Mutual exclusion statistics and the infinite-range Hubbard model: critical behavior in the quantum regime

Abstract We investigate the critical behavior in the quantum regime of the Hubbard model in d -dimensional hypercubic lattices with infinite-range Coulomb interaction. The Haldane–Wu mutual exclusion statistics approach, which comprises three species of exclusions, allows us to derive the singular part of the free energy, entropy, and specific heat in the quantum regime g &gt; T , where g is the scaling interaction parameter and T is the temperature. We present results from the analytic series expansions of observables near the quantum critical point: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mrow> </mml:math> , where µ is the chemical potential. The signature of the mutual exclusion statistics manifests through the appearance of polylogarithms related to Lerch functions in the coefficients of the free energy series in powers of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>g</mml:mi> </mml:mrow> </mml:math> . The calculated critical exponents from the analytic series are consistent with our scaling analysis in the quantum regime. In particular, the specific heat is shown to display a variety of critical behaviors, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> , depending on the lattice dimension: from a singular behavior in d = 1 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> ) to a constant value in d = 2 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ), and regular behavior for d &gt; 2 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mrow> </mml:math> ).