Bilayer Heisenberg Model Research Articles

Unsupervised mapping of phase diagrams of 2D systems from infinite projected entangled-pair states via deep anomaly detection

We demonstrate how to map out the phase diagram of a two dimensional quantum many body system with no prior physical knowledge by applying deep anomaly detection to ground states from infinite projected entangled pair state simulations. As a benchmark, the phase diagram of the 2D frustrated bilayer Heisenberg model is analyzed, which exhibits a second-order and two first-order quantum phase transitions. We show that in order to get a good qualitative picture of the transition lines, it suffices to use data from the cost-efficient simple update optimization. Results are further improved by post-selecting ground-states based on their energy at the cost of contracting the tensor network once. Moreover, we show that the mantra of ``more training data leads to better results'' is not true for the learning task at hand and that, in principle, one training example suffices for this learning task. This puts the necessity of neural network optimizations for these learning tasks in question and we show that, at least for the model and data at hand, a simple geometric analysis suffices.

Universal scaled Higgs-mass gap for the bilayer Heisenberg model in the ordered phase

The spectral properties for the bilayer quantum Heisenberg model were investigated with the numerical diagonalization method. In the ordered phase, there appears the massive Higgs excitation embedded in the continuum of the Goldstone excitations. Recently, it was claimed that the properly scaled Higgs mass is a universal constant in proximity to the critical point. Diagonalizing the finite-size cluster with $N \le 36$ spins, we calculated the dynamical scalar susceptibility $\chi_s"(\omega)$, which is rather insensitive to the Goldstone mode. A finite-size-scaling analysis of $\chi_s "(\omega)$ is made, and the universal (properly scaled) Higgs mass is estimated.

Dynamical structure factors and excitation modes of the bilayer Heisenberg model

Using quantum Monte Carlo simulations along with higher-order spin-wave theory, bond-operator and strong-coupling expansions, we analyse the dynamical spin structure factor of the spin-half Heisenberg model on the square-lattice bilayer. We identify distinct contributions from the low-energy Goldstone modes in the magnetically ordered phase and the gapped triplon modes in the quantum disordered phase. In the antisymmetric (with respect to layer inversion) channel, the dynamical spin structure factor exhibits a continuous evolution of spectral features across the quantum phase transition, connecting the two types of modes. Instead, in the symmetric channel we find a depletion of the spectral weight when moving from the ordered to the disordered phase. While the dynamical spin structure factor does not exhibit a well-defined distinct contribution from the amplitude (or Higgs) mode in the ordered phase, we identify an only marginally-damped amplitude mode in the dynamical singlet structure factor, obtained from interlayer bond correlations, in the vicinity of the quantum critical point. These findings provide quantitative information in direct relation to possible neutron or light scattering experiments in a fundamental two-dimensional quantum-critical spin system.

Velocity of excitations in ordered, disordered, and critical antiferromagnets

We test three different approaches, based on quantum Monte Carlo simulations, for computing the velocity $c$ of triplet excitations in antiferromagnets. We consider the standard $S=1/2$ one- and two-dimensional Heisenberg models, as well as a bilayer Heisenberg model at its critical point. Computing correlation functions in imaginary time and using their long-time behavior, we extract the lowest excitation energy versus momentum using improved fitting procedures and a generalized moment method. The velocity is then obtained from the dispersion relation. We also exploit winding numbers to define a cubic space-time geometry, where the velocity is obtained as the ratio of the spatial and temporal lengths of the system when all winding number fluctuations are equal. The two methods give consistent results for both ordered and critical systems, but the winding-number estimator is more precise. For the Heisenberg chain, we accurately reproduce the exactly known velocity. For the two-dimensional Heisenberg model, our results are consistent with other recent calculations, but with an improved statistical precision; $c=1.65847(4)$. We also use the hydrodynamic relation $c^2=\rho_s/\chi_\perp(q\to 0)$ between $c$, the spin stiffness $\rho_s$, and the transversal susceptibility $\chi_\perp$, using the smallest non-zero momentum $q=2\pi/L$. This method also is well controlled in two dimensions, but the cubic criterion for winding numbers delivers better numerical precision. In one dimension the hydrodynamic relation is affected by logarithmic corrections which make accurate extrapolations difficult. As an application of the winding-number method, for the quantum-critical bilayer model our high-precision determination of the velocity enables us to quantitatively test, at an unprecedented level, field-theoretic predictions for low-temperature scaling forms where $c$ enters.

