Quantum Monte Carlo study of the quasiparticle effective mass of the two-dimensional uniform electron liquid

The real-space variation quantum Monte Carlo (VMC) and diffusion quantum Monte Carlo (DMC) are used to calculate the energy bands and the effective mass of the paramagnetic and ferromagnetic two-dimensional uniform electron liquid (2D-UEL). The many-body finite-size errors are minimized by performing simulations for three system sizes with the number of electrons, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>N</a:mi><a:mo>=</a:mo><a:mn>146</a:mn></a:mrow></a:math>, 218, and 302, for paramagnetic and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>N</b:mi><b:mo>=</b:mo><b:mn>91</b:mn></b:mrow></b:math>, 139, and 151 for ferromagnetic systems. We consider 2D-UEL to be within the density range <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mn>1</c:mn><c:mo>≤</c:mo><c:msub><c:mi>r</c:mi><c:mi>s</c:mi></c:msub><c:mo>≤</c:mo><c:mn>10</c:mn></c:mrow></c:math>. The VMC and DMC results predict that the effective mass <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msup><d:mi>m</d:mi><d:mo>*</d:mo></d:msup></d:math> of the paramagnetic 2D-UEL at high density <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:msub><e:mi>r</e:mi><e:mi>s</e:mi></e:msub><e:mo>=</e:mo><e:mn>1</e:mn></e:mrow></e:math> is very close to 1, suggesting that the effective mass renormalization due to electron-electron interaction is negligible. We find that <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msup><f:mi>m</f:mi><f:mo>*</f:mo></f:msup></f:math> of the paramagnetic 2D-UEL obtained by the VMC and DMC methods increases by <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>r</g:mi><g:mi>s</g:mi></g:msub></g:math> but with different slopes. Our VMC and DMC results for ferromagnetic 2D-UEL indicate that <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msup><h:mi>m</h:mi><h:mo>*</h:mo></h:msup></h:math> decreases rapidly by reducing the density due to the strong suppression of the electron-electron interaction.