The spin-1 Blume–Capel model on the Bethe lattice in ± J distribution with an adjustable parameter between FM and AFM phases

The spin-1 Blume–Capel (BC) model is studied on the Bethe lattice (BL) for the ?±? J distribution with a competing adjustable parameter α which alters the strength of bilinear exchange interaction parameter for the ferromagnetic phase (J?>?0) with respect to antiferromagnetic phase (J?<?0). The J?>?0 and αJ?<?0 values are also distributed throughout the BL with probabilities p and $1?p,$ respectively. The order-parameters are obtained on the BL in terms of exact recursion relations (ERR’s) and their temperature (T) variations are studied to calculate the phase diagrams on the (α, T) planes for given values of p, crystal field (D) and coordination number $q=3$ corresponding to honeycomb lattice. It is found that the model gives both first- and second-order phase transitions and also tricritical points. In addition to the well known ordinary phases and TCP’s, the spin glass phase and two more special points are also observed.     

The single-ion anisotropy effects in the mixed-spin ternary-alloy

The effect of single-ion anisotropy on the thermal properties of the ternary-alloy in the form of $A{B}_p{C}_{1?p}$ is investigated on the Bethe lattice (BL) in terms of exact recursion relations. The simulation on the BL consists of placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2) or C (spin-5/2) atoms with concentrations p and $1?p$, respectively, on the even shells. The phase diagrams are calculated in possible planes spanned by the system parameters: temperature, single-ion anisotropy, concentration and ratio of the bilinear interaction parameters for $z=3$ corresponding to the honeycomb lattice. It is found that the crystal field drives the system to the lowest possible state therefore reducing the temperatures of the critical lines in agreement with the literature.     

Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere

Tsallis [Physica A 340 (2004) 1) identified a set of numbers, the “q-triplet” ≡ {q_stat, q_sen, q_rel}, for a system described by non-extensive statistical mechanics. The deviation of the q's from unity is a measure of the departure from thermodynamic equilibrium. We present observations of the q-triplets derived from two sets of daily averages of the magnetic field strength B observed by Voyager 1 in the solar wind near 40 A.U. during 1989 and near 85 A.U. during 2002, respectively. The results for 1989 do not differ significantly from those for 2002. We find $q_{\mathrm{stat}}=1.75\pm 0.06$, $q_{\mathrm{sen}}=-0.6\pm 0.2$, and $q_{\mathrm{rel}}=3.8\pm 0.3$.     

Multiple phase transitions in the XY model with nematic-like couplings

Critical behavior of the two-dimensional generalized XY model involving solely nematic-like terms of the second, third and fourth orders is studied by Monte Carlo method. We find that such a system can undergo three successive phase transitions. At higher temperatures there is a phase transition of the Berezinskii–Kosterlitz–Thouless type to the $q=4$ nematic-like phase, followed by two more transitions of the Ising type to the $q=2$ nematic-like and ferromagnetic phases, respectively. The q nematic-like phases are characterized by spin alignments with angles $2k\pi /q$, where $k\le q$ is an integer. The ferromagnetic phase appears at low temperatures even without the presence of magnetic interactions owing to a synergic effect of the nematic-like terms.     

Magnetocaloric properties of the spin-S (S ≥ 1) Ising model on a honeycomb lattice

Using effective field theory, we have investigated magnetocaloric features of the spin-S Ising model for some spin values $S=1,3/2,2,5/2,3$ and 7/2 on a honeycomb lattice. Effects of the external magnetic field on the isothermal entropy change, its half-width and also cooling capacity have been determined and discussed in detail. The numerical results show that the isothermal entropy change decreases whereas its half-width and cooling capacity tends to increase when the spin magnitude of the system is increased starting from $S=1$ to 7/2 for a fixed value of the external field. Finally, we have also performed some calculations to check the applicability of magnetocaloric scaling procedure for the present system. It is found that the rescaled numerical data collapse onto the same universal curve for all considered values of spin component S.     

