Showing papers on "Canonical transformation published in 2023"

Effective-one-body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

Abstract: We determine the general local-in-time effective-one-body (EOB) Hamiltonian for massless Scalar-Tensor (ST) theories at third post-Newtonian (PN) order. Starting from the Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)], we map it to the corresponding ordinary Hamiltonian describing the two-body interaction in ST theories at 3PN level. Using a canonical transformation, we then map this onto an EOB Hamiltonian so as to determine the ST corrections to the 3PN-accurate EOB potentials $(A,B,Q_e)$ at 3PN. We then focus on circular orbits and compare the effect of the newly computed 3PN terms, also completed with finite-size and nonlocal-in-time contributions, on predictions for the frequency at the innermost stable circular orbit. Our results will be useful to build high-accuracy waveform models in ST theory, which could be used to perform precise tests against General Relativity using gravitational wave data from coalescing compact binaries.

Canonical Hamiltonian Theory, Hamiltonian Symmetries and Hamilton-Jacobi Theory

Abstract: In this chapter we shall discuss more advanced topics in Hamiltonian Mechanics. We will introduce the theory of canonical transformations and the Poisson bracket to study the relationship between symmetries and conservation laws in Hamilton’s formulation. Together with the canonical transformations of coordinates we will introduce a special atlas on phase spacetime that extends the one of natural coordinates. Using that, we shall reformulate Liouville’s theorem and deduce the Poincaré “recurrence” theorem. In the last part we will return to canonical transformations from a novel point of view which will allows us to introduce the Hamilton-Jacobi theory.

Exact solution for quantum strong long-range models via a generalized Hubbard-Stratonovich transformation

Abstract: We present an exact analytical solution for quantum strong long-range models in the canonical ensemble by extending the classical solution proposed in Campa et al., J. Phys. A 36, 6897 (2003). Specifically, we utilize the equivalence between generalized Dicke models and interacting quantum models as a generalization of the Hubbard-Stratonovich transformation. To demonstrate our method, we apply it to the Ising chain in transverse field and discuss its potential application to other models, such as the Fermi-Hubbard model, combined short and long-range models and models with antiferromagnetic interactions. Our findings indicate that the critical behaviour of a model is independent of the range of interactions, within the strong long-range regime, and the dimensionality of the model. Moreover, we show that the order parameter expression is equivalent to that provided by mean-field theory, thus confirming the exactness of the latter. Finally, we examine the algebraic decay of correlations and characterize its dependence on the range of interactions in the full phase diagram.

Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

Abstract: We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The extended Hamiltonian $H_{1}$ and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian $H$ will be derived from a generalized formulation of Hamilton's variational principle. The extended canonical transformation theory then naturally permits transformations that also map the time scales of original and destination system, while preserving the extended Hamiltonian $H_{1}$, and hence the form of the canonical equations derived from $H_{1}$. The Lorentz transformation, as well as time scaling transformations in celestial mechanics, will be shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows to directly map explicitly time-dependent Hamiltonians into time-independent ones. An ``extended'' generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we will reestablish the proper form of the extended Hamiltonian $H_{1}$ by means of a Legendre transformation of the extended Lagrangian $L_{1}$.

Projective cluster-additive transformation for quantum lattice models

Abstract: We construct a projection-based cluster-additive transformation that block-diagonalizes wide classes of lattice Hamiltonians $\mathcal{H}=\mathcal{H}_0 +V$. Its cluster additivity is an essential ingredient to set up perturbative or non-perturbative linked-cluster expansions for degenerate excitation subspaces of $\mathcal{H}_0$. Our transformation generalizes the minimal transformation known amongst others under the names Takahashi's transformation, Schrieffer-Wolff transformation, des Cloiseaux effective Hamiltonian, canonical van Vleck effective Hamiltonian or two-block orthogonalization method. The effective cluster-additive Hamiltonian and the transformation for a given subspace of $\mathcal{H}$, that is adiabatically connected to the eigenspace of $\mathcal{H}_0$ with eigenvalue $e_0^n$, solely depends on the eigenspaces of $\mathcal{H}$ connected to $e_0^m$ with $e_0^m\leq e_0^n$. In contrast, other cluster-additive transformations like the multi-block orthognalization method or perturbative continuous unitary transformations need a larger basis. This can be exploited to implement the transformation efficiently both perturbatively and non-perturbatively. As a benchmark, we perform perturbative and non-perturbative linked-cluster expansions in the low-field ordered phase of the transverse-field Ising model on the square lattice for single spin-flips and two spin-flip bound-states.

Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications

Abstract: We construct so-called Darboux matrices and fundamental solutions in the important case of generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in this way. Interesting results for dynamical systems depending on several variables and their explicit solutions follow. For these purposes we use our version of Bäcklund-Darboux transformation and square roots of the corresponding generalised matrix eigenvalues. Some new auxiliary results on the roots of matrices are included as well. An appendix is added to make the paper self-contained.