Three-color Model Research Articles

A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials p<sub>n</sub>

By specializing the parameters in the partition function of the 8VSOS model with domain wall boundary conditions and diagonal reflecting end, we find connections between the three-color model and certain polynomials $p_n(z)$, which are conjectured to be equal to certain polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the Hamiltonian of the supersymmetric XYZ spin chain. This article is a continuation of a previous paper where we investigated the related polynomials $q_n(z)$, also conjectured to be equal to polynomials of Bazhanov and Mangazeev, appearing in the eigenvectors of the supersymmetric XYZ spin chain.

Frustration-induced emergent Hilbert space fragmentation

Although most quantum systems thermalize locally on short time scales independent of initial conditions, recent developments have shown this is not always the case. Lattice geometry and quantum mechanics can conspire to produce constrained quantum dynamics and associated glassy behavior, a phenomenon that falls outside the rubric of the eigenstate thermalization hypothesis. Constraints "fragment" the many-body Hilbert space due to which some states remain insulated from others and the system fails to attain thermal equilibrium. Such fragmentation is a hallmark of geometrically frustrated magnets with low-energy "icelike manifolds" exhibiting a broad range of relaxation times for different initial states. Focusing on the highly frustrated kagome lattice, we demonstrate these phenomena in the Balents-Fisher-Girvin Hamiltonian (easy-axis regime), and a three-coloring model (easy-plane regime), both with constrained Hilbert spaces. We study their level statistics and relaxation dynamics to develop a coherent picture of fragmentation in various limits of the XXZ model on the kagome lattice.

A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain

We study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for $q_n(z)$ in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$ have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.

Abstract 438: The CFP nude mouse as a host for color-coded imaging of the tumor microenvironment

Abstract We have developed a color-coded model to study the tumor microenvironment using cancer cells expressing RFP or GFP implanted in nude mice expressing cyan fluorescent protein (CFP). When RFP or GFP HCT-116 were injected subcutaneously in CFP expressing nude mice, CFP expressing cancer-associated fibroblasts migrated to the tumor site. CFP-expressing tumor blood vessels were observed in the tumor. Other CFP host stromal cells were observed throughout the tumor mass. When RFP HCT-116 cells were transplanted in the spleen of CFP nude mice, they formed tumors in the liver. When RFP HCT-116 cancer cells were transplanted in the tail vein of CFP nude mice and they formed tumors in the lung. CFP-expressing host cells were observed in these tumors. When the RFP mouse mammary cancer cell line (RFP MMT) was injected subcutaneously in the CFP nude mouse, numerous CFP-expressing fibroblasts and CFP-expressing blood vessels was observed in the resulting tumor. A large inflammatory population of different CFP-expressing white blood cells surrounded the necrotic area of the tumor suggesting a relationship of these cells to tumor necrosis. GFP and RFP HCT-116 cancer cells were co-injected subcutaneously in CFP-nude mice and tumor growth was followed for 21 days. This three-color model enabled visualization of the tumor mass and its neovasculature in addition to a detailed visualization of other host reactions. GFP HCT-116 cancer cells were initially injected subcutaneously in RFP-nude mice and tumors formed within 14 days. At this point, the tumor mass which consisted of GFP cancer cells and RFP stroma derived from the RFP nude mouse, was harvested and transplanted to a CFP-nude mouse. CFP host cells invaded the growing transplanted tumor containing dual-color cancer cells and RFP stroma. Thus the CFP-nude mouse adds another color to the pallet to study the human tumor microenvironment allowing 4 types of cells to be simultaneously imaged in tumors. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr 438. doi:10.1158/1538-7445.AM2011-438

Gauge symmetry and non-Abelian topological sectors in a geometrically constrained model on the honeycomb lattice

We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions. They are (i) the Ising model on the honeycomb lattice, constrained to have three up spins and three down spins on every hexagon, (ii) the three-color and fully packed loop model on the links of the honeycomb lattice, with loops around a single hexagon forbidden, and (iii) three Ising models on interleaved triangular lattices, with domain walls of the different Ising models not allowed to cross. Unlike the three-color model, the configuration space on the sphere or plane is connected under local moves. On higher-genus surfaces there are infinitely many dynamical sectors, labeled by a noncontractible set of nonintersecting loops. We demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice. This enables us to diagonalize the original transfer matrix for up to 36 sites, finding an entropy per plaquette S/k{B} approximately 0.3661 ... centered and substantial evidence that the model is not critical. We also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of Fibonacci numbers. We comment on the possibility of a topological phase, with infinite topological degeneracy, in an associated two-dimensional quantum model.

Local-structural diversity and protein folding: Application to all-β off-lattice protein models

Global measures of structural diversity within a distribution of biopolymers, such as the radius of gyration and percent native contacts, have proven useful in the analysis of simulation data for protein folding. In this paper we describe a statistical-based methodology to quantify the local structural variability of a distribution of biopolymers, applied to 46- and 69-"residue" off-lattice, three-color model proteins. Each folds into beta-barrel structures. First we perform a principal component analysis of all interbead distance variables for a large number of independent, converged Boltzmann-distributed samples of conformations collected at each of a wide range of temperatures. Next, the principal component vectors are subjected to orthogonal (varimax) rotation. The results are displayed on so-called "squared-loading" plots. These provide a quantitative measure of the contribution to the sample variance of the position of each residue relative to the others. Dominant structural elements, those having the largest structural diversity within the sampled distribution, are responsible for peaks and shoulders observed in the specific heat versus temperature curves, generated using the weighted histogram analysis method. The loading plots indicate that the local-structural diversity of these systems changes gradually with temperature through the folding transition but radically changes near the collapse transition temperature. The analysis of the structural overlap order statistic suggests that the 46-mer thermodynamic folding transition involves the native state and at least three other nearly native intermediates. In the case of the 46-mer protein model, data are generated at sufficiently low temperatures that squared-loading plots, coupled with cluster analysis, provide a local and energetic description of its glassy state.

