<p>We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>, vanishes like <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\exp (-2\pi \sqrt{t/U})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>exp</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>2</mn> <mi>π</mi> <msqrt> <mrow> <mi>t</mi> <mo stretchy="false">/</mo> <mi>U</mi> </mrow> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in the weak-coupling limit <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$U/t\downarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">/</mo> <mi>t</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.</p>

On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature

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2026年4月20日

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E. Langmann,
J. Lenells

摘要

We consider the antiferromagnetic gap for the half-filled two-dimensional (2D) Hubbard model (on a square lattice) at zero temperature in Hartree–Fock theory. It was conjectured by Hirsch in 1985 that this gap, $\Delta $ $Δ$ , vanishes like $\exp (-2\pi \sqrt{t/U})$ $exp (- 2 π \sqrt{t / U})$ in the weak-coupling limit $U/t\downarrow 0$ $U / t ↓ 0$ ( $U>0$ $U > 0$ and $t>0$ $t > 0$ are the usual Hubbard model parameters). We give a proof of this conjecture based on recent mathematical results about Hartree-Fock theory for the 2D Hubbard model. The key step is the exact computation of an integral involving the density of states of the 2D tight binding band relation.