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arXiv:1007.2517 · Physical Review B

Variational cluster approach to ferromagnetism in infinite dimensions and in one-dimensional chains

Matthias Balzer, Michael Potthoff

Published Nov 30, 2010• 21 citations
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Abstract

The variational cluster approach (VCA) is applied to study spontaneous ferromagnetism in the Hubbard model at zero temperature. We discuss several technical improvements of the numerical implementation of the VCA which become necessary for studies of a ferromagnetically ordered phase, e.g., more accurate techniques to evaluate the variational ground-state energy, improved local as well as global algorithms to find stationary points, and different methods to locate the magnetic phase transition. Using the single-site VCA, i.e., the dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the model in infinite dimensions is worked out. The results are compared with previous dynamical mean-field studies for benchmarking purposes. The DIA results provide a unified picture of ferromagnetism in the infinite-dimensional model by interlinking different parameter regimes that are governed by different mechanisms for ferromagnetic order. Using the DIA and the VCA, we then study ferromagnetism in one-dimensional Hubbard chains with nearest- and next-nearest-neighbor hopping ${t}_{2}$. In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard-$U$, the electron filling, and ${t}_{2}$. The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.

DOI
10.1103/physrevb.82.174441

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The variational cluster approach (VCA) is applied to study spontaneous ferromagnetism in the Hubbard model at zero temperature. We discuss several technical improvements of the numerical implementation of the VCA which become necessary for studies of a ferromagnetically ordered phase, e.g., more accurate techniques to evaluate the variational ground-state energy, improved local as well as global algorithms to find stationary points, and different methods to locate the magnetic phase transition. Using the single-site VCA, i.e., the dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the model in infinite dimensions is worked out. The results are compared with previous dynamical mean-field studies for benchmarking purposes. The DIA results provide a unified picture of ferromagnetism in the infinite-dimensional model by interlinking different parameter regimes that are governed by different mechanisms for ferromagnetic order. Using the DIA and the VCA, we then study ferromagnetism in one-dimensional Hubbard chains with nearest- and next-nearest-neighbor hopping t2{t}_{2}t2​. In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard-UUU, the electron filling, and t2{t}_{2}t2​. The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.
The variational cluster approach (VCA) is applied to the one-dimensional Hubbard model at zero temperature using clusters (chains) of up to ten sites with full diagonalization and the Lanczos method as cluster solver. Within the framework of the self-energy-functional theory (SFT), different cluster reference systems with and without bath degrees of freedom, in different topologies and with different sets of variational parameters, are considered. Static and one-particle dynamical quantities are calculated for half-filling as a function of UUU as well as for fixed UUU as a function of the chemical potential to study the interaction- and filling-dependent metal-insulator (Mott) transition. The recently developed QQQ-matrix technique is used to compute the SFT grand potential. For benchmarking purposes we compare the VCA results with exact results available from the Bethe ansatz, with essentially exact dynamical density-matrix renormalization-group data, with (cellular) dynamical mean-field theory and full diagonalization of isolated Hubbard chains. Several issues are discussed including convergence of the results with cluster size, the ability of cluster approaches to access the critical regime of the Mott transition, efficiency in the optimization of correlated-site vs bath-site parameters, and of multidimensional parameter optimization. We also study the role of bath sites for the description of excitation properties and as charge reservoirs for the description of filling dependencies. The VCA turns out to be a computationally cheap method which is competitive with established cluster approaches.
We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion UUU and temperature TTT. We also calculate the TTT dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the TTT and/or UUU derivative of the grand potential, we obtain a crossover diagram in the (U,T)(U,T)(U,T) plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the (U,T)(U,T)(U,T) plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.
We study the phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping using exact diagonalization, the density-matrix renormalization group, the Edwards variational ansatz, and an adaptation of weak-coupling calculations on the two-chain Hubbard model. We find that a substantial region of the strong-coupling phase diagram is ferromagnetic, and that three physically different limiting cases are connected in one ferromagnetic phase. At a point in the phase diagram at which there are two Fermi points at weak coupling, we study carefully the phase transition from the paramagnetic state to the fully polarized one as a function of the on-site Coulomb repulsion. We present evidence that the transition is second order and determine the critical exponents numerically. In this parameter regime, the system can be described as a Luttinger liquid at weak coupling. We extract the Luttinger-liquid parameters and show how their behavior differs from that of the nearest-neighbor Hubbard model. The general weak-coupling phase diagram can be mapped onto that of the two-chain Hubbard model. We exhibit explicitly the adapted phase diagram and determine its validity by numerically calculating spin and charge gaps using the density-matrix renormalization group.
The stability of the ferromagnetic state with complete spin alignment against a single spin reversal is studied in the square-lattice Hubbard model with nearest-neighbour hopping. A variational ansatz is used, which is exact in one dimension and by comparison with accurate results for small clusters yields almost exact results in the two-dimensional case. The ferromagnetic region in the (W/U, delta ) phase diagram is mapped out, where W is the bandwidth, U the on-site electron interaction and delta is the number of holes per atom. This region is considerably smaller than in previous variational calculations and it is rigorously concluded that the state of complete spin alignment is unstable when delta >0.29, for all U, and when W/U>0.19, for all delta . The nature of the instability, and of the low-lying excitations in the ferromagnetic state for varying delta and W/V, is discussed.
The variational cluster approximation (VCA) proposed by M. Potthoff et al. [Phys. Rev. Lett. 91, 206402 (2003)] is extended to electron or spin systems with nonlocal interactions. By introducing more than one source field in the action and employing the Legendre transformation, we derive a generalized self-energy functional with stationary properties. Applying this functional to a proper reference system, we construct the extended VCA (EVCA). In the limit of continuous degrees of freedom for the reference system, EVCA can recover the cluster extension of the extended dynamical mean-field theory (EDMFT). For a system with correlated hopping, the EVCA recovers the cluster extension of the dynamical mean-field theory for correlated hopping. Using a discrete reference system composed of decoupled three-site single impurities, we test the theory for the extended Hubbard model. Quantitatively good results as compared with EDMFT are obtained. We also propose VCA (EVCA) based on clusters with periodic boundary conditions. It has the (extended) dynamical cluster approximation as the continuous limit. A number of related issues are discussed.
Using the variational cluster approach, we study the transition from the antiferromagnetic to the superconducting phase of the two-dimensional Hubbard model at zero temperature. Our calculations are based on a method to evaluate the VCA grand potential which employs a modified Lanczos algorithm and avoids integrations over the real or imaginary frequency axis. Thereby, very accurate results are possible for cluster sizes not accessible to full diagonalization. This is important for an improved treatment of short-range correlations, including correlations between Cooper pairs in particular. We apply this improved method in order to investigate the cluster-size dependence of the phase-separation tendency that has been proposed recently on the basis of calculations for smaller clusters. While the energy barrier associated with phase separation rapidly decreases with increasing cluster size for both hole and electron doping, the extension of the phase-separation region behaves differently in the two cases. More specifically, our results suggest that phase separation remains persistent in the hole-doped case and disappears in the electron-doped case. We also study the evolution of the single-particle spectrum as a function of doping and point out the relevance of our results for experimental findings in electron and hole-doped materials.