Invariant Green Functions Research Articles

Power-duality in path integral formulation of quantum mechanics

Power duality in Feynman's path integral formulation of quantum mechanics is investigated. The power duality transformation consists of a change in coordinate and time variables, an exchange of energy and coupling, and a classical angular momentum replacement. Two physical systems connected by the transformation form a power-dual pair. The propagator (Feynman's kernel) expressed by Feynman's path integral cannot be form-invariant under the transformation, whereas the promotor constructed by modifying Feynman's path integral is found form-invariant insofar as the angular momentum is classical. Upon angular quantization, the power duality breaks down. To save the notion of power duality, the idea of quasi power duality is proposed, which constitutes of an ad hoc angular momentum replacement. The power-dual invariant promotor leads to the quasi-dual invariant Green function. A formula is proposed, which determines the Green function for one of a dual pair by knowing the Green function for the other. As examples, the Coulomb-Hooke dual pair and a family of two-term confinement potentials for a zero-energy state are discussed.

Lead Green's functions from quadratic eigenvalue problems without mode velocity calculations.

In quantum transport calculations, the proper handling of incoming and outgoing modes for retarded Green's functions is achieved via the lead self-energies. Computationally efficient and accurate methods to calculate the self-energies are thus very important. Here we present an alternative method for calculating lead self-energies which improves on a standard approach to solving quadratic eigenvalue problems that arise in quantum transport modeling. The method is based on a perturbative analysis of the generalized Schur decomposition to determine the relevant set of eigenvalues for transmitting modes. This allows us to circumvent finding the velocities of the modes (left- or right-moving) that are needed in order to calculate the lead Green's function from translationally invariant Green's functions. This saves computational time irrespective of the value of the imaginary part added to the energy. We compare our method with two existing methods-a popular iterative method and a standard eigenvalue method that explicitly calculates the velocities of the propagating modes. Our comparison shows that both eigenvalue methods are more robust than the iterative method. Furthermore, the comparison also shows that above a small threshold of propagating modes, the standard eigenvalue method requires extra computation time over our perturbation method. This excess of computation time grows linearly with the number of propagating modes.

Thermodynamics of the kagome-lattice Heisenberg antiferromagnet with arbitrary spin S

We use a second-order rotational invariant Green's function method (RGM) and the high-temperature expansion (HTE) to calculate the thermodynamic properties, of the kagome-lattice spin-$S$ Heisenberg antiferromagnet with nearest-neighbor exchange $J$. While the HTE yields accurate results down to temperatures of about $T/S(S+1) \sim J$, the RGM provides data for arbitrary $T \ge 0$. For the ground state we use the RGM data to analyze the $S$-dependence of the excitation spectrum, the excitation velocity, the uniform susceptibility, the spin-spin correlation functions, the correlation length, and the structure factor. We found that the so-called $\sqrt{3}\times\sqrt{3}$ ordering is more pronounced than the $q=0$ ordering for all values of $S$. In the extreme quantum case $S=1/2$ the zero-temperature correlation length is only of the order of the nearest-neighbor separation. Then we study the temperature dependence of several physical quantities for spin quantum numbers $S=1/2,1,\dots,7/2$. As increasing $S$ the typical maximum in the specific heat and in the uniform susceptibility are shifted towards lower values of $T/S(S+1)$ and the height of the maximum is growing. The structure factor ${\cal S}(\mathbf{q})$ exhibits two maxima at magnetic wave vectors $\mathbf{q}={\mathbf{Q}_i}, i=0,1,$ corresponding to the $q=0$ and $\sqrt{3}\times\sqrt{3}$ state. We find that the $\sqrt{3}\times \sqrt{3}$ short-range order is more pronounced than the $q=0$ short-range order for all temperatures $T \ge 0$. For the spin-spin correlation functions, the correlation lengths and the structure factors, we find a finite low-temperature region $0 \le T < T^*\approx a/S(S+1)$, $a \approx 0.2$, where these quantities are almost independent of $T$.

Coherent phonon dynamics of normal metal in ultrafast spectroscopy: Non-equilibrium gauge invariant Green's function approach

The phonon dynamics of normal metal in the coherent regime of ultrafast spectroscopy is studied based on the non-equilibrium gauge invariant Green's function method. The non-equilibrium phonon self-energy is computed explicitly as a function of time in a gauge invariant way up to the second order of electric field of applied laser pulse. The extension beyond the coherent regime and the incorporation of correlation effects are discussed.

The Block-P 3M algorithm

The Block-P 3M algorithm is a new development of the P 3M (particle–particle/particle–mesh) method [J. Eastwood, Optimal P 3M algorithms for molecular dynamics simulations, in: M. Hooper (Ed.), Computational Methods in Classical and Quantum Physics, Advance Publications Ltd, 1976, pp. 206–228; R. Hockney, J. Eastwood, Computer Simulation using Particles, McGraw-Hill, 1981 (Student Ed., Adam-Hilger, 1988)] for boundary integral and N-body calculations. It gives O ( N log N ) complexity and O ( N ) storage of action-at-a-distance convolution sums for non-uniform distributions of sources and for any displacement invariant Green's function. It gives optimal accuracy and, unlike the Fast Multipole Method, it does not suffer from low frequency breakdown problems.

