Symmetric Hamiltonians Research Articles

Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction

A PT -symmetric, non-Hermitian Hamiltonian in the PT -unbroken regime can lead to unitary dynamics under the appropriate choice of the Hilbert space. The Hilbert space is determined by a Hamiltonian-compatible inner product map on the underlying vector space, facilitated by a ‘metric operator’. A more traditional method, however, involves treating the evolution as open system dynamics, and the state is constructed through normalization at each time step. In this work, we present a comparative study of the two methods of constructing the reduced dynamics of a system evolving under a PT -symmetric Hamiltonian. Our system is a one-dimensional quantum walk with the spin and position degrees of freedom forming its two subsystems. We compare the information flow between the subsystems under the two methods. We find that under the metric formalism, a power law decay of the information backflow to the subsystem gives a clear indication of the transition from PT -unbroken to the broken phase. This is unlike the information backflow under the normalized state method. We also note that even though non-Hermiticity models open system dynamics, pseudo-Hermiticity can increase entanglement between the subsystem in the metric Hilbert space, thus indicating that pseudo-Hermiticity cases can be seen as a resource in quantum mechanics.

Mode-shell correspondence, a unifying phase space theory in topological physics - Part I: Chiral number of zero-modes

We propose a theory, that we call the , which relates the topological zero-modes localised in phase space to a invariant defined on the surface forming a shell enclosing these zero-modes. We show that the mode-shell formalism provides a general framework unifying important results of topological physics, such as the bulk-edge correspondence, higher-order topological insulators, but also the Atiyah-Singer and the Callias index theories. In this paper, we discuss the already rich phenomenology of chiral symmetric Hamiltonians where the topological quantity is the chiral number of zero-dimensional zero-energy modes. We explain how, in a lot of cases, the shell-invariant has a semi-classical limit expressed as a generalised winding number on the shell, which makes it accessible to analytical computations.

Power-Law Entanglement and Hilbert Space Fragmentation in Nonreciprocal Quantum Circuits.

Quantum circuits utilizing measurement to evolve a quantum wave function offer a new and rich playground to engineer unconventional entanglement dynamics. Here, we introduce a hybrid, nonreciprocal setup featuring a quantum circuit, whose updates are conditioned on the state of a classical dynamical agent. In our example the circuit is represented by a Majorana quantum chain controlled by a classical N-state Potts chain undergoing pair flips. The local orientation of the classical spins controls whether randomly drawn local measurements on the quantum chain are allowed or not. This imposes a dynamical kinetic constraint on the entanglement growth, described by the transfer matrix of an N-colored loop model. It yields an equivalent description of the circuit by an SU(N)-symmetric Temperley-Lieb Hamiltonian or by a kinetically constrained surface growth model for an N-component height field. For N=2, we find a diffusive growth of the half-chain entanglement toward a stationary profile S(L)∼L^{1/2} for L sites. For N≥3, the kinetic constraints impose Hilbert space fragmentation, yielding subdiffusive growth toward S(L)∼L^{0.57}. This showcases how the control by a classical dynamical agent can enrich the entanglement dynamics in quantum circuits, paving a route toward novel entanglement dynamics in nonreciprocal hybrid circuit architectures.

Nonreciprocal routing of microwave photons with broad bandwidth via magnon-cavity chiral coupling.

We propose a scheme for realizing nonreciprocal microwave photon routing with two cascaded magnon-cavity coupled systems, which work around the exceptional points of a parity-time (PT)-symmetric Hamiltonian. An almost perfect nonreciprocal transmission can be achieved with a broad bandwidth, where the transmission for a forward-propagating photon can be flexibly controlled with the backpropagating photon being isolated. The transmission or isolated direction can be reversed via simply controlling the magnetic field direction applied to the magnons. The isolation bandwidth is improved by almost three times in comparison with the device based on a single PT-symmetric system. Moreover, the effect of intrinsic cavity loss and added thermal noises is considered, confirming the experimental feasibility of the nonreciprocal device and potential applications in quantum information processing.

