The basic question is addressed of how the space dimension $d$ is encoded in the Hilbert space of $N$ identical fermions. There appears a finite number $N{!}^{d\ensuremath{-}1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonized into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the $N$ -fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.The basic question is addressed of how the space dimension $d$ is encoded in the Hilbert space of $N$ identical fermions. There appears a finite number $N{!}^{d\ensuremath{-}1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonized into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the $N$-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.

arXiv:1605.06002Physical Review AJun 7, 20168 citations

A Fixed Universal Determinant is Variationally Complete for Continuum Fermions

Giuseppe Carleo, Riccardo Rossi

How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.How many Slater determinants does an accurate variational description of interacting fermions require? Exact expansions in a finite basis need combinatorially many, and state-of-the-art fermionic neural quantum states stack growing numbers of them. We prove that, in the norms that govern variational calculations, at most two are needed, independently of the number of particles and of the target accuracy. A single universal Slater determinant-specified in advance, independent of both the system and the state-multiplied by a smooth bosonic wave function approximates any fermionic wave function in up to three spatial dimensions in the first-order Sobolev norm, which controls the variational energy. Reaching the second-order Sobolev norm-for Coulomb interactions, the domain of the Hamiltonian, which bounds the variance of the local energy at the core of variational Monte Carlo-requires at most one additional fixed determinant, and only in three dimensions. Antisymmetry therefore costs at most two universal determinants and no expressiveness: generalized Slater-Jastrow neural quantum states are variationally complete.

cond-mat.str-elCross-listedarXiv:2608.14476Aug 17, 20260 citations

A new stochastic method superposing multiple Slater determinants

Kazuhiro Yabana, Satoshi Shinohara, Hirofumi Ohta +1 more

Bulletin of the American Physical SocietySep 22, 20050 citations

New Approach to the Calculation of Schrödinger Determinants

Michael Bordag B

Jan 1, 19970 citations

Some properties of the eigenstates in the many-electron problem

Jacob Szeftel, A. Khater

A general Hamiltonian H of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H= ${\mathit{H}}_{\mathit{D}}$ +${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where ${\mathit{H}}_{\mathit{D}}$ consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that ( ${\mathit{H}}_{\mathit{D}}$ +${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills ( ${\mathit{H}}_{\mathit{D}}$ -${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.A general Hamiltonian H of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H=${\mathit{H}}_{\mathit{D}}$+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where ${\mathit{H}}_{\mathit{D}}$ consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (${\mathit{H}}_{\mathit{D}}$+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (${\mathit{H}}_{\mathit{D}}$-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.

arXiv:2006.14292Physical review. B, Condensed matterNov 15, 19968 citations

The basic question is addressed of how the space dimension

d

is encoded in the Hilbert space of

N

identical fermions. There appears a finite number $N{!}^{d\ensuremath{-}1}$ of many-body wave functions, called shapes, which cannot be generated by trivial combinatorial extension of the one-dimensional ones. A general algorithm is given to list them all in terms of standard Slater determinants. Conversely, excitations which can be induced from the one-dimensional case are bosonized into a system of distinguishable bosons, called Euler bosons, much like the electromagnetic field is quantized in terms of photons distinguishable by their wave numbers. Their wave functions are given explicitly in terms of elementary symmetric functions, reflecting the fact that the fermion sign problem is trivial in one dimension. The shapes act as vacua for the Euler bosons. They are the natural generalization of the single-Slater-determinant form for the ground state to more than one dimension. In terms of algebraic invariant theory, the shapes are antisymmetric invariants which finitely generate the

N

-fermion Hilbert space as a graded algebra over the ring of symmetric polynomials. Analogous results hold for identical bosons.

{\mathit{H}}_{\mathit{D}}

+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where

{\mathit{H}}_{\mathit{D}}

consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (

{\mathit{H}}_{\mathit{D}}

+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (

{\mathit{H}}_{\mathit{D}}

-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=0. We prove also that the eigenvectors of class 1 have off-diagonal long-range order, whereas those of class 2 do not. Finally, our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity. \textcopyright{} 1996 The American Physical Society.