Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9307067

Variational bound for the energy of two-dimensional quantum antiferromagnet

А. А. Овчинников

Published Aug 1, 1993• 0 citations
arXivHTMLPDF

Abstract

We obtain the variational upper bound for the ground- state energy of two-dimensional antiferromagnetic Heisenberg model on a square lattice at arbitrary value of the anisotropy parameter using the two-dimensional generalization of Jordan-Wigner transformation. Our result can be considered as an upper bound for the perturbation theory series about the Ising limit.

DOI
10.48550/arxiv.cond-mat/9307067

Related papers

A New Variational Ansatz for the Ground State of the 2 d -Heisenberg Antiferromagnet

Claudio Albanese, Christoph Bruder

We propose a new variational ansatz for the ground state of the (isotropic) antiferromagnetic Heisenberg model on the square lattice. Our motivation for considering this ansatz is that the ground state of the anisotropic Heisenberg model with small transversal coupling has precisely this form. The evaluation of the expectation value of the energy is reduced to the problem of measuring the statistical expectation value of a certain correlation function of an Ising model on the square lattice itself. Hence we can deal with this problem by using a Monte Carlo method.

Europhysics Letters (EPL)•Mar 1, 1989•0 citations

A numerical study for two-dimensional spin 1/2 antiferromagnets: a generalization of Entanglement Perturbation Theory to two-dimensional systems

Lihua Wang, Tomonori Shirakawa, Hiroshi Watanabe +1 more

Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.

Journal of Physics Conference Series•Jan 1, 2011•0 citations

Numerical Jordan-Wigner approach for two-dimensional spin systems

D. C. Cabra, G. L. Rossini

We present a numerical self-consistent variational approach based on the Jordan-Wigner transformation for two-dimensional spin systems. We apply it to the study of the well-known quantum $(S=1/2)$ antiferromagnetic $\mathrm{XXZ}$ system as a function of the easy-axis anisotropy $\ensuremath{\Delta}$ on a periodic square lattice. For the $\mathrm{SU}(2)$ case the method converges to a N\'eel ordered ground state irrespective of the input density profile used and in accordance with other studies. This shows the potential utility of the proposed method to investigate more complicated situations such as frustrated or disordered systems.

arXiv:cond-mat/0310131•Physical Review B•May 28, 2004•9 citations

Ground-state properties of the spin-\frac{1}{2} antiferromagnetic Heisenberg model on the triangular lattice: a variational study based on entangled-plaquette states

Fabio Mezzacapo, J. I. Cirac

We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).

arXiv:1006.4480•New Journal of Physics•Oct 26, 2010•33 citations

Ground state of the two-dimensional antiferromagnetic Heisenberg model studied using an extended Wigner-Jordon transformation

Y. R. Wang

A two-dimensional (2D) Wigner-Jordon transformation which maps spin variables into spinless fermion variables is found and is used to study the 2D spin-1/2 antiferromagnetic Heisenberg model. The transformation generates a fictitious gauge field in the XY component of the Heisenberg model, and hence induces an in-phase orbital current in the Ising component flowing around each elementary plaquette of the underlying lattice. The ground state of the Heisenberg model in a 2D square lattice is found to be an in-phase N\'eel flux phase, i.e., a coexisting state of the flux phase with in-phase orbital currents and a long-range antiferromagnetic spin order. The zero-temperature mean-field energy of the in-phase N\'eel flux state, ${\mathit{E}}_{0}$=-0.33J per bond, is only 1% higher than the best-estimated ground-state energy, -0.334J per bond.

Physical review. B, Condensed matter•Feb 1, 1991•92 citations

A variational lower bound on the ground state of a many-body system and the squaring parametrization of density matrices

F. Uskov, O. Lychkovskiy

A variational upper bound on the ground state energy E gs of a quantum system, E gs ⩽ 〈Ψ| H |Ψ〉, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term H i is supported on the i 'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 /tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.

arXiv:1902.09246•Journal of Physics Conference Series•Feb 1, 2019•9 citations
We obtain the variational upper bound for the ground- state energy of two-dimensional antiferromagnetic Heisenberg model on a square lattice at arbitrary value of the anisotropy parameter using the two-dimensional generalization of Jordan-Wigner transformation. Our result can be considered as an upper bound for the perturbation theory series about the Ising limit.
We propose a new variational ansatz for the ground state of the (isotropic) antiferromagnetic Heisenberg model on the square lattice. Our motivation for considering this ansatz is that the ground state of the anisotropic Heisenberg model with small transversal coupling has precisely this form. The evaluation of the expectation value of the energy is reduced to the problem of measuring the statistical expectation value of a certain correlation function of an Ising model on the square lattice itself. Hence we can deal with this problem by using a Monte Carlo method.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
We present a numerical self-consistent variational approach based on the Jordan-Wigner transformation for two-dimensional spin systems. We apply it to the study of the well-known quantum (S=1/2)(S=1/2)(S=1/2) antiferromagnetic XXZ\mathrm{XXZ}XXZ system as a function of the easy-axis anisotropy $\ensuremath{\Delta}$ on a periodic square lattice. For the SU(2)\mathrm{SU}(2)SU(2) case the method converges to a N\'eel ordered ground state irrespective of the input density profile used and in accordance with other studies. This shows the potential utility of the proposed method to investigate more complicated situations such as frustrated or disordered systems.
We study, on the basis of the general entangled-plaquette variational ansatz, the ground-state (GS) properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice. Our numerical estimates are in good agreement with available exact results and comparable, for large system sizes, to those computed via the best alternative numerical approaches, or by means of variational schemes based on specific (i.e. incorporating problem-dependent terms) trial wave functions (WFs). The extrapolation to the thermodynamic limit of our results for lattices comprising up to N =324 spins yields an upper bound of the GS energy per site (in units of the exchange coupling) of − 0.5458(2) [−0.4074(1) for the XX model], while the estimated infinite-lattice order parameter is 0.3178(5) (i.e. ∼64% of the classical value).
A two-dimensional (2D) Wigner-Jordon transformation which maps spin variables into spinless fermion variables is found and is used to study the 2D spin-1/2 antiferromagnetic Heisenberg model. The transformation generates a fictitious gauge field in the XY component of the Heisenberg model, and hence induces an in-phase orbital current in the Ising component flowing around each elementary plaquette of the underlying lattice. The ground state of the Heisenberg model in a 2D square lattice is found to be an in-phase N\'eel flux phase, i.e., a coexisting state of the flux phase with in-phase orbital currents and a long-range antiferromagnetic spin order. The zero-temperature mean-field energy of the in-phase N\'eel flux state, E0{\mathit{E}}_{0}E0​=-0.33J per bond, is only 1% higher than the best-estimated ground-state energy, -0.334J per bond.
A variational upper bound on the ground state energy E gs of a quantum system, E gs ⩽ 〈Ψ| H |Ψ〉, is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as , where a term H i is supported on the i 'th cluster. The bound reads , where is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set , which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 /tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.