Bosonic Partition Function Research Articles

Fermionic and Bosonic Partition Functions at Imaginary Chemical Potential as Bloch Functions

In this work it is pointed out that the phase transitions of the <i>d</i>+1 Gross-Neveu (fermionic) and <I>CP<SUP>N</SUP></I><sup>−1</sup> (bosonic) models at finite temperature and imaginary chemical potential can be mapped to transformations of Hubbard-like regular hexagonal to square lattice with the intermediate steps to be specific surfaces (irregular hexagonal kind) with an ordered construction based on the even indexed Bloch-Wigner-Ramakrishnan polylogarithm function. The zeros and extrema of the Clausen <i>Cl<sub>d</sub></i>(<i>θ</i>) function play an important role to the analysis since they allow us not only to study the fermionic and bosonic theories and their phase transitions but also the possibility to explore the existence of conductors arising from the correspondence between the partition functions of the two models and the Bloch and Wannier functions that play a crucial role in the tight-binding approximation in solid state physics. The main aim of this work is not only to unveil the relevance of the canonical partition functions of a fermionic and a bosonic model to Bloch states by using an imaginary chemical potential but also to examine the overlap between two Bloch wave-functions that differ by a lattice momentum that calculates the momentum transfer of a Bloch wave during the interaction with a lattice point of a hexagonal construction.

Second quantization and the spectral action

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential μ approaches 0, can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.

Convergence from divergence

We show how to convert divergent series, which typically occur in many applications in physics, into rapidly convergent inverse factorial series. This can be interpreted physically as a novel resummation of perturbative series. Being convergent, these new series allow rigorous extrapolation from an asymptotic region with a large parameter, to the opposite region where the parameter is small. We illustrate the method with various physical examples, and discuss how these convergent series relate to standard methods such as Borel summation, and also how they incorporate the physical Stokes phenomenon. We comment on the relation of these results to Dyson’s physical argument for the divergence of perturbation theory. This approach also leads naturally to a wide class of relations between bosonic and fermionic partition functions, and Klein–Gordon and Dirac determinants.

Bosonic partition functions at nonzero (imaginary) chemical potential

We consider bosonic random matrix partition functions at nonzero chemical potential and compare the chiral condensate, the baryon number density and the baryon number susceptibility to the result of the corresponding fermionic partition function. We find that as long as results are finite, the phase transition of the fermionic theory persists in the bosonic theory. However, in case that the bosonic partition function diverges and has to be regularized, the phase transition of the fermionic theory does not occur in the bosonic theory, and the bosonic theory is always in the broken phase.

Asymptotic prime partitions of integers.

In this paper, we discuss P(n), the number of ways a given integer n may be written as a sum of primes. In particular, an asymptotic form P_{as}(n) valid for n→∞ is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This gives directly the asymptotic number of prime partitions P_{as}(n). The leading term in the asymptotic expression grows exponentially as sqrt[n/ln(n)] and agrees with previous estimates. We calculate the next-to-leading-order term in the exponent, proportional to ln[ln(n)]/ln(n), and we show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher-order correction, proportional to 1/ln(n) and given in Eq.(43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n∼8×10^{6}. For the highest values, the remaining error between the exact P(n) and our P_{as}(n) is only about half of that obtained with the leading-order approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n∼10^{7}.

Factorization of the bosonic partition function

The factorization formula in the non-interacting quantum field theories that relates the fermionic partition function to the bosonic partition function considered recently by Chair (2013) [3] is obtained for the harmonic oscillator using the path integral formulation. By using the latter, the fermionic partition function turns out to be the ratio of two determinants of the same operator (∂τ+ω), whose eigenmodes being both periodic on the imaginary time intervals [0,2β], [0,β]. The natural generalization of the factorization formula when β→2mβ is derived, such a factorization implies that the bosonic oscillator at temperature β can be seen as a non-interacting mixture of a bosonic oscillator at temperature 2mβ and m-fermionic oscillators at different temperatures 2m−kβ, k=1,2,…,m. As a consequence, a general relationship between the bosonic and fermionic thermal zeta functions is deduced.

Comparaison des résultats cliniques et radiologiques d’une réparation arthroscopique de la coiffe des rotateurs chez des patients de moins de 50 ans et de plus de 70 ans sur une série prospective continue multicentrique de 312 cas

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

On partition function and Weyl anomaly of conformal higher-spin fields

We study 4-dimensional higher-derivative conformal higher-spin (CHS) fields generalizing Weyl graviton and conformal gravitino. They appear, in particular, as “induced” theories in the AdS/CFT context. We consider their partition function on curved Einstein-space backgrounds like (A)dS or sphere and Ricci-flat spaces. Remarkably, the bosonic (integer spin s) CHS partition function appears to be given by a product of partition functions of the standard 2nd-derivative “partially massless” spin s fields, generalizing the previously known expression for the 1-loop Weyl graviton (s=2) partition function. We compute the corresponding spin s Weyl anomaly coefficients as and cs. Our result for as reproduces the expression found recently in arXiv:1306.5242 by an indirect method implied by AdS/CFT (which relates the partition function of a CHS field on S4 to a ratio of known partition functions of massless higher-spin field in AdS5 with alternate boundary conditions). We also obtain similar results for the fermionic CHS fields. In the half-integer s case the CHS partition function on (A)dS background is given by the product of squares of “partially massless” spin s partition functions and one extra factor corresponding to a special massive conformally invariant spin s field. It was noticed in arXiv:1306.5242 that the sum of the bosonic as coefficients over all s is zero when computed using the ζ-function regularization, and we observe that the same property is true also in the fermionic case.

