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• W3170249145 abstract "We study N <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>N</mml:mi></mml:math> spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>β</mml:mi></mml:math> . In the fermion model beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>β</mml:mi></mml:math> controls the strength of the interaction, beta=2 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>β</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_D <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>N</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> in a domain D <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>D</mml:mi></mml:math> of macroscopic size in the bulk of the Fermi gas. We predict that for general beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>β</mml:mi></mml:math> the variance of N_D <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>N</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> grows as A_{beta} log N + B_{beta} <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>β</mml:mi></mml:msub><mml:mo>log</mml:mo><mml:mi>N</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:mrow></mml:math> for N gg 1 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>N</mml:mi><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> and we obtain a formula for A_beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>A</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:math> and B_beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>B</mml:mi><mml:mi>β</mml:mi></mml:msub></mml:math> . This is based on an explicit calculation for betainleft{ 1,2,4right} <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>β</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo stretchy=true form=prefix>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy=true form=postfix>}</mml:mo></mml:mrow></mml:mrow></mml:math> and on a conjecture that we formulate for general beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>β</mml:mi></mml:math> . This conjecture further allows us to obtain a universal formula for the higher cumulants of N_D <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:msub><mml:mi>N</mml:mi><mml:mi>D</mml:mi></mml:msub></mml:math> . Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>/</mml:mi><mml:mi>β</mml:mi></mml:mrow></mml:math> , and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter beta <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline><mml:mi>β</mml:mi></mml:math> . In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model." @default.
• W3170249145 created "2021-06-22" @default.
• W3170249145 creator A5012513905 @default.
• W3170249145 creator A5031449563 @default.
• W3170249145 creator A5041706495 @default.
• W3170249145 creator A5047748977 @default.
• W3170249145 date "2021-12-22" @default.
• W3170249145 modified "2023-10-16" @default.
• W3170249145 title "Full counting statistics for interacting trapped fermions" @default.
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