We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems with critical properties equivalent to those of the class of one-dimensional quantum systems discussed in a companion paper (J.
Hutchinson, J. P. Keating, and F. Mezzadri, arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki mapping; the method of coherent states; and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in our previous article for the classical systems identified.
We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups $U(N)$, $O(N)$ and $Sp(2N)$. In p
articular we calculate critical exponents $s$, $ u$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators $leftlangle sigma^{x}_{i} sigma^{x}_{i+n} rightrangle_{g}$, $leftlangle sigma^{y}_{i} sigma^{y}_{i+n} rightrangle_{g}$ and $leftlangle prod^{n}_{i=1} sigma^{z}_{i} rightrangle_{g}$, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.
We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and appl
ies to all the symmetry classes beta=1,2,4 of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for beta=1,4, thus proving a number of conjectures of Khoruzhenko et al. (Phys. Rev. B, Vol. 80 (2009), 125301). We derive differential equations that characterize the cumulant generating functions for all beta=1,2,4. Furthermore, we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painleve III transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit n -> infinity. Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.
We systematically study the first three terms in the asymptotic expansions of the moments of the transmission eigenvalues and proper delay times as the number of quantum channels n in the leads goes to infinity. The computations are based on the assu
mption that the Landauer-Butticker scattering matrix for chaotic ballistic cavities can be modelled by the circular ensembles of Random Matrix Theory (RMT). The starting points are the finite-n formulae that we recently discovered (Mezzadri and Simm, J. Math. Phys. 52 (2011), 103511). Our analysis includes all the symmetry classes beta=1,2,4; in addition, it applies to the transmission eigenvalues of Andreev billiards, whose symmetry classes were classified by Zirnbauer (J. Math. Phys. 37 (1996), 4986-5018) and Altland and Zirnbauer (Phys. Rev. B. 55 (1997), 1142-1161). Where applicable, our results are in complete agreement with the semiclassical theory of mesoscopic systems developed by Berkolaiko et al. (J. Phys. A.: Math. Theor. 41 (2008), 365102) and Berkolaiko and Kuipers (J. Phys. A: Math. Theor. 43 (2010), 035101 and New J. Phys. 13 (2011), 063020). Our approach also applies to the Selberg-like integrals. We calculate the first two terms in their asymptotic expansion explicitly.
By means of neutron powder diffraction, we investigated the effect of the polar Bi$^{3+}$ ion on the magnetic ordering of the Mn$^{3+}$ ions in BiMn$_3$Mn$_4$O$_{12}$, the counterpart with textit{quadruple} perovskite structure of the textit{simple}
perovskite BiMnO$_3$. The data are consistent with a textit{noncentrosymmetric} spacegroup $Im$ which contrasts the textit{centrosymmetric} one $I2/m$ previously reported for the isovalent and isomorphic compound LaMn$_3$Mn$_4$O$_{12}$, which gives evidence of a Bi$^{3+}$-induced polarization of the lattice. At low temperature, the two Mn$^{3+}$ sublattices of the $A$ and $B$ sites order antiferromagnetically (AFM) in an independent manner at 25 and 55 K, similarly to the case of LaMn$_3$Mn$_4$O$_{12}$. However, both magnetic structures of BiMn$_3$Mn$_4$O$_{12}$ radically differ from those of LaMn$_3$Mn$_4$O$_{12}$. In BiMn$_3$Mn$_4$O$_{12}$ the moments $textbf{M}_{A}$ of the $A$ sites form an anti-body AFM structure, whilst the moments textbf{M}$_{B}$ of the $B$ sites result from a large and textit{uniform} modulation $pm textbf{M}_{B,b}$ along the b-axis of the moments textbf{M}$_{B,ac}$ in the $ac$-plane. The modulation is strikingly correlated with the displacements of the Mn$^{3+}$ ions induced by the Bi$^{3+}$ ions. Our analysis unveils a strong magnetoelastic coupling between the internal strain created by the Bi$^{3+}$ ions and the moment of the Mn$^{3+}$ ions in the $B$ sites. This is ascribed to the high symmetry of the oxygen sites and to the absence of oxygen defects, two characteristics of quadruple perovskites not found in simple ones, which prevent the release of the Bi$^{3+}$-induced strain through distortions or disorder. This demonstrates the possibility of a large magnetoelectric coupling in proper ferroelectrics and suggests a novel concept of internal strain engineering for multiferroics design.
We report on the high pressure synthesis of BiMn$_7$O$_{12}$, a manganite displaying a quadruple perovskite structure. Structural characterization of single crystal samples shows a distorted and asymmetrical coordination around the Bi atom, due to pr
esence of the $6s^{2}$ lone pair, resulting in non-centrosymmetric space group Im, leading to a permanent electrical dipole moment and ferroelectric properties. On the other hand, magnetic characterization reveals antiferromagnetic transitions, in agreement with the isostructural compounds, thus evidencing two intrinsic properties that make BiMn$_7$O$_{12}$ a promising multiferroic material.
