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Physical Review B

Topologically correct phase boundaries and transition temperatures for Ising Hamiltonians via self-consistent coarse-grained cluster-lattice models

Teck Leong Tan, Duane D. Johnson

Published Apr 27, 2011• 12 citations
Publisher

Abstract

We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size ${N}_{\mathrm{c}}$ and a coarse-grained (CG) lattice into cells of size ${N}_{\mathrm{cell}}$. We explore forms with ${N}_{\mathrm{cell}}\ensuremath{\geqslant}{N}_{\mathrm{c}}$, including a non-CG (NCG) version with ${N}_{\mathrm{cell}}\ensuremath{\rightarrow}\ensuremath{\infty}$. For ${N}_{\text{c}}={N}_{\mathrm{cell}}$, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational ${N}_{\mathrm{c}}$-site cluster grand potential based on ${N}_{\mathrm{c}}={N}_{\mathrm{cell}}$ CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the ${N}_{\mathrm{c}}=1$ CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed ${N}_{\mathrm{c}}$ for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.

DOI
10.1103/physrevb.83.144427

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We derive a cluster mean-field theory for an Ising Hamiltonian using a cluster-lattice Fourier transform with a cluster of size Nc{N}_{\mathrm{c}}Nc​ and a coarse-grained (CG) lattice into cells of size Ncell{N}_{\mathrm{cell}}Ncell​. We explore forms with Ncell\ensuremath⩾Nc{N}_{\mathrm{cell}}\ensuremath{\geqslant}{N}_{\mathrm{c}}Ncell​\ensuremath⩾Nc​, including a non-CG (NCG) version with Ncell\ensuremath→\ensuremath∞{N}_{\mathrm{cell}}\ensuremath{\rightarrow}\ensuremath{\infty}Ncell​\ensuremath→\ensuremath∞. For Nc=Ncell{N}_{\text{c}}={N}_{\mathrm{cell}}Nc​=Ncell​, the set of static, self-consistent equations relating cluster and CG lattice correlations is analogous to that in dynamical cluster approximation and cellular dynamical mean-field theory used in correlated electron physics. A variational Nc{N}_{\mathrm{c}}Nc​-site cluster grand potential based on Nc=Ncell{N}_{\mathrm{c}}={N}_{\mathrm{cell}}Nc​=Ncell​ CG lattice maintains thermodynamic consistency and improves predictions, recovering Monte Carlo and series expansion results upon finite-size scaling; notably, the Nc=1{N}_{\mathrm{c}}=1Nc​=1 CG results already predict well the first- and second-order phase boundary topology and transition temperatures for frustrated lattices. The NCG version is significantly faster computationally than the CG case and more accurate at fixed Nc{N}_{\mathrm{c}}Nc​ for ferromagnetism, which is potentially useful for cluster expansion and quantum cluster applications.
The dynamical cluster approximation (DCA) and its DCA+{\mathrm{DCA}}^{+}DCA+ extension use coarse-graining of the momentum space to reduce the complexity of quantum many-body problems, thereby mapping the bulk lattice to a cluster embedded in a dynamical mean-field host. Here, we introduce a new form of an interlaced coarse-graining and compare it with the traditional coarse-graining. While it gives a more localized self-energy for a given cluster size, we show that it leads to more controlled results with weaker cluster shape and smoother cluster size dependence, which converge to the results obtained from the standard coarse-graining with increasing cluster size. Most importantly, the new coarse-graining reduces the severity of the fermionic sign problem of the underlying quantum Monte Carlo cluster solver and thus allows for calculations on larger clusters. This enables the treatment of correlations longer ranged than those accessible with the standard coarse-graining and thus can allow for the evaluation of the exact infinite cluster size result via finite size scaling. As a demonstration, we study the hole-doped two-dimensional Hubbard model and show that the interlaced coarse-graining in combination with the extended DCA+{\mathrm{DCA}}^{+}DCA+ algorithm permits the determination of the superconducting Tc{T}_{c}Tc​ on cluster sizes for which the results can be fit with a Kosterlitz-Thouless scaling law.
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger UUU than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.
We examine the cluster-size dependence of the cellular dynamical mean-field theory (CDMFT) applied to the two-dimensional Hubbard model. Employing the continuous-time quantum Monte Carlo method as the solver for the effective cluster model, we obtain CDMFT solutions for 4-, 8-, 12-, and 16-site clusters at a low temperature. Comparing various periodization schemes, which are used to construct the infinite-lattice quantities from the cluster results, we find that the cumulant periodization yields the fastest convergence for the hole-doped Mott insulator where the most severe size dependence is expected. We also find that the convergence is much faster around (0,0)(0,0)(0,0) and $(\frac{\ensuremath{\pi}}{2},\frac{\ensuremath{\pi}}{2})$ than around $(\ensuremath{\pi},0)$ and $(\ensuremath{\pi},\ensuremath{\pi})$. The cumulant-periodized self-energy seems to be close to its thermodynamic limit already for a 16-site cluster in the range of parameters studied. The 4-site results remarkably agree well with the 16-site results, indicating that the previous studies based on the 4-site cluster capture the essence of the physics of doped Mott insulators.
Mean-field theory (MFT) is one of the main available tools for analytical calculations entailed in investigations regarding many-body systems. Recently, there has been a surge of interest in ameliorating this kind of method, mainly with the aim of incorporating geometric and correlation properties of these systems. The correlated cluster MFT (CCMFT) is an improvement that succeeded quite well in doing that for classical spin systems. Nevertheless, even the CCMFT presents some deficiencies when applied to quantum systems. In this article, we address this issue by proposing the quantum CCMFT (QCCMFT), which, in contrast to its former approach, uses general quantum states in its self-consistent mean-field equations. We apply the introduced QCCMFT to the transverse Ising model in honeycomb, square, and simple cubic lattices and obtain fairly good results both for the Curie temperature of thermal phase transition and for the critical field of quantum phase transition. Actually, our results match those obtained via exact solutions, series expansions or Monte Carlo simulations.
This article reviews quantum cluster theories, a set of approximations for infinite lattice models which treat correlations within the cluster explicitly, and correlations at longer length scales either perturbatively or within a mean-field approximation. These methods become exact when the cluster size diverges, and most recover the corresponding mean-field approximation when the cluster size becomes 1. Although quantum cluster theories were originally developed to treat disordered systems, they have more recently been applied to the study of ordered and disordered correlated systems, which will be the focus of this review. After a brief historical review, the authors provide detailed derivations of three cluster formalisms: the cluster perturbation theory, the dynamical cluster approximation, and the cellular dynamical mean-field theory. They compare their advantages and review their applications to common models of correlated electron systems.