Rate function modeling of quantum many-body adiabaticity

The quantum adiabatic theorem is a fundamental result in quantum mechanics, with a multitude of applications, both theoretical and practical. Here, we investigate the dynamics of adiabatic processes for quantum many-body systems by analyzing the properties of observable-free intensive quantities. In particular, we study the adiabatic rate function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>f</a:mi><a:mo>(</a:mo><a:mi>T</a:mi><a:mo>,</a:mo><a:mi mathvariant="normal">Δ</a:mi><a:mi>λ</a:mi><a:mo>)</a:mo></a:mrow></a:math> in dependence of the ramp time <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>T</c:mi></c:math>, which gives us a complete characterization of the many-body adiabatic fidelity as a function of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>T</d:mi></d:math> and the strength of the parameter displacement <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi mathvariant="normal">Δ</e:mi><e:mi>λ</e:mi></e:mrow></e:math>. <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>f</g:mi><g:mo>(</g:mo><g:mi>T</g:mi><g:mo>,</g:mo><g:mi mathvariant="normal">Δ</g:mi><g:mi>λ</g:mi><g:mo>)</g:mo></g:mrow></g:math> quantifies the deviation from adiabaticity for a given process and therefore allows us to control and define the notion of adiabaticity in many-body systems. First we study <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>f</i:mi><i:mo>(</i:mo><i:mi>T</i:mi><i:mo>,</i:mo><i:mi mathvariant="normal">Δ</i:mi><i:mi>λ</i:mi><i:mo>)</i:mo></i:mrow></i:math> for the one-dimensional transverse field Ising model and the Luttinger liquid, both of which are quadratic systems and therefore allow us to look at the thermodynamic limit. For ramps across gapped phases, we relate <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>f</k:mi><k:mo>(</k:mo><k:mi>T</k:mi><k:mo>,</k:mo><k:mi mathvariant="normal">Δ</k:mi><k:mi>λ</k:mi><k:mo>)</k:mo></k:mrow></k:math> to the transition probability of the system and for ramps across a gapless point, or gapless phase we relate it to the excitation density of the relevant quasiparticles. Then we investigate the XXZ model, which allows us to see the qualitative features that survive when interactions are turned on. Several key results in the literature regarding the interplay of the thermodynamic and the adiabatic limit are obtained as inferences from the properties of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>f</m:mi><m:mo>(</m:mo><m:mi>T</m:mi><m:mo>,</m:mo><m:mi mathvariant="normal">Δ</m:mi><m:mi>λ</m:mi><m:mo>)</m:mo></m:mrow></m:math> in the large-<o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:mi>T</o:mi></o:math> limit.