Nonlinear bond-operator theory and1/dexpansion for coupled-dimer magnets. I. Paramagnetic phase

For coupled-dimer Heisenberg magnets, a paradigm of magnetic quantum phase transitions, we develop a systematic expansion in 1/d, the inverse number of space dimensions. The expansion employs a formulation of the bond-operator technique and is based on the observation that a suitably chosen product-state wavefunction yields exact zero-temperature expectation values of local observables in the d->infty limit, with corrections vanishing as 1/d. We demonstrate the approach for a model of dimers on a hypercubic lattice, which generalizes the square-lattice bilayer Heisenberg model to arbitrary d. In this paper, we use the 1/d expansion to calculate static and dynamic observables at zero temperature in the paramagnetic singlet phase, up to the quantum phase transition, and compare the results with numerical data available for d=2. Contact is also made with previously proposed refinements of bond-operator theory as well as with a perturbative expansion in the inter-dimer coupling. In a companion paper, the present 1/d expansion will be extended to the ordered phase, where it is shown to consistently describe the entire phase diagram including the quantum critical point.

Corner contribution to the entanglement entropy of an O(3) quantum critical point in 2 + 1 dimensions

.The entanglement entropy for a quantum critical system across a boundary with a corner exhibits a subleading logarithmic scaling term with a scale-invariant coefficient. Using a Numerical Linked Cluster Expansion, we calculate this universal quantity for a square-lattice bilayer Heisenberg model at its quantum critical point. We find, for this 2 + 1 dimensional O(3) universality class, that it is thrice the value calculated previously for the Ising universality class. This relation gives substantial evidence that this coefficient provides a measure of the number of degrees of freedom of the theory, analogous to the central charge in a 1 + 1 dimensional conformal field theory.

Cluster functional renormalization group

Functional renormalization group (FRG) has become a diverse and powerful tool to derive effective low-energy scattering vertices of interacting many-body systems. Starting from a free expansion point of the action, the flow of the RG parameter $\ensuremath{\Lambda}$ allows us to trace the evolution of the effective one- and two-particle vertices towards low energies by taking into account the vertex corrections between all parquet channels in an unbiased fashion. In this work, we generalize the expansion point at which the diagrammatic resummation procedure is initiated from a free UV limit to a cluster product state. We formulate a cluster FRG scheme where the noninteracting building blocks (i.e., decoupled spin clusters) are treated exactly, and the intercluster couplings are addressed via RG. As a benchmark study, we apply our cluster FRG scheme to the spin-$\frac{1}{2}$ bilayer Heisenberg model (BHM) on a square lattice where the neighboring sites in the two layers form the individual two-site clusters. Comparing with existing numerical evidence for the BHM, we obtain reasonable findings for the spin susceptibility, the spin-triplet excitation energy, and quasiparticle weight even in coupling regimes close to antiferromagnetic order. The concept of cluster FRG promises applications to a large class of interacting electron systems.

Excitation Spectra of Disordered Dimer Magnets Near Quantum Criticality

For coupled-dimer magnets with quenched disorder, we introduce a generalization of the bond-operator method, appropriate to describe both singlet and magnetically ordered phases. This allows for a numerical calculation of the magnetic excitations at all energies across the phase diagram, including the strongly inhomogeneous Griffiths regime near quantum criticality. We apply the method to the bilayer Heisenberg model with bond randomness and characterize both the broadening of excitations and the transfer of spectral weight induced by disorder. Inside the antiferromagnetic phase this model features the remarkable combination of sharp magnetic Bragg peaks and broad magnons, the latter arising from the tendency to localization of low-energy excitations.

Dynamics of a single exciton in strongly correlated bilayers

We formulated an effective theory for a single interlayer exciton in a bilayer quantum antiferromagnet, in the limit when the holon and doublon are strongly bound onto one interlayer rung by the Coulomb force. Upon using a rung linear spin-wave approximation of the bilayer Heisenberg model, we calculated the spectral function of the exciton for a wide range of the interlayer Heisenberg coupling α = J⊥/Jz. In the disordered phase at large α, a coherent quasi-particle peak appears, representing free motion of the exciton in a spin singlet background. In the Néel phase, which applies to more realistic model parameters, a ladder spectrum arises due to Ising confinement of the exciton. The exciton spectrum is visible in measurements of the dielectric function, such as c-axis optical conductivity measurements.

