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Journal of Physics Conference Series

On the quantum critical behaviour of a model of structural phase transitions with long-range interaction

Ekaterina S. Pisanova, N. S. Tonchev

Published Nov 1, 2010• 2 citations
Publisher

Abstract

The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.

DOI
10.1088/1742-6596/253/1/012020

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The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.
Usually, the critical phenomena properties at their upper critical dimensions are characterized by leading logarithmic terms in the corresponding equations and the Lambert W‐function can be used to obtain exact results. A class of models with long‐range interactions and developed classical and quantum fluctuations is considered at the upper quantum critical dimension The exact results, obtained here in the Lambert W‐function terms, and the asymptotic ones for the inverse susceptibility of a lattice model of structural phase transitions in an anharmonic crystal are compared on the basis of their graphic presentations.
The entropy and the specific heat of a quantum spherical model with a long-range interaction (decreasing at large distances r as r –d–σ where d is the space dimensionality and 0 < σ ≤ 2) are studied in the quantum critical region at the upper quantum critical dimension d = 3σ/2. The problem of obtaining the temperature-dependent corrections to the ground state free energy involves the solution of a transcendental equation, the exact solution of which is expressed in terms of the Lambert W-function. The free energy, the entropy and the specific heat in the quantum critical region are derived in terms of the Lambert W-function. For systems in real physical dimensions (chains, thin layers, i.e. films and three-dimensional systems) the exact results for the entropy and the specific heat obtained in terms of the Lambert W-function and the leading asymptotic ones are compared on the basis of the calculated relative errors. It can be concluded that for a class of quantum models at the upper critical dimension the Lambert W-function is a very effective tool for an exact computation of low-temperature critical properties, especially in the finite-temperature quantum critical region.
An exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction (decreasing at large distances r as r−d−σ, where d is the space dimensionality and 0 < σ ≤ 2) is considered near to its zero-temperature critical point. The low-temperature behavior of the bulk specific heat capacity at the lower classical critical dimension (d = σ) is studied in different regions of the (T, λ)-phase diagram, where T is the temperature and λ is a parameter which switches on quantum fluctuations. From the results obtained one can see that when T → 0+ the specific heat capacity tends to zero in a different way in the regions: (a) renormalized classical region - as T raised to the second power and (b) quantum disordered region - exponentially.
A model of quantum particles performing D -dimensional anharmonic oscillations around their equilibrium positions which form the d -dimensional simple cubic lattice Zd is considered. The model undergoes a structural phase transition when the fluctuations of displacements of particles become macroscopic. This phenomenon is described by susceptibilities depending on Matsubara frequencies ωn , n ∈ Z . We prove two theorems concerning the thermodynamic limits of these susceptibilities. The first theorem states that the susceptibilities with nonzero ωn remain bounded at all temperatures, which means that the macroscopic fluctuations in the model are always non-quantum. The second theorem gives a sufficient condition for the static susceptibility (i.e. corresponding to ωn = 0 ) to be bounded at all temperatures. This condition involves the particle mass, the anharmonicity parameters and the interaction intensity. The physical meaning of this result is that, for all D and all values of the temperature, strong quantum effects suppress critical points and the long range order. The proof is performed in the approach where the susceptibilities are represented as functional integrals. A brief description of the main features of this approach is delivered.
The behaviour of the finite-temperature C -function, defined by Neto and Fradkin (1993 Nucl. Phys. B 400 525), is analysed within a d -dimensional exactly solvable lattice model, recently considered by Vojta (1996 Phys. Rev. B 53 710), which is of the same universality class as the quantum nonlinear O( n ) sigma model in the limit n . The scaling functions of C for the cases d = 1 (absence of long-range order), d = 2 (existence of a quantum critical point), d = 4 (existence of a line of finite-temperature critical points that ends up with a quantum critical point) are derived and analysed. The locations of regions where C is monotonically increasing (which depend significantly on d ) are exactly determined. The results are interpreted within the finite-size scaling theory that has to be modified for d = 4.
In this work we present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighbourhood of a critical point keeping, at the same time, fuli information on the non universal properties of the model. A simple approximation to our formally exact equations is studied for the spin-S Heisenberg model where explicit results for critical exponents, critical temperature and coexistence curve are obtained. In three dimensions the results are in good agreement with available quantum Monte Carlo simulations and series expansions.