Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTwo-dimensional polymer networks at a mixed boundary: Surface and wedge exponents
113
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We provide general formulae for the configurational exponents of an arbitrary polymer network connected to the surface of an arbitrary wedge of the two-dimensional plane, where the surface is allowed to assume a general mixture of boundary conditions on either side of the wedge. We report on a comprehensive study of a linear chain by exact enumeration, with various attachments of the walks ends to the surface, in wedges of angles $pi/2$ and $pi$, with general mixed boundary conditions.
rate research
Read More
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some $rho > 0$, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility $chi_{1,1}$ has been evaluated exactly. For $rho =1$ we find that $chi_{1,1}$ is finite at the bulk critical temperature $T_c$, in contrast with the recently derived value $gamma_{1,1}=1$ in the case of just one global spherical constraint. The result $gamma_{1,1}=1$ is recovered only if $rho =rho_c= 2-(12 K_c)^{-1}$, where $K_c$ is the dimensionless critical coupling. When $rho > rho_c$, $chi_{1,1}$ diverges exponentially as $Tto T_c^{+}$. An effective hamiltonian which leads to an exactly solvable model with $gamma_{1,1}=2$, the value for the $nto infty $ limit of the corresponding O(n) model, is proposed too.
Quasicritical exponents of one-dimensional models displaying a quasitransition at finite temperatures are examined in detail. The quasitransition is characterized by intense sharp peaks in physical quantities such as specific heat and magnetic susceptibility, which are reminiscent of divergences accompanying a continuous (second-order) phase transition. The question whether these robust finite peaks follow some power law around the quasicritical temperature is addressed. Although there is no actual divergence of these quantities at a quasicritical temperature, a power-law behavior fits precisely both ascending as well as descending part of the peaks in the vicinity but not too close to a quasicritical temperature. The specific values of the quasicritical exponents are rigorously calculated for a class of one-dimensional models (e.g. Ising-XYZ diamond chain, coupled spin-electron double-tetrahedral chain, Ising-XXZ two-leg ladder, and Ising-XXZ three-leg tube), whereas the same set of quasicritical exponents implies a certain `universality of quasitransitions of one-dimensional models. Specifically, the values of the quasicritical exponents for one-dimensional models are: $alpha=alpha=3$ for the specific heat, $gamma=gamma=3$ for the susceptibility and $ u= u=1$ for the correlation length.
The critical behaviour of semi-infinite $d$-dimensional systems with short-range interactions and an O(n) invariant Hamiltonian is investigated at an $m$-axial Lifshitz point with an isotropic wave-vector instability in an $m$-dimensional subspace of $mathbb{R}^d$ parallel to the surface. Continuum $|bphi|^4$ models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant $lambda$) must be included in addition to the familiar ones $proptophi^2$. Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in $d=4+frac{m}{2}-epsilon$ dimensions (with $epsilon>0$) are located at $lambda=lambda^*=Or(epsilon)$. At second order in $epsilon$, the surface critical exponents of both the ordinary and the special transitions start to deviate from their $m=0$ analogues. Results to order $epsilon^2$ are presented for the surface critical exponent $beta_1^{rm ord}$ of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (theta-1)$, where $theta= u_{l4}/ u_{l2}$ is the bulk anisotropy exponent.
We numerically investigate the influence of self-attraction on the critical behaviour of a polymer in two dimensions, by means of an analysis of finite-size results of transfer-matrix calculations. The transfer matrix is constructed on the basis of the O($n$) loop model in the limit $n to 0$. It yields finite-size results for the magnetic correlation length of systems with a cylindrical geometry. A comparison with the predictions of finite-size scaling enables us to obtain information about the phase diagram as a function of the chemical potential of the loop segments and the strength of the attractive potential. Results for the magnetic scaling dimension can be interpreted in terms of known universality classes. In particular, when the attractive potential is increased, we observe the crossover between polymer critical behaviour of the self-avoiding walk type to behaviour described earlier for the theta point.
Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives an understanding of the expressive power of RBM and DBM using entanglement structures of the tensor networks, also provides an efficient tensor network contraction algorithm for the computing partition function of RBM and DBM. Using numerical experiments, we demonstrate that the proposed algorithm is much more accurate than the state-of-the-art machine learning methods in estimating the partition function of restricted Boltzmann machines and deep Boltzmann machines, and have potential applications in training deep Boltzmann machines for general machine learning tasks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
