We present both the gauge theoretic description and the numerical calculations of the Berry phases with the real eigenstates, involving one with a many-body system as a background and the other with no such background. We demonstrate that for the former the sign of the Berry phase factor for a spin $\f{1}{2}$ particle (hole) coupled to a slow subsystem (phonon) depends on both the strength of electron correlations and the characteristics of the closed paths, unlike the cases for the latter.
The Lie group adiabatic evolution determined by a Lie algebra parameter dependent Hamiltonian is considered. It is demonstrated that in the case when the parameter space of the Hamiltonian is a homogeneous Kähler manifold its fundamental Kähler potentials completely determine Berry geometrical phase factor. Explicit expressions for Berry vector potentials (Berry connections) and Berry curvatures are obtained using the complex parametrization of the Hamiltonian parameter space. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.
The relation between level crossings, entanglement, and Berry phases is investigated for the Breit-Rabi Hamiltonian of hydrogen and sodium atoms, describing a hyperfine interaction of electron and nuclear spins in a magnetic field. It is shown that the entanglement between nuclear and electron spins is maximum at avoided crossings. An entangled state encircling avoided crossings acquires a marginal Berry phase of a subsystem like an instantaneous eigenstate moving around real crossings accumulates a Berry phase. Especially, the nodal points of a marginal Berry phase correspond to the avoided c…
For a time-dependent $τ$-periodic harmonic oscillator of two linearly independent homogeneous solutions of classical equation of motion which are bounded all over the time (stable), it is shown, there is a representation of states cyclic up to multiplicative constants under $τ$-evolution or $2τ$-evolution depending on the model. The set of the wave functions is complete. Berry's phase which could depend on the choice of representation can be defined under the $τ$- or $2τ$-evolution in this representation. If a homogeneous solution diverges as the time goes to infinity, it is shown that, Berry'…
A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang-Baxterization approach, we obtain a unitary solution $\breve{R}(θ,φ_{1},φ_{2})$ of Yang-Baxter Equation. It is shown that any pure two-qutrit entangled states can be generated via the universal $\breve{R}$-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the $\breve{R}$-matrix, and Berry phase of the Yang-Baxter system is investigated. Specifically, for $ φ_{1}=φ_{2}$, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians …
In the framework of the single-field slow-roll inflation, we derive the Hamiltonian of the linear primordial scalar and tensor perturbations in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes in terms of the Lewis-Riesenfeld phase. We conclude by discussing the discrepancy in the results of Pal et. al [Class. Quant. Grav. 30 (2013) 12] for these Berry phases, which is resolved to yield agreement with our results.
The dynamics of observables which are matrices depending on \hbar and taking values in classical phase space is defined retaining the terms up to the first order in \hbar of the Moyal bracket. Within this semiclassical approach a first order lagrangian involving gauge fields is studied as a constrained hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the T…
In this paper we develop a general method to obtain the Berry phase of time-dependent Hamiltonians with a linear structure given in terms of the $SU(1,1)$ and $SU(2)$ groups. This method is based on the similarity transformations of the displacement operator performed to the generators of each group, and let us diagonalize these Hamiltonians. Then, we introduce a trilinear form of the Tavis-Cummings model to compute the $SU(1,1)$ and $SU(2)$ Berry phases of this model.
The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.
We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) introduced for one-dimensional lattice models to two dimensions. In contrast to the geometric phase for density matrices suggested by Uhlmann, the EGP leads a proper Chern number for Gaussian, finite-temperature or non-equilibrium steady states. The Chern number can be expressed as an integral of the Berry curvature of the so-called fictitious Hamiltonian, constructed from single-particle correlations, over the two-dimensional Brillouin zone. For the Chern number to be non-zero the …
Spin interaction Hamiltonians are obtained from the unitary Yang--Baxter $\breve{R}$-matrix. Based on which, we study Berry phase and quantum criticality in the Yang--Baxter systems.
