The effective potential and effective Hamiltonian in quantum statistical mechanics

Cuccoli, Alessandro; Giachetti, R.; Tognetti, V.; Vaia, Ruggero; Verrucchi, Paola

doi:10.1088/0953-8984/7/41/003

Cited by 107 publications

(142 citation statements)

References 130 publications

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“…(8) is well behaved and negligible in the critical region. Alternatively, the same information can be also obtained from the total k-dependent susceptibility, (8), and the total one satisfy the following relation…”

Section: B Thermodynamic Observablesmentioning

confidence: 99%

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Phase transitions induced by easy-plane anisotropy in the classical Heisenberg antiferromagnet on a triangular lattice: A Monte Carlo simulation

et al. 1998

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We present the results of Monte Carlo simulations for the antiferromagnetic classical XXZ model with easy-plane exchange anisotropy on the triangular lattice, which causes frustration of the spin alignment. The behaviour of this system is similar to that of the antiferromagnetic XY model on the same lattice, showing the signature of a Berezinskii-Kosterlitz-Thouless transition, associated to vortex-antivortex unbinding, and of an Ising-like one due to the chirality, the latter occurring at a slightly higher temperature. Data for internal energy, specific heat, magnetic susceptibility, correlation length, and some properties associated with the chirality are reported in a broad temperature range, for lattice sizes ranging from 24×24 to 120×120; four values of the easy-plane anisotropy are considered. Moving from the strongest towards the weakest anisotropy (1 %) the thermodynamic quantities tend to the isotropic model behaviour, and the two transition temperatures decrease by about 25 % and 22 %, respectively.

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Section: B Thermodynamic Observablesmentioning

confidence: 99%

“…In addition, to treat three-dimensional spins is a necessary step for studying the corresponding quantum system by means of the pure-quantum self-consistent harmonic approximation 8,9,10,11 . In the XXZ TAF the easy-plane anisotropy results in a double degeneracy of the ground state.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions induced by easy-plane anisotropy in the classical Heisenberg antiferromagnet on a triangular lattice: A Monte Carlo simulation

et al. 1998

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“…It should be mentioned as well the simulation techniques widely used in recent years, especially a Monte Carlo formalism in which the quantum mechanical effects are included by making a modification of the potential energy. This formalism has its origin in a previous illustration by Feynman [22] of the use of the path-integral form of the partition function in statistical mechanics, and is known as the method of the effective potential and effective Hamiltonian [23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

An overview of the interatomic correlation moments and the mean square relative atomic displacements in anharmonic crystals

2006

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“…One of the main steps in deriving the PQSCHA for a magnetic system is the construction of an effective classical spin Hamiltonian from that written in terms of canonical variables [14]. For the sake of clarity, we use (p, q)={p i , q i } and s={s i } to indicate the set of classical canonical variables and unit vectors, respectively, needed to describe the classical system (see Ref.…”

mentioning

confidence: 99%

“…As for increasing |r| the coefficient θ r rapidly converges to (1+1/2S) −1 ζ 0 , the PQSCHA effective exchange for the low-T correlation length reads JS 2 ζ 0 ζ 1 , consistent with Eq. (14).…”

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confidence: 99%

Quantum two-dimensional Heisenberg antiferromagnet: Bridging the gap between field-theoretical and semiclassical approaches

et al. 2003

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The field-theoretical result for the low-T behaviour of the correlation length of the quantum Heisenberg antiferromagnet on the square lattice was recently improved by Hasenfratz [Eur. Phys. J. B 13, 11 (2000)], who corrected for cutoff effects. We show that starting from his expression, and exploiting our knowledge of the classical thermodynamics of the model, it is possible to take into account non-linear effects which are responsible for the main features of the correlation length at intermediate temperature. Moreover, we find that cutoff effects lead to the appearance of an effective exchange integral depending on the very same renormalization coefficients derived in the framework of the semiclassical pure-quantum self-consistent harmonic approximation: The gap between quantum field-theoretical and semiclassical results is here eventually bridged.PACS numbers: 05.50.+q, 75.10.Jm In the last two decades, thermodynamic properties of the quantum Heisenberg antiferromagnet on the squarelattice (QHAF) have been determined by a number of substantially different methods. Theoretical predictions were compared with experimental data for real compounds [1], as well as with numerical simulations obtained by different methods [2,3], and also with high-T series expansions [4]. Despite this effort, however, a comprehensive picture of the subject has not yet been formulated.Much of the analysis and debate on the QHAF has focused on the temperature-and spin-dependence of the staggered correlation length ξ(T, S). Goal of this paper is to show that ξ(T, S) can be expressed aswhere the effective exchange integral J eff (T, S) embodies quantum effects and is defined in terms of purely quantum spin-fluctuations, while ξ cl (T /J cl ) is the correlation length of the classical HAF. In particular, we find that Eq. (1), besides being the outcome of the semiclassical pure-quantum self-consistent harmonic approximation (PQSCHA) [5], remarkably holds also for the quantum field-theoretical prediction [6], as recently improved by Hasenfratz [7]. On the other hand, we show that the PQSCHA, when properly designed to such purpose, allows for a correct description of the low-T regime, via the appearance of the very same J eff (T, S) implicitly entering Hasenfratz's expression. It is a definite suprise that a typical semiclassical expression such as Eq. (1), come out of a field-theoretical result, especially in the case of the QHAF, where semiclassical and field-theoretical approaches seemed destined to describe different regions of the (T, S) plane, with no possible overlap.The QHAF is defined by the spin Hamiltonianwhere J>0, i=(i 1 , i 2 ), d=(±1, ±1), and lengths are in lattice units; the spinThe correlation length ξ(T, S) is defined via the asymptotic behaviour, lim |r|→∞ G(r) ∝ e −|r|/ξ , of the staggered correlation function G(r)≡(−1) r1+r2 Ŝ z iŜ z i+r , with r=(r 1 , r 2 ) any vector on the square lattice.The classical counterpart of the QHAF corresponds to the limit S→∞ with JS 2 →J cl , that gives the classical Hamiltonian H=...

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The effective potential and effective Hamiltonian in quantum statistical mechanics

Cited by 107 publications

References 130 publications

Phase transitions induced by easy-plane anisotropy in the classical Heisenberg antiferromagnet on a triangular lattice: A Monte Carlo simulation

Phase transitions induced by easy-plane anisotropy in the classical Heisenberg antiferromagnet on a triangular lattice: A Monte Carlo simulation

An overview of the interatomic correlation moments and the mean square relative atomic displacements in anharmonic crystals

Quantum two-dimensional Heisenberg antiferromagnet: Bridging the gap between field-theoretical and semiclassical approaches

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