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arXiv:2007.06112 · Journal of Mathematical Physics

Computing Floquet Hamiltonians with symmetries

Terry A. Loring, Fredy Vides

Published Nov 1, 2020• 3 citations
arXivHTMLPDF

Abstract

Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system, one frequently wishes to compute a Floquet Hamiltonian that should be a Hermitian operator H such that e−iTH = U(T), where U(T) is the time evolution operator at time corresponding to the period of the system. That is, we want H to be equal to −i times a matrix logarithm of U(T). If the system has a symmetry, such as time reversal symmetry, one can expect H to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries, which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator U(T) can lead to a symmetry in a basis of Floquet eigenstates.

DOI
10.1063/5.0023028

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Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system, one frequently wishes to compute a Floquet Hamiltonian that should be a Hermitian operator H such that e−iTH = U(T), where U(T) is the time evolution operator at time corresponding to the period of the system. That is, we want H to be equal to −i times a matrix logarithm of U(T). If the system has a symmetry, such as time reversal symmetry, one can expect H to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries, which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator U(T) can lead to a symmetry in a basis of Floquet eigenstates.
Periodically driven (Floquet) systems can exhibit possibilities beyond what can be obtained in equilibrium. Both in Floquet systems and in the related problems of discrete-time quantum walks and quantum cellular automata, a basic distinction arises among unitary time evolution operators: while all physical operators are local, not all are locally generated (i.e., generated by some local Hamiltonian). In this paper, we define the notion of equivalence up to a locally generated unitary in all Altland-Zirnbauer symmetry classes. We then classify noninteracting unitaries in all dimensions on this basis by showing that equivalence up to a locally generated unitary is identical to homotopy equivalence.
We investigate the role of symmetries in determining the random matrix class describing quantum thermalization in a periodically driven many-body quantum system. Using a combination of analytical arguments and numerical exact diagonalization, we establish that a periodically driven ``Floquet'' system can be in a different random matrix class from the instantaneous Hamiltonian. A periodically driven system can thermalize even when the instantaneous Hamiltonian is integrable. A Floquet system that thermalizes in general can display integrable behavior at commensurate driving frequencies. When the instantaneous Hamiltonian and the Floquet operator both thermalize, the Floquet problem can be in the unitary class while the instantaneous Hamiltonian is always in the orthogonal class, and vice versa. We extract general principles regarding when a Floquet problem can thermalize to a different symmetry class from the instantaneous Hamiltonian. A (finite-sized) Floquet system can even display crossovers between different random matrix classes as a function of driving frequency.
Abstract We study quantum Floquet (periodically-driven) systems having continuous dynamical symmetry (CDS) consisting of a time translation and a unitary transformation on the Hilbert space. Unlike the discrete ones, the CDS strongly constrains the possible Hamiltonians H(t) and allows us to obtain all the Floquet states by solving a finite-dimensional eigenvalue problem. Besides, Noether’s theorem leads to a time-dependent conservation charge, whose expectation value is time-independent throughout evolution. We exemplify these consequences of CDS in the seminal Rabi model, an effective model of a nitrogen-vacancy center in diamonds without strain terms, and Heisenberg spin models in rotating fields. Our results provide a systematic way of solving for Floquet states and explain how they avoid hybridization in quasienergy diagrams.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA†AHA^\daggerAHA† that leaves the Hamiltonian HHH unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0[H,A]=0[H,A]=0, and a conservation law, namely the invariance of expectation values of AAA. For non-Hermitian Hamiltonians, H†H^\daggerH† comes into play as a distinct operator that complements HHH in generalized unitarity relations. The above description of symmetries has to be extended to include also AAA-pseudohermiticity relations of the form AH=H†AAH=H^\dagger AAH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of AAA operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.