In this paper, we show that the Laughlin wave function is a Hamiltonian and its associated Berry connection as the Schrödinger equation by transforming the Schrödinger equation into the Kirchhoff equation which describes the evolution of $n$ point vortices in Hydrodynamics. This helps us to view the Berry connection associated with Laughlin wave function or Schrödinger equation is not Hermitian, therefore we propose a self adjoint model Hamiltonian for the fractional quantum Hall effect, from the study of Kirchhoff equation, using superymmetric quantum mechanics whose solutions are complex Her…
We investigate the adiabatic evolution of thermal state in non-reciprocal many-body systems coupled to their environment and subject to periodic drivings. In such systems we show that besides the dynamical phase a geometrical phase can exist and it drives the relaxation dynamic of the system. On the contrary to the dynamical phase which always pushes the system toward its equilibrium state we show that the geometric phase can speed up or reduce the speed of relaxation process. These results could have applications in the field of thermal management of complex systems.
Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is immediately extensible to any interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.
In this paper, a (3\times3)-matrix representation of the Birman-Wenzl-Murakami(BWM) algebra has been presented. Based on which, unitary matrices (A(θ,ϕ_1,ϕ_2), B(θ,ϕ_1,ϕ_2)) are generated via the Yang-Baxterization approach. A hamiltonian is constructed from the unitary (B(θ,ϕ)) matrix. We then study the Berry phase of the Yang-Baxter system and find the topological parameter d has relationship with berry phase.
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $π/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases…
In this paper we define a non-dynamical phase for a spin-1/2 particle in a rotating magnetic field in the non-adiabatic non-cyclic case, and this phase can be considered as a generalized Berry phase. We show that this phase reduces to the geometric Berry phase, in the adiabatic limit, up to a factor independent of the parameters of the system. We could add an arbitrary phase to the eigenstates of the Hamiltonian due to the gauge freedom. Then, we fix this arbitrary phase by comparing our Berry phase in the adiabatic limit with the Berry's result for the same system. Also, in the extreme non-ad…
Time-dependent supersymmetry allows one to delete quasienergy levels for time-periodic Hamiltonians and to create new ones. We illustrate this by examining an exactly solvable model related to the simple harmonic oscillator with a time-varying frequency. For an interesting nonharmonic example we present the change of the Berry phase due to a supersymmetry transformation.
The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian $PT$-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.
By studying the topological invariance andBerry phase in non-Hermitian systems, we reveal the basic properties of the complex Berry phase and generalize the global Berry phases Q to identify the topological invariance for non-Hermitian systems.We find thatQcan identify topological invariance in two kinds of non-Hermitian model, the two-level non-Hermitian Hamiltonian and the bipartite dissipative model. For the bipartite dissipative model, an abrupt change of the Berry phase in the parameter space reveals a quantum phase transition and is related to the exceptional points. These results give t…
The eigenvalues of a parameter-dependent Hamiltonian matrix form a band structure in parameter space. In such $N$-band systems, the quantum geometric tensor (QGT), consisting of the Berry curvature and quantum metric tensors, is usually computed from numerically obtained energy eigenstates. Here, an alternative approach to the QGT based on eigenprojectors and (generalized) Bloch vectors is exposed. It offers more analytical insight than the eigenstate approach. In particular, the full QGT of each band can be obtained without computing eigenstates, using only the Hamiltonian matrix and the resp…
We show that braiding transformation is a natural approach to describe quantum entanglement, by using the unitary braiding operators to realize entanglement swapping and generate the GHZ states as well as the linear cluster states. A Hamiltonian is constructed from the unitary $\check{R}_{i,i+1}(θ,ϕ)$-matrix, where $ϕ=ωt$ is time-dependent while $θ$ is time-independent. This in turn allows us to investigate the Berry phase in the entanglement space.
The paper aims to spell out the relevance of the Berry phase in view of the question what the minimal mathematical structure is that accounts for all observable quantum phenomena. The question is both of conceptual and of ontological interest. While common wisdom tells us that the quantum structure is represented by the structure of the projective Hilbert space, the appropriate structure rich enough to account for the Berry phase is the U(1) bundle over that projective space. The Berry phase is ultimately rooted in the curvature of this quantum bundle, it cannot be traced back to the Hamiltoni…
We study the connection between Berry phases and quantum phase transitions of generic quantum many-body systems. Consider sequences of Berry phases associated to sequences of loops in the parameter space whose limit is a point. If the sequence of Berry phases does not converge to zero, then the limit point is a quantum critical point. Quantum critical points are associated to failures of adiabaticity. We discuss the remarkable example of the anisotropic XY spin chain in a transverse magnetic field and detect the XX region of criticality.
