Forty-year-old physics puzzle finally solved with mathematical proof
Researchers have proven a 1985 conjecture about how magnetic gaps form in quantum materials, resolving a foundational question in condensed matter physics. The result could accelerate development of quantum computers and advanced materials by validating theoretical models used to design new electronics.
Originaltitel: On the mean-field antiferromagnetic gap for the half-filled 2D Hubbard model at zero temperature
Abstract 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, $$\Delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> , vanishes like $$\exp (-2\pi \sqrt{t/U})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msqrt> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mi>U</mml:mi> </mml:mrow> </mml:msqrt> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in the weak-coupling limit $$U/t\downarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>/</mml:mo> <mml:mi>t</mml:mi> <mml:mo>↓</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ( $$U>0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and $$t>0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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.