Ground-state properties, excitation spectra, and phase transitions in theS=12andS=32bilayer Heisenberg models on the honeycomb lattice

Motivated by the observation of a disordered spin ground state in the $S=3/2$ material Bi$_3$Mn$_4$O$_{12}$NO$_3$, we study the ground state properties and excitation spectra of the $S=3/2$ (and for comparison $S=1/2$) bilayer Heisenberg model on the honeycomb lattice, with and without frustrating further neighbor interactions. We use series expansions around the N\'eel state to calculate properties of the magnetically ordered phase. Furthermore, series expansions in $1/\lambda=J_1/J_{\perp}$, where $J_1$ is an in-plane exchange constant and $J_\perp$ is the exchange constant between the layers are used to study properties of the spin singlet phase. For the unfrustrated case, our results for the phase transitions are in very good agreement with recent Quantum Monte Carlo studies. We also obtain the excitation spectra in the disordered phase and study the change in the critical $\lambda$ when frustrating exchange interactions are added to the $S=3/2$ system and find a rapid suppression of the ordered phase with frustration. Implications for the material Bi$_3$Mn$_4$O$_{12}$NO$_3$ are discussed.

Variational study of the quantum phase transition in the bilayer Heisenberg model with bosonic RVB wavefunction

We study the ground state phase diagram of the bilayer Heisenberg model on a square lattice with a bosonic resonating valence bond (RVB) wavefunction. The wavefunction has the form of a Gutzwiller projected Schwinger boson mean-field ground state and involves two variational parameters. We find the wavefunction provides an accurate description of the system on both sides of the quantum phase transition. In particular, through the analysis of the spin structure factor, ground state fidelity susceptibility and the Binder moment ratio Q2, a continuous transition from the antiferromagnetic ordered state to the quantum disordered state is found at the critical coupling of αc = J⊥/J∥ ≈ 2.62, in good agreement with the result of quantum Monte Carlo simulation. The critical exponent estimated from the finite size scaling analysis (1/ν ≈ 1.4) is consistent with that of the classical 3D Heisenberg universality class.

Critical behaviour of structure factors at a quantum phase transition

We review the theoretical behaviour of the total and one-particlestructure factors at a quantum phase transition for temperatureT = 0. The predictions are compared with exact or numerical results for the transverse Isingmodel, the alternating Heisenberg chain, and the bilayer Heisenberg model. At the criticalwavevector, the results are generally in accord with theoretical expectations. Away fromthe critical wavevector, however, different models display quite different behaviours for theone-particle residues and structure factors.

Single-hole dynamics in the Kondo necklace and bilayer Heisenberg models on a square lattice

We study single hole dynamics in the bilayer Heisenberg and Kondo Necklace models. Those models exhibit a magnetic order-disorder quantum phase transition as a function of the interlayer coupling J_perp. At strong coupling in the disordered phase, both models have a single-hole dispersion relation with band maximum at p = (\pi,\pi) and an effective mass at this p-point which scales as the hopping matrix element t. In the Kondo Necklace model, we show that the effective mass at p = (\pi,\pi) remains finite for all considered values of J_perp such that the strong coupling features of the dispersion relation are apparent down to weak coupling. In contrast, in the bilayer Heisenberg model, the effective mass diverges at a finite value of J_perp. This divergence of the effective mass is unrelated to the magnetic quantum phase transition and at weak coupling the dispersion relation maps onto that of a single hole doped in a planar antiferromagnet with band maximum at p = (\pi/2,\pi/2). We equally study the behavior of the quasiparticle residue in the vicinity of the magnetic quantum phase transition both for a mobile and static hole. In contrast to analytical approaches, our numerical results do not unambiguously support the fact that the quasiparticle residue of the static hole vanishes in the vicinity of the critical point. The above results are obtained with a generalized version of the loop algorithm to include single hole dynamics on lattice sizes up to 20 X 20.