The variational quantum eigensolver (VQE) is an algorithm to find eigenenergies and eigenstates of systems in quantum chemistry and quantum many-body physics. The VQE is one of the most promising applications of near-term quantum devices to investigate such systems. Here we propose an extension of the VQE to calculate the nonadiabatic couplings of molecules in quantum chemical systems and Berry's phase in quantum many-body systems. Both quantities play an important role to understand the properties of a system beyond the naive adiabatic picture, e.g., nonadiabatic dynamics and topological phas…
Motivated by the Nahm's construction, in this paper we present a systematic construction of Schrödinger Hamiltonians for a spin-1/2 particle where the Berry connection in the ground-state sector becomes the Bogomolny-Prasad-Sommerfield (BPS) monopole of $SU(2)$ Yang-Mills-Higgs theory. Our construction enjoys a single arbitrary monotonic function, thereby creating infinitely many quantum-mechanical models that simulate the BPS monopole in the space of model parameters.
In the holonomic approach to quantum computation information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible, for a specific model, to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multi-partite system unitary transformations can be implemented efficiently by sequences…
We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by $\mathcal{PT}$-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional $\mathcal{PT}$-symmetric lattice system and verify our numerical results by an analytical calculation.
The evolution of a two level system with a slowly varying Hamiltonian, modeled as s spin 1/2 in a slowly varying magnetic field, and interacting with a quantum environment, modeled as a bath of harmonic oscillators is analyzed using a quantum Langevin approach. This allows to easily obtain the dissipation time and the correction to the Berry phase in the case of an adiabatic cyclic evolution.
The Berry curvature and its descendant, the Berry phase, play an important role in quantum mechanics. They can be used to understand the Aharonov-Bohm effect, define topological Chern numbers, and generally to investigate the geometric properties of a quantum ground state manifold. While Berry curvature has been well-studied in the regimes of few-body physics and non-interacting particles, its use in the regime of strong interactions is hindered by the lack of numerical methods to solve it. In this paper we fill this gap by implementing a quantum Monte Carlo method to solve for the Berry curva…
We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time-dependent transformation operator has to be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain analytically the wave functions and associated non-adiabatic Berry phase $γ_{n}$ for the $n$th eigenstate. The classical version of the non-Hermiti…
The properties that quantify photonic topological insulators (PTIs), Berry phase, Berry connection, and Chern number, are typically obtained by making analogies between classical Maxwell's equations and the quantum mechanical Schrödinger equation, writing both in Hamiltonian form. However, the aforementioned quantities are not necessarily quantum in nature, and for photonic systems they can be explained using only classical concepts. Here we provide a derivation and description of PTI quantities using classical Maxwell's equations, we demonstrate how an electromagnetic mode can acquire Berry p…
In a nondegenerate syste, the abelian Berry's phase will never cause transitions among the Hamiltonian's eigenstate. However, in a degenerate syatem, it is well known that the state transition can be caused by the non-abelian Berry phase. Actually, in a such a system, the phase factor is not always nonabelian. Even in the case that the phase is a nondiagonal matrix, it still can be an abelian phase and can cause the state transitions. We then propose a simple scheme to detect such an effect of the Berry phase to the degenerate states by using the optical entangled pair state. In this scheme, w…
Based on the method which is given in Ref. [Sun et.al. arXiv:0904.0092v1], we present another $9\times 9$ unitary $\breve{R}-$matrix, solution of the Yang-Baxter Equation, is obtained in this paper. The entanglement properties of $\breve{R}-$matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via $\breve{R}$-matrix acting on the standard basis. A Yang-Baxter Hamiltonian can be constructed from unitary $\breve{R}-$matrix. Then the geometric properties of this system is studied. The results showed that the Berry phase of this system c…
In this paper we construct a new $8\times8$ $\mathbb{M}$ matrix from the $4\times4$ $M$ matrix, where $\mathbb{M}$ / $M$ is the image of the braid group representation. The $ 8\times8 $ $\mathbb{M}$ matrix and the $4\times4$ $M$ matrix both satisfy extraspecial 2-groups algebra relations. By Yang-Baxteration approach, we derive a unitary $ \breve{R}(θ, ϕ)$ matrix from the $\mathbb{M}$ matrix with parameters $ϕ$ and $θ$. Three-qubit entangled states can be generated by using the $\breve{R}(θ,ϕ)$ matrix. A Hamiltonian for 3 qubits is constructed from the unitary $\breve{R}(θ,ϕ)$ matrix. We then …
