We analyze the finite-size scaling exponents in the Lipkin-Meshkov-Glick model by means of the Holstein-Primakoff representation of the spin operators and the continuous unitary transformations method. This combination allows us to exactly compute the leading corrections to the groundstate energy, the gap, the magnetization, and the two-spin correlation functions. We also present numerical calculations for large system size which confirm the validity of this approach. Finally, we use these results to discuss the entanglement properties of the ground state focusing on the (rescaled) concurrence that we compute in the thermodynamical limit.
We study the ground state properties of the critical Lipkin-Meshkov-Glick model. Using the Holstein-Primakoff boson representation, and the continuous unitary transformation technique, we compute explicitly the finite-size scaling exponents for the energy gap, the ground state energy, the magnetization, and the spin-spin correlation functions. Finally, we discuss the behavior of the two-spin entanglement in the vicinity of the phase transition.
We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong-and weak-field limit, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.
We establish a relation between several entanglement properties in the Lipkin-Meshkov-Glick model, which is a system of mutually interacting spins embedded in a magnetic field. We provide analytical proofs that the single-copy entanglement and the global geometric entanglement of the ground state close to and at criticality behave as the entanglement entropy. These results are in deep contrast to what is found in one- dimensional spin systems where these three entanglement measures behave differently.
We consider the finite-size corrections in the Dicke model and determine the scaling exponents at the critical point for several quantities such as the ground state energy or the gap. Therefore, we use the Holstein-Primakoff representation of the angular momentum and introduce a nonlinear transformation to diagonalize the Hamiltonian in the normal phase. As already observed in several systems, these corrections turn out to be singular at the transition point and thus lead to nontrivial exponents. We show that for the atomic observables, these exponents are the same as in the Lipkin-Meshkov-Glick model, in agreement with numerical results. We also investigate the behavior of the order parameter related to the radiation mode and show that it is driven by the same scaling variable as the atomic one. PACS numbers: 42.50.Fx, 05.30.Jp, 73.43.Nq Superradiance is the collective decay of an excited population of atoms via spontaneous emission of photons. This phenomenon first predicted by Dicke in 1954 [1] has, since then, been observed experimentally in several quantum optical as well as solid-state systems (for a review see Ref.[2]). The phase diagram of the Dicke model, which is the subject of the present study, has been established in the thermodynamical limit by Hepp and Lieb [3] revealing the existence of a second-order quantum phase transition. This transition has been shown to be associated to a crossover between Poisson and Wigner-Dyson level statistics for a finite number of atoms N , thus raising the question of the finite-size corrections in this system [4,5]. These corrections have also been shown to be crucial in the understanding of entanglement properties [6,7] which become trivial if one directly considers the thermodynamical limit [6,7]. In these latter studies, nontrivial finite-size scaling exponents have been numerically found at the critical point and further been compared to those obtained in the Lipkin-Meshkov-Glick model [8]. The aim of the present work is to determine these exponents.To achieve this goal, we proceed in several steps. First, we use the Holstein-Primakoff boson representation [9] for the atomic degrees of freedom which is well adapted for a 1/N expansion of the Hamiltonian, N being the number of atoms. Second, we exactly diagonalize the expanded (quartic) Hamiltonian at order 1/N . In a recent series of papers [10,11,12], this diagonalization was performed using the Continuous Unitary Transformations (CUTs) methods [13] but here, the problem is more complicated for several reasons: (i) it involves two different degrees of freedom; (ii) the parameter space is two-dimensional; and (iii) the total number of particle is not fixed. These complications render the analytical resolution of the flow equations coming from CUTs approach difficult [14]. We are thus led to use an alternative approach relying on a canonical transformation of the initial bosonic operators. This transformation provides both the eigenstates and the eigenspectrum of H, and thus allows one to compute any matrix el...
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