In this chapter we consider the unit sphere \(\mathbb S^2\) of the Euclidean space \(\mathbb R^3\) with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on \(\mathbb S^2\) and allow us to build a quantization of the two-sphere \(\mathbb S^2\). They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of \(\mathbb S^2\) and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.
In this chapter we consider the unit sphere (\mathbb S^2) of the Euclidean space (\mathbb R^3) with its canonical symplectic structure is a phase space. Then coherent states are labelled by points on (\mathbb S^2) and allow us to build a quantization of the two-sphere (\mathbb S^2). They are defined in each finite dimensional space of an irreducible unitary representation of the symmetry rotation group SO(3) (or its covering SU(2)) of (\mathbb S^2) and give a semiclassical interpretation for the spin. As an application we state the Berezin-Lieb inequalities and compute the thermodynamic limit for large spin systems.
We have seen before that spin coherent states are strongly linked with the algebraic and geometric properties of the Euclidean 2-sphere S2. We shall now consider analogue setting when the sphere is replaced by the 2-pseudosphere i.e. the Euclidean metric in R3 is replaced by the Minkowski metric and the group SO(3) by the symmetry group SO(2, 1). The main big difference with the Euclidean case is that in the Minkowski case the pseudosphere and its symmetry groupe SO(2, 1) are not compact and that all irreducible unitary representations of SO(2, 1) are infinite dimensional.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn. So that in the Hamiltonian formalism, the phase space is Rn×Rn with its canonical symplectic form σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
The aim of this chapter is to present a construction of a set of coherent states for the hydrogen atom proposed by Villegas-Blas (The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. thesis, Mathematics Department, University of Virginia, 1996, [251]), Thomas and Villegas-Blas (Commun Math Phys 187:623–645, 1997, [244]). We show that in a semiclassical sense they concentrate essentially around the Kepler orbits (in configuration space) of the classical motion. A suitable unitary transformation (the Fock operator) maps the pure point subspace of the hydrogen atom Hamitonian onto the Hilbert space L2(S3) for the S3 sphere. We study the coherent states for the S3 sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group SO(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).
Using Isham's group-theoretic quantization scheme, we construct the canonical groups of the systems on the two-dimensional sphere and one-dimensional complex projective space, which are homeomorphic. In the first case, we take SO(3) as the natural canonical Lie group of rotations of the two-sphere and find all the possible Hamiltonian vector fields, and followed by verifying the commutator and Poisson bracket algebra correspondences with the Lie algebra of the group. In the second case, the same technique is resumed to define the Lie group, in this case SU (2), of CP'.We show that one can simply use a coordinate transformation from S 2 to CP 1 to obtain all the Hamiltonian vector fields of CP 1 . We explicitly show that the Lie algebra structures of both canonical groups are locally homomorphic. On the other hand, globally their corresponding canonical groups are acting on different geometries, the latter of which is almost complex. Thus the canonical group for CP 1 is the double-covering group of SO(3), namely SU(2). The relevance of the proposed formalism is to understand the idea of CP 1 as a space of where the qubit lives which is known as a Bloch sphere.
We introduce a phase-space representation for qubits and spin models. The technique uses an SU(n) coherent-state basis and can equally be used for either static or dynamical simulations. We review previously known definitions and operator identities, and show how these can be used to define an off-diagonal, positive phase-space representation analogous to the positive-P function. As an illustration of the phase-space method, we use the example of the Ising model, which has exact solutions for the finite-temperature canonical ensemble in two dimensions. We show how a canonical ensemble for an Ising model of arbitrary structure can be efficiently simulated using SU(2) or atomic coherent states. The technique utilizes a transformation from a canonical (imaginary-time) weighted simulation to an equivalent unweighted real-time simulation. The results are compared to the exactly soluble two-dimensional case. We note that Ising models in one, two, or three dimensions are potentially achievable experimentally as a lattice gas of ultracold atoms in optical lattices. The technique is not restricted to canonical ensembles or to Ising-like couplings. It is also able to be used for real-time evolution and for systems whose time evolution follows a master equation describing decoherence and coupling to external reservoirs. The case of SU(n) phase space is used to describe n-level systems. In general, the requirement that time evolution be stochastic corresponds to a restriction to Hamiltonians and master equations that are quadratic in the group generators or generalized spin operators.
The study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.