Finite-size behavior of quantum collective spin systems

Liberti, Giuseppe Di; Piperno, F.; Plastina, Francesco

doi:10.1103/physreva.81.013818

Cited by 15 publications

(15 citation statements)

References 55 publications

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“…For the ground state of the Dicke model the finite-size corrections and scaling exponents are known [24], and show a good agreement with the critical scaling behavior of infinitely coordinated spin systems described by the Lipkin-Meskhov-Glick model [25]. This correspondence has been investigated both analytically [26] and by numerical calculations [27]. It is well known that for infinitely coordinated systems the mean-field approach is exact in the thermodynamic limit and the large-size critical behavior is related to the upper critical dimensionality of the corresponding short range system [28].…”

Section: Introductionmentioning

confidence: 57%

Finite-size scaling in the quantum phase transition of the open-system Dicke model

et al. 2012

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Laser-driven Bose-Einstein condensate of ultracold atoms loaded into a lossy high-finesse optical resonator exhibits critical behavior and, in the thermodynamic limit, a phase transition between stationary states of different symmetries. The system realizes an open-system variant of the celebrated Dicke-model. We study the transition for a finite number of atoms by means of a Hartree-Fock-Bogoliubov method adapted to a damped-driven open system. The finite-size scaling exponents are determined and a clear distinction between the non-equilibrium and the equilibrium quantum criticality is found.

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Section: Introductionmentioning

confidence: 57%

Finite-size scaling in the quantum phase transition of the open-system Dicke model

et al. 2012

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“…Similar singularities arise when using Holstein-Primakoff description of the rotor in more complex systems, as for example in the Dicke model [12], where N two-level atoms are coupled with one electromagnetic mode in a cavity [13,14,15]. Its energy eigenstates in the adiabatic limit, where the field frequency is much faster than the atomic excitation energy, are closely related with the collective spin model [16]. While the limitations of truncated Hamiltonian and the need to include the next order corrections has been exposed previously [10,17,18], it seems necessary to discuss in detail the situations in which the prediction of the truncated Hamiltonian cannot be employed to describe finite systems, and also how useful information can be extracted from the divergences through the use of renormalisation techniques [10].…”

Section: Introductionmentioning

confidence: 87%

Virtues and limitations of the truncated Holstein–Primakoff description of quantum rotors

Hirsch¹,

Castaños²,

López‐Peña³

et al. 2013

Phys. Scr.

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A Hamiltonian describing the collective behaviour of N interacting spins can be mapped to a bosonic one employing the Holstein-Primakoff realisation, at the expense of having an infinite series in powers of the boson creation and annihilation operators. Truncating this series up to quadratic terms allows for the obtention of analytic solutions through a Bogoliubov transformation, which becomes exact in the limit N → ∞. The Hamiltonian exhibits a phase transition from single spin excitations to a collective mode. In a vicinity of this phase transition the truncated solutions predict the existence of singularities for finite number of spins, which have no counterpart in the exact diagonalization. Renormalisation allows to extract from these divergences the exact behaviour of relevant observables with the number of spins around the phase transition, and relate it with the class of universality to which the model belongs. In the present work a detailed analysis of these aspects is presented for the Lipkin model.

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“…At zero temperature, these finite-size corrections have been shown to be very sensitive to the transition 20,21,30,31 . For instance, corrections to the thermodynamical value of the order parameter m behave as N −1/3 at the critical point and as N −1/2 otherwise.…”

Section: Finite-size Correctionsmentioning

confidence: 99%

Finite-temperature mutual information in a simple phase transition

Wilms¹,

Vidal²,

Verstraete³

et al. 2012

J. Stat. Mech.

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We study the finite-temperature behavior of the Lipkin-Meshkov-Glick model with a focus on correlation properties as measured by the mutual information. The latter, which quantifies the amount of both classical and quantum correlations, is computed exactly in the two limiting cases of vanishing magnetic field and vanishing temperature. For all other situations , numerical results provide evidence of a finite mutual information at all temperatures except at criticality. There, it diverges as the logarithm of the system size, with a prefactor that can take only two values, depending on whether the critical temperature vanishes or not. Our work provides a simple example in which the mutual information appears as a powerful tool to detect finite-temperature phase transitions, contrary to entanglement measures such as the concurrence.

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Finite-size behavior of quantum collective spin systems

Cited by 15 publications

References 55 publications

Finite-size scaling in the quantum phase transition of the open-system Dicke model

Finite-size scaling in the quantum phase transition of the open-system Dicke model

Virtues and limitations of the truncated Holstein–Primakoff description of quantum rotors

Finite-temperature mutual information in a simple phase transition

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