Dimer Correlation Functions Research Articles

Thermodynamic Scaling of Interfering Hemoglobin Strain Field Waves.

Hemoglobin (Hgb) forms tetramers (dimerized α-β dimers), which enhance its globular stability and may also facilitate small gas molecule transport, as shown by recent all-atom Newtonian solvated simulations. Hydropathic bioinformatic thermodynamic scaling enables close comparisons of hemoglobin dimers with myoglobin and neuroglobin, and reveals many nonlocal wave-like features of strained Hgb structures at the coarse-grained amino acid level. The thermodynamic analysis employs two hydropathic scales, one describing abrupt first-order unfolding transitions, the other continuous second-order transitions. Small molecule exchange at hemes is a first-order process. Wave-like collective tetrameric features appropriate to ligand absorption and release, seen in optical experiments (short times), are identified thermodynamically at long times. Strain fields localized near hemes interfere with extended strain fields associated with dimer interfacial misfit, resulting in novel wavelength dependent dimer correlation function Fano antiresonances.

Dimer correlation amplitudes and dimer excitation gap in spin- 12 XXZ and Heisenberg chains

Correlation functions of dimer operators, the product operators of spins on two adjacent sites, are studied in the spin-$\frac{1}{2}$ XXZ chain in the critical regime. The amplitudes of the leading oscillating terms in the dimer correlation functions are determined with high accuracy as functions of the exchange anisotropy parameter and the external magnetic field, through the combined use of bosonization and density-matrix renormalization group methods. In particular, for the antiferromagnetic Heisenberg model with SU(2) symmetry, logarithmic corrections to the dimer correlations due to the marginally-irrelevant operator are studied, and the asymptotic form of the dimer correlation function is obtained. The asymptotic form of the spin-Peierls excitation gap including logarithmic corrections is also derived.

Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics

In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model via the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predicts particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms.

Resonating valence-bond physics on the honeycomb lattice

We study bond and spin correlations of the nearest-neighbour resonating valence bond (RVB) wavefunction for a SU($2$) symmetric $S=1/2$ antiferromagnet on the honeycomb lattice. We find that spin correlations in this wavefunction are short-ranged, while the bond energy correlation function takes on an oscillatory power-law form $D(\vec{r}) \sim \cos({\mathbf Q}\cdot {\vec{r}}) /|{\vec{r}}|^{\eta_w(2)}$, where ${\mathbf Q} = (2\pi/3, -2\pi/3)$ is the wavevector corresponding to "columnar" valence-bond solid order on the honeycomb lattice, and $\eta_w(2) \approx 1.49(3)$. We use a recently introduced large-$g$ expansion approach to relate bond-energy correlators of the SU($g$) wavefunction to dimer correlations of an interacting fully-packed dimer model with a three-dimer interaction of strength $V(g)=-\log(1+1/g^2)$. Putting $g=2$, we find numerically that the dimer correlation function $D^{d}(\vec{r})$ of this dimer model has power-law behaviour $D^{d}(\vec{r}) \sim \cos({\mathbf Q}\cdot {\vec{r}}) /|{\vec{r}}|^{\eta_d(2)}$ with $\eta_d(2) \approx 1.520(15)$, in rather good agreement with the wavefunction results. We also study the same quantities for $g=3,4,10$ and find that the bond-energy correlations in the SU($g$) wavefunction are consistently well-reproduced by the corresponding dimer correlations in the interacting dimer model.

Percolation of interacting classical dimers on the square lattice

We study the percolation properties of the interacting classical dimer model on the square lattice by means of Monte Carlo simulations and finite-size scaling analysis. We define Ising clusters based on the dimer configuration; the percolation point of the clusters coincides with the critical point of the Kosterlitz–Thouless transition of the dimer model, which is Tc=0.654(2). Furthermore, we find that the largest cluster at the Kosterlitz–Thouless point is a fractal, with fractal dimension Dc=1.874(2), which coincides with the critical exponent describing the critical behavior of the dimer–dimer correlation function, which is theoretically predicted to be 15/8.

Constructing a Gapless Spin-Liquid State for the Spin-1/2J1−J2Heisenberg Model on a Square Lattice

We construct a class of projected entangled pair states which is exactly the resonating valence bond wave functions endowed with both short range and long range valence bonds. With an energetically preferred resonating valence bond pattern, the wave function is simplified to live in a one-parameter variational space. We tune this variational parameter to minimize the energy for the frustrated spin-1/2 J(1)-J(2) antiferromagnetic Heisenberg model on the square lattice. Taking a cylindrical geometry, we are able to construct four topological sectors with an even or odd number of fluxes penetrating the cylinder and an even or odd number of spinons on the boundary. The energy splitting in different topological sectors is exponentially small with the cylinder perimeter. We find a power law decay of the dimer correlation function on a torus, and a lnL correction to the entanglement entropy, indicating a gapless spin-liquid phase at the optimum parameter.

