Correlation effects in one-dimensional metallic quantum wires under various confinements

Dynamical response theory is employed to investigate the effects of various transverse confinements on electron correlations in the ground state of a ferromagnetic one-dimensional quantum wire for different wire widths <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>b</a:mi> </a:math> and density parameters <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msub> <b:mi>r</b:mi> <b:mtext>s</b:mtext> </b:msub> </b:math> . In the regime of a thin quantum wire, electrons are treated as a one-dimensional gas under different confinement models via effective electron-electron interaction potentials. Using the first-order random phase approximation (FRPA) including self-energy and exchange contribution, which provides the ground-state structure beyond the random phase approximation, we numerically compute the structure factor, pair-correlation function, correlation energy, and ground-state energy for various values of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>b</c:mi> </c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>r</d:mi> <d:mtext>s</d:mtext> </d:msub> </d:math> . Our results reveal that the correlation energy depends on the choice of confinement model. For the ultrathin wire <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mo>(</e:mo> <e:mi>b</e:mi> <e:mo>→</e:mo> <e:mn>0</e:mn> <e:mo>)</e:mo> </e:mrow> </e:math> in the high-density limit, we find that the correlation energy for transverse confinement models <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:msub> <f:mi>V</f:mi> <f:mn>1</f:mn> </f:msub> <f:mrow> <f:mo>(</f:mo> <f:mi>q</f:mi> <f:mo>)</f:mo> </f:mrow> </f:mrow> </f:math> (harmonic), <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:msub> <g:mi>V</g:mi> <g:mn>2</g:mn> </g:msub> <g:mrow> <g:mo>(</g:mo> <g:mi>q</g:mi> <g:mo>)</g:mo> </g:mrow> </g:mrow> </g:math> (cylindrical), and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:msub> <h:mi>V</h:mi> <h:mn>5</h:mn> </h:msub> <h:mrow> <h:mo>(</h:mo> <h:mi>q</h:mi> <h:mo>)</h:mo> </h:mrow> </h:mrow> </h:math> (harmonic-delta) approaches <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:msub> <i:mi>ε</i:mi> <i:mtext>c</i:mtext> </i:msub> <i:mrow> <i:mo>(</i:mo> <i:msub> <i:mi>r</i:mi> <i:mtext>s</i:mtext> </i:msub> <i:mo>)</i:mo> </i:mrow> <i:mo>=</i:mo> <i:mo>−</i:mo> <i:msup> <i:mi>π</i:mi> <i:mn>2</i:mn> </i:msup> <i:mo>/</i:mo> <i:mn>360</i:mn> <i:mo>∼</i:mo> <i:mo>−</i:mo> <i:mn>0.02741</i:mn> </i:mrow> </i:math> a.u., which agrees with the exact results in this limit [P.-F. Loos, ; V. Ashokan, , ]. This clearly illustrates that for at least these three confinement potentials, the one-dimensional Coulomb potential can be regularized at interparticle distance <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>x</j:mi> <j:mo>=</j:mo> <j:mn>0</j:mn> </j:mrow> </j:math> to yield the same correlation energy. In contrast, other confinement potentials, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:msub> <k:mi>V</k:mi> <k:mn>3</k:mn> </k:msub> <k:mrow> <k:mo>(</k:mo> <k:mi>q</k:mi> <k:mo>)</k:mo> </k:mrow> </k:mrow> </k:math> (infinite square well), <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msub> <l:mi>V</l:mi> <l:mn>4</l:mn> </l:msub> <l:mrow> <l:mo>(</l:mo> <l:mi>q</l:mi> <l:mo>)</l:mo> </l:mrow> </l:mrow> </l:math> (infinite square-infinite triangular well), and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mn>6</m:mn> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>q</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> (infinite square-delta well), do not approach the same high-density limit; instead, the correlation energy tends to <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:msub> <n:mi>ε</n:mi> <n:mtext>c</n:mtext> </n:msub> <n:mo>∼</n:mo> <n:mo>−</n:mo> <n:mn>0.03002</n:mn> </n:mrow> </n:math> a.u. for these potentials. The percentage difference in correlation energy between the confinement models <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msub> <o:mi>V</o:mi> <o:mn>1</o:mn> </o:msub> <o:mrow> <o:mo>(</o:mo> <o:mi>q</o:mi> <o:mo>)</o:mo> </o:mrow> </o:mrow> <o:mo>,</o:mo> <o:mo> </o:mo> <o:mrow> <o:msub> <o:mi>V</o:mi> <o:mn>2</o:mn> </o:msub> <o:mrow> <o:mo>(</o:mo> <o:mi>q</o:mi> <o:mo>)</o:mo> </o:mrow> </o:mrow> <o:mo>,</o:mo> <o:mo> </o:mo> <o:mrow> <o:msub> <o:mi>V</o:mi> <o:mn>5</o:mn> </o:msub> <o:mrow> <o:mo>(</o:mo> <o:mi>q</o:mi> <o:mo>)</o:mo> </o:mrow> </o:mrow> </o:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:msub> <p:mi>V</p:mi> <p:mn>3</p:mn> </p:msub> <p:mrow> <p:mo>(</p:mo> <p:mi>q</p:mi> <p:mo>)</p:mo> </p:mrow> </p:mrow> <p:mo>,</p:mo> <p:mo> </p:mo> <p:mrow>