operators, and can be diagonalized using a Bogoliubov transformation. We have hence shown that there is a formal equivalence between the eigenstates and energy eigenvalues of the Kitaev model and the Ising model. 3 The transverse eld Ising model The transverse eld Ising model (TFIM) was rst introduced by de Gennes in 1963 [11] as a pseudo spin model to describe the tunneling of protons in ferroelectric crystalls. ... [11] to simulate transition probabilities and scattering in the one dimensional transverse Ising model. Such a momentum-space entanglement has been discussed in non-interacting fermions [18] and superfluids [19]. Quantum Ising Models Andrew Tan1 1Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7. Figure 1.1: Schematic depiction of the one dimensional Ising model (Ising chain). Let us now perform the following transformation on a, called Bogoliubov translation for coherent states or boson translation: a ¡! Thus we first undo Bogoliubov transformation. Figure 1.2: Illustration of the two dimensional Ising model on a rectangular lattice. But the eigenmode of and are special. Interestingly, by using a Bogoliubov transformation to construct the ground state of the transverse field Ising model, we identify another bipartite structure in momentum space as well. Quantum phase transition takes place at the critical ... lowed by Bogoliubov transformation [14]. The Hamiltonian of the one dimensional quantum Ising model is given by H I = Jg å i sˆx i J å sˆ z i sˆ j . [1]. We can make the correspondence more explicit by choosing the Ising spins in the x-direction, while the quantization axis remains the z-axis. Since then the model has become a famous example for studying low dimensional strongly in … It is solvable by the Bogoliubov transformation and, therefore, provides analytic insights into the problem. (Dated: December 9, 2018) I. Since Bogoliubov transformation of a pair of fermions will not change the fermion number parity, one would naively expect that is even for APBC and is odd for PBC. If its energy is already positive, then the Bogoliubov transformation did nothing. Rabi model in the Bargmann-Fock space of analytic functions, deriving conditions for determining the energy spectrum [38, 39, 40]. Alba Cervera-Lierta 1,2. The Bogoliubov transformation can be decomposed into a single-particle basis change to momentum space and quasi-particle operators connecting particles of opposite momenta [29–31] and we obtain H Ising = X k∈Z N (k) γ† k γ k − 1 2 + O(1/N), (2) where the operators γ The former one is equivalent, via the Jordan-Wigner transformation, to the long-range Kitaev model. Application of the Bogoliubov inequality in the Ising model. (c) In the region of ° where Bogoliubov transformation is not helpful, directly solve the equation of motion. In (15) we have as many n j’s as there are single-particle eigenstates, i.e. Copyright © 1969 Published by Elsevier B.V. https://doi.org/10.1016/0375-9601(69)90783-X. It was further found that the analytic solution for the quantum Rabi model can be given in terms of confluent Heun functions B n. k gates written in Eq. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Bogoliubov transformation is implemented using. Exact Ising model simulation on a quantum computer. We use cookies to help provide and enhance our service and tailor content and ads. in Two-Dimensional Ising Model V.N. The entanglement present in the gen-eralized model is also calculated, showing the absence of concurrence, but large localizable entanglement, for spins that are not nearest neigh-bours. Subsequently this condition was reproduced via Bogoliubov transformation [41]. It has been shown that exact values for the spectral functions of the Ising model and Bogoliubov inequality lead to some non-trivial inequalities for the correlation functions. In the image notation, S i is the spin at the i-th position, and J ij is the (site dependent) coupling between two adjacent spins [1]. The block entropy is shown to be fully equivalent to that of the corresponding quantum Ising model. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. (a) Show that without a loss of generality (up to a simple canonical transformation) one may choose ° to be real and positive. (b) Explore the possibility of solving for the dynamics of this model by Bogoliubov transformation. The discussion includes the trans- The n j’s tell us how many particles are in the single-particle state jji.By construction the … By continuing you agree to the use of cookies. Fourier trans-form consistent with the antiperiodic … quantum Ising type of behaviour. dimH1.For fermions we have n j= 0;1 only as a consequence of the Pauli principle.The representation (15) is called occupation number representation. Plechko Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Abstract The anticommuting analysis with Grassmann variables is applied to the two-dimensional Ising model in statistical mechanics.

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