Z2 classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

Degenerate subspace localization and local symmetries

Domain specific localization of eigenstates has been a persistent observation for systems with local symmetries. The underlying mechanism for this localization behavior has, however, remained elusive. We provide here an analysis of a local reflection symmetric tight-binding Hamiltonian which attempts at identifying the key features that lead to the localized eigenstates. A weak coupling expansion of closed-form expressions for the eigenvectors demonstrates that the degeneracy of on-site energies occurring at the center of the locally symmetric domains represents the nucleus for eigenstates spreading across the domain. Since the symmetry-related subdomains constituting a locally symmetric domain are isospectral, we encounter pairwise degenerate eigenvalues that split linearly with an increasing coupling strength of the subdomains. The coupling to the (nonsymmetric) environment in an extended setup then leads to the survival of a certain system specific fraction of linearly splitting eigenvalues. The latter go hand in hand with the eigenstate localization on the locally symmetric domain. We provide a brief outlook addressing possible generalizations of local symmetry transformations while maintaining isospectrality. Published by the American Physical Society 2024

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field.

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

Spread complexity for measurement-induced non-unitary dynamics and Zeno effect

Using spread complexity and spread entropy, we study non-unitary quantum dynamics. For non-hermitian Hamiltonians, we extend the bi-Lanczos construction for the Krylov basis to the Schrödinger picture. Moreover, we implement an algorithm adapted to complex symmetric Hamiltonians. This reduces the computational memory requirements by half compared to the bi-Lanczos construction. We apply this construction to the one-dimensional tight-binding Hamiltonian subject to repeated measurements at fixed small time intervals, resulting in effective non-unitary dynamics. We find that the spread complexity initially grows with time, followed by an extended decay period and saturation. The choice of initial state determines the saturation value of complexity and entropy. In analogy to measurement-induced phase transitions, we consider a quench between hermitian and non-hermitian Hamiltonian evolution induced by turning on regular measurements at different frequencies. We find that as a function of the measurement frequency, the time at which the spread complexity starts growing increases. This time asymptotes to infinity when the time gap between measurements is taken to zero, indicating the onset of the quantum Zeno effect, according to which measurements impede time evolution.

Non-standard quantum algebras and finite dimensional -symmetric systems

In this work, -symmetric Hamiltonians defined on quantum algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard Hopf algebra deformation of . By making use of a particular boson representation of the generators of , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the -transformation. In terms of these operators, we construct several finite dimensional -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 -symmetry and the broken -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.

Realizing quantum speed limit in open system with a PT -symmetric trapped-ion qubit

Quantum speed limit (QSL), the lower bound of the time for transferring an initial state to a target one, is of fundamental interest in quantum information processing. Despite that the speed limit of a unitary evolution could be well analyzed by either the Mandelstam–Tamm or the Margolus–Levitin bound, there are still many unknowns for the QSL in open systems. A particularly exciting result is about that the evolution time can be made arbitrarily small without violating the time-energy uncertainty principle, whenever the dynamics is governed by a parity-time ( PT ) symmetric Hamiltonian. Here we study the QSLs with both PT and anti- PT Hamiltonians, and pose the QSL as a brachistochrone problem on a non-Hermitian Bloch sphere. We then use dissipative trapped-ion qubits to construct the Hamiltonians, where the state evolutions reach the QSL governed by a generalized Margolus-Levitin bound of the non-Hermitian system. We find that the evolution time monotonously decreases with the increase of the dissipation strength and exhibits chiral dependence on the Bloch sphere. These results enable a well-controlled knob for speeding up the state manipulation in open quantum systems, which could be used for quantum control and simulation with non-unitary dynamics.

The Unified Hamiltonian Formalism of Spin-Orbit Jahn-Teller and Pseudo-Jahn-Teller Problems in All Axial Symmetries.

Spatial degeneracy of electronic states closely connects spin-orbit coupling and vibronic coupling, which together determine properties of materials, especially heavy element compounds. Accurate description of those materials entails accurate mathematical formulas for spin-orbit vibronic Hamiltonians. For the first time ever, we in this work derive the Hamiltonian formalism to describe all spin-orbit Jahn-Teller and pseudo-Jahn-Teller vibronic problems in all axial symmetries. The conventional one-electron approximation of spin-orbit coupling, which was the foundation of all previous studies in this field, is not involved in the present work. Actually, the present formalism is applicable to all time-reversal symmetric hermitian Hamiltonian that has a Rank-1 dependence on the spin operator, without any restriction on the type and the number of term symbols and vibrational modes.

