Hadronic density of states from string theory

We present an exact calculation of the finite temperature partition function for the hadronic states corresponding to a Penrose-Güven limit of the Maldacena-Nù?ez embedding of the N=1 super Yang-Mills (SYM) into string theory. It is established that the theory exhibits a Hagedorn density of states. We propose a semiclassical string approximation to the finite temperature partition function for confining gauge theories admitting a supergravity dual, by performing an expansion around classical solutions characterized by temporal windings. This semiclassical approximation reveals a hadronic energy density of states of a Hagedorn type, with the coefficient determined by the gauge theory string tension as expected for confining theories. We argue that our proposal captures primarily information about states of pure N=1 SYM theory, given that this semiclassical approximation does not entail a projection onto states of large U(1) charge.     

Fractal structure of zeros in hierarchical models

We consider an Ising and aq-state Potts model on a diamond hierarchical lattice. We give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq. These zeros are just the Julia set corresponding to the renormalization group transformation.     

Polymers on disordered hierarchical lattices: A nonlinear combination of random variables

The problem of directed polymers on disordered hierarchical and hypercubic lattices is considered. For the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way. We present the results of numerical simulations of two hierarchical lattices, finding evidence of a phase transition in one case. For a limiting case we extend the perturbation theory developed by Derrida and Griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties (overlaps) of the model. In this limit we obtain an interpolation formula, allowing one to obtain the noninteger moments of the partition function from the integer moments. We obtain bounds for the transition temperature for hierarchical and hypercubic lattices, and some similarities between the problem on the two different types of lattice are discussed.     

Oscillatory critical amplitudes in hierarchical models

We study the oscillatory critical amplitudes of theq-states Potts model on a diamond hierarchical lattice. We consider an example of the generic case (finite critical index), as well as the degenerate case (essential singularity). In both cases, we compare the magnitude of the oscillations with geometrical characteristics of the Julia set of zeroes of the partition function.     

Effective Hamiltonians for holes in antiferromagnets: a new approach to implement forbidden double occupancy

A coherent state representation for the electrons of ordered antiferromagnets is used to derive effective Hamiltonians for the dynamics of holes in such systems. By an appropriate choice of these states, the constraint of forbidden double occupancy can be implemented rigorously. Using these coherent states, one arrives at a path integral representation of the partition function of the systems, from which the effective Hamiltonians can be read off. We apply this method to the t-J model on the square lattice and on the triangular lattice. In the former case, we reproduce the well-known fermion-boson Hamiltonian for a hole in a collinear antiferromagnet. We demonstrate that our method also works for non-collinear antiferromagnets by calculating the spectrum of a hole in the triangular antiferromagnet in the self-consistent Born approximation and by comparing it with numerically exact results. Received: 23 December 1997 / Accepted: 17 March 1998     

Wall-Crossing,Free Fermions and Crystal Melting

We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years.     

An alternative approach to calculate the density of states in nonextensive statistical mechanics

A relation between the generalized partition function (Tsallis) and density of states is established by using the method of integral transform which enables reducing some integral equations into the algebraic equations. Inverse Mellin transformation of this equation gives the density of states. Similar relation is also hold the for standard partition function (Boltzmann-Gibbs) and the density of states. Using these relations, we recover the density of states for the classical ideal gas within both statistics.     

Decorated ising model with classical vector spins

A decorated Ising model with classical vector spins on a square lattice is investigated in detail. The partition function is reduced to the one of the Ising model with effective exchange integrals. Three successive phase-transition temperatures are obtained and four states, namely, paramagnetic, antiferromagnetic, again paramagnetic and ferromagnetic states are realized as the temperature is decreased. For systems on other two- and three-dimensional loose-packed lattices, the situation is the same as the system on the square lattice.     

Partition function for an electron in a random potential

We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.Work supported by the National Science Foundation.     

Generating functional models for a statistical system near a phase transition

Starting from an assumed form of the distribution function near the phase transition point, an expression for the generating functional of a statistical system suitable for describing two-phase states of matter is derived. We then obtain formulas for the partition function and correlation functions by the standard procedure.     

Low depth quantum circuits for Ising models

A scheme for measuring complex temperature partition functions of Ising models is introduced. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimated error are provided through a central-limit theorem whose validity extends beyond the present context; it holds for example for estimations of the Jones polynomial. The kind of state preparations and measurements involved in this application can be made independent of the system size or the parameters of the system being simulated. Second, the scheme allows to accurately estimate non-trivial invariants of links. Another result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models. We provide conditions under which estimating such partition functions allows to reconstruct scattering amplitudes of quantum circuits, making the problem BQP-hard. We also show fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we discuss how accurate corner magnetisation measurements on thermal states of two-dimensional Ising models lead to fully polynomial random approximation schemes (FPRAS) for the partition function.     

Partition functions of atomic and diatomic species in high-temperature atmospheric plasmas

Accurate atomic and diatomic partition functions are required to determine the level populations for the calculations of radiative properties in thermodynamic equilibrium and nonequilibrium plasmas produced by various atmospheric re-entries. In this work, a reliable partition functions database was rebuilt for some atomic and diatomic species from 100 K to 50000 K. The atomic partition functions were obtained by a four-level model, while the diatomic partition functions were predicted based upon a more rigorous approach for the computation of the energy levels. Compared with previous publications, the number of diatomic electronic states considered in our work is as large as possible. Estimates are made for the contributions of each electronic state of the diatomic molecule to the partition function. Moreover, the effect of the number of electronic states on the partition function was also evaluated. Finally, we calculated the specific heat based upon the obtained partition functions. All the results were validated by the available data in recent references and the relative errors were systematically analysed.     

Renormalization of Möbius infinities and partition function representation for the string theory effective action

We show that subtraction of the Möbius group volume in the open string massless amplitudes can be realized as a renormalization of linear 2D ultraviolet divergences in the generating functional (“partition function”). This implies that the vector field effective action can be represented as a renormalized partition function (i.e. as a path integral of the “Wilson factor”). We check this by computing several leading terms in the non-abelian effective action.     

Construction of a natural partition of incomplete horseshoes

We present a method for constructing a partition of an incomplete horseshoe in a Poincare map. The partition is based only on the unstable manifolds of the outermost fixed points and eventually their limits. Consequently, this partition becomes natural from the point of view of asymptotic scattering observations. The symbolic dynamics derived from this partition coincides with the one derived from the hierarchical structure of the singularities of the scattering functions.     

Statistical characterization of a 1D random potential problem—With applications in score statistics of MS-based peptide sequencing

We provide a complete thermodynamic solution of a 1D hopping model in the presence of a random potential by obtaining the density of states. Since the partition function is related to the density of states by a Laplace transform, the density of states determines completely the thermodynamic behavior of the system. We have also shown that the transfer matrix technique, or the so-called dynamic programming, used to obtain the density of states in the 1D hopping model may be generalized to tackle a long-standing problem in statistical significance assessment for one of the most important proteomic tasks—peptide sequencing using tandem mass spectrometry data.     

The O(n) Model on the Annulus

We use Coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n→0 also gives explicit examples of partition functions in logarithmic conformal field theory.