Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation

We say of an isolated macroscopic quantum system in a pure state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ψ</mml:mi></mml:math> that it is in macroscopic thermal equilibrium (MATE) if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ψ</mml:mi></mml:math> lies in or close to a suitable subspace <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi> <mml:mtext>eq</mml:mtext></mml:msub> </mml:math> of Hilbert space. It is known that every initial state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ψ</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> will eventually reach and stay there most of the time ("thermalize") if the Hamiltonian is non-degenerate and satisfies the appropriate version of the eigenstate thermalization hypothesis (ETH), i.e., that every eigenvector is in MATE. Tasaki recently proved the ETH for a certain perturbation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi> <mml:mi>θ</mml:mi> <mml:mtext>fF</mml:mtext></mml:msubsup> </mml:math> of the Hamiltonian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mtext>fF</mml:mtext></mml:msubsup> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> free fermions on a one-dimensional lattice. The perturbation is needed to remove the high degeneracies of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mtext>fF</mml:mtext></mml:msubsup> </mml:math> . Here, we first point out that also for degenerate Hamiltonians all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ψ</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> thermalize if the ETH holds, i.e., if <i>every</i> eigenbasis lies in MATE, and we prove that this is the case for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mtext>fF</mml:mtext></mml:msubsup> </mml:math> . Inspired by the fact that there is <i>one</i> eigenbasis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mtext>fF</mml:mtext></mml:msubsup> </mml:math> for which MATE can be proved more easily than for the others, with smaller error bounds, and also in higher spatial dimensions, we show for any given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> that the existence of one eigenbasis in MATE implies quite generally that <i>most</i> eigenbases of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> lie in MATE. We also show that, as a consequence, after adding a small generic perturbation, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub><mml:mi>H</mml:mi> <mml:mn>0</mml:mn></mml:msub> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>V</mml:mi></mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>λ</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , for most perturbations <i>V</i> the perturbed Hamiltonian <i>H</i> satisfies ETH and all states thermalize.