Strong-coupling Series Research Articles

Universal large-order asymptotic behavior of the strong-coupling and high-temperature series expansions

For theories that exhibit second-order phase transition, we conjecture that the large-order asymptotic behavior of the strong-coupling (high-temperature) series expansion takes the form ${\ensuremath{\sigma}}^{n}{n}^{b}$ where $b$ is a universal parameter. The associated critical exponent is then given by $b+1$. The series itself can be approximated by the hypergeometric approximants ${_{p}F}_{p\ensuremath{-}1}$ which can mimic the same large-order behavior of the given series. Near the tip of the branch cut, the hypergeometric function ${_{p}F}_{p\ensuremath{-}1}$ has a power-law behavior from which the critical exponent and critical coupling can be extracted. We test the conjecture in this work for the perturbation series of the ground state energy of the Yang-Lee model as a strong-coupling form of the $\mathcal{P}\mathcal{T}$-symmetric $i{\ensuremath{\phi}}^{3}$ theory and the high-temperature expansion within the Ising model. From the known $b$ parameter for the Yang-Lee model, we obtain the exact critical exponents, which reflects the universality of $b$. Very accurate prediction for $b$ has been obtained from the many orders available for the high-temperature series expansion of the Ising model, which in turn predicts accurate critical exponents. Apart from critical exponents, the hypergeometric approximants for the Yang-Lee model show almost exact predictions for the ground state energy from low orders of perturbation series as input.

Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb-Liniger and Yang-Gaudin models

In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb-Liniger Bose gas and the Yang-Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders

Abelian color cycles: A new approach to strong coupling expansion and dual representations for non-abelian lattice gauge theory

We propose a new approach to strong coupling series and dual representations for non-abelian lattice gauge theories using the SU(2) case as an example. The Wilson gauge action is written as a sum over “abelian color cycles” (ACC) which correspond to loops in color space around plaquettes. The ACCs are complex numbers which can be commuted freely such that the strong coupling series and the dual representation can be obtained as in the abelian case. Using a suitable representation of the SU(2) gauge variables we integrate out all original gauge links and identify the constraints for the dual variables in the SU(2) case. We show that the construction can be generalized to the case of SU(2) gauge fields with staggered fermions. The result is a strong coupling series where all gauge integrals are known in closed form and we discuss its applicability for possible dual simulations. The abelian color cycle concept can be generalized to other non-abelian gauge groups such as SU(3).

Strong coupling results in the AdS 5 /CF T 4 correspondence from the numerical solution of the quantum spectral curve

In this paper, we solved numerically the Quantum Spectral Curve (QSC) equations corresponding to some twist-2 single trace operators with even spin from the $sl(2)$ sector of $AdS_5/CFT_4$ correspondence. We describe all technical details of the numerical method which are necessary to implement it in C++ language. In the $S=2,4,6,8$ cases, our numerical results confirm the analytical results, known in the literature for the first 4 coefficients of the strong coupling expansion for the anomalous dimensions of twist-2 operators. In the case of the Konishi operator, due to the high precision of the numerical data we could give numerical predictions to the values of two further coefficients, as well. The strong coupling behaviour of the coefficients $c_{a,n}$ in the power series representation of the ${\bf P}_{\!a}$-functions is also investigated. Based on our numerical data, in the regime, where the index of the coefficients is much smaller than $\lambda^{1/4}$, we conjecture that the coefficients have polynomial index dependence at strong coupling. This allows one to propose a strong coupling series representation for the ${\bf P}$-functions being valid far enough from the real short cut. In the paper the qualitative strong coupling behaviour of the ${\bf P}$-functions at the branch points is also discussed.

Numerical corrections to the strong coupling effective Polyakov-line action for finite T Yang-Mills theory

We consider a three-dimensional eective theory of Polyakov lines derived previously from lattice Yang-Mills theory and QCD by means of a resummed strong coup- ling expansion. The eective theory is useful for investigations of the phase structure, with a sign problem mild enough to allow simulations also at finite density. In this work we present a numerical method to determine improved values for the eective couplings directly from correlators of 4d Yang-Mills theory. For values of the gauge coupling up to the vicinity of the phase transition, the dominant short range eective coupling are well described by their corresponding strong coupling series. We provide numerical results also for the longer range interactions, Polyakov lines in higher representations as well as four- point interactions, and discuss the growing significance of non-local contributions as the lattice gets finer. Within this approach the critical Yang-Mills coupling c is reproduced to better than one percent from a one-coupling eective theory on N = 4 lattices while up to five couplings are needed on N = 8 for the same accuracy.

