Corresponding author

Ali Alavi

Max Planck Institute for Solid State Research


Booth, G.H.; Thom, A.J.W.; Alavi, A.
Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space
Stein, T.; Henderson, T.M.; Scuseria, G.M.
Seniority zero pair coupled cluster doubles theory
Helgaker, T.; Jürgensen, P.; Olsen, J.
Molecular Electronic-Structure Theory

Department "Electronic Structure Theory"

Insight into the many-electron wavefunction of Mott-insulating antiferromagnets


L. R. Schwarz and A. Alavi


Electronic Structure Theory (Ali Alavi)

We investigate the structure of full configuration-interaction wave functions of a prototypical Mott-insulating anti-ferromagnet using a state-of-the-art methodology (FCIQMC). We show that the compactness of the wavefunctions depends dramatically and counterintuitively on the nature of the single-particle representation, with restricted Hartree-Fock (HF) yielding a much more compact solution than unrestricted HF, and which furthermore may be amenable to approximation. Our study suggests that in a suitable basis, powerful quantum chemical methodologies may be employed in describing such systems.

Introduction: Mott-insulating antiferromagnets are ubiquitously found in the parent compounds of many strongly-correlated materials, including cuprates and 3d transition metal mono-oxides. In order to describe such systems, there is growing interest in wavefunction methods, including configuration-interaction (CI) methods, such as full configuration interaction Quantum Monte Carlo, (i)-FCIQMC [1], applied to periodic clusters. Whilst wavefunctions of strongly-correlated molecular systems primarily build on restricted one-particle bases, the general approach in condensed matter physics communities is to start from a qualitatively correct broken-symmetry solution. We therefore ask how the structure of the many-electron wavefunction of a typical Mott-insulator depends on the representation of the single-particle basis used to describe the configuration space of the system. On the face it, such a question appears to be more of a mathematical rather than a physical one. However, we will show that the choice of basis impacts the convergence of the exact full configuration interaction (FCI) expansion of the many-electron wavefunction in a dramatic and counterintuitive manner. Thus a broken-symmetry mean-field basis (UHF), which produces a qualitatively correct description of the antiferromagnet, is shown to provide an extremely poor basis to construct the exact FCI wavefunction: the CI coefficients decay very slowly with increasing particle-hole excitation of the reference, resulting in a highly complex wavefunction which is very difficult to approximate. 

In contrast, we find that the qualitatively incorrect band-structure of spin-restricted mean-field theory (RHF) provides a rapidly convergent CI expansion, which is much more amenable to approximation. Natural orbitals provide yet more rapidly converging, compact, wavefunctions. Our results suggest that with the correct single-particle basis, such strongly-correlated systems may be amenable to powerful single-reference wavefunction methods, opening a new direction of theoretical research for these systems [2].

Model: The three-band (p–d) Hubbard Hamiltonian describes the dynamics of holes in a CuO2 plane represented by a Cu 3dx2–y2 orbital and two O 2pσ orbitals. Its Hamiltonian comprises kinetic energy, tij, and hole interaction terms, Uij,

where ai,σ (ai,σ) creates (annihilates) a hole with spin σ in the Cu 3d or O 2pσ orbital at site i. We present results for a tilted lattice with 10 CuO2 units and N = 10 holes, an undoped system at half-filling with a full Hilbert space of 20.3×109 determinants. 

Method: In (i)-FCIQMC [1] the ground state wavefunction |ψ>, expressed as an FCI expansion |ψ> = ∑i Ci |Di>, is sampled by an ensemble of Nw signed walkers which stochastically evolve in a combinatorially large Hilbert space of N-particle Slater determinants constructed from an orthonormal one-particle basis set of size M, acording to

The discrete basis allows for effective cancellation algorithms, such that for sufficiently large walker populations Nw, the fermion sign problem, an exponential increase in noise, can be controlled and (i)-FCIQMC converges to the FCI limit within stochastic errors, with a reduced scaling compared to traditional FCI (exact diagonalization).

The choice of one-particle basis set affects the nature of the Ci coefficients and may lead to a compact and sparse FCI representation of |ψ> that is more amenable to treatment with configuration-based methods. A natural measure for the sparsity of a wavefunction is the L1-norm, L1 = ∑i |Ci|  and we therefore seek a representation of |ψ> in which the L1-norm and level of complexity is small.  For this purpose, we investigate two widely available sources of single-particle orbitals, restricted and unrestricted Hartree-Fock spin orbitals (RHF, UHF) which we also compare with restricted and unrestricted natural orbitals (RNO, UNO) which diagonalize the exact one-particle density matrix (γqp = <Ψ|apaq|Ψ>) and are known to give rapidly converging FCI expansions. Whereas restricted HF and NO orbitals restrict the spatial distributions ψi(r) to be equal for α and β channels, unrestricted spin orbitals, are characterized by relaxation of this constraint, ψiα (r) ≠ ψiβ (r) [3].