Low-energy effective theory and two distinct critical phases in a spin-12frustrated three-leg spin tube

Motivated by the crystal structures of [(CuCl${}_{2}$tachH)${}_{3}$Cl]Cl${}_{2}$ and Ca${}_{3}$Co${}_{2}$O${}_{6}$, we develop a low-energy effective theory using the bosonization technique for a spin-$\frac{1}{2}$ frustrated three-leg spin tube with trigonal prism units in two limit cases. The features obtained with the effective theory are numerically elucidated by the density matrix renormalization group method. Three different quantum phases in the ground state of the system, say, one gapped dimerized phase and two distinct gapless phases, are identified, where the two gapless phases are found to have the conformal central charge $c=1$ and $\frac{3}{2}$, respectively. Spin gaps, spin and dimer correlation functions, and the entanglement entropy are obtained. In particular, it is disclosed that the critical phase with $c=\frac{3}{2}$ is the consequence of spin frustrations, which might belong to the SU(2)${}_{k=2}$ Wess-Zumino-Witten-Novikov universality class, and is induced by the twist term in the bosonized Hamiltonian density.

Nature of the Spin-Liquid Ground State of theS=1/2Heisenberg Model on the Kagome Lattice

We perform a density-matrix renormalization group (DMRG) study of the S=1/2 Heisenberg antiferromagnet on the kagome lattice to identify the conjectured spin liquid ground state. Exploiting SU(2) spin symmetry, which allows us to keep up to 16,000 DMRG states, we consider cylinders with circumferences up to 17 lattice spacings and find a spin liquid ground state with an estimated per site energy of -0.4386(5), a spin gap of 0.13(1), very short-range decay in spin, dimer and chiral correlation functions, and finite topological entanglement γ consistent with γ=log(2)2, ruling out gapless, chiral, or nontopological spin liquids in favor of a topological spin liquid of quantum dimension 2, with strong evidence for a gapped topological Z(2) spin liquid.

Dimer and fermionic formulations of a class of colouring problems

We show that the number Z of q-edge-colourings of a simple regular graph of degree q is deducible from functions describing dimers on the same graph, namely the dimer generating function or equivalently the set of connected dimer correlation functions. Using this relationship to the dimer problem, we derive fermionic representations for Z in terms of Grassmann integrals with quartic actions. Expressions are given for planar graphs and nonplanar graphs embeddable (without edge crossings) on a torus. We discuss exact numerical evaluations of the Grassmann integrals using an algorithm by Creutz and present an application to the 4-edge-colouring problem on toroidal square lattices, comparing the results with numerical transfer matrix calculations and a previous Bethe ansatz study.

Dimerized ferromagnetic Heisenberg chain

Ferromagnetic, in contrast to antiferromagnetic, Heisenberg chains can undergo a spin-Peierls dimerization only at finite temperatures. They show reentrant behavior as a function of temperature, which might play a role for systems with small effective elastic constants as, for example, monatomic chains on surfaces. We investigate the physical properties of the dimerized ferromagnetic Heisenberg chain using a modified spin-wave theory. We calculate the exponentially decaying spin- and dimer-correlation functions, analyze the temperature dependence of the corresponding coherence lengths, the susceptibility, as well as the static and dynamic spin-structure factors. By comparing with numerical data obtained by the density-matrix renormalization group applied to transfer matrices, we find that the modified spin-wave theory yields excellent results for all these quantities for a wide range of dimerizations and temperatures.

Dynamical dimer correlations at bipartite and non-bipartite Rokhsar–Kivelsonpoints

We determine the dynamical dimer correlation functions of quantum dimer models at theRokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartitetriangular lattice. On the basis of an algorithmic idea by Henley, we simulate astochastic process of classical dimer configurations in continuous time and perform astochastic analytical continuation to obtain the dynamical correlations in momentumspace and the frequency domain. This approach allows us to observe directlythe dispersion relations and the evolution of the spectral intensity within theBrillouin zone beyond the single-mode approximation. On the square lattice,we confirm analytical predictions related to soft modes close to the wavevectors(π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector(0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at(π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadraticdispersion at very long wavelength, but crosses over to a linear behavior very rapidly. Webelieve this to be the remnant of the linearly dispersing ‘photon’ of the Coulomb phase.Finally the triangular lattice is in a fully gapped liquid phase where the bottom of thedimer spectrum exhibits a rich structure. At the M point the gap is minimal and thespectral response is dominated by a sharp quasiparticle peak. On the other hand, at the Xpoint the spectral function is much broader. We sketch a possible explanation basedon the crossing of the coherent dimer excitations into the two-vison continuum.

