Cluster Series Expansion Research Articles

Correlations and entanglement in quantum critical bilayer and necklace XY models

We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on the two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional $\phi^4$-theory. The second R\'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of $0.0674(7)$ [$0.0664(4)$] for the bilayer [necklace] model, respectively. Furthermore, the presence of $90^{\circ}$ corners leads to an additive logarithmic term in both models. We estimate a contribution of $-0.010(2)$ [$-0.009(2)$] due to each $90^{\circ}$ corner to the logarithmic correction for the bilayer [necklace] model, and compare our findings to recent numerical linked cluster calculations and series expansion results on related models.

Stochastic cluster series expansion for quantum spin systems

In this paper we develop the stochastic cluster series expansion (SCSE) which is a (cluster) variant of the stochastic series expansion (SSE) method. For certain systems with longer-range interactions the SCSE is considerably more efficient than the standard implementation of the SSE, at low temperatures. As an application of this method we calculated the $T=0$ conductance for a linear Heisenberg chain with a (diagonal) next-nearest-neighbor interaction.

Magnetic excitations and exchange interactions in the spin-gap system TlCuCl 3

The magnetic excitations from the gapped ground state in TlCuCl3 have been investigated by means of inelastic neutron scattering experiments. The excitation data were collected along four different directions in the a* – c* plane. A well-defined single magnetic excitation mode was observed. The lowest excitation occurs at Q=(h,0,l) with integer h and odd l, as observed in KCuCl3. The dispersion relations were analyzed by the cluster-series expansion up to the sixth order, so that the individual exchange interactions were evaluated. It was demonstrated that TlCuCl3 is a strongly coupled spin-dimer system.

Magnetic excitations in the spin-gap systemTlCuCl3

${\mathrm{TlCuCl}}_{3}$ has a singlet ground state with the excitation gap $\ensuremath{\Delta}=0.65 \mathrm{meV}.$ The magnetic excitations in ${\mathrm{TlCuCl}}_{3}$ have been investigated by means of neutron inelastic-scattering experiments. The constant-$\mathit{Q}$ energy scan profiles were collected in the ${a}^{*}\ensuremath{-}{c}^{*}$ plane. A well-defined single magnetic excitation mode was observed. The dispersion relations along four different directions were determined. The lowest excitation occurs at $\mathit{Q}=(h,0,l)$ with integer h and odd l, as observed in ${\mathrm{KCuCl}}_{3}.$ A cluster series expansion up to the sixth order was applied to analyze the dispersion relations, and the individual exchange interactions were evaluated. It was demonstrated that ${\mathrm{TlCuCl}}_{3}$ is a strongly coupled spin-dimer system.

Thermodynamic properties of the ferromagnetic spinel Zn pCd 1− pCr 2Se 4

The magnetic properties of the B-spinel ferromagnetic Zn p Cd 1− p Cr 2Se 4 compound are studied via a cluster series expansion approximation with nearest and next-nearest exchange integrals J 1 and J 2, respectively. Using the minimization of the free-energy expressions, the magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat are obtained and computed numerically as a function of temperature and for each composition of the system. The magnetization curves are used to determine the critical temperatures T c. Using the power laws in the vicinity of the critical regions the critical exponents β, γ and α associated, respectively, with the magnetization, the magnetic susceptibility and the specific heat are numerically calculated. The critical temperatures T c obtained are in very good agreement with those predicted by the magnetic measurements and the values of the critical exponents may be compared with other theoretical results based on the 3D Heisenberg model.

A CLUSTER SERIES EXPANSION TECHNIQUE FOR SOLVING VIBRATION PROBLEMS

A cluster series expansion technique for spectral solution of vibration problems is presented. In this technique the eigenfunction approximations are composed of clusters of single basis functions instead of the classical Fourier series of single basis functions. To realize the cluster expansion an alternating fixed-free subspace strategy is used to improve the coefficients of the single basis functions inside the clusters. This strategy generates a family of m -step iterative algorithms solving large dense matrix eigenproblem of order m. n as a few dense matrix eigenproblems of ordern +(m−1)ℓ, where ℓ denotes the number of computed eigenvalues. The computational behaviour of the proposed m -step algorithms is investigated by solving matrix eigenproblems up to the order 3600 by using them -step algorithms for m=2, 3, 4, 5.

