Berry phase and quantum criticality in Yang-Baxter systems

Spin interaction Hamiltonians are obtained from the unitary Yang-Baxter -matrix. Based on which, we study Berry phase and quantum criticality in the Yang-Baxter systems.     

On the classical limit of Berry's phase integrable systems

Berry's Phase is given by integration of a characteristic two form. We consider integrable systems defined by Weyl quantized classical Hamiltonians. It is shown that the limit of /i times this tow form is the curvature of the classical connection whose holonomy is the Hannay angels. A result of this type was derived by Berry [B2].     

Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains

We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ ₂×Z ₂ symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of diagonally dominant Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ ₂×Z ₂ symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.     

Berry phase in the gravitational quantum well and the Seiberg-Witten map

We explicitly compute the geometrical Berry phase for the noncommutative gravitational quantum well for different SW maps. We find that they lead to different partial contributions to the Berry phase. For the most general map we obtain that Δγ(S)∼η³, in a segment S of the path in the configuration space where is the fundamental momentum scale for the noncommutative gravitational quantum well. For the full closed path, we find, through an explicit computation, that γ(C)=0. This result is consistent with the fact that physical properties are independent of the SW map and shows that these maps do not introduce degeneracies or level crossing in the noncommutative extensions of the gravitational quantum well.     

Quantized Berry phase in twisted Bloch momentum space as a topological order parameter for spin chains

We show that a quantized Berry phase in Bloch momentum space can serve as a topological order parameter to the quantum phases of a gapped spin chain system with time-reversal invariance. Specifically, we study this approach analytically in a class of XY spin-1/2 chain with multiple sites interactions in a transverse field. In order to derive a proper definition of the Berry curvature in a two-dimensional parameter space, we performed a local gauge transformation to the spin chain system by a twist operator, which endows the Hamiltonian of the system with the topology of a torus T²$T^2$ without changing its energy spectrum. We show that a topological _Z2$Z_2$ order parameter can be obtained as a quantized Berry phase by a loop integral of the Berry gauge potential along quarter of the Brillouin zone, which determines the zero-temperature phase diagram of the system.     

Topological invariants for phase transition points of one-dimensional Z<Subscript>2</Subscript> topological systems

We study topological properties of phase transition points of two topologicallynon-trivial Z₂ classes (D and DIII) in one dimension byassigning a Berry phase defined on closed circles around the gap closing points in theparameter space of momentum and a transition driving parameter. While the topologicalproperty of the Z₂ system is generally characterized by aZ₂topological invariant, we identify that it has a correspondence to the quantized Berryphase protected by the particle-hole symmetry, and then give a proper definition of Berryphase to the phase transition point. By applying our scheme to some specific models ofclass D and DIII, we demonstrate that the topological phase transition can be wellcharacterized by the Berry phase of the transition point, which reflects the change ofBerry phases of topologically different phases across the phase transition point.     

Gauge-theoretic invariants for topological insulators: a bridge between Berry,Wess–Zumino,and Fu–Kane–Mele

We establish a connection between two recently proposed approaches to the understanding of the geometric origin of the Fu–Kane–Mele invariant \(\mathrm {FKM}\in \mathbb {Z}_2\), arising in the context of two-dimensional time-reversal symmetric topological insulators. On the one hand, the \(\mathbb {Z}_2\) invariant can be formulated in terms of the Berry connection and the Berry curvature of the Bloch bundle of occupied states over the Brillouin torus. On the other, using techniques from the theory of bundle gerbes, it is possible to provide an expression for \(\mathrm {FKM}\) containing the square root of the Wess–Zumino amplitude for a certain U(N)-valued field over the Brillouin torus. We link the two formulas by showing directly the equality between the above-mentioned Wess–Zumino amplitude and the Berry phase, as well as between their square roots. An essential tool of independent interest is an equivariant version of the adjoint Polyakov–Wiegmann formula for fields \(\mathbb {T}^2 \rightarrow U(N)\), of which we provide a proof employing only basic homotopy theory and circumventing the language of bundle gerbes.     

