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Journal of Physics A Mathematical and General

Fluctuations in an asymmetric anharmonic crystal

M Van Canneyt, Jacek Wojtkiewicz

Published Sep 7, 1997• 1 citation
Publisher

Abstract

In this paper we continue the investigation of the structural phase transition in a quantum anharmonic crystal. We calculate the influence of breaking the symmetry of the interaction potential on the behaviour of fluctuations and their critical exponents. More precisely, it is shown that when the low-momentum asymptotics of the phonon spectrum (i.e. the Fourier transform of the interaction matrix) have a quartic asymptotic in one direction, the critical exponents for the momentum and position fluctuations increase on the critical line. The quantum nature of the fluctuations in T = 0 is, however, preserved.

DOI
10.1088/0305-4470/30/17/007

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In this paper we continue the investigation of the structural phase transition in a quantum anharmonic crystal. We calculate the influence of breaking the symmetry of the interaction potential on the behaviour of fluctuations and their critical exponents. More precisely, it is shown that when the low-momentum asymptotics of the phonon spectrum (i.e. the Fourier transform of the interaction matrix) have a quartic asymptotic in one direction, the critical exponents for the momentum and position fluctuations increase on the critical line. The quantum nature of the fluctuations in T = 0 is, however, preserved.
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.
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.
Effects of cubic and quartic anharmonic phonons on the polaronic properties are investigated in the one-dimensional Hubbard–Holstein model at half filling, using the variable-displacement Lang–Firsov canonical transformation and the exact solution of Lieb and Wu. Although the quartic anharmonicity always reduces the effect of the Coulomb repulsion U , the cubic one mainly responsible for the thermal expansion of lattice brings about the effect of strong asymmetric nature in the sign of the electron–phonon coupling constant g ; it enhances the effect of U for positive g , while it suppresses for negative g . As a result, the overall features of the anharmonic system are basically similar to those in the harmonic one for positive g , but even in the range of the realistic magnitudes for those anharmonicities, they may become qualitatively so different for negative g as to provide a first-order phase transition.
We investigate the effect of anharmonicity on the one-dimensional half-filled Holstein model by using the determinant quantum Monte Carlo method. By calculating the order parameters we find that with and without anharmonicity there is always an transition from a disorder phase to a dimerized phase. Moreover, in the dimerized phase a lattice dimerization and a charge density wave coexist. The anharmonicity represented by the quartic term suppresses the dimerization as well as the charge density wave, while a double-well potential favors the dimerization. In addition, by calculating the correlation exponents we show that the disorder phase is metallic with gapless charge excitations and gapful spin excitations while in the dimerized phase both excitations are gapful.
The definition of the fluctuation terms in the matrix representations of operators, especially function operators, brings a lot of facilitations to the numerical analysis of matrix representation issues. This work considers a one‐dimensional quantum oscillator with purely quartic anharmonicity and focuses on its position and momentum expectations. By using certain level commutator operation it is possible to get two ODEs as equations of motion, which are in classical nature when the fluctuation terms are omitted. To understand how fluctuation terms affect expectation dynamical analysis we add extra temporal unknowns and construct extra ODEs over them. There seem to be an infinite ODEs chain for fluctuation altough their truncations seem to be utilizable as approximations. Here basic theory is summarized. Further details will be prasented at the conference presentation.
Long-range interactions can fundamentally reshape the low-energy properties of low-dimensional quantum matter, altering both continuous symmetry breaking and topological phenomena. However, their impact on the quantum criticality separating these regimes remains poorly understood. We determine the ground-state phase diagram and critical properties of the spin-one Heisenberg chain with single-ion anisotropy and staggered antiferromagnetic power-law interactions, using matrix-product state (MPS) calculations complemented by high-order series expansions (pCUT+MC). Such long-range, non-frustrated interactions circumvent the Hohenberg-Mermin-Wagner theorem, thereby stabilizing continuous symmetry breaking (CSB) phases in direct competition with the Haldane phase. We map out the resulting phase diagram and analyze the entanglement entropy scaling behavior in the U(1) and SU(2) CSB phases, finding logarithmic corrections beyond the short-range, universal contributions expected from linearly dispersed Goldstone modes. We further characterize all critical boundaries through finite-size scaling of either the entanglement entropy or the staggered magnetization. In particular, the large-D-to-U(1)-CSB transition exhibits unconventional, continuously varying critical exponents as a function of the long-range decay exponent with a strong dependence on the imposed boundary conditions leading to distinct finite-size scalings for sufficiently long-range potentials. Remarkably, the Haldane-to-U(1)-CSB transition likewise displays unconventional quantum criticality with distinct continuously varying critical exponents. Our work positions this model as a target for near-term atomic platforms with tunable long-range couplings and exhibiting natural single-ion anisotropy, offering a minimal playground for exploring the interplay between long-range interactions, continuous symmetry breaking, and topology.