents at the isotropic point. Along the decay exponent axis, we demonstrate mean-field behavior for strong long-range couplings, the aforementioned three universality classes for sufficiently weak interactions, and continuously varying critical exponents in-between. Using extrapolations we are able to determine the one-triplon dispersion in a quantitative fashion up to the quantum-critical breakdown of the rung-singlet phase. This allows to extract the dynamical critical exponent zzz as a function of the decay exponent, displaying a universal behavior. The detected z<1z<1z<1 for small decay exponents is in agreement with the expected properties of the anomalous Goldstone modes in the ordered phases with broken continuous symmetry.
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 $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
The d=1 Ising ferromagnet and spin glass with long-range power-law interactions J r^-a is studied for all interaction range exponents a by a renormalization-group transformation that simultaneously projects local ferromagnetism and antiferromagnetism. In the ferromagnetic case, J>0, a finite-temperature ferromagnetic phase occurs for interaction range 0.742, the phase transition temperature discontinuously drops to zero and for a>2 there is no ordered phase above zero temperature, also as predicted by rigorous results. At the other end, on approaching a=0.74 from above, namely increasing the range of the interaction, the phase transition temperature diverges to infinity, meaning that, at all non-infinite temperatures, the system is ferromagnetically ordered. Thus, the equivalent-neighbor interactions regime is entered before (a > 0) the neighbors become equivalent, namely before the interactions become equal for all separations. The critical exponents alpha,beta, gamma,delta,eta,nu are calculated, from a large recursion matrix, varying as function of a. For antiferromagnetic J<0, all triplets of spins at all ranges have competing interactions and this highly frustrated system does not have an ordered phase. In the spin-glass system, where all couplings for all separations are randomly ferromagnetic or antiferromagnetic (with probability p), a finite-temperatures spin-glass phase is obtained in the absence of antiferromagnetic phase. In the spin-glass phase, the signature chaotic behavior under scale change occurs in a richer version than previously: In the long-range interaction of this system, the interactions at every separation become chaotic, yielding a piecewise chaotic interaction function.
We study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as ∝r−α\propto r^{-α}∝r−α using a quantum Monte Carlo approach based on the split-spin representation. This formulation enables efficient large-scale simulations by mapping the spin-1 model onto spin-1/21/21/2 degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large ααα (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small ααα (LR regime), where the continuous SU(2) symmetry is broken. From finite-size scaling and crossing point analyses, we determine the critical point to be at αc=2.49(1)α_c = 2.49(1)αc​=2.49(1) and extract the associated critical exponents, which indicate unconventional criticality. In particular, the transition is found to be nonconformal, characterized by a dynamical exponent z≠1z \neq 1z=1. We further analyze the scaling of entanglement entropy and bipartite fluctuations across the transition, and determine the corresponding universal scalings in both phases and at criticality.