BRST-driven cancellations and gauge invariant Green's functions

We study a fundamental, all order cancellation operating between graphs of distinct kinematic nature, which allows for the construction of gauge-independent effective self-energies, vertices, and boxes at arbitrary order.

Sp(2M)-invariant Green functions

Explicit form of two-point and three-point Sp(2M)-invariant Green functions is found.

Sp(2M)-invariant Green functions

Explicit form of two-point and three-point Sp(2M)-invariant Green functions is found.

Variation of Potentials in the Unit Ball of $ {\\shadC}^n$

In this paper, we study the variation of invariant Green potentials G w in the unit ball B of $ {\\shadC}^n$ , which for suitable measures w are defined by $$ G_{\\mu}(z) = \\int_{B}G(z,w)\\, d\\mu(w), $$ where G is the invariant Green function for the Laplace-Beltrami operator ¨ j on B . The main result of the paper is as follows. Let w be a non-negative regular Borel measure on B satisfying $$ \\int_{B}(1-|w|^2)^n\\log {1 \\over (1-|w|^2)}\\, \\d\\mu(w) ] B , { z denotes the holomorphic automorphism of B satisfying { z (0) = z , { z ( z ) = 0 and ( { z { z )( w ) = w for every w ] B .

Thermally Generated Vortices, Gauge Invariance, and Electron Spectral Function in the Pseudogap Regime

Starting from the classical vortex fluctuation picture, we study the single electron properties in the pseudogap regime. We show that it is the gauge invariant Green function of spinon which is directly related to ARPES data in the pseudogap regime instead of the nongauge invariant one. We find that the random gauge field from the thermally generated vortices destroys the coherent spinon motion. The energy distribution curves show broad peaks, while the momentum distribution curves show sharp peaks with Lorenz form. The local density of state at zero energy scales as the inverse of Kosterlitz-Thouless length. These results are qualitatively consistent with the ARPES data in the pseudogap regime.

Spin-polaron damping in the spin-fermion model for cuprate superconductors

A self-consistent, spin rotational invariant Green's function procedure has been developed to calculate the spectral function of carrier excitations in the spin-fermion model for the CuO2 plane. We start from the mean field description of a spin polaron in the Mori-Zwanzig projection method. In order to determine the spin polaron lifetime in the self-consistent Born approximation, the self-energy is expressed by an irreducible Green's function. Both, spin polaron and bare hole spectral functions are calculated. The numerical results show a well pronounced quasiparticle peak near the bottom of the dispersion at (pi/2,pi/2), the absence of the quasiparticle at the Gamma-point, a rather large damping away from the minimum and an asymmetry of the spectral function with respect to the antiferromagnetic Brillouin zone. These findings are in qualitative agreement with photoemission data for undoped cuprates. The direct oxygen-oxygen hopping is responsible for a more isotropic minimum at (pi/2,pi/2).

The Luttinger–Schwinger Model

We study the Luttinger–Schwinger model, i.e. the (1+1) dimensional model of massless Dirac fermions with a non-local 4-point interaction coupled to a U(1)-gauge field. We work within the Hamiltonian framework on the cylinder, and construct the field operators and observables as well-defined operators on the physical Hilbert space. The complete solution of the model is found using the boson-fermion correspondence, and the formalism for calculating all gauge invariant Green functions is provided. We discuss the role of anomalies and show how the existence of large gauge transformations implies a fermion condensate in all physical states. The meaning of regularization and renormalization in our well-defined Hilbert space setting is discussed. We illustrate the latter by performing the limit to the Thirring–Schwinger model where the interaction becomes local.

Superinstantons and the Reliability of Perturbation Theory in Non-Abelian Models

In dimension $D\leq 2$ the low temperature behavior of systems enjoying a continuous symmetry is dominated by super-instantons: classical configurations of arbitrarily low energy. Perturbation theory in the background of a super-instanton produces thermodynamic answers for the invariant Green's functions that differ from the standard ones, but only in non-Abelian models and only starting at $O(1/\beta^2)$. This effect modifies the $\beta$-function of the $O(N)$ models and persists in the large $N$ limit of the $O(N)$ models.

ON THE IR-FINITENESS OF THE NONLINEAR O(N) σ-MODEL

Using the proper definition1 of the massless 2D-propagator – depending on an arbitrary ir-scale μ – we check Elitzur’s conjecture on the ir finiteness of O(N)-invariant Green functions. Explicit computations to the second order in the coupling constant show that non-invariant correlators become finite but μ-dependent objects, whereas the correlators of invariant operators, which are possible candidates for “physical” observables, do not depend on the ir-scale.