Quantum three-coloring dimer model and the disruptive effect of quantum glassiness on its line of critical points

We construct a quantum extension of the (classical) three-coloring model introduced by Baxter [J. Math. Phys. 11, 784 (1970)] for which the ground state can be computed exactly along a continuous line of Rokhsar-Kivelson solvable points. The quantum model, which admits a local spin representation, displays at least three different phases; an antiferromagnetic phase, a line of quantum critical points, and a ferromagnetic phase. We argue that, in the ferromagnetic phase, the system cannot reach dynamically the quantum ground state when coupled to a bath through local interactions, and thus lingers in a state of quantum glassiness.

Geometric criticality for transitions between plaquette phases in integer-spin kagome XXZ antiferromagnets

The phase diagram of the uniaxially anisotropic $s=1$ antiferromagnet on the kagom\'e lattice includes a critical line exactly described by the classical three-color model. This line is distinct from the standard geometric classical criticality that appears in the classical limit ($s \to \infty$) of the 2D XY model; the $s=1$ geometric T=0 critical line separates two unconventional plaquette-ordered phases that survive to nonzero temperature. The experimentally important correlations at finite temperature and the nature of the transitions into these ordered phases are obtained using the mapping to the three-color model and a combination of perturbation theory and a variational ansatz for the ordered phases. The ordered phases show sixfold symmetry breaking and are similar to phases proposed for the honeycomb lattice dimer model and $s=1/2$ $XXZ$ model. The same mapping and phase transition can be realized also for integer spins $s \geq 2$ but then require strong on-site anisotropy in the Hamiltonian.

From quantum mechanics to classical statistical physics: Generalized Rokhsar–Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar–Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.

Dynamical obstruction in a constrained system and its realization in lattices of superconducting devices

Hard constraints imposed in statistical mechanics models can lead to interesting thermodynamical behaviors, but may at the same time raise obstructions in the thoroughfare to thermal equilibration. Here we study a variant of Baxter's three-color model in which local interactions and defects are included, and discuss its connection to triangular arrays of Josephson junctions of superconductors with broken time-reversal symmetry and kagom\'e networks of superconducting wires. The model is equivalent to an Ising model in a hexagonal lattice with the additional constraint that the magnetization of each hexagon is $\ifmmode\pm\else\textpm\fi{}6$ or $0.$ Defects in the superconducting models correspond to violations of this constraint, and include fractional and integer vortices, as well as open strings within two-color loops. In the absence of defects, and for ferromagnetic interactions, we find that the system is critical for a range of temperatures (critical line) that terminates when it undergoes an exotic first-order phase transition with a jump from a zero magnetization state into the fully magnetized state at finite temperature. Dynamically, however, we find that the system becomes frozen into domains. The domain walls are made of perfectly straight segments, and domain growth appears frozen within the time scales studied with Monte Carlo simulations, with the system trapped into a ``polycrystalline'' phase. This dynamical obstruction has its origin in the topology of the allowed reconfigurations in phase space, which consist of updates of closed loops of spins. Only an extreme rare-event dominated proliferation of confined defects may overcome this obstruction, at much longer time scales. Also as a consequence of the dynamical obstruction, there exists a dynamical temperature, lower than the (avoided) static critical temperature, at which the system is seen to jump from a ``supercooled liquid'' to the polycrystalline phase within our Monte Carlo time scale. In contrast, for antiferromagnetic interactions, we argue that the system orders for infinitesimal coupling because of the constraint, and we observe no interesting dynamical effects.

Geometric effects onT-breaking inp+ipandd+idsuperconducting arrays

Superconducting order parameters that change phase around the Fermi surface modify Josephson tunneling behavior, as in the phase-sensitive measurements that confirmed $d$ order in the cuprates. This paper studies Josephson coupling when the individual grains break time-reversal symmetry; the specific cases considered are $p \pm ip$ and $d \pm id$, which may appear in Sr$_2$RuO$_4$ and Na$_x$CoO$_2 \cdot $(H$_2$O)$_y$ respectively. $T$-breaking order parameters lead to frustrating phases when not all grains have the same sign of time-reversal symmetry breaking, and the effects of these frustrating phases depend sensitively on geometry for 2D arrays of coupled grains. These systems can show perfect superconducting order with or without macroscopic $T$-breaking. The honeycomb lattice of superconducting grains has a superconducting phase with no spontaneous breaking of $T$ but instead power-law correlations. The superconducting transition in this case is driven by binding of fractional vortices, and the zero-temperature criticality realizes a generalization of Baxter's three-color model.

Simulation of mutation: Influence of a “side group” on global minimum structure and dynamics of a protein model

The 46-bead, three-color model of a β-barrel-forming protein is modified by the addition of a single side group, represented by a bead which may be hydrophilic or hydrophobic. Molecular dynamics and quenching simulations show how the nature and location of the bead influence both the structure at the global minimum of internal energy and the relaxation processes by which the system finds its minima. The most drastic effects occur with a hydrophobic side group in the middle of a sequence of hydrophobes.

Kac-Moody symmetries of critical ground states

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su(2) k=1 , su(3) k=1 , and the su(4) k=1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.

Radiative Decays ofηandη′as Probes of Quark Charges

Radiative decays of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ allow us to test the color hypothesis and to distinguish between fractional- and integral-charge quark models. Accurate measurements of ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\ensuremath{\gamma}$ are essential to test the analysis and the preliminary conclusion, which favors the three-color model with fractional charges.