Results and discussions: Counterintuitively, we observe that working with RHF orbitals in i-FCIQMC leads to a |ψ> with a smaller L1-norm (L1 = 723.6) as opposed to UHF spin orbitals (L1 = 1059.5).

Ground states: In order to shed light on the reasons behind this, we examine the i-FCIQMC, RHF and UHF ground states (Tab. 1). Like previous studies, we find that the exact i-FCIQMC ground-state establishes an antiferromagnetic long-range order across the copper sites. |ψRHF> and |ψUHF> constitute a clear example of the symmetry dilemma [3]. While the UHF basis provides a physically closer single-determinant description of the antiferromagnetic ground-state, it breaks spin symmetry by separating α and β orbitals on two sublattices thereby leading to an inherent spin contamination. In contrast, the spin symmetry conserving RHF basis yields a qualitatively incorrect metallic ground state, but is still found to be a much more effective basis for correlated calculations within i-FCIQMC.

<strong>Tab. 1:</strong> Ground-state energies <em>E</em><sub>0 </sub>(eV/hole), energy of the lowest-energy determinant <em>E</em><sub>D </sub>(eV/hole), percentage of correlation energy <em>p</em><sub>corr</sub> = (<em>E</em><sub>0 </sub>&ndash; <em>E</em><sub>D</sub>)/(<em>E</em><sub>exact </sub>&ndash; <em>E</em><sub>D</sub>) (%) captured by |&Psi;<sub>CAS</sub>&gt;, average hole densities per atom &lt;<em>n</em><sub>at</sub>&gt; (holes/atom), staggered magnetization &lt;<em>M</em><sup>2</sup>&gt; = N<sup>&ndash;1</sup> &sum;<sub>ij</sub> (&ndash;1)<sup>(x<sub>i</sub>+y<sub>i</sub>)+(x<sub>j</sub>+y<sub>j</sub>)</sup> &lt;&Psi;<sub>FCI</sub>|<strong>S</strong><sub>i</sub> &middot; <strong>S</strong><sub>j</sub>|&Psi;<sub>FCI</sub>&gt;, square magnitude of spin &lt;<em>S</em><sup>2</sup>&gt;. Errors in the previous digit are presented in parentheses. Local spin-spin correlation function&nbsp;&lt;&Psi;<sub>FCI</sub>|<strong>S</strong><sub>1</sub> &middot; <strong>S</strong><sub>j</sub>|&Psi;<sub>FCI</sub>&gt; of the correlated <em>i</em>-FCIQMC wavefunction&nbsp;|&Psi;<sub>FCI</sub>&gt; in the metallic RHF one-particle basis. Zoom Image
Tab. 1: Ground-state energies E0 (eV/hole), energy of the lowest-energy determinant ED (eV/hole), percentage of correlation energy pcorr = (E0 ED)/(Eexact ED) (%) captured by |ΨCAS>, average hole densities per atom <nat> (holes/atom), staggered magnetization <M2> = N–1ij (–1)(xi+yi)+(xj+yj)FCI|Si · SjFCI>, square magnitude of spin <S2>. Errors in the previous digit are presented in parentheses. Local spin-spin correlation function <ΨFCI|S1 · SjFCI> of the correlated i-FCIQMC wavefunction |ΨFCI> in the metallic RHF one-particle basis.

Orbital occupations numbers: A clue with the difficulty introduced by the UHF basis can be obtained by considering the orbital occupation numbers γpp in the four bases (Fig. 1). Whilst for the RHF, RNO and UNO bases the γpp decay roughly monotonically with mean-field orbital energy εp (Fig. 1), a sharp increase is observed for the N highest UHF virtuals, which are also far higher in energy than those of any other basis. These virtuals correspond to spin-flipped counterparts of the occupied UHF orbitals which introduces anti-boding character and splits the Hubbard bands far apart.

