Non-standard quantum algebras and finite dimensional $\mathcal{PT}$ -symmetric systemsNon-standard quantum algebras and finite dimensional $\mathcal{PT}$-symmetric systems

Ángel Ballesteros, Romina Ramírez, M. Reboiro

In this work, $\mathcal{PT}$ -symmetric Hamiltonians defined on quantum $sl(2, \mathbb R)$ algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2, \mathbb R)$ . By making use of a particular boson representation of the generators of $U_{z}(sl(2, \mathbb R))$ , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the $\mathcal{PT}$ -transformation. In terms of these operators, we construct several finite dimensional $\mathcal{PT}$ -symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$ -symmetry and the broken $\mathcal{PT}$ -symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter $z$ has to be identified with the detuning parameter of the system.In this work, $\mathcal{PT}$-symmetric Hamiltonians defined on quantum $sl(2, \mathbb R)$ algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2, \mathbb R)$. By making use of a particular boson representation of the generators of $U_{z}(sl(2, \mathbb R))$, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the $\mathcal{PT}$-transformation. In terms of these operators, we construct several finite dimensional $\mathcal{PT}$-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$-symmetry and the broken $\mathcal{PT}$-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter $z$ has to be identified with the detuning parameter of the system.

arXiv:2309.15305Sep 26, 20230 citations

Representations of non-commutative quantum mechanics and symmetries

Larisa Jonke, Stjepan Meljanac

arXiv:hep-th/0210042The European Physical Journal CJul 1, 200349 citations

Non-standard quantum algebras and finite dimensional P T -symmetric systems

Ángel Ballesteros, Romina Ramírez, M. Reboiro

Abstract In this work, P T -symmetric Hamiltonians defined on quantum s l ( 2 , R ) algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard U z ( s l ( 2 , R ) ) Hopf algebra deformation of s l ( 2 , R ) . By making use of a particular boson representation of the generators of U z ( s l ( 2 , R ) ) , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the P T -transformation. In terms of these operators, we construct several finite dimensional P T -symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact P T -symmetry and the broken P T -symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter z has to be identified with the detuning parameter of the system.Abstract In this work, P T -symmetric Hamiltonians defined on quantum s l ( 2 , R ) algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard U z ( s l ( 2 , R ) ) Hopf algebra deformation of s l ( 2 , R ) . By making use of a particular boson representation of the generators of U z ( s l ( 2 , R ) ) , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the P T -transformation. In terms of these operators, we construct several finite dimensional P T -symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact P T -symmetry and the broken P T -symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter z has to be identified with the detuning parameter of the system.

Journal of Physics A Mathematical and TheoreticalDec 19, 20233 citations

Symmetries and Symmetry Groups in Quantum Physics

Springer eBooksApr 16, 20070 citations

Symmetries and Symmetry Groups in Quantum Physics

Florian A. Scheck

Springer eBooksJan 1, 20131 citation

Exhaustive Characterization of Quantum Many-Body Scars Using Commutant Algebras

Sanjay Moudgalya, Olexei I. Motrunich

We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024We study quantum many-body scars (QMBS) in the language of commutant algebras, which are defined as symmetry algebras of of local Hamiltonians. This framework explains the origin of dynamically disconnected subspaces seen in models with exact QMBS, i.e., the large “thermal” subspace and the small “nonthermal” subspace, which are attributed to the existence of unconventional nonlocal conserved quantities in the commutant; hence, it unifies the study of conventional symmetries and weak ergodicity-breaking phenomena into a single framework. Furthermore, this language enables us to use the von Neumann double commutant theorem to formally write down the exhaustive algebra of Hamiltonians with a desired set of QMBS, which demonstrates that QMBS survive under large classes of local perturbations. We illustrate this using several standard examples of QMBS, including the spin- 1 / 2 ferromagnetic, AKLT, spin-1 XY π -bimagnon, and the electronic η -pairing towers of states; in each of these cases, we explicitly write down a set of generators for the full algebra of Hamiltonians with these QMBS. Understanding this hidden structure in QMBS Hamiltonians also allows us to recover results of previous “brute-force” numerical searches for such Hamiltonians. In addition, this language clearly demonstrates the equivalence of several unified formalisms for QMBS proposed in the literature and also illustrates the connection between two apparently distinct classes of QMBS Hamiltonians—those that are captured by the so-called Shiraishi-Mori construction and those that lie beyond. Finally, we show that this framework motivates a precise definition for QMBS that automatically implies that they violate the conventional eigenstate thermalization hypothesis, and we discuss its implications to dynamics. Published by the American Physical Society 2024

arXiv:2209.03377Physical Review XDec 26, 202438 citations

Non-standard quantum algebras and finite dimensional

\mathcal{PT}

-symmetric systems

In this work,

\mathcal{PT}

-symmetric Hamiltonians defined on quantum

sl(2, \mathbb R)

algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard

U_{z}(sl(2, \mathbb R))

Hopf algebra deformation of

sl(2, \mathbb R)

. By making use of a particular boson representation of the generators of

U_{z}(sl(2, \mathbb R))

, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the

\mathcal{PT}

-transformation. In terms of these operators, we construct several finite dimensional

\mathcal{PT}

-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact

\mathcal{PT}

-symmetry and the broken

\mathcal{PT}

-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter

z

has to be identified with the detuning parameter of the system.