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Linked-cluster Expansions for Lattice Spin Models

Yuyi Wan

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

Similar to various series expansions that are used to approximate mathematical func- tions, the linked-cluster expansion is an approximation method that allows us to approach the actual values of a very large physical system’s different physical quan- tities by systematically studying smaller systems embedded in this larger system. The main concept in linked-cluster expansion, weight, represents the additional con- tribution to a certain physical quantity by increasing the system size by one unit. These weights are used to eventually build up the result on a larger system. In our case, we focus on the partition function, a quantity that can be used to calculate several essential thermodynamic aspects of the system such as average energy, spe- cific heat, magnetic susceptibility, etc. We study these weights for the Ising and Potts models on a one dimensional lattice and Bethe lattice, with and without an external magnetic field. Previous studies have shown that in the one dimensional Ising model, adding more than two lattice sites to the system does not create any ad- ditional contribution to the partition function (i.e. their weights are all zero), giving us the result for an infinitely large system with just two lattice sites, a remarkable simplification. In our study, we prove that this property holds not only for the Ising model, but also its generalization, the Potts model, and show that it is a result of a spin flip symmetry inherent in the system. In order to test this, we break this symmetry using an externally applied magnetic field and show that in this case, for magnetic dipole energies comparable with the exchange energy between neighboring spins, this special property vanishes.

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The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.

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Einar Halvorsen, M. Bartkowiak

We consider $1/d$ expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate $1/d$ expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in $1/d.$ The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in $1/d.$

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Similar to various series expansions that are used to approximate mathematical func- tions, the linked-cluster expansion is an approximation method that allows us to approach the actual values of a very large physical system’s different physical quan- tities by systematically studying smaller systems embedded in this larger system. The main concept in linked-cluster expansion, weight, represents the additional con- tribution to a certain physical quantity by increasing the system size by one unit. These weights are used to eventually build up the result on a larger system. In our case, we focus on the partition function, a quantity that can be used to calculate several essential thermodynamic aspects of the system such as average energy, spe- cific heat, magnetic susceptibility, etc. We study these weights for the Ising and Potts models on a one dimensional lattice and Bethe lattice, with and without an external magnetic field. Previous studies have shown that in the one dimensional Ising model, adding more than two lattice sites to the system does not create any ad- ditional contribution to the partition function (i.e. their weights are all zero), giving us the result for an infinitely large system with just two lattice sites, a remarkable simplification. In our study, we prove that this property holds not only for the Ising model, but also its generalization, the Potts model, and show that it is a result of a spin flip symmetry inherent in the system. In order to test this, we break this symmetry using an externally applied magnetic field and show that in this case, for magnetic dipole energies comparable with the exchange energy between neighboring spins, this special property vanishes.
The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
We show that numerical linked cluster expansions (NLCEs) based on sufficiently large building blocks allow one to obtain accurate low-temperature results for the thermodynamic properties of spin lattice models with continuous disorder distributions. Specifically, we show that such results can be obtained computing the disorder averages in the NLCE clusters before calculating their weights. We provide a proof of concept using three different NLCEs based on L, square, and rectangle building blocks. We consider both classical (Ising) and quantum (Heisenberg) spin-1/2 models and show that convergence can be achieved down to temperatures that are up to two orders of magnitude lower than the relevant energy scale in the model. Additionally, we provide evidence that in one dimension one can obtain accurate results for observables such as the energy down to their ground-state values.
We propose a generalization of the linked-cluster expansions to study driven-dissipative quantum lattice models, directly accessing the thermodynamic limit of the system. Our method leads to the evaluation of the desired extensive property onto small connected clusters of a given size and topology. We first test this approach on the isotropic spin-1/2 Hamiltonian in two dimensions, where each spin is coupled to an independent environment that induces incoherent spin flips. Then we apply it to the study of an anisotropic model displaying a dissipative phase transition from a magnetically ordered to a disordered phase. By means of a Pad\'e analysis on the series expansions for the average magnetization, we provide a viable route to locate the phase transition and to extrapolate the critical exponent for the magnetic susceptibility.
We use numerical linked cluster expansions (NLCEs) to study correlated lattice systems in one and two dimensions.For fermions in the honeycomb lattice, we study its finite-temperature properties and short-range spin correlations using NLCEs and determinantal quantum Monte Carlo (DQMC) simulations.For the homogeneous system, we analyze a number of thermodynamic quantities, including the entropy, the specific heat, uniform and staggered spin susceptibilities, short-range spin correlations, and the double occupancy at and away from half filling.Employing a local density approximation (LDA), we examine the viability of adiabatic cooling by increasing the interaction strength for homogeneous as well as for trapped systems.We also use NLCEs to study thermodynamic properties of the two-dimensional spin-1/2 Ising, XY, and Heisenberg models with bimodal random-bond disorder on the square and honeycomb lattices.In all cases, the nearest-neighbor coupling between the spins takes values ±J with equal probability.We obtain the disorder averaged (over all disorder configurations) energy, entropy, specific heat, and uniform magnetic susceptibility in each case.These results are compared with the corresponding ones in the clean models.Analytic expressions are obtained for low orders in the expansion of these thermodynamic quantities in inverse temperature.For many-body localization in disordered isolated systems, we study quantum quenches in the thermodynamic limit.By a quantum quench it is meant that the initial state is stationary with respect iii to an initial Hamiltonian, which is suddenly changed to a new (time-independent) Hamiltonian.The latter then drives the (unitary) dynamics of the system.We are interested in the time average of observables after relaxation following the quench, which can be computed under the so-called diagonal ensemble.
We consider 1/d1/d1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate 1/d1/d1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in 1/d.1/d.1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in 1/d.1/d.1/d.