Supersymmetric extensions of Schrödinger-invariance

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization of the Schrödinger algebra that may be extended into a representation of the conformal algebra in $d+2$ dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-$\frac{1}{2}$ Lévy-Leblond equation. An $N=2$ supersymmetric extension of these equations leads, respectively, to a ‘super-Schrödinger’ model and to the $(3|2)$-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras $\mathfrak{osp}(2|2)⋉\mathfrak{sh}(2|2)$ and $\mathfrak{osp}(2|4)$, respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schrödinger–Neveu–Schwarz algebra ${\mathfrak{sns}}^{(N)}$ with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of $\mathfrak{osp}(2|4)$. If one includes both $N=2$ supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.     

Simulating quantum systems on the Bethe lattice by translationally invariant infinite-tree tensor network

We construct an algorithm to simulate imaginary time evolution of translationally invariant spin systems with local interactions on an infinite, symmetric tree. We describe the state by symmetric infinite-Tree Tensor Network (iTTN) and use translation-invariant operators for the updates at each time step. The contraction of this tree tensor network can be computed efficiently by recursion without approximations and one can then truncate all the iTTN tensors at the same time. The translational symmetry is preserved at each time step that makes the algorithm very well conditioned and stable. The computational cost scales like $O\left(D^{q+1}\right)$ with the bond dimension $D$ and coordination number $q$, much favorable than that of the iTEBD on trees [D. Nagaj, E. Farhi, J. Goldstone, P. Shor, I. Sylvester, Phys. Rev. B 77 (2008) 214431]. Studying the transverse-field Ising model on the Bethe lattice, the numerics indicate a ferromagnetic-paramagnetic phase transition, with a finite correlation length even at the transition point.     

Classification of “Real” Bloch-bundles: Topological quantum systems of type AI

We provide a classification of type AI topological quantum systems in dimension $d=1,2,3,4$ which is based on the equivariant homotopy properties of “Real” vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via $K$-theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies “Real” line bundles. We also show that the classification of “Real” line bundles suffices for the complete classification of AI topological quantum systems in dimension $d⩽3$. In dimension $d=4$ the determination of different topological phases (for free or periodic systems) is fixed by the second “Real” Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.     

The antiferromagnetic transition for the square-lattice Potts model

We solve in this paper the problem of the antiferromagnetic transition for the Q-state Potts model (defined geometrically for Q generic using the loop/cluster expansion) on the square lattice. This solution is based on the detailed analysis of the Bethe ansatz equations (which involve staggered source terms of the type “real” and “anti-string”) and on extensive numerical diagonalization of transfer matrices. It involves subtle distinctions between the loop/cluster version of the model, and the associated RSOS and (twisted) vertex models. The essential result is that the twisted vertex model on the transition line has a continuum limit described by two bosons, one which is compact and twisted, and the other which is not, with a total central charge $c=2-\frac{6}{t}$, for $\sqrt{Q}=2\cos \frac{\pi }{t}$. The non-compact boson contributes a continuum component to the spectrum of critical exponents. For Q generic, these properties are shared by the Potts model. For Q a Beraha number, i.e., $Q=4{\cos }^2\frac{\pi }{n}$ with n integer, and in particular Q integer, the continuum limit is given by a “truncation” of the two boson theory, and coincides essentially with the critical point of parafermions $Z_{n-2}$.Moreover, the vertex model, and, for Q generic, the Potts model, exhibit a first-order critical point on the transition line—that is, the antiferromagnetic critical point is not only a point where correlations decay algebraically, but is also the locus of level crossings where the derivatives of the free energy are discontinuous. In that sense, the thermal exponent of the Potts model is generically equal to $\nu =\frac{1}{2}$. Things are however profoundly different for Q a Beraha number. In this case, the antiferromagnetic transition is second order, with the thermal exponent determined by the dimension of the ${\psi }_1$ parafermion, $\nu =\frac{t-2}{2}$. As one enters the adjacent “Berker–Kadanoff” phase, the model flows, for t odd, to a minimal model of CFT with central charge $c=1-\frac{6}{(t-1)t}$, while for t even it becomes massive. This provides a physical realization of a flow conjectured long ago by Fateev and Zamolodchikov in the context of $Z_N$ integrable perturbations.Finally, though the bulk of the paper concentrates on the square-lattice model, we present arguments and numerical evidence that the antiferromagnetic transition occurs as well on other two-dimensional lattices.