Europhysics Letters (EPL)

A Variational Method for Many-Body Systems Using a Separation into a Difference of Hamiltonians

Manfred Hader, Franz G. Mertens

Published Feb 1, 1988• 2 citations

Abstract

A particular ansatz for the wave function leads to an upper bound for the exact ground-state energy. This allows a variation with respect to a separation parameter. The method is tested for a one-dimensional lattice with Morse interactions where the Toda subsystems can be solved by the Bethe ansatz. In two limiting cases our results are exact, otherwise they are in excellent agreement with the quantum transfer integral method.

DOI: 10.1209/0295-5075/5/3/003

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Europhysics Letters (EPL)

A Variational Method for Many-Body Systems Using a Separation into a Difference of Hamiltonians

Manfred Hader, Franz G. Mertens

Published Feb 1, 1988• 2 citations

Publisher

Abstract

DOI: 10.1209/0295-5075/5/3/003

Related papers

A perturbative and variational approach to quantum lattice Hamiltonians

José G. Esteve, Germán Sierra

We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form ${\mathit{H}}_{0}$ +\ensuremath{\lambda} ${\mathit{H}}_{1}$ , where \ensuremath{\lambda} is a coupling constant and ${\mathit{H}}_{0}$ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form ${\mathit{H}}_{0}$+\ensuremath{\lambda}${\mathit{H}}_{1}$, where \ensuremath{\lambda} is a coupling constant and ${\mathit{H}}_{0}$ is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.

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arXiv:2001.06510Jan 17, 20204 citations

Quantizing the Toda lattice

Rahul Siddharthan, B. Sriram Shastry

In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as \ensuremath{\hbar} grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small \ensuremath{\hbar}, and remain similar all the way up to \ensuremath{\hbar}=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various \ensuremath{\hbar} obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/ ${\mathrm{N}}^{2}$ ). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as \ensuremath{\hbar} grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small \ensuremath{\hbar}, and remain similar all the way up to \ensuremath{\hbar}=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various \ensuremath{\hbar} obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/${\mathrm{N}}^{2}$). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.

Physical review. B, Condensed matterMay 1, 19977 citations

We propose a method to construct the ground state \ensuremath{\psi}(\ensuremath{\lambda}) of local lattice Hamiltonians with the generic form

{\mathit{H}}_{0}

+\ensuremath{\lambda}

{\mathit{H}}_{1}

, where \ensuremath{\lambda} is a coupling constant and

{\mathit{H}}_{0}

is a Hamiltonian with a nondegenerate ground state ${\mathrm{\ensuremath{\psi}}}_{0}$. The method is based on the choice of an exponential ansatz \ensuremath{\psi}(\ensuremath{\lambda})=exp[U(\ensuremath{\lambda})]${\mathrm{\ensuremath{\psi}}}_{0}$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get successive approximations of the operator U(\ensuremath{\lambda}). Perturbation theory is used to set up a variational method which in turn produces nonperturbative results. The computation with this kind of ansatz leads us to associate to the original quantum-mechanical problem a statistical-mechanical system defined in the same spatial dimension. In some cases these statistical-mechanical systems turn out to be integrable, which allows us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.

\hbar

grows large. We calculate ground state and excitation energies for finite sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small

\hbar

, and remain similar all the way up to

\hbar=1

, but then deviate substantially as we go further into the quantum regime. On comparing the sound velocities for various

\hbar

obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(

1/N^2

). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.

1/2

Hamiltonians, we demonstrate the ability of this approach to produce long-range correlations, and a ground-state energy that converges to the exact result. Possible extensions, including to higher-excited states are discussed.

{\mathrm{N}}^{2}

). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.