Solvable limits of a class of generalized vector nonlocal nonlinear schrödinger equation with balanced loss-gain

We consider a class of one dimensional Vector Nonlocal Non-linear Schrödinger equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schrödinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power, -symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The coordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.

Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners

Dualities in One-Dimensional Quantum Lattice Models: Symmetric Hamiltonians and Matrix Product Operator Intertwiners

Shape isomers of α -like nuclei in terms of the multiconfigurational dynamical symmetry

Background: The shape isomers of the $N=Z=\mathrm{even}$ nuclei are known from the energy-surface calculations within the Bloch-Brink (BB) $\ensuremath{\alpha}$-cluster model and Nilsson model. As an alternative, a new method (called SCS) has been proposed recently for determining the stable shapes, which is based on the investigation of the stability and consistency of the $\mathrm{SU}(3)$ symmetry (or quadrupole deformation).Purpose: We wish to derive the shape isomers of the $\ensuremath{\alpha}$-like nuclei from the SCS method and compare them with the results of the energy-surface calculation. Furthermore, we intend to study the consequences of the stable symmetries. In particular, we investigate (i) what kind of binary clusterizations are structurally allowed for the shape isomers and (ii) if energy spectra similar to those of the BB model can be obtained with a simple dynamically symmetric Hamiltonian.Methods: We determine the stability and self-consistency of the quadrupole deformation from a systematic calculation with the Nilsson model by applying the concept of the quasidynamical $\mathrm{U}(3)$ symmetry. The allowed cluster configurations are determined from the application of the U(3) and ${\mathrm{U}}^{\mathrm{ST}}(4)$ selection rules. The forbidden ones are characterized quantitatively. The energy spectrum is calculated with a simple dynamically symmetric Hamiltonian of the multiconfigurational dynamical symmetry (MUSY), which proved to be useful for the description of some experimental spectra.Results: The shape isomers found from the SCS method are in good agreement with the results of the energy surface method (BB and Nilsson models). Their allowed binary clusterizations are obtained from structural selection rules, but they give a hint for the available reaction channels, too. Last, a very simple energy functional is able to reproduce the gross features of their energy spectra in a large range of excitation energy and deformation.Conclusions: The similarity of the shape isomers from two very different methods gives a strong support to these predictions. The reasonable reproduction of the energy distribution of the ``ground-band heads'' is especially remarkable, considering the fact that MUSY is able to produce the complete spectra in detail, not only the energy minima.

Connection problem of the first Painlevé transcendents with large initial data

In previous work, Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a -symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions located on a sequence of curves , , will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation for the curves as is derived, where A is a positive constant. The discrete set could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of matches the numerical results remarkably well, even for small n. The main tool is the method of uniform asymptotics introduced by Bassom et al (1998 Arch. Rational Mech. Anal. 143 241–71) in the studies of the second Painlevé (PII) equation.

Frequency-Stable Robust Wireless Power Transfer Based on High-Order Pseudo-Hermitian Physics.

Nonradiative wireless power transfer (WPT) technology has made considerable progress with the application of the parity-time (PT) symmetry concept. In this Letter, we extend the standard second-order PT-symmetric Hamiltonian to a high-order symmetric tridiagonal pseudo-Hermitian Hamiltonian, relaxing the limitation of multisource/multiload systems based on non-Hermitian physics. We propose a three-mode pseudo-Hermitian dual-transmitter-single-receiver circuit and demonstrate that robust efficiency and stable frequency WPT can be attained despite the absence of PT symmetry. In addition, no active tuning is required when the coupling coefficient between the intermediate transmitter and the receiver is changed. The application of pseudo-Hermitian theory to classical circuit systems opens up an avenue for expanding the application of coupled multicoil systems.

Multipartite Entangled States in Dipolar Quantum Simulators.

