Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.
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 study the ground-state phase diagram of a spin-1 Heisenberg chain with staggered long-range (LR) interactions decaying as
∝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-
1/2 degrees of freedom with local projection constraints. We resolve the continuous quantum phase transition between the gapped Haldane phase at large
α (short-range regime) and a gapless antiferromagnetically ordered Néel phase at small
α (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) 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=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.
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-
D phase occurs for slowly decaying antiferromagnetic interactions.
We study spin-
1/2 chains with long-range power-law decaying unfrustrated (bipartite) Heisenberg exchange
Jr∝r−α and multi-spin interactions
Q favoring a valence-bond solid (VBS) ground state. Employing quantum Monte Carlo techniques and Lanczos diagonalization, we analyze order parameters and excited-state level crossings to characterize quantum states and phase transitions in the
(α,Q) plane. For weak
Q and sufficiently slowly decaying Heisenberg interactions (small
α), the system has a long-range-ordered antiferromagnetic (AFM) ground state, and upon increasing
α there is a continuous transition into a quasi long-range ordered (QLRO) critical state of the type in the standard Heisenberg chain. For rapidly decaying long-range interactions, there is transition between QLRO and VBS ground states of the same kind as in the frustrated
J1-
J2 Heisenberg chain. Our most important finding is a direct continuous quantum phase transition between the AFM and VBS states - a close analogy to the 2D deconfined quantum-critical point. In previous 1D analogies the ordered phases both have gapped fractional excitations, and the critical point is a conventional Luttinger Liquid. In our model the excitations fractionalize upon transitioning from the AFM state, changing from spin waves to deconfined spinons. We extract critical exponents at the AFM-VBS transition and use order-parameter distributions to study emergent symmetries. We find emergent O(
4) symmetry of the O(
3) AFM and scalar VBS order parameters. Thus, the order parameter fluctuations exhibit the covariance of a uniaxially deformed O(
4) sphere (an "elliptical" symmetry). This unusual quantum phase transition does not yet have any known field theory description, and our detailed results can serve to guide its construction. We discuss possible experimental realizations.
We study the fidelity susceptibility of a quantum antiferromagnetic Ising chain with a long-range power-law interaction $1/{r}^{\ensuremath{\alpha}}$ using the large-scale density matrix renormalization group method. We find that the critical adiabatic dimension $\ensuremath{\mu}=2$ and the critical exponent of the correlation length $\ensuremath{\nu}=1$ for arbitrary $\ensuremath{\alpha}>0$, indicating all quantum phase transitions are second-order Ising transitions. In addition, we numerically determine the complete phase diagram for $0<\ensuremath{\alpha}\ensuremath{\le}3$ from the data collapse of the fidelity susceptibility and show that the critical point
hc changes monotonously with respect to $\ensuremath{\alpha}$. This work will shed light on the nature of phase transitions in the quantum long-range antiferromagnetic Ising chain from a quantum information perspective.
The quantum phase transition between a spin gap state and an antiferromagnetic phase is investigated. We study S =1/2 antiferromagnetic Heisenberg chains coupled by antiferromagnetic interchain interaction. The intrachain exchanges have alternating strength. The phase boundary between the antiferromagnetically ordered phase and a spin gap phase is also obtained in a parameter space of the amplitude of the interchain coupling and the dimerization. The spin-wave approximation substantially overestimates the antiferromagnetic phase. The competition between the long range order and the spin gap is examined in detail. We estimate a variety of critical exponents at the transition, namely, exponents ν, θ and z defined as the exponent of the correlation length, the magnetization curve and the dynamical exponent, respectively. From the quantum Monte Carlo simulation, the exponents ν, θ and z are estimated to be unity. The exponents ν and θ are different from the estimated values in one dimension. It suggests that the universality changes due to the dimensionality change. In our estimates, the exponent ν does not agree with the prediction from three dimensional classical Heisenberg model. We also discuss the relevance of our result to spin-Peierls systems with lattice distortion.