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization II

We give a characterization of the class of gapped Hamiltonians introduced in Part I (Ogata, A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015). The Hamiltonians in this class are given as MPS (Matrix product state) Hamiltonians. In Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), we list up properties of ground state structures of Hamiltonians in this class. In this Part II, we show the converse. Namely, if a (not necessarily MPS) Hamiltonian H satisfies five of the listed properties, there is a Hamiltonian H′ from the class by Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), satisfying the following: The ground state spaces of the two Hamiltonians on the infinite interval coincide. The spectral projections onto the ground state space of H on each finite intervals are approximated by that of H′ exponentially well, with respect to the interval size. The latter property has an application to the classification problem with open boundary conditions.     

Semiclassical diagonalization of quantum Hamiltonian and equations of motion with Berry phase corrections

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order ħ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.     

Non-Hermitian Hamiltonians with unitary and antiunitary symmetries

We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary for the diagonalization of the Hamiltonian in a given basis set. We can also classify the solutions according to the irreducible representations of the point group and thus analyse their properties separately. One of the main results of this paper is that some PT-symmetric Hamiltonians with point-group symmetry C_2v$C_{2v}$ exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit which suggests that the phenomenon is not robust. Point-group symmetry enables us to explain such anomalous behaviour and to choose a suitable antiunitary operator for the PT symmetry.     

Derivation of the Geometrical Phase from the Evans Phase Law of Generally Covariant Unified Field Theory

The phase law of generally covariant electrodynamics is used to explain straightforwardly the origin of the geometrical and Berry phase effects, exemplified by the Tomita-Chiao effect. Both effects are described by a phase factor that is constructed from the generally covariant Stokes formula of differential geometry, a phase factor in which the contour integral over the potential field A ⁽³⁾ is equated to the area integral over the gauge invariant field B ⁽³⁾, the Evans-Vigier field. The latter is the fundamental spin Casimir invariant of the Einstein group of general relativity applied to electrodynamics. General relativity as extended in the Evans unified field theory is needed for a correct understanding of all phase effects in physics, an understanding that is forged through the Evans phase law, the origin both of the Berry phase and the geometrical phase of electrodynamics observed in the Sagnac and Tomita-Chiao effects.     

Rates of Convergence in the Blume–Emery–Griffiths Model

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature $\beta $ and the interaction strength $K$ . The rates of convergence results are obtained as $(\beta ,K)$ converges along appropriate sequences $(\beta _n,K_n)$ to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.     

Entanglement and Berry Phase in a Parameterized Three-Qubit System

In this paper, we construct a parameterized form of unitary \(\breve {R}_{123}(\theta _{1},\theta _{2},\varphi )\) matrix through the Yang-Baxterization method. Acting such matrix on three-qubit natural basis as a quantum gate, we can obtain a set of entangled states, which possess the same entanglement value depending on the parameters ?? ₁ and ?? ₂. Particularly, such entangled states can produce a set of maximally entangled bases Greenberger-Horne-Zeilinger (GHZ) states with respect to ?? ₁ = ?? ₂ = π/2. Choosing a useful Hamiltonian, one can study the evolution of the eigenstates and investigate the result of Berry phase. It is not difficult to find that the Berry phase for this new three-qubit system consistent with the solid angle on the Bloch sphere.     

The Hilbert-Space Structure of Non-Hermitian Theories with Real Spectra

We investigate the quantum-mechanical interpretation of models with non-Hermitian Hamiltonians and real spectra. After describing a general framework to reformulate such models in terms of Hermitian Hamiltonians defined on the Hilbert space L ₂(-, ), we discuss the significance of the algebra of physical observables.     

Bethe Subalgebras in Hecke Algebra and Gaudin models

The generating function for elements of the Bethe subalgebra of the Hecke algebra is constructed as Sklyanin’s transfer-matrix operator for the Hecke chain. We show that in a special classical limit ${q \to 1}$ the Hamiltonians of the Gaudin model can be derived from the transfer-matrix operator of the Hecke chain. We construct a non-local analog of the Gaudin Hamiltonians in the case of the Hecke algebras.     