Two-spin cluster theory applied to the ferromagnetic spinel CdCr 2pIn 2–2pS 4

The magnetic properties of the B-spinel ferromagnetic CdCr 2pIn 2–2pS 4 compound are studied via a cluster series expansion approximation with nearest and next-nearest exchange integrals J 1 and J 2, respectively. Using the minimization of the free-energy expressions, the magnetization, the magnetic susceptibility, the two-spin correlation functions and the specific heat are obtained and computed numerically as a function of temperature and for each composition of the system. The magnetization curves are used to determine the critical temperatures T c. Using the power laws in the vicinity of the critical regions the critical exponents β, γ and α associated, respectively, with the magnetization, the magnetic susceptibility and the specific heat are numerically calculated. The critical temperatures T c obtained are in very good agreement with those predicted by the magnetic measurements and the values of the critical exponents may be compared with other theoretical results based on the 3D Heisenberg model.

Local spin correlations in Heisenberg antiferromagnets

We use linked cluster series expansion methods to estimate the values of various short distance correlation functions in $S=1/2$ Heisenberg antiferromagnets at T=0, for dimension $d=1,2,3$. The method incorporates the possibility of spontaneous symmetry breaking, which is manifest in $d=2,3$. The results are important in providing a test for approximate theories of the antiferromagnetic ground state.

A cluster series expansion technique for the spectral solution of differential equations

A cluster series expansion technique for the spectral solution of differential equations is presented. An alternating fixed-free subspace strategy used for improving the accuracy of the cluster series expansion generates a family of m-step iterative algorithms solving large dense systems of linear equations of order m×n as a few partial systems of order n+m−1. The computational behaviour of the proposed m-step algorithms is examined by solving dense systems of linear equations up to the order 40 000 using the m-step algorithms for m = 2, 4, 6, 8, and 10.

Magnetic properties of spin-one ferromagnets with an arbitrary crystal-field potential

The magnetic properties of spin-one Heisenberg ferromagnets with general crystal field potential are investigated using the linked cluster expansion. A modified Wick reduction theorem is used to calculate the τ integrals occuring in the series expansion. The results of the linked cluster series expansion cover both the ordered and disordered phases of a system. The dependence of the critical temperature on the crystal-field potentials and the magnetization as a function of temperature and crystal-field potentials are shown. A comparison is made with the prediction in the mean-field approximation.

Quantum fluctuations in easy-plane spin-one ferromagnet

The magnetic properties of easy-plane spin-one Heisenberg ferromagnet, H=D∑i(Six)2 −J∑(i,j)Si⋅Sj−h∑iSiz, are investigated using the linked cluster series expansion. The results of the linked cluster series expansion method cover the whole range of temperatures, both magnetically ordered and disordered phases. The order parameter is included in correlations of spin fluctuation and solved self-consistently to high orders. The magnetization as a function of temperature for a given value of easy-plane anisotropy is shown. The observed values of magnetization from the linked cluster series expansion are appreciably depressed from their mean-field approximation values due to the quantum and thermal fluctuations. The effects of the quantum fluctuation on the magnetic ordering in three-dimensional (3D) systems are discussed.

Linked cluster series expansion and phase transitions in complex models

We show that the use of the linked-cluster series expansion aided by the extrapolation techniques is an effective method in the study of complex models which display first-order phase transitions as well as second-order phase transitions. The Blume–Capel model is used for the demonstration.

Methods in hamiltonian lattice field theory (II). Linked-cluster expansions

The linked cluster series expansion proposed by Nickel is extended to the ground state and axial string tension of lattice gauge theory. Proofs of these expansions and applications to Z 2 and U(1) gauge theory in 2 + 1 dimensions are presented. We also propose a new finite cluster scaling method based on the linked-cluster expansion and test it against known results for Z 2 gauge theory. The utility of the method in studying more complicated lattice gauge theories is emphasized.