Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids
Abstract
The light-cone spreading of entanglement and correlation is a fundamental and ubiquitous feature of homogeneous extended quantum systems. Here we point out that a class of inhomogenous Luttinger liquids (those with a uniform Luttinger parameter K <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>K</mml:mi> </mml:math> ) at low energy display the universal phenomenon of curved light cones: gapless excitations propagate along the null geodesics of the metric ds^2 = dx^2 - v(x)^2 dt^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , with v(x) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> being the calculable spatial dependent velocity induced by the inhomogeneity. We confirm our findings with explicit analytic and numerical calculations both in- and out-of-equilibrium for a Tonks-Girardeau gas in a harmonic potential and in lattice systems with artificially tuned hamiltonian density.