The entanglement spectrum (ES) is a powerful tool for probing topological phases. While its behavior in gapped systems is well understood, its properties in gapless regimes remain unclear. In this work, we employ a quantum Monte Carlo method to study the ES of a two-dimensional square-octagon lattice Heisenberg model at quantum criticality and in the Néel phase. We find that the ES exhibits an M-shape magnon mode with a distinct sublinear dispersion, deviating from the conventional linear magnon. This behavior, similar to that of a one-dimensional long-range Heisenberg chain, reveals the emergence of relevant long-range interactions in the entanglement Hamiltonian. We demonstrate that the mechanism underlying short- and long-range interactions in the entanglement Hamiltonian can be interpreted as the confinement/deconfinement of worldlines in the path integral formulation. Our results reveal that gapless modes can fundamentally change the entanglement Hamiltonian and its spectrum, thereby offering insight into this general phenomenon.
The entanglement spectrum (ES) is a powerful tool for probing topological phases. While its behavior in gapped systems is well understood, its properties in gapless regimes remain unclear. In this work, we employ a quantum Monte Carlo method to study the ES of a two-dimensional square-octagon lattice Heisenberg model at quantum criticality and in the Néel phase. We find that the ES exhibits an M-shape magnon mode with a distinct sublinear dispersion, deviating from the conventional linear magnon. This behavior, similar to that of a one-dimensional long-range Heisenberg chain, reveals the emergence of relevant long-range interactions in the entanglement Hamiltonian. We demonstrate that the mechanism underlying short- and long-range interactions in the entanglement Hamiltonian can be interpreted as the confinement/deconfinement of worldlines in the path integral formulation. Our results reveal that gapless modes can fundamentally change the entanglement Hamiltonian and its spectrum, thereby offering insight into this general phenomenon.
Beyond the Li-Haldane-Poilblanc conjecture, we find the entanglement Hamiltonian (EH) is actually not closely similar to the original Hamiltonian on the virtual edge. Unexpectedly, the EH has some relevant long-range interacting terms which hugely affect the physics. Without loss of generality, we study a spin-1/2 Heisenberg bilayer to obtain the entanglement information between the two layers through our newly developed quantum Monte Carlo scheme, which can simulate large-scale EH. Although the entanglement spectrum carrying the Goldstone mode seems like a Heisenberg model on a single layer, which is consistent with Li-Haldane-Poilblanc conjecture, we demonstrate that there actually exists a finite-temperature phase transition of the EH. The results violate the Mermin-Wagner theorem, which means there should be relevant long-range terms in the EH. It reveals that the Li-Haldane-Poilblanc conjecture ignores necessary corrections for the EH which may lead totally different physics.
We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled nonchiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled nonchiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.
Shannon-Rényi entropies and associated participation spectra quantify how much a many-body wave-function is localized in a given configuration basis. Using these tools, we present an analysis of the ground-state wave functions of various quantum spin systems in one and two dimensions. General ideas and a review of the current status of this field are first given, with a particular emphasis on universal subleading terms characterizing different quantum phases of matter, and associated transitions. We highlight the connection with the related entanglement entropies and spectra when this is possible. In a second part, new results are presented for the participation spectra of interacting spin models, mostly based on quantum Monte Carlo simulations, but also using perturbation theory in some cases. For full antiferromagnetic systems, participation spectra are analyzed in terms of ferromagnetic domain walls which experience a repulsive pairwise interaction. This confinement potential is either linear for long-range Néel order, or logarithmic for quasi-long-range order. The case of subsystems is also analyzed in great detail for a 2d dimerized Heisenberg model undergoing a quantum phase transition between a gapped paramagnet and a Néel phase. Participation spectra of line shaped (1d) sub-systems are quantitatively compared with finite temperature participation spectra of ansatz effective boundary (1d) entanglement Hamiltonians. While short-range models describe almost perfectly the gapped side, the Néel regime is best compared using long-range effective Hamiltonians. Spectral comparisons performed using Kullback-Leibler divergences, a tool potentially useful for entanglement spectra, provide a quantitative way to identify both the best boundary entanglement Hamiltonian and temperature.
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.
The study of quantum criticality and entanglement in systems with long-range (LR) interactions is still in its early stages, with many open questions remaining. In this work, we investigate critical exponents and scaling of entanglement entropies (EE) in the LR bilayer Heisenberg model using large-scale quantum Monte Carlo (QMC) simulations and the recently developed nonequilibrium increment algorithm for measuring EE. By applying modified (standard) finite-size scaling (FSS) above (below) the upper critical dimension and field theory analysis, we obtain precise critical exponents in three regimes: the LR Gaussian regime with a Gaussian fixed point, the short-range (SR) regime with Wilson-Fisher (WF) exponents, and a LR non-Gaussian regime where the critical exponents vary continuously from LR Gaussian to SR values. We compute the Rényi EE both along the critical line and in the Néel phase and observe that as the LR interaction is enhanced, the area-law contribution in EE gradually vanishes both at quantum critical points (QCPs) and in the Néel phase. The log-correction in EE arising from sharp corners at the QCPs also decays to zero as LR interaction grows, whereas the log-correction for Néel states, caused by the interplay of Goldstone modes and restoration of the symmetry in a finite system, is enhanced as LR interaction becomes stronger. We also discuss relevant experimental settings to detect these nontrivial properties in critical behavior and entanglement information for quantum many-body systems with LR interactions.
Employing large-scale quantum Monte Carlo simulations, we systematically compute the energy spectra of the two-dimensional (2D) spin-1/2 Heisenberg model with long-range interactions. With the $1/{r}^{\ensuremath{\alpha}}$ ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in $\ensuremath{\alpha}$ for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless), and Higgs-type (gapped) spectra. Accompanied by the spin-wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find the explicit case where the gapped excitation emerges at a noninteger decay exponent $\ensuremath{\alpha}$ for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones, which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moir\'e materials.