EXACTLY SOLVABLE CONFORMALLY INVARIANT QUANTUM FIELD MODELS IN D DIMENSIONS

A class of exactly solvable models of the conformally invariant quantum field theory in D dimensions is proposed. It is shown that in any conformal theory of the field φ(x) with the scale dimension d there exists an infinite collection of the tensor fields Ps of the ranks and the dimensions ds=d+s independently of the type of interaction. These tensor fields appear in the product Tµν(x1)φ(x2) operator expansion where Tµν is the energy-momentum tensor. The fields Ps are analogues to the certain superpositions of the secondary fields of D=2 models. The existence of the fields Ps follows from the structure of the Ward identities for the energy-momentum tensor conformally invariant Green functions. Each model of the above-mentioned class is defined by the operator equation Ps(x)=0. The method of solving these models is proposed. Some of the models coincide with the certain lagrangian models. The method allows us to obtain closed differential equations for each Green function of fundamental and composite fields, and also algebraic equations for scale dimensions of the fields. The derivation of all these equations is based on the energy momentum conformal Ward identities’ specific property. Detailed analysis of the Ward identities is given in the paper. It is shown that each D&gt;2-model involves a special scalar field PD−2 with the scale dimension dP=D−2. This field is an analogue of the central charge of D=2 models and becomes the constant coinciding with the central charge when D=2. A new class of D=2 models with broken infinite parametric symmetry is obtained. In each model the closed differential equations for the highest Green functions are derived and the central charge is calculated. The general method of solving the D≥2 models is illustrated on examples of Thirring and Wess-Zumino-Witten models and on a trivial model in four-dimension space.

Boundary behavior of invariant Green’s potentials on the unit ball in 𝐶ⁿ

Let p ( z ) = ∫ B G ( z , w ) d μ ( w ) p(z) = \int _B {G(z,\,w)\,d\mu (w)} be an invariant Green’s potential on the unit ball B B in C n ( n ⩾ 1 ) {{\mathbf {C}}^n}\;(n \geqslant 1) , where G G is the invariant Green’s function and μ \mu is a positive measure with ∫ B ( 1 − | w | 2 ) n d μ ( w ) &gt; ∞ \int _B {{{(1 - |w{|^2})}^n}\,d\mu (w) &gt; \infty } . In this paper, a necessary and sufficient condition on a subset E E of B B such that for every invariant Green’s potential p p , \[ lim z → e inf ( 1 − | z | 2 ) n p ( z ) = 0 , e = ( 1 , 0 , … , 0 ) ∈ ∂ B , z ∈ E , \lim \limits _{z \to e} \,\inf {(1 - |z{|^2})^n}p(z) = 0,\qquad e = (1,\,0,\, \ldots ,\,0)\; \in \partial B,\;z \in E, \] is given. The condition is that the capacity of the sets E ∩ { z ∈ B | | z − e | &gt; ε } E \cap \{ z \in B|\;|z - e| &gt; \varepsilon \} , ε &gt; 0 \varepsilon &gt; 0 , is bounded away from 0 0 . The result obtained here generalizes Luecking’s result, see [L], on the unit disc in C {\mathbf {C}} .

Weak gauge coupling expansion in the lattice-regularized standard SU(2) Higgs model

The weak gauge coupling expansion is derived in the lattice-regularized SU(2) Higgs model with a scalar doublet field at an arbitrary point of the parameter space boundary with vanishing gauge coupling. Consequences of the triviality of the φ 4 component on gauge invariant Green's functions are formulated.

Conformal gauges and renormalized equations of motion in massless quantum electrodynamics

A formulation of massless QED is studied with a non-singular Lagrangian and conformal invariant equations of motion. It makes use of non-decomposable representations of the conformal groupG and involves two dimensionless scalar fields (in addition to the conventional charged field and electromagnetic potential) but gauge invariant Green functions are shown to coincide with those of standard (massless) QED. Assuming that the (non-elementary) representation ofG for the 5-potential which leaves the equations of motion invariant and leads to the free photon propagator of Johnson-Baker-Adler (JBA) conformal QED remains unaltered by renormalization, we prove that consistency requirements for conformal invariant 2-, 3-, and 4-point Green functions satisfying (renormalized) equations of motion and standard Ward identities lead to either a trivial solution (witheψ=0) or to a subcanonical dimensiond=1/2 for the charged field.

Gaussian dominance on compact spin manifolds

The low temperature regime of continuous spin models is discussed. The relevance of the weak coupling expansion for the calculation of invariant Green's functions is analyzed. Notably it is found that in two dimensions Green's functions of invariant operators cannot be computed perturbatively.

The applicability of perturbation expansion to two-dimensional goldstone systems

In the two-dimensional O( n) non-linear σ model, O( n) invariant Green functions are calculated perturbatively with lattice regularization and found to be infrared finite to two loops. This finiteness makes it possible to obtain the correct renormalization group parameters of the theory directly in the symmetric theory without symmetry-breaking terms in the action. It is conjectured that any invariant functional of the fields has an infrared-finite expectation value to any order in the perturbation expansion.