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arXiv:1009.3718 · International Journal of Modern Physics B

NEW RANDOM ORDERED PHASE IN ISOTROPIC MODELS WITH MANY-BODY INTERACTIONS

Yoichiro Hashizume, Masuo Suzuki

Published Jan 10, 2011• 10 citations
arXivHTMLPDF

Abstract

In this study, we have found a new random ordered phase in isotropic models with many-body interactions. Spin correlations between neighboring planes are rigorously shown to form a long-range order, namely coplanar spin-pair order, using a unitary transformation, and the phase transition of this new order has been analyzed on the bases of the mean-field theory and correlation identities. In the systems with regular 4-body interactions, the transition temperature T c is obtained as T c = (z-2)J/k B , and the field conjugate to this new order parameter is found to be H2. In contrast, the corresponding physical quantities in the systems with random 4-body interactions are given by [Formula: see text] and H4, respectively. Scaling forms of order parameters for regular or random 4-body interactions are expressed by the same scaling functions in the systems with regular or random 2-body interactions, respectively. Furthermore, we have obtained the nonlinear susceptibilities in the regular and random systems, where the coefficient χnl of H3 in the magnetization shows positive divergence in the regular model, while the coefficient χ7 of H7 in the magnetization shows negative divergence in the random model.

DOI
10.1142/s0217979211057724

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In this study, we have found a new random ordered phase in isotropic models with many-body interactions. Spin correlations between neighboring planes are rigorously shown to form a long-range order, namely coplanar spin-pair order, using a unitary transformation, and the phase transition of this new order has been analyzed on the bases of the mean-field theory and correlation identities. In the systems with regular 4-body interactions, the transition temperature T c is obtained as T c = (z-2)J/k B , and the field conjugate to this new order parameter is found to be H2. In contrast, the corresponding physical quantities in the systems with random 4-body interactions are given by [Formula: see text] and H4, respectively. Scaling forms of order parameters for regular or random 4-body interactions are expressed by the same scaling functions in the systems with regular or random 2-body interactions, respectively. Furthermore, we have obtained the nonlinear susceptibilities in the regular and random systems, where the coefficient χnl of H3 in the magnetization shows positive divergence in the regular model, while the coefficient χ7 of H7 in the magnetization shows negative divergence in the random model.
The random field and random anisotropy N -vector models are studied with the functional renormalization group in 4−ε dimensions. The random anisotropy Heisenberg ( N =3) model has a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law 〈 m ( r 1 ) m ( r 2 )〉∼¦ r 1 − r 2 ¦ − 0.62ε . The magnetic susceptibility diverges at low fields as χ∼ H −1+0.15ε . In the random field N -vector model the correlation length is finite at arbitrarily weak disorder for any N >3.
The large-distance behaviors of the random field and random anisotropy O(N)O(N)O(N) models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg (N=3)(N=3)(N=3) model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field O(N)O(N)O(N) model the correlation length is found to be finite at the arbitrarily weak disorder for any N>3.N>3.N>3. The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.
We have considered a classical spin system, consisting of n-component unit vectors (n = 2, 3), associated with a semi-infinite lattice in one dimension {uk, k ∈ N+}, and interacting via inhomogeneous pair potentials, anisotropic in spin space, and of the long-range ferromagnetic form [Formula: see text] here ∊ is a positive constant setting energy and temperature scales (i.e. T* = k B T/∊), a > 0, b ≥ 0, and the symbols uj,α denote Cartesian components of the spins. On the basis of available theoretical results, one can prove the existence of an ordering transition at a finite temperature for n ≥ 2, b = 0, as well as upper and lower bounds on the transition temperatures. We have studied the two cases n = 2, 3, b = 0, by computer simulation, and made a comparison with the Mean Field treatment.
We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by complex many-body systems such as liquids or amorphous solids, classical spin chains with certain long-range interactions have been shown to demonstrate the same phenomenon. Such spin-chain systems are ideal models for exploring the effects of quantum mechanics on hyperuniformity. It is well-known that the transverse field Ising model shows a quantum phase transition (QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform an extensive analysis of these systems using density matrix renormalization group simulations to study the possibilities of phase transitions and the mechanism by which they lose hyperuniformity. Even for a spin chain of length 30, we see discontinuous changes in properties like the ``$\ensuremath{\tau}$ order metric'' of the ground state, the measure of hyperuniformity, and the second cumulant of the total magnetization along the xxx-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of a transverse magnetic field may lead remarkably to an increase in the order of the ground state as measured by the ``$\ensuremath{\tau}$ order metric,'' even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping onto a system of spinless fermions, showing that under the pairwise-interaction approximation and a mean-field treatment, there can be at most two quantum phase transitions. Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of two quantum phase transitions at zero temperature.
For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.