We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

arXiv:2212.00605PRX QuantumAug 29, 202373 citations

Symmetries in Quantum Mechanics

Andreas Wipf

Springer eBooksJan 1, 20260 citations

Are degenerate groundstates induced by spontaneous symmetry breakings in quantum phase transitions?

Mei He, Qian‐Qian Shi, Sam Young Cho

Recently, emergent symmetry is one of fast-growing intriguing issues in many-body systems. Its roles and consequential physics have not been well understood in quantum phase transitions. Emergent symmetry of degenerate groundstates is discussed in possible connection to spontaneous symmetry breaking within the Landau theory. For a clear discussion, a quantum spin- $1/2$ plaquette chain system is shown to have rich emergent symmetry phenomena in its groundstates. A covering symmetry group over all emergent symmetries responsible for degenerate groundstates in the plaquette chain system is found to correspond to a largest common symmetry group of constituent Hamiltonians describing the plaquette system. Consequently, this result suggests that, as a guiding symmetry principle in quantum phase transitions, {\it degenerate groundstates are induced by a spontaneous breaking of symmetries belonging to a largest common symmetry group of continent Hamiltonians describing a given system but can have more symmetries than the largest common symmetry}.Recently, emergent symmetry is one of fast-growing intriguing issues in many-body systems. Its roles and consequential physics have not been well understood in quantum phase transitions. Emergent symmetry of degenerate groundstates is discussed in possible connection to spontaneous symmetry breaking within the Landau theory. For a clear discussion, a quantum spin-$1/2$ plaquette chain system is shown to have rich emergent symmetry phenomena in its groundstates. A covering symmetry group over all emergent symmetries responsible for degenerate groundstates in the plaquette chain system is found to correspond to a largest common symmetry group of constituent Hamiltonians describing the plaquette system. Consequently, this result suggests that, as a guiding symmetry principle in quantum phase transitions, {\it degenerate groundstates are induced by a spontaneous breaking of symmetries belonging to a largest common symmetry group of continent Hamiltonians describing a given system but can have more symmetries than the largest common symmetry}.

arXiv:1505.00948May 5, 20150 citations

Quantum Symmetries

Valter Moretti

Springer eBooksJan 1, 20190 citations

A local description of strong symmetries and strong-to-weak symmetry breaking in quantum many-body systems

Ruizhi Liu, Jinmin Yi, Dominic V. Else

In mixed states of quantum systems, symmetries come in two types: strong and weak. Furthermore, it has been argued that in quantum many-body systems, strong symmetries can be "spontaneously broken" down to weak symmetries. An issue is that as previously formulated, such "strong-to-weak symmetry breaking" appears to be a fairly non-local effect. In this paper, we show how to understand and diagnose strong symmetries and strong-to-weak symmetry breaking in an explicitly local way. Our main technical tool is a rigorous definition of strong symmetry in the limit of infinite volume, which generalizes the conventional finite-volume definitions, and for which we give several equivalent formulations, including one involving the concept of "local charge coherence". Finally, we introduce von Neumann systems, which in infinite-volume symmetries are intermediate between strong and weak symmetries. We derive a Lieb-Schultz-Mattis type anomaly constraint for von Neumann symmetries (and therefore, in particular, strong symmetries) in quantum spin chains.In mixed states of quantum systems, symmetries come in two types: strong and weak. Furthermore, it has been argued that in quantum many-body systems, strong symmetries can be "spontaneously broken" down to weak symmetries. An issue is that as previously formulated, such "strong-to-weak symmetry breaking" appears to be a fairly non-local effect. In this paper, we show how to understand and diagnose strong symmetries and strong-to-weak symmetry breaking in an explicitly local way. Our main technical tool is a rigorous definition of strong symmetry in the limit of infinite volume, which generalizes the conventional finite-volume definitions, and for which we give several equivalent formulations, including one involving the concept of "local charge coherence". Finally, we introduce von Neumann systems, which in infinite-volume symmetries are intermediate between strong and weak symmetries. We derive a Lieb-Schultz-Mattis type anomaly constraint for von Neumann symmetries (and therefore, in particular, strong symmetries) in quantum spin chains.

quant-pharXiv:2605.28925May 29, 20260 citations

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plaquette chain system is shown to have rich emergent symmetry phenomena in its groundstates. A covering symmetry group over all emergent symmetries responsible for degenerate groundstates in the plaquette chain system is found to correspond to a largest common symmetry group of constituent Hamiltonians describing the plaquette system. Consequently, this result suggests that, as a guiding symmetry principle in quantum phase transitions, {\it degenerate groundstates are induced by a spontaneous breaking of symmetries belonging to a largest common symmetry group of continent Hamiltonians describing a given system but can have more symmetries than the largest common symmetry}.