Heat Capacity and Antiferromagnetic Phase Transition of the Organic Free Radical Magnet, 2-tert-Butylaminoxylbenzimidazole (BABI)

Heat capacities of the organic free radical magnet 2-tert-butylaminoxylbenzimidazole (BABI) have been measured in the 0.2−300 K range by adiabatic calorimetry for investigation of its magnetic properties from a thermodynamic viewpoint. An antiferromagnetic phase transition was observed at TN = 1.7 K. A heat capacity hump arising from short-range spin ordering characteristic of low-dimensional magnets was found around 2 K. The magnetic enthalpy and entropy were determined to be 17.4 J mol-1 and 5.34 J K-1 mol-1, respectively. The magnitude of the magnetic entropy is in good agreement with the theoretical value for the magnetic ordering of spin quantum number S = 1/2 systems, R ln 2 = 5.76 J K-1 mol-1, where R is the gas constant. The heat capacity hump above TN is well reproduced by the S = 1/2 two-dimensional antiferromagnetic bilayer Heisenberg model with the intralayer interaction J1/kB = −1.2 K and the interlayer interaction J2/kB = −1.9 K, where kB is the Boltzmann constant. The bilayer model gives a ...

Studies of the exactly dimerized bilayer Heisenberg model via strong coupling expansions

The one-triplet excitation spectra and thermodynamic properties for the dimerized phase of the frustrated bilayer Heisenberg model are studied using strong-coupling expansion theory. The model has an exact dimerized ground state as well as exact one-triplet excitations in a special case that the frustration J2 is equal to the in-plane coupling J1. We demonstrate that the models with and without frustrations have distinct excitation spectra, so their thermodynamic properties exhibit quite different behaviors. Especially, the low-temperature behaviors of the frustrated model with J1=J2 are independent of the inter-dimer couplings, due to the exact one-triplet excitations.

Order-disorder transitions in bilayer Heisenberg models on the triangular lattice

The triangular-lattice bilayer Heisenberg model with antiferromagnetic interlayer coupling ${J}_{0}$ is studied using strong-coupling perturbation theory and bond-operator mean-field theory. It is found that the model with ferromagnetic intralayer coupling $J<0$ undergoes a quantum order-disorder transition at a strong-${J}_{0}$ critical point ${J}_{0c}=\ensuremath{-}3.83J.$ While for the model with $J>0,$ the transition point is very small, ${J}_{0c}=0.15J.$ The result implies that the single-layer triangular Heisenberg antiferromagnet only has small long-range order parameter (if ordered) so that it is very sensitive to the interlayer perturbation.

Quantum Phase Transitions in the Triangular-Lattice Bilayer Heisenberg Model

We study the triangular lattice bilayer Heisenberg model with antiferromagnetic interplane coupling $J_\perp$ and nearest neighbour intraplane coupling $J= \lambda J_\perp$, which can be ferro- or antiferromagnetic, by expansions in $\lambda$. For negative $\lambda$ a phase transition is found to an ordered phase at a critical $\lambda_c=-0.2636 \pm 0.0001$ which is in the 3D classical Heisenberg universality class. For $\lambda>0$, we find a transition at a rather large $\lambda_c\approx 1.2$. The universality class of the transition is consistent with that of Kawamura's 3D antiferromagnetic stacked triangular lattice. The spectral weight for the triplet excitations, at the ordering wavevector, remains finite at the transition, suggesting that a phase with free spinons does not exist in this model.

Field-dependent thermodynamics and quantum critical phenomena in the dimerized spin systemCu2(C5H12N2)2Cl4

Experimental data for the uniform susceptibility, magnetization, and specific heat for the material ${\mathrm{Cu}}_{2}{(\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}$ (CuHpCl) as a function of temperature and external field are compared with those of three different dimerized spin models: alternating spin chains, spin ladders, and the bilayer Heisenberg model. It is shown that because this material consists of weakly coupled spin dimers, much of the data are insensitive to how the dimers are coupled together and what the effective dimensionality of the system is. When such a system is tuned to the quantum critical point by application of a field, dimensionality shows up in the power-law dependences of thermodynamic quantities on the temperature. We discuss the temperature window for such a quantum critical behavior in CuHpCl.

Bond-Operator Mean Field Theory for the Bilayer Heisenberg Model

Bond-operator mean field equations for the square-lattice, S=1/2 bilayer Heisenberg model are developed and solved numerically. In the vicinity of both the zero-field critical point and the field-induced transitions, comparisons are made with T=0 and finite-temperature strong coupling expansions. The mean-field theory suggests that the quantum critical region for the field-induced transitions is restricted to significantly lower temperatures than one might have concluded based on strong-coupling expansions or other numerical studies.

Strong-coupling expansions at finite temperatures: Application to quantum disordered and quantum critical phases

By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.