Toward a classification of PT-symmetric quantum systems: From dissipative dynamics to topology and wormholes

Abstract

Studies of many-body non-Hermitian parity-time (PT)-symmetric quantum systems are attracting a lot of interest due to their relevance in research areas ranging from quantum optics and continuously monitored dynamics to Euclidean wormholes in quantum gravity and dissipative quantum chaos. While a symmetry classification of non-Hermitian systems leads to 38 universality classes, we show that, under certain conditions, PT-symmetric systems are grouped into 24 universality classes. We identify 14 of them in a coupled two-site Sachdev-Ye-Kitaev (SYK) model and confirm the classification by spectral analysis using exact diagonalization techniques. Intriguingly, in 4 of these 14 universality classes, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mrow><a:mi>AIII</a:mi></a:mrow><a:mi>ν</a:mi></a:msub></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msubsup><c:mrow><c:mi>BDI</c:mi></c:mrow><c:mrow><c:mi>ν</c:mi></c:mrow><c:mrow><c:mi>†</c:mi></c:mrow></c:msubsup></c:mrow></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:msub><e:mrow><e:mi>BDI</e:mi></e:mrow><e:mrow><e:mo>+</e:mo><e:mo>+</e:mo><e:mi>ν</e:mi></e:mrow></e:msub></e:mrow></e:math>, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:msub><g:mrow><g:mi>CI</g:mi></g:mrow><g:mrow><g:mo>−</g:mo><g:mo>−</g:mo><g:mi>ν</g:mi></g:mrow></g:msub></g:mrow></g:math>, we identify a basis in which the SYK Hamiltonian has a block structure in which some blocks are rectangular, with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>ν</i:mi><i:mo>∈</i:mo><i:mi mathvariant="double-struck">N</i:mi></i:math> the difference between the number of rows and columns. We show analytically that this feature leads to the existence of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>ν</l:mi></l:math> robust purely eigenvalues, whose level statistics follow the predictions of Hermitian random matrix theory for classes A, AI, BDI, and CI, respectively. We have recently found that this <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>ν</n:mi></n:math> is a topological invariant, so these classes are topological. By contrast, nontopological real eigenvalues display a crossover between Hermitian and non-Hermitian level statistics. Similarly to the case of Lindbladian dynamics, the reduction of universality classes leads to unexpected results, such as the absence of Kramers degeneracy in a given sector of the theory. Another novel feature of the classification scheme is that different sectors of the PT-symmetric Hamiltonian may have different symmetries. Published by the American Physical Society 2024

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