Why are all dualities conformal? Theory and practical consequences

We relate duality mappings to the “Babbage equation” F(F(z))=z, with F a map linking weak- to strong-coupling theories and demonstrate that, under fairly general conditions, F may only be a specific conformal transformation of the fractional linear type. This general result has enormous practical consequences. For example, one can establish that weak- and strong-coupling series expansions of arbitrarily large finite size systems are trivially related, i.e., after generating one of those series the other is automatically determined through a set of linear constraints between the series coefficients. This latter relation partially solves or, equivalently, localizes the computational complexity of evaluating the series expansion to a simple fraction of those coefficients. As a bonus, those relations also encode non-trivial equalities between different geometric constructions in general dimensions, and connect derived coefficients to polytope volumes. We illustrate our findings by examining various models including, but not limited to, ferromagnetic and spin-glass Ising, and Ising gauge type theories on hypercubic lattices in 1<D<9 dimensions.

Electronic spectral properties of the two-dimensional infinite-UHubbard model

A strong-coupling series expansion for the Green's function and the extremely-correlated Fermi liquid (ECFL) theory are used to calculate the moments of the electronic spectral functions of the infinite-U Hubbard model. Results from these two complementary methods agree very well at both, low densities, where the ECFL solution is the most accurate, and at high to intermediate temperatures, where the series converge. We find that a modified first moment, which underestimates the contributions from the occupied states and is accessible in the series through the time-dependent Green's function, best describes the quasiparticle peak location in the strongly-correlated regime. This is examined by the ECFL results at low temperatures, where it is shown that the spectral function is largely skewed towards the occupied states.

Erratum: centre symmetric 3d effective actions for thermal SU(N) Yang-Mills from strong coupling series

Erratum: centre symmetric 3d effective actions for thermal SU(N) Yang-Mills from strong coupling series

Centre symmetric 3d effective actions for thermal SU(N) Yang-Mills from strong coupling series

We derive three-dimensional, Z(N)-symmetric effective actions in terms of Polyakov loops by means of strong coupling expansions, starting from thermal SU(N) Yang-Mills theory in four dimensions on the lattice. An earlier action in the literature, corresponding to the (spatial) strong coupling limit, is thus extended by several higher orders, as well as by additional interaction terms. We provide analytic mappings between the couplings of the effective theory and the parameters $N_\tau,\beta$ of the original thermal lattice theory, which can be systematically improved. We then investigate the deconfinement transition for the cases SU(2) and SU(3) by means of Monte Carlo simulations of the effective theory. Our effective models correctly reproduce second order 3d Ising and first order phase transitions, respectively. Furthermore, we calculate the critical couplings $\beta_c(N_\tau)$ and find agreement with results from simulations of the 4d theory at the few percent level for $N_\tau=4-16$.

Strong Coupling Series for QCD at Finite Temperature and Density

We discuss the use of strong coupling expansions for Yang-Mills theory and QCD at finite temperature and density. In particular we consider the onset of temperature effects for the free energy and screening masses, derive the hadron resonance gas model from first principles and compute the weakening of the deconfinement transition with chemical potential.

The deconfinement transition of finite density QCD with heavy quarks from strong coupling series

Starting from Wilson's action, we calculate strong coupling series for the Polyakov loop susceptibility in lattice gauge theories for various small N_\tau in the thermodynamic limit. Analysing the series with Pad\'e approximants, we estimate critical couplings and exponents for the deconfinement phase transition. For SU(2) pure gauge theory our results agree with those from Monte-Carlo simulations within errors, which for the coarser N_\tau=1,2 lattices are at the percent level. For QCD we include dynamical fermions via a hopping parameter expansion. On a N_\tau=1 lattice with N_f=1,2,3, we locate the second order critical point where the deconfinement transition turns into a crossover. We furthermore determine the behaviour of the critical parameters with finite chemical potential and find the first order region to shrink with growing \mu. Our series moreover correctly reflects the known Z(N) transition at imaginary chemical potential.

Test of the Anti–de Sitter-Space/Conformal-Field-Theory Correspondence Using High-Spin Operators

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resumes the perturbative series. In a confirmation of the anti-de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.

Spin nematics correlations in bilinear-biquadraticS=1spin chains

We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as Exact Diagonalization, Density Matrix Renormalization Group and strong coupling series expansions. For the single chain we clarify the dominant correlation function in the enigmatic gapless period-three phase for theta in (pi/4,pi/2), which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for theta in (-3pi,4,-pi/2) and -- suprisingly -- seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg S=1 ladder and therefore also with the Haldane phase of isolated chains.