SU(N) quantum spin models: a variational wavefunction study

SU(N) quantum spin systems may be realized in a variety of physical systems includingultracold atoms in optical lattices. The study of such models also leadsto insights into possible novel quantum phases and phase transitions ofSU(2) spin models. Here we use Gutzwiller projected fermionic variationalwavefunctions to explore the phase diagram and correlation functions ofSU(N) quantum spin models in the self-conjugate representation. In one dimension, the ground state of theSU(4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examininginstabilities of the Gutzwiller projected free fermion ground state to various broken symmetries.The variational phase diagram so obtained agrees well with exact results. The spin–spinand dimer–dimer correlation functions of the Gutzwiller projected free fermion state withN flavours of fermions are in good agreement with exact and1/N calculations for thecritical points of SU(N) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenbergmodel on the square lattice is obtained by finding instabilities of the Gutzwiller projectedπ-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order forN = 2 and 4, and spin Peierls (columnar dimer) order forN>4. Upon including biquadratic interactions in theSU(4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes andeventually disappears, giving way to an extended valence bond crystal. In the case of theSU(6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projectedπ-flux spin liquid phase which in turn undergoes a transition into an extended valence bondcrystal at even larger biquadratic interaction. The spin correlations of the projectedπ-flux state atN = 4 are in goodagreement with 1/N calculations. We find that the state shows strongly enhanced dimer correlations,in qualitative agreement with recent theoretical predictions. We alsocompare our results with a recent quantum Monte Carlo study of theSU(4) Heisenberg model.

Exact evaluation of density matrix elements for the Heisenberg chain

We have obtained all the density matrix elements on six lattice sites for the spin-1/2Heisenberg chain via the algebraic method based on the quantum Knizhnik–Zamolodchikovequations. Several interesting correlation functions, such as chiral correlation functions,dimer–dimer correlation functions, etc, have been analytically evaluated. Furthermore, wehave calculated all the eigenvalues of the density matrix and analyse the eigenvaluedistribution. As a result, the exact von Neumann entropy for the reduced density matrix onsix lattice sites has been obtained.

Absence of dimer order and the Tomonaga–Luttinger liquid properties of the quarter-filled one-dimensional Kondo-lattice model

A novel spin dimer order is recently reported in the ground state of the one-dimensional Kondo lattice model at quarter-filling. Being dubious about such long-range order in the absence of a spin gap, we perform the density matrix renormalization group calculation at under several boundary conditions. It turns out that dimer spin structure is sensitive to the boundary conditions, and that the dimer correlation function decays in small power law at long distances, indicating the absence of dimer long-range order. This together with a possible absence of charge gap suggests that the true ground state remains to be a Tomonaga–Luttinger liquid with small K ρ .

Suppression of dimer correlations in the two-dimensionalJ1−J2Heisenberg model: An exact diagonalization study

We present an exact diagonalization study of the ground state of the spin-half $J_1{-}J_2$ model. Dimer correlation functions and the susceptibility associated to the breaking of the translational invariance are calculated for the $4\times 4$ and the $6\times 6$ clusters. These results -- especially when compared to the one dimensional case, where the occurrence of a dimerized phase for large enough frustration is well established -- suggest either a homogeneous spin liquid or, possibly, a dimerized state with a rather small order parameter.

Decoupling of theS=1/2antiferromagnetic zig-zag ladder with anisotropy

The spin-1/2 antiferromagnetic zig-zag ladder is studied by exact diagonalization of small systems in the regime of weak inter-chain coupling. A gapless phase with quasi long-range spiral correlations has been predicted to occur in this regime if easy-plane (XY) anisotropy is present. We find in general that the finite zig-zag ladder shows three phases: a gapless collinear phase, a dimer phase and a spiral phase. We study the level crossings of the spectrum,the dimer correlation function, the structure factor and the spin stiffness within these phases, as well as at the transition points. As the inter-chain coupling decreases we observe a transition in the anisotropic XY case from a phase with a gap to a gapless phase that is best described by two decoupled antiferromagnetic chains. The isotropic and the anisotropic XY cases are found to be qualitatively the same, however, in the regime of weak inter-chain coupling for the small systems studied here. We attribute this to a finite-size effect in the isotropic zig-zag case that results from exponentially diverging antiferromagnetic correlations in the weak-coupling limit.

Spin gap in low-dimensional Mott insulators with orbital degeneracy

We consider the exchanged Hamiltonian HST=−J∑〈rr′〉(2Sr⋅Sr′−12)(2Tr⋅Tr′−12), describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on equivalent bonds. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit. To investigate the ground state properties we use a Green’s Function Monte Carlo, calculating energy gaps and correlation functions, the latter through the forward walking technique. In one dimension we find that the ground state is a “crystal” of valence bond dimers. In two dimensions, the spin gap appears to remain finite in the thermodynamic limit, and, consistently, the staggered magnetization—signal of Néel long range order—seems to vanish. From the analysis of dimer–dimer correlation functions, however, we find no sign of a valance bond crystal. A spin liquid appears as a plausible scenario compatible with our findings.

Ground-State Properties of the One-Dimensional Isotropic Spin-1/2 Heisenberg Antiferromagnet with Competing Interactions

The ground-state properties of the one-dimensional isotropic spin-1/2 Heisenberg antiferromagnet with antiferromagnetic first- and second-neighbor exchange interactions are discussed. The ground-state energy, the singlet-triplet energy gap, the Néel correlation function, the dimer correlation function, and the magnetization are calculated exactly for finite-size systems up to 20 spins. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. In particular, it is found that the limiting ground state is in the gapless or finite-gap phase depending on whether 0≦ j ≦ j c or j > j c , where j is the ratio of the second- to the first-neighbor interaction constant and j c =0.30±0.01. Dimer long-range order does exist in the finite-gap phase and it does not in the gapless phase, while no Néel long-range order exists in both phases.