We demonstrate that Berry phases may greatly affect the dynamics of spin-orbit coupled Bose-Einstein condensates. The effective model Hamiltonian under consideration is shown to be equivalent to the Exe Jahn-Teller model first introduced in molecular physics. The corresponding conical intersection is identified and the Berry phase acquired for a wave packet encircling the intersection studied. It is found that this phase manifests itself in the density profile of the condensate, making it a directly measurable quantity via time-of-flight detection. Moreover, the non-Abelian gauge structure of …
The dynamical effects of topological charge in two-dimensional QED can be expressed in terms of a topological order parameter via a Berry phase construction. The Berry phase describes the electric charge polarization of the vacuum in a manner similar to the theory of polarization in topological insulators. The topological order parameter labels discrete vacua which differ by units of electric flux. Here the associated Berry connection is explicitly constructed from the Dirac Hamiltonian eigenstates by introducing a small attractive Thirring coupling, so that there is still a stable boson in th…
We investigate the effect of the environment on a Berry phase measurement involving a spin-half. We model the spin+environment using a biased spin-boson Hamiltonian with a time-dependent magnetic field. We find that, contrary to naive expectations, the Berry phase acquired by the spin can be observed, but only on timescales which are neither too short nor very long. However this Berry phase is not the same as for the isolated spin-half. It does not have a simple geometric interpretation in terms of the adiabatic evolution of either bare spin-states or the dressed spin-resonances that remain on…
The Berry phase for a variety of systems comprising of two angular momenta is discussed. These include the electron and proton in the ground state of the hydrogen atom (taking into account the hyperfine interaction), the positronium atom, the spin and orbital angular momenta of a single electron, muon capture by the deuteron in an external magnetic field, etc. Though the time scales involved, the underlying intrinsic Hamiltonians are quite different, as also the possible experimental probes, the geometric nature of the results for the Berry phase due to a time varying externally imposed magnet…
We analyse the Kitaev honeycomb model, by means of the Berry curvature with respect to Hamiltonian parameters. We concentrate on the ground-state vortex-free sector, which allows us to exploit an appropriate Fermionisation technique. The parameter space includes a time-reversal breaking term which provides an analytical headway to study the curvature in phases in which it would otherwise vanish. The curvature is then analysed in the limit in which the time-reversal-symmetry-breaking perturbation vanishes. This provides remarkable information about the topological phase transitions of the model…
By using a second quantized formulation of level crossing, which does not assume adiabatic approximation, a convenient formula for geometric terms including off-diagonal terms is derived. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian in the present formulation. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial for any finite time interval $T$. The topological interpretation of Berry's phase such as the topological proof of phase-change rule thus fails in the practical Bor…
For two-dimensional holographic CFTs, we demonstrate the role of Berry phases for relating the non-factorization of the Hilbert space to the presence of wormholes. The wormholes are characterized by a non-exact symplectic form that gives rise to the Berry phase. For wormholes connecting two spacelike regions in gravitational spacetimes, we find that the non-exactness is linked to a variable appearing in the phase space of the boundary CFT. This variable corresponds to a loop integral in the bulk. Through this loop integral, non-factorization becomes apparent in the dual entangled CFTs. Further…
We examine Berry phase pertaining to purely quadrupolar state ($\langle ψ| \vec{S} | ψ\rangle = 0$) of a spin-$1$ system. Using the Majorana stellar representation of these states, we provide a visualization for the topological (zero or $π$) nature of such quadrupolar Berry phase. We demonstrates that the $π$ Berry phase of quadrupolar state is induced by the Majorana stars collectively tracing out a closed path (a great circle) by exchanging their respective positions on the Bloch sphere. We also analyse the problem from the perspective of dynamics where a state from the quadrupolar subspace …
