Symmetry and quantum systems

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

Symmetry and quantum mechanics

William Martin McClain

Springer eBooksJan 1, 20090 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

Symmetry Breaking in Quantum Systems

Franco Strocchi

Lecture notes in physicsOct 30, 20070 citations

Symmetries in open quantum systems

Juzar Thingna

Controlling nonequilibrium properties of quantum systems holds the key to many promising quantum technologies. Here I'll introduce the power of symmetry as a resource to manipulate quantum Non-Equilibrium Steady States (NESS). In the first part of the talk I'll introduce some preliminaries about the role of symmetries in steady state quantum transport and how this concept can be exploited to control quantum currents. I’ll introduce a novel scheme to identify symmetries in presence of a weak disorder. The scheme will identify key signatures in the dynamical quantum currents that are unique to symmetric systems and give us vital information about the underlying molecular symmetries. In the second part, I'll introduce magnetic fields that can be used to break the open system symmetries in single-particle simple cubic lattices. In the absence of a magnetic field the open system symmetries are preserved and help modulate the NESS based on the symmetry decomposition of the initial state. The presence of a magnetic field breaks symmetries in preferential directions with all symmetries being broken only in the presence of an anisotropic field. These results demonstrate the importance of symmetry not only as an organizing principle in physics but also as a tool to control quantum systems.Controlling nonequilibrium properties of quantum systems holds the key to many promising quantum technologies. Here I'll introduce the power of symmetry as a resource to manipulate quantum Non-Equilibrium Steady States (NESS). In the first part of the talk I'll introduce some preliminaries about the role of symmetries in steady state quantum transport and how this concept can be exploited to control quantum currents. I’ll introduce a novel scheme to identify symmetries in presence of a weak disorder. The scheme will identify key signatures in the dynamical quantum currents that are unique to symmetric systems and give us vital information about the underlying molecular symmetries. In the second part, I'll introduce magnetic fields that can be used to break the open system symmetries in single-particle simple cubic lattices. In the absence of a magnetic field the open system symmetries are preserved and help modulate the NESS based on the symmetry decomposition of the initial state. The presence of a magnetic field breaks symmetries in preferential directions with all symmetries being broken only in the presence of an anisotropic field. These results demonstrate the importance of symmetry not only as an organizing principle in physics but also as a tool to control quantum systems.

Proceedings of the Samahang Pisika ng PilipinasApr 6, 20190 citations

Remarks on quantum symmetry

Karl-Henning Rehren

Czechoslovak Journal of PhysicsDec 1, 19920 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}.