Fubini-Study Research Articles

The quantum brachistochrone problem for an arbitrary spin in a magnetic field

We consider quantum brachistochrone evolution for a spin-s system on rotational manifolds. Such manifolds are determined by the rotation of the eigenstates of the operator of projection of spin-s on some direction. The Fubini–Study metrics of these manifolds are those of spheres with radii dependent on the value of the spin and on the value of the spin projection. The conditions for optimal evolution of the spin-s system on rotational manifolds are obtained.

Band geometry of fractional topological insulators

Recent numerical simulations of flat band models with interactions which show clear evidence of fractionalized topological phases in the absence of a net magnetic field have generated a great deal of interest. We provide an explanation for these observations by showing that the physics of these systems is the same as that of conventional fractional quantum Hall phases in the lowest Landau level under certain ideal conditions which can be specified in terms of the Berry curvature and the Fubini study metric of the topological band. In particular, we show that when these ideal conditions hold, the density operators projected to the topological band obey the celebrated $W_{\infty}$ algebra. Our approach provides a quantitative way of testing the suitability of topological bands for hosting fractionalized phases.

Conformally Kähler surfaces and orthogonal holomorphic bisectional curvature

We show that a compact complex surface which admits a conformally K\"ahler metric g of positive orthogonal holomorphic bisectional curvature is biholomorphic to the complex projective plane. In addition, if g is a Hermitian metric which is Einstein, then the biholomorphism can be chosen to be an isometry via which g becomes a multiple of the Fubini-Study metric.

Holomorphic Isometry from a Kähler Manifold into a Product of Complex Projective Manifolds

We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.

Emergent Newtonian dynamics and the geometric origin of mass

We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dynamics near the adiabatic limit we find the emergent Newton’s second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Coriolis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor. The mass tensor is related to the non-equal time correlation functions in equilibrium and describes the dressing of the slow degree of freedom by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini–Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space highlighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples.

Einstein Hermitian metrics of positive sectional curvature

It is shown that, up to scaling and isometry, the only complete 4-manifold with an Einstein metric of positive sectional curvature which is also Hermitian with respect to some complex structure is the complex projective plane C P 2 \mathbb {CP}_2 , equipped with its Fubini-Study metric.

Geometric Analysis of Grover’s Search Algorithmin the Presence of Perturbation

For an initial uniform superposition over all possible computational basis states, we explore the performance of Grover’s search algorithm geometrically when imposing a perturbation on the Walsh-Hadamard transformation contained in the Grover iteration. We give the geometric picture to visualize the quantum search process in the three-dimensional space and show that Grover’s search algorithm can work well with an appropriately chosen perturbation. Thereby we corroborate Grover’s conclusion that if the perturbation is small, then it will have little impact of an impact on the implementation of this algorithm. We also prove that Grover’s path cannot achieve a geodesic under a perturbation of the Fubini-Study metric.

Critical values of random analytic functions on complex manifolds

We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical point of the norm. The metric and Kahler form endow the polynomials with a Hilbert space structure and we consider the associated Gaussian random polynomials and the spherical ensemble where ||s_n|| = 1 is chosen from Haar measure. Our main result is that the expected limit distribution of critical values as n tends to infinity in the spherical ensemble is universal (i.e. is independent of the choice of h), and we give an explicit formula for it. The limit distribution is the same as for suitably normalized Gaussian measures.

Vortices and the Abel–Jacobi map

The abelian Higgs model on a compact Riemann surface Σ supports vortex solutions for any positive vortex number d∈Z. Moreover, the vortex moduli space for fixed d has long been known to be the symmetrized d-th power of Σ, in symbols, Symd(Σ). This moduli space is Kähler with respect to the physically motivated metric whose geodesics describe slow vortex motion.In this paper we appeal to classical properties of Symd(Σ) to obtain new results for the moduli space metric. Our main tool is the Abel–Jacobi map, which maps Symd(Σ) into the Jacobian of Σ. Fibres of the Abel–Jacobi map are complex projective spaces, and the first theorem we prove states that near the Bradlow limit the moduli space metric restricted to these fibres is a multiple of the Fubini–Study metric. Additional significance is given to the fibres of the Abel–Jacobi map by our second result: we show that if Σ is a hyperelliptic surface, there exist two special fibres which are geodesic submanifolds of the moduli space. Even more is true: the Abel–Jacobi map has a number of fibres which contain complex projective subspaces that are geodesic.

Geometrical description of fractional Chern insulators based on static structure factor calculations

We study the static structure factor of the fractional Chern insulator Laughlin-like state and provide analytical forms for this quantity in the long-distance limit. In the course of this we identify averaged over Brillouin zone Fubini Study metric as the relevant metric in the long-distance limit. We discuss under which conditions the static structure factor will assume the usual behavior of Laughlin-like fractional quantum Hall system i.e. the scenario of Girvin, MacDonald, and Platzman [Phys. Rev. B 33, 2481 (1986)]. We study the influence of the departure of the averaged over Brillouin zone Fubini Study metric from its fractional quantum Hall value which appears in the long-distance analysis as an effective change of the filling factor. According to our exact diagonalization results on the Haldane model and analytical considerations we find persistence of fractional Chern insulator state even in this region of the parameter space.

Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology

We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an ``entanglement Fermi surface.'' Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and ${\mathbb{Z}}_{2}$ topological insulators.

MELTING OF THE EUCLIDEAN METRIC TO NEGATIVE SCALAR CURVATURE

We find a C ∞ -continuous path of Riemannian metrics gt on Rk, k � 3, for 0 � t � for some number > 0 with the following property: g0 is the Euclidean metric on Rk, the scalar curvatures of gt are strictly decreasing in t in the open unit ball and gt is isometric to the Euclidean metric in the complement of the ball. Furthermore we extend the discussion to the Fubini-Study metric in a similar way.

CONVERGENCE OF FUBINI–STUDY CURRENTS FOR ORBIFOLD LINE BUNDLES

We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.

Zero-energy fields on complex projective space

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

Moduli of Vortices and Grassmann Manifolds

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface Σ with target \({{\rm Mat}_{r \times n}(\mathbb{C})}\), where n ≥ r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kähler metrics of Fubini–Study type. These spaces are smooth at least in the local case r = n. For abelian local vortices we prove that, if a certain “quantization” condition is satisfied, the embedding can be chosen in such a way that the induced Fubini–Study structure realizes the Kähler class of the usual L 2 metric of gauged vortices.

COHERENT STATE OF α-DEFORMED WEYL–HEISENBERG ALGEBRA

At first, we introduce α-deformed algebra as a kind of generalization of the Weyl–Heisenberg algebra so that we get the su(2)- and su(1, 1)-algebras whenever α has specific values. After that, we construct coherent states of this algebra. Third, a realization of this algebra is given in the system of a harmonic oscillator confined at the center of a potential well. Then, we introduce two-boson realization of the α-deformed Weyl–Heisenberg algebra and use this representation to write α-deformed coherent states in terms of the two modes number states. Following these points, we consider mean number of excitations (we call them in general photons) and Mandel parameter as statistical properties of the α-deformed coherent states. Finally, the Fubini–Study metric is calculated for the α-coherent states manifold.

Non-abelian vortices on $\mathbb {CP}^{1}$CP1 and Grassmannians

Many properties of the moduli space of abelian vortices on a compact Riemann surface Σ are known. For non-abelian vortices the moduli space is less well understood. Here we consider non-abelian vortices on the Riemann sphere $\mathbb {CP}^{1}$CP1, and we study their moduli spaces near the Bradlow limit. We give an explicit description of the moduli space as a Kähler quotient of a finite-dimensional linear space. The dimensions of some of these moduli spaces are derived. Strikingly, there exist non-abelian vortex configurations on $\mathbb {CP}^{1}$CP1, with non-trivial vortex number, for which the moduli space is a point. This is in stark contrast to the moduli space of abelian vortices. For a special class of non-abelian vortices the moduli space is a Grassmannian, and the metric near the Bradlow limit is a natural generalization of the Fubini–Study metric on complex projective space. We use this metric to investigate the statistical mechanics of non-abelian vortices. The partition function is found to be analogous to the one for abelian vortices.

A note on sub-Riemannian structures associated with complex Hopf fibrations

Sub-Riemannian structures on odd-dimensional spheres respecting the Hopf fibration naturally appear in quantum mechanics. We study the curvature maps for such a sub-Riemannian structure and express them using the Riemannian curvature tensor of the Fubini-Study metric of the complex projective space and the curvature form of the Hopf fibration. We also estimate the number of conjugate points of a sub-Riemannian extremal in terms of the bounds of the sectional curvature and the curvature form. It presents a typical example for the study of curvature maps and comparison theorems for a general corank 1 sub-Riemannian structure with symmetries done by C. Li and I. Zelenko (2011) in [2].

Information metric from Riemannian superspaces

The Fisher information metric is introduced in order to find the meaning of the probability distribution in classical and quantum systems described by Riemannian non-degenerated superspaces. In particular, the physical rôle played by the coefficients a and a⁎ of the fermionic part of an emergent metric solution, obtained previously (Cirilo-Lombardo, 2012 [1]) is explored. We find that the metric solution of the superspace establishes a connexion between the Fisher metric and its quantum counterpart, corroborating early conjectures by Caianiello et al. This quantum mechanical extension of the Fisher metric is described by the CP1 structure of the Fubini–Study metric, with coordinates a and a⁎.

Restricted numerical shadow and the geometry of quantum entanglement

The restricted numerical range WR(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A—a normalized probability distribution on the complex plane supported in WR(A). Its value at point is equal to the probability that the expectation value 〈ψ|A|ψ〉 is equal to z, where |ψ〉 represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size D = NANB we analyse the geometry of sets of separable and maximally entangled states of the NA × NB composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow, we study the dynamics of quantum entanglement. A similar analysis extended for operators on D = 23-dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three-qubit system.