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Physics Letters A

A new technique for the construction of effective Hamiltonians

JOSIP SEKE, Andrey Vladimirovich Soldatov, Nikolay N. Bogolubov

Published Dec 1, 1997• 4 citations
Publisher
DOI
10.1016/s0375-9601(97)00827-x

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Seke's self-consistent projection-operator method has been developed for deriving non-Markovian equations of motion for probability amplitudes of a relevant set of state vectors. This method, in a Born-like approximation, leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method in particular applications, its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Unlike most perturbative techniques, in the present case each order of the approximation determines its own effective (approximating) Hamiltonian ensuring self-consistency and formal exactness of all results in the corresponding approximation order.
Constructing matrix product operators (MPOs) is at the core of the modern density matrix renormalization group (DMRG) and its time dependent formulation. For the DMRG to be conveniently used in different problems described by different Hamiltonians, in this work, we propose a new generic algorithm to construct the MPO of an arbitrary operator with a sum-of-products form based on the bipartite graph theory. We show that the method has the following advantages: (i) it is automatic in that only the definition of the operator is required; (ii) it is symbolic thus free of any numerical error; (iii) the complementary operator technique can be fully employed so that the resulting MPO is globally optimal for any given order of degrees of freedom; and (iv) the symmetry of the system could be fully employed to reduce the dimension of MPO. To demonstrate the effectiveness of the new algorithm, the MPOs of Hamiltonians ranging from the prototypical spin-boson model and the Holstein model to the more complicated ab initio electronic Hamiltonian and the anharmonic vibrational Hamiltonian with the sextic force field are constructed. It is found that for the former three cases, our automatic algorithm can reproduce exactly the same MPOs as the optimally hand-crafted ones already known in the literature.
We propose a novel quantum technique to search for unmodelled anomalies in multi-dimensional binned collider data.We propose associating an Ising lattice spin site with each bin, with the Ising Hamiltonian suitably constructed from the observed data and a corresponding theoretical expectation.In order to capture spatially correlated anomalies in the data, we introduce spin-spin interactions between neighboring sites, as well as self-interactions.The ground state energy of the resulting Ising Hamiltonian can be used as a new test statistic, which can be computed either classically or via adiabatic quantum optimization.We demonstrate that our test statistic outperforms some of the most commonly used goodness-of-fit tests.The new approach greatly reduces the look-elsewhere effect by exploiting the typical differences between statistical noise and genuine new physics signals.