Integrability of zero-dimensional replica field theories atβ=1

Building on insights from the theory of integrable lattices, the integrability is claimed for nonlinear replica σ models derived in the context of real symmetric random matrices. Specifically, the fermionic and the bosonic replica partition functions are proven to form a single (supersymmetric) Pfaff-KP hierarchy whose replica limit is shown to reproduce the celebrated nonperturbative formula for the density-density eigenvalue correlation function in the infinite-dimensional Gaussian orthogonal ensemble. Implications of the formalism outlined are briefly discussed.

The Euler–Riemann gases, and partition identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given.

Bosonization of QED fermions on the (3+1)D lattice

The fermionic quantum electrodynamical determinant is expressed by a bosonic partition function for configurations of worldsheets. The Pauli minus sign of fermionic loops is obtained from the local structure of a worldsheet. The action is local and gauge independent. The 1–1 correspondence is discussed and proved in an asymptotic limit.

Supersymmetric quenching of the Toda lattice equation

The average of the ratio of powers of the spectral determinants of the Dirac operator in the ϵ-regime of QCD is shown to satisfy a Toda lattice equation. The quenched limit of this Toda lattice equation is obtained using the supersymmetric method. This supersymmetric approach is then shown to be equivalent to taking the replica limit of the Toda lattice equation. Among others, the factorization of the microscopic spectral correlation functions of the QCD Dirac operator into fermionic and bosonic partition functions follows naturally from both approaches. While the replica approach relies on an analytic continuation in the number of flavors no such assumptions are made in the present approach where the numbers of flavors in the Toda lattice equation are strictly integer.

Factorization of correlation functions and the replica limit of the Toda lattice equation

Exact microscopic spectral correlation functions are derived by means of the replica limit of the Toda lattice equation. We consider both Hermitian and non-Hermitian theories in the Wigner–Dyson universality class (class A) and in the chiral universality class (class AIII). In the Hermitian case we rederive two-point correlation functions for class A and class AIII as well as several one-point correlation functions in class AIII. In the non-Hermitian case the average spectral density of non-Hermitian complex random matrices in the weak non-Hermiticity limit is obtained directly from the replica limit of the Toda lattice equation. In the case of class A, this result describes the spectral density of a disordered system in a constant imaginary vector potential (the Hatano–Nelson model) which is known from earlier work. New results are obtained for the average spectral density in the weak non-Hermiticity limit of a quenched chiral random matrix model at non-zero chemical potential. These results apply to the ergodic or ϵ domain of the quenched QCD partition function at non-zero chemical potential. Our results have been checked against numerical results obtained from a large ensemble of random matrices. The spectral density obtained is different from the result derived by Akemann for a closely related model, which is given by the leading order asymptotic expansion of our result. In all cases, the replica limit of the Toda lattice equation explains the factorization of spectral one- and two-point functions into a product of a bosonic (non-compact integral) and a fermionic (compact integral) partition function. We conclude that the fermionic partition functions, the bosonic partition functions and the supersymmetric partition function are all part of a single integrable hierarchy. This is the reason that it is possible to obtain the supersymmetric partition function, and its derivatives, from the replica limit of the Toda lattice equation.

Thermodynamic fermion–boson symmetry in harmonic oscillator potentials

A remarkable thermodynamic fermion–boson symmetry is found for the canonical ensemble of ideal quantum gases in harmonic oscillator potentials of odd dimensions. The bosonic partition function is related to the fermionic one extended to negative temperatures, and vice versa.

BOSONIC AND FERMIONIC PARTITION FUNCTIONS ON S3

Using the ζ-function regularization method, we compute expressions for the bosonic and fermionic partition functions for a field theory defined on a sphere S3. We also analyze their behavior with the radius r0 of this manifold, showing that for specific values of the field's masses closed expressions can be obtained, which give a completely different behavior as r0→∞, indicating isolated discontinuities of these partition functions with the masses of the matter fields.

Divergences in the moduli space integral and accumulating handles in the infinite-genus limit

The symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived through integration over a subset of the fundamental region.

Genus-dependent divergences in bosonic string perturbation theory

The divergence in the bosonic string partition function found by Gross and Periwal (1988) is analysed using the Schottky parametrisation of the Polyakov measure for moduli space. It is show how the problem of the integration region can be solved and how lower limits on the measure may be obtained in this domain. The factorial growth of the bound on the partition function with respect to the genus can thus be derived by translating the cut-off for closed geodesic lengths to a cut-off for Schottky group parameters.

Modular invariance of the partition function for the bosonic string in background fields

In the present paper we investigate the modular properties of the bosonic string partition function with massless background fields and show that the requirement of modular invariance on the said functional implies the vanishing of the dilaton β-function at the one-loop level in string perturbation theory.

Supersymmetric Bogomolny bounds at finite temperature

The authors expresses the first quantum terms contributing to SUSY Bogomolny bounds at finite temperature. In particular the 'collective coordinates' treatment is mandatory in order to avoid an 'infrared catastrophe' associated with the bosonic partition function while the conventional saturation when T=0 is clearly recovered.