LINKED CLUSTER SERIES EXPANSIONS FOR TWO-PARTICLE STATES IN QUANTUM LATTICE MODELS

We have developed strong-coupling series expansion methods to study the two-particle spectra in quantum lattice models. The properties of bound states and multiparticle excitations can reveal important information about the dynamics of a given model. At the heart of this method lies the calculation of an effective Hamiltonian in the two-particle subspace. We use an orthogonal transformation to perform this block diagonalising, and find that maintaining orthogonality is crucial for cases where the ground state and the two-particle subspace have identical quantum numbers. The two-particle Schrödinger equation is solved by using a finite lattice approach in coordinate space or an integral equation in momentum space. These methods allow us to determine precisely the low-lying excitation spectra and dispersion relations for the two-particle bound states. The method has been tested for the (1+1) D transverse Ising model, and applied to the two-leg spin-1/2 Heisenberg ladder. We study the coherence lengths of the bound states, and how they merge with the two-particle continuum. Finally, these techniques are applied to the frustrated alternating Heisenberg chain, which has been of considerable recent interest due to its relevance to spin-Peierls systems such as CuGeO 3. Starting from a limit corresponding to weakly-coupled dimers, we develop high-order series expansions for the effective Hamiltonian in the two-particle subspace. In the regime of strong dimerisation, various properties of the singlet and triplet bound states, and the quintet antibound states, can be accurately calculated. We also study the behaviour as the external bond alternation vanishes, and the way in which the bound states of triplet dimer excitations make the transition to a soliton-antisoliton continuum.

Application of Feynman-Kleinert approximants to the massive Schwinger model on a lattice

A trial application of the method of Feynman-Kleinert approximants is made to perturbation series arising in connection with the lattice Schwinger model. In extrapolating the lattice strong-coupling series to the weak-coupling continuum limit, the approximants do not converge well. In interpolating between the continuum perturbation series at large fermion mass and small fermion mass, however, the approximants do give good results. In the course of the calculations, we picked up and rectified an error in an earlier derivation of the continuum series coefficients.

Boundary-layer theory, strong-coupling series, and large-order behavior

The introduction of a lattice converts a singular boundary-layer problem in the continuum into a regular perturbation problem. However, the continuum limit of the discrete problem is extremely nontrivial and is not completely understood. This article examines two singular boundary-layer problems taken from mathematical physics, the instanton problem and the Blasius equation, and in each case examines two strategies, Padé resummation and variational perturbation theory, to recover the solution to the continuum problem from the solution to the associated discrete problem. Both resummation procedures produce good and interesting results for the two cases, but the results still deviate from the exact solutions. To understand the discrepancy a comprehensive large-order behavior analysis of the strong-coupling lattice expansions for each of the two problems is done.

Linked cluster series expansions for two-particle bound states

We develop strong-coupling series expansion methods to study two-particle spectra of quantum lattice models. At the heart of the method lies the calculation of an effective Hamiltonian in the two-particle subspace. We explicitly consider an orthogonality transformation to generate this block diagonalization, and find that maintaining orthogonality is crucial for systems where the ground state and the two-particle subspace are characterized by identical quantum numbers. We discuss the solution of the two-particle Schrodinger equation by using a finite lattice approach in coordinate space or by an integral equation in momentum space. These methods allow us to precisely determine the low-lying excitation spectra of the models at hand, including all two-particle bound/antibound states. Further, we discuss how to generate series expansions for the dispersions of the bound/antibound states. These allow us to employ series extrapolation techniques, whereby binding energies can be determined even when the expansion parameters are not small. We apply the method to the ~111!-dimensional transverse Ising model and the two-leg spin- 1 2 Heisenberg ladder. For the latter model, we also calculate the coherence lengths and determine the critical properties where bound states merge with the two-particle continuum.

The strong-coupling expansion in simplicial quantum gravity

We construct the strong-coupling series in 4d simplicial quantum gravity up to volume 38. It is used to calculate estimates for the string susceptibility exponent γ for various modifications of the theory. It provides a very efficient way to get a first view of the phase structure of the models.

Large-order behavior of the convergent perturbation theory for anharmonic oscillators

Using the large-order formula for the coefficients of the divergent weak-coupling series for the energy of the anharmonic oscillators, we derive a simple analytic large-order formula for the coefficients of the convergent renormalized strong-coupling series. This formula is valid for all the states of the anharmonic oscillators defined by the Hamiltonians ${H=p}^{2}{+x}^{2}+\ensuremath{\beta}{x}^{2m}$ with $m&gt;~2.$ A further generalization of this formula is also proposed. Numerical tests of the formula are performed for the quartic, sextic, octic, and decadic oscillator with the help of asymptotic analysis. Further it is shown that the renormalized strong-coupling perturbation expansion converges for all the states of these oscillators and for all physically relevant $\ensuremath{\beta}\ensuremath{\in}[0,\ensuremath{\infty}).$

Series expansions for three-dimensional QED

Strong-coupling series expansions are calculated for the Hamiltonian version of compact lattice electrodynamics in (2+1) dimensions, with 4-component fermions. Series are calculated for the ground-state energy per site, the chiral condensate, and the masses of `glueball' and positronium states. Comparisons are made with results obtained by other techniques.