The scalable production of multipartite entangled states in ensembles of qubits is a crucial function of quantum devices, as such states are an essential resource both for fundamental studies on entanglement, as well as for applied tasks. Here we focus on the U(1) symmetric Hamiltonians for qubits with dipolar interactions-a model realized in several state-of-the-art quantum simulation platforms for lattice spin models, including Rydberg-atom arrays with resonant interactions. Making use of exact and variational simulations, we theoretically show that the nonequilibrium dynamics generated by this Hamiltonian shares fundamental features with that of the one-axis-twisting model, namely, the simplest interacting collective-spin model with U(1) symmetry. The evolution governed by the dipolar Hamiltonian generates a cascade of multipartite entangled states-spin-squeezed states, Schrödinger's cat states, and multicomponent superpositions of coherent spin states. Investigating systems with up to N=144 qubits, we observe full scalability of the entanglement features of these states directly related to metrology, namely, scalable spin squeezing at an evolution time O(N^{1/3}) and Heisenberg scaling of sensitivity of the spin parity to global rotations for cat states reached at times O(N). Our results suggest that the native Hamiltonian dynamics of state-of-the-art quantum simulation platforms, such as Rydberg-atom arrays, can act as a robust source of multipartite entanglement.

Complex Spatiotemporal Modulations and Non-Hermitian Degeneracies in PT -Symmetric Phononic Materials

Unraveling real eigenfrequencies in non-Hermitian $\mathcal{PT}$-symmetric Hamiltonians has opened new avenues in quantum physics, photonics, and most recently, phononics. However, the existing literature squarely focuses on exploiting such systems in the context of scattering profiles (i.e., transmission and reflection) at the boundaries of a modulated waveguide, rather than the rich dynamics of the non-Hermitian medium itself. In this work, we investigate the wave propagation behavior of a one-dimensional non-Hermitian elastic medium with a universal complex stiffness modulation which encompasses a static term in addition to real and imaginary harmonic variations in both space and time. Using plane wave expansion, we conduct a comprehensive dispersion analysis for a wide set of sub-scenarios to quantify the onset of complex conjugate eigenfrequencies, and set forth the existence conditions for gaps which emerge along the wavenumber space. Upon defining the hierarchy and examining the asymmetry of these wavenumber gaps, we show that both the position with respect to the wavenumber axis and the imaginary component of the oscillatory frequency largely depend on the modulation type and gap order. Finally, we demonstrate the coalescence of multiple Bloch-wave modes at the emergent exceptional points where significant direction-dependent amplification can be realized by triggering specific harmonics through a process which is detailed herein.

Signatures of a quantum stabilized fluctuating phase and critical dynamics in a kinetically constrained open many-body system with two absorbing states

We introduce and investigate an open many-body quantum system in which kinetically constrained coherent and dissipative processes compete. The form of the incoherent dissipative dynamics is inspired by that of epidemic spreading or cellular-automaton-based computation related to the density-classification problem. It features two non-fluctuating absorbing states as well as a $\mathcal{Z}_2$-symmetric point in parameter space. The coherent evolution is governed by a kinetically constrained $\mathcal{Z}_2$-symmetric many-body Hamiltonian which is related to the quantum XOR-Fredrickson-Andersen model. We show that the quantum coherent dynamics can stabilize a fluctuating state and we characterize the transition between this active phase and the absorbing states. We also identify a rather peculiar behavior at the $\mathcal{Z}_2$-symmetric point. Here the system approaches the absorbing-state manifold with a dynamics that follows a power-law whose exponent continuously varies with the relative strength of the coherent dynamics. Our work shows how the interplay between coherent and dissipative processes as well as symmetry constraints may lead to a highly intricate non-equilibrium evolution and may stabilize phases that are absent in related classical problems.

Bootstrapping PT symmetric quantum mechanics

Bootstrapping in Quantum Mechanics uses positivity condition to derive the eigenspectum. For non-hermitian systems usual positivity condition does not work. In this paper we define positivity condition for special class of non-hermitian Hamiltonian, the PT symmetric Hamiltonian. We illustrate this modified positivity condition with several examples and obtain eigenspectrum.