Topological entanglement properties of disconnected partitions in the Su-Schrieffer-Heeger model
Tommaso Micallo, Vittorio Vitale, Marcello Dalmonte, Pierre Fromholz
Abstract
We study the disconnected entanglement entropy, S^\mathrm{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>D</mml:mi> </mml:mstyle> </mml:msup> </mml:math> , of the Su-Schrieffer-Heeger model. S^\mathrm{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>D</mml:mi> </mml:mstyle> </mml:msup> </mml:math> is a combination of both connected and disconnected bipartite entanglement entropies that removes all area and volume law contributions and is thus only sensitive to the non-local entanglement stored within the ground state manifold. Using analytical and numerical computations, we show that S^\mathrm{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>D</mml:mi> </mml:mstyle> </mml:msup> </mml:math> behaves like a topological invariant, i.e., it is quantized to either 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>0</mml:mn> </mml:math> or 2\log(2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>log</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> in the topologically trivial and non-trivial phases, respectively. These results also hold in the presence of symmetry-preserving disorder. At the second-order phase transition separating the two phases, S^\mathrm{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>D</mml:mi> </mml:mstyle> </mml:msup> </mml:math> displays a finite-size scaling behavior akin to those of conventional order parameters, that allows us to compute entanglement critical exponents. To corroborate the topological origin of the quantized values of S^\mathrm{D} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mi>D</mml:mi> </mml:mstyle> </mml:msup> </mml:math> , we show how the latter remain quantized after applying unitary time evolution in the form of a quantum quench, a characteristic feature of topological invariants associated with particle-hole symmetry.