Fermionic logarithmic negativity in the Krawtchouk chain

Abstract The entanglement of non-complementary regions is investigated in an inhomogeneous free-fermion chain through the lens of the fermionic logarithmic negativity. Focus is on the Krawtchouk chain, whose relation to the eponymous orthogonal polynomials allows for exact diagonalization and analytical calculations of certain correlation functions. For adjacent regions, the negativity scaling corresponds to that of a conformal field theory with central charge c = 1, in agreement with previous studies on bipartite entanglement in the Krawtchouk chain. For disjoint regions, we focus on the skeletal regime where each region reduces to a single site. This regime is sufficient to extract the leading behaviour at large distances. In the bulk, the negativity decays as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>d</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , where d is the separation between the regions. This is in agreement with the homogeneous result of free Dirac fermions in one dimension. Surprisingly, when one site is close to the boundary, this exponent changes and depends on the parity of the boundary site <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:mrow> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mtext>even</mml:mtext> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>f</mml:mi> <mml:mrow> <mml:mtext>odd</mml:mtext> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> . The results are supported by numerics and analytical calculations.