Three-point phase,symplectic measure,and Berry phase

It is shown that the geometric phase (Berry phase) around a cycle in the complex projective space of pure states of a quantum mechanical system can be expressed in terms of an elementary three-point phase function which is the simplest manifestation of the complexity of the underlying Hilbert space. In terms of this three-point phase it is possible to construct a geometrically relevant phase function defined mod 4 on the cycles and closely related to the natural symplectic structure of the state space.     

Toda Systems,Cluster Characters,and Spectral Networks

We show that the Hamiltonians of the open relativistic Toda system are elements of the generic basis of a cluster algebra, and in particular are cluster characters of nonrigid representations of a quiver with potential. Using cluster coordinates defined via spectral networks, we identify the phase space of this system with the wild character variety related to the periodic nonrelativistic Toda system by the wild nonabelian Hodge correspondence. We show that this identification takes the relativistic Toda Hamiltonians to traces of holonomies around a simple closed curve. In particular, this provides nontrivial examples of cluster coordinates on SL_n-character varieties for n > 2 where canonical functions associated to simple closed curves can be computed in terms of quivers with potential, extending known results in the SL₂ case.     

Quantum information processing with large nuclear spins in GaAs semiconductors

We propose an implementation for quantum information processing based on coherent manipulations of nuclear spins I=3/2 in GaAs semiconductors. We describe theoretically an NMR method which involves multiphoton transitions and which exploits the nonequidistance of nuclear spin levels due to quadrupolar splittings. Starting from known spin anisotropies we derive effective Hamiltonians in a generalized rotating frame, valid for arbitrary I, which allow us to describe the nonperturbative time evolution of spin states generated by magnetic rf fields. We identify an experimentally observable regime for multiphoton Rabi oscillations. In the nonlinear regime, we find Berry phase interference.     

Integrable spin–boson models descending from rational six-vertex models

We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain inhomogeneous rational vertex models combining bosonic and spin representations of SU(2)$\mathit{SU}(2)$, subject to non-diagonal toroidal and open boundary conditions. Only open boundary conditions are found to lead to integrable Hamiltonians combining both rotating and counter-rotating terms in the interaction. If the boundary matrices can be brought to triangular form simultaneously, the spectrum of the model can be obtained by means of the algebraic Bethe ansatz after a suitable gauge transformation; the corresponding Hamiltonians are found to be non-Hermitian. Alternatively, a certain quasi-classical limit of the transfer matrix is considered where Hermitian Hamiltonians are obtained as members of a family of commuting operators; their diagonalization, however, remains an unsolved problem.     

Electronic transport in graphene nanoribbons with disorder look at the pseudo-spin polarization: Dirac versus tight-binding model

We have compared results of electronic transport using two different approaches: Dirac vs tight-binding (TB) Hamiltonians to assesses disorder-induced effects in graphene nanoribbons. We apply the proposed Hamiltonians to calculate the density of states, the transmission along the ribbon, and the pseudo-spin polarization (P(E)) in metallic armchair graphene nanoribbons. We clearly show differences between these two approaches in the interference processes, especially in the low-lying energy limit, when the systems are found in the presence of random impurities (disorder). This allows us to find fingerprints associated with each model used. As the disorder increases, more robust electronic transmission (through polarized states in a given sublattice) arises when one is dealing with the Dirac model only. We also find with this model unexpected peaks in the P(E) far from the Dirac point for wider nanoribbons. In the other hand, the model TB show the Dirac limit with disturbances of the hyperboloid subbands for certain potentials of the impurities. In general, our study is indicating that a P(E) spectroscopy (analyzing the line width and intensity) can be used to detect fingerprints of the increase of asymmetry in the scattering processes and the transport limits where hyperboloid subbands are important.