<strong>Fig. 1:</strong> The magnitude of coefficients |<em>C</em><sub>i</sub>| &ge; 0.0001 in the FCI expansion |&Psi;&gt; = &sum;<sub>i&nbsp;</sub><em>C</em><sub>i </sub>|<em>D</em><sub>i</sub>&gt; shown against the respective determinant energy <em>E</em><sub>i</sub> for RHF (upper left), RNO (lower left), UHF (upper middle) and UNO (lower middle) basis sets. The |<em>C</em><sub>i</sub> coefficients of the full space |&Psi;&gt; are depicted in the lower panels while those of the (10,10)-CAS space |&Psi;<sub>CAS</sub>&gt; are shown in the top panels. The colours distinguish the <em>x</em>-fold excitations (<em>x</em> &isin; {1,2,...,<em>N</em>}) of the reference. The orbital occupation numbers &gamma;<sub>p</sub><sup>p</sup> (upper right) and mean-field orbital energies &epsilon;<sub>i</sub> (diagonal elements of mean-field generalized Fock matrix) (lower right) [3]. The numbers indicate degeneracies which are exact for HF and approximate for NO spin orbitals. Zoom Image
Fig. 1: The magnitude of coefficients |Ci| ≥ 0.0001 in the FCI expansion |Ψ> = ∑Ci |Di> shown against the respective determinant energy Ei for RHF (upper left), RNO (lower left), UHF (upper middle) and UNO (lower middle) basis sets. The |Ci coefficients of the full space |Ψ> are depicted in the lower panels while those of the (10,10)-CAS space |ΨCAS> are shown in the top panels. The colours distinguish the x-fold excitations (x ∈ {1,2,...,N}) of the reference. The orbital occupation numbers γpp (upper right) and mean-field orbital energies εi (diagonal elements of mean-field generalized Fock matrix) (lower right) [3]. The numbers indicate degeneracies which are exact for HF and approximate for NO spin orbitals. [less]

By breaking spin symmetry, UHF theory leads to a set of single-particle states characterized by localized spatial distribution ψiσ(r) which strongly differ in their extent. In contrast, the metallic RHF orbitals are very delocalized and hence exhibit similar ψi(r). This facilitates correlation of the single-particle states, thereby favouring a more rapidly converging CI expansion in comparison to UHF orbitals. Additionally, the RHF orbital and determinant energies cover a smaller energy range (Fig. 1), a characteristic also shared by both NO bases. A quantitative measure of the configurational mixing present in the |ψ> representations is provided by the correlation entropy, SCE = –(N)–1p γpp ln γpp. With SCE = 0.6421 for RNOs and SCE = 0.6115 for UNOs, the entanglement in |ψ> is smallest in NO bases, followed by the RHF (SCE = 0.7635) and UHF (SCE = 0.8846) orbitals.

Subspace diagonalizations: The FCIQMC descriptions of |ψ> (Fig. 1) are all highly multiconfigurational with many single- to N-fold particle-hole excitations of the reference with |Ci| decaying exponentially with determinant energy. In particular, in the UHF expansion a plethora of 10-fold excitations contribute to |ψ>. By contrast, the RHF, and more so RNO and UNO bases, are both sparser and strongly weighted towards the low particle-hole excitations, which are more amenable to accurate treatment of correlations via a compact set of explicit configurations.

Remarkably, |ψFCI> in the RHF basis can be well approximated by simple subspace diagonalizations in a (10,10)-CAS space. This is also the case for RNO and UNO spaces where |ψCAS> captures a majority of the respective correlation energy (Tab. 1) and basic structure of |ψ>. However, in the UHF space barely any correlation energy is captured by this subspace. This is a consequence of the fact that determinant weight is entirely absent from high particle-hole excitation when compared to |ψ> despite the fact that many of the significant high-excitations determinants are included in the CAS space. This suggests that orbitals outside the CAS space, especially the N highest-energy virtuals, are essential for establishing the basic structure of |ψ> in the UHF basis.

Conclusions: We have analysed the FCI wavefunction representation in different single-particle basis sets and their amenability to accurate correlation treatments using i-FCIQMC and the strongly-correlated three-band Hubbard model. Counterintuitively, the effectiveness of single-particle basis sets for rapidly converging CI expansions is not necessarily paralleled by their ability to reproduce the physics of the system within a single-determinant description. Whilst the UHF determinant represents qualitatively the correct insulating antiferromagnet, imposing spin symmetry in the RHF basis gives an RHF determinant describing a qualitatively incorrect metal. Yet, in this basis the FCI representation of |ψ> is sparser, and converges rapidly with particle-hole excitations of the reference. Our results therefore suggest that with an appropriate single-particle description, it may be possible to describe the many-electron wavefunction of strongly-correlated materials based on single-reference quantum chemical methodologies [2,3], which opens up a vast array of powerful many-body techniques for the study of such systems.

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