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arXiv:cond-mat/0702080 · Physical Review B

Density functional formulations for quantum chains

Francisco Castilho Alcaraz, Klaus Capelle

Published Jul 12, 2007• 13 citations
arXivHTMLPDF

Abstract

We show that a lattice formulation of density functional theory (DFT), guided by renormalization-group concepts, can be used to obtain numerical predictions of energy gaps, spin-density profiles, critical exponents, sound velocities, surface energies, and conformal anomalies of spatially inhomogeneous quantum-spin chains. To this end, we (i) cast the formalism of DFT in the notation of quantum-spin chains to make the powerful methods and concepts developed in ab initio DFT available to workers in this field, (ii) explore to what extent simple local approximations in the spirit of the local-density approximation (LDA) can be used to predict critical exponents and conformal anomalies of quantum-spin models, and (iii) propose and explore various nonlocal approximations depending on the size of the system or on its average density in addition to the local density. These nonlocal functionals turn out to be superior to LDA functionals.

DOI
10.1103/physrevb.76.035109

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We show that a lattice formulation of density functional theory (DFT), guided by renormalization-group concepts, can be used to obtain numerical predictions of energy gaps, spin-density profiles, critical exponents, sound velocities, surface energies, and conformal anomalies of spatially inhomogeneous quantum-spin chains. To this end, we (i) cast the formalism of DFT in the notation of quantum-spin chains to make the powerful methods and concepts developed in ab initio DFT available to workers in this field, (ii) explore to what extent simple local approximations in the spirit of the local-density approximation (LDA) can be used to predict critical exponents and conformal anomalies of quantum-spin models, and (iii) propose and explore various nonlocal approximations depending on the size of the system or on its average density in addition to the local density. These nonlocal functionals turn out to be superior to LDA functionals.
Most of the basic ideas of the density-functional theory (DFT) are shown to be unrelated to the fact that the particle density is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, we discuss various approaches to density-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-density approximation (LDA) are compared to the exact results for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approximation is needed to formulate the LDA.
By using the quantum Ising chain as a test bed and treating the spin polarization along the external transverse field as the generalized density, we examine the performance of different levels of density functional approximations parallel to those widely used for interacting electrons, such as local density approximation (LDA) and generalized-gradient approximation (GGA). We show that by adding the lowest-order and nearest-neighbor density variation correction to the simple LDA, a semilocal energy functional in the spirit of GGA is almost exact over a wide range of inhomogeneous density distribution. In addition, the LDA and GGA error structures bear a high level of resemblance to the quantum phase diagram of the system. These results provide insights into the triumph and failure of these approximations in a general context.
Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.
Abstract Context: In the realm of quantum chemistry, the accurate prediction of electronic structure and properties of nanostructures remains a formidable challenge. Density Functional Theory (DFT) and Density Matrix Renormalization Group (DMRG) have emerged as two powerful computational methods for addressing electronic correlation effects in diverse molecular systems. We compare ground-state energies (e0), density profiles (n) and average entanglement entropies (Sˉ\bar SSˉ) in metals, insulators and at the transition from metal to insulator, in homogeneous, superlattices and harmonically confined chains described by the fermionic one-dimensional Hubbard model. While for the homogeneous systems there is a clear hierarchy between the deviations, DD%(\bar S)<D%(e_0)< \bar D%(n)D, and all the deviations decrease with the chain size; for superlattices and harmonical confinement the relation among the deviations is less trivial and strongly dependent on the superlattice structure and the confinement strength considered. For the superlattices, in general increasing the number of impurities in the unit cell represents less precision on the DFT calculations. For the confined chains, DFT performs better for metallic phases, while the highest deviations appear for the Mott and band-insulator phases. This work provides a comprehensive comparative analysis of these methodologies, shedding light on their respective strengths, limitations, and applications. Methods: The DFT calculations were performed using the standard Kohn-Sham scheme within the BALDA approach. It integrated the numerical Bethe-Ansatz (BA) solution of the Hubbard model as the homogeneous density functional within a local-density approximation (LDA) for the exchange-correlation energy. The DMRG algorithms were implemented using the ITensor library, which is based on the Matrix Product States (MPS) ansatz. The calculations were performed until the energy reaches convergence of at least 10-8.
Abstract Spin‐density functional theory (SDFT) is the most widely implemented and applied formulation of density functional theory. However, it is still finding novel applications, and occasionally encounters unexpected problems. In this article, we first briefly describe a few of the latter, related to issues such as nonuniqueness, noncollinearity, and currents. We then turn to an example of the former, namely SDFT for the Heisenberg model. It is shown that time‐honored concepts of Coulomb DFT, such as the local‐density approximation, can be applied to this (and other) model Hamiltonians, as well, once the concept of “density” has been suitably reinterpreted. Local‐density‐type approximations for the inhomogeneous Heisenberg model are constructed. Numerical applications to finite‐size and impurity systems demonstrate that DFT is a computationally efficient and reasonably accurate alternative to conventional methods of statistical mechanics for the Heisenberg model. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
We present a simple method, combining the density-matrix renormalization-group algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-1/21/21/2 Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-1 chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.