arXiv subjects
Publications and source records attributed to Edwin Langmann.
We review the construction of a conformal field theory model which describes anyons on a circle and at finite temperature, including previously unpublished results. This anyon model is closely related to the quantum elliptic Calogero-Sutherland (eCS) system. We describe this relation and how it has led to an explicit construction of the eigenvalues and eigenfunctions of the eCS system.
Contribution to the Encyclopedia of Mathematical Physics (Elsevier, 2006): a brief and (hopefully) pedagogical introduction to quantum field theory models describing particles in external fields is presented. Following the instructions, the only references are a few recommendations for further reading in the first and last sections.
As is well-known, there exists a four parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum dependent terms, and we determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe Ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta- and (so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the other model we write down explicit formulas for all eigenfunctions.
We present explicit formulas for the eigenvalues and eigenfunctions of the elliptic Calogero-Sutherland (eCS) model as formal power series to all orders in the nome of the elliptic functions, for arbitrary values of the (positive) coupling constant and particle number. Our solution gives explicit formulas for an elliptic deformation of the Jack polynomials.
It is well-known from unrestricted Hartree-Fock computations that the 2D Hubbard model does not have homogeneous mean field states in significant regions of parameter space away from half filling. This is incompatible with standard mean field theory. We present a simple extension of the mean field method that avoids this problem. As in standard mean field theory, we restrict Hartree-Fock theory to simple translation invariant states describing antiferromagnetism (AF), ferromagnetism (F) and paramagnetism (P), but we use an improved method to implement the doping constraint allowing us to detect when a phase separated state is energetically preferred, e.g. AF and F coexisting at the same time. We find that such mixed phases occur in significant parts of the phase diagrams, making them much richer than the ones from standard mean field theory. Our results for the 2D t-t'-U Hubbard model demonstrate the importance of band structure effects.
We consider two particular 1D quantum many-body systems with local interactions related to the root system $C_N$. Both models describe identical particles moving on the half-line with non-trivial boundary conditions at the origin, and they are in many ways complementary to each other. We discuss the Bethe Ansatz solution for the first model where the interaction potentials are delta-functions, and we find that this provides an exact solution not only in the boson case but even for the generalized model where the particles are distinguishable. In the second model the particles have particular momentum dependent interactions, and we find that it is non-trivial and exactly solvable by Bethe Ansatz only in case the particles are fermions. This latter model has a natural physical interpretation as the non-relativistic limit of the massive Thirring model on the half-line. We establish a duality relation between the bosonic delta-interaction model and the fermionic model with local momentum dependent interactions. We also elaborate on the physical interpretation of these models. In our discussion the Yang-Baxter relations and the Reflection equation play a central role.
We discuss a 1D many-body model of distinguishable particles with local, momentum dependent two-body interactions. We show that the restriction of this model to fermions corresponds to the non-relativistic limit of the massive Thirring model. This fermion model can be solved exactly by a mapping to the 1D boson gas with inverse coupling constant. We provide evidence that this mapping is the non-relativistic limit of the duality between the massive Thirring model and the quantum sine-Gordon model. We also investigate the question if the generalization of this model to distinguishable particles is exactly solvable by the coordinate Bethe ansatz and find that this is not the case.
I discuss many-body models for interacting fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: 2D fermions in a constant magnetic field and a particular non-local 4-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra $\gl_\infty\oplus \gl_\infty$. There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.
I present a novel class of exactly solvable quantum field theories. They describe non-relativistic fermions on even dimensional flat space, coupled to a constant external magnetic field and a four point interaction defined with the Groenewold-Moyal star product. Using Hamiltonian quantization and a suitable regularization, I show that these models have a dynamical symmetry corresponding to $\gl_\infty\oplus \gl_\infty$ at the special points where the magnetic field $B$ is related to the matrix $θ$ defining the star product as $Bθ=\pm I$. I construct all eigenvalues and eigenstates of the many-body Hamiltonian at these special points. I argue that this solution cannot be obtained by any mean-field theory, i.e. the models describe correlated fermions. I also mention other possible interpretations of these models in solid state physics.
We describe a novel duality symmetry of Phi(4)-theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of non-interacting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regualarization we show, to all orders of perturbation theory, that that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories.
We study dimensional reductions of noncommutative electrodynamics on flat space which lead to gauge theories of gravitation. For a general class of such reductions, we show that the noncommutative gauge fields naturally yield a Weitzenbock geometry on spacetime and that the induced diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity which macroscopically describes general relativity. The Planck length is determined in this setting by the Yang-Mills coupling constant and the noncommutativity scale. The effective field theory can also contain higher-curvature and non-local terms which are characteristic of string theory. Some applications to D-brane dynamics and generalizations to include the coupling of ordinary Yang-Mills theory to gravity are also described.
We give a self-contained presentation and comparison of two different algorithms to explicitly solve quantum many body models of indistinguishable particles moving on a circle and interacting with two-body potentials of $1/\sin^2$-type. The first algorithm is due to Sutherland and well-known; the second one is a limiting case of a novel algorithm to solve the elliptic generalization of the Sutherland model. These two algorithms are different in several details. We show that they are equivalent, i.e., they yield the same solution and are equally simple.
We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension.
We use loop group techniques to construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian (this algorithm is elaborated elsewhere). This paper contains a detailed introduction, technical details and proofs.
We obtain a second quantization of the elliptic Calogero-Sutherland (eCS) model by constructing a quantum field theory model of anyons on a circle and at a finite temperature. This yields a remarkable identity involving anyon correlation functions and providing an algorithm for solving of the eCS model. The eigenfunctions obtained define an elliptic generalization of the Jack polynomials.
We find a large class of quantum gauge models with massless fermions where the coupling to the gauge fields is not chirally symmetric and which nevertheless do not suffer from gauge anomalies. To be specific we study two dimensional Abelian models in the Hamiltonian framework which can be constructed and solved by standard techniques. The general model describes $\Np$ photon fields and $\Nf$ flavors of Dirac fermions with $2\Nf\Np$ different coupling constants i.e. the chiral component of each fermion can be coupled to the gauge fields differently. We construct these models and find conditions so that no gauge anomaly appears. If these conditions hold it is possible to construct and solve the model explicitly, so that gauge- and Lorentz invariance are manifest.
Yang-Mills gauge theory models on a cylinder coupled to external matter charges provide powerful means to find and solve certain non-linear integrable systems. We show that, depending on the choice of gauge group and matter charges, such a Yang-Mills model is equivalent to trigonometric Calogero-Moser systems and certain known spin generalizations thereof. Choosing a more general ansatz for the matter charges allows us to obtain and solve novel integrable systems. The key property we use to prove integrability and to solve these systems is gauge invariance of the corresponding Yang-Mills model.
We study a lattice fermion model for superconductivity in the presence of an antiferromagnetic background, described as a fixed external staggered magnetic field. We discuss the possibility that our model provides an effective description of coexistence of antiferromagnetic correlations and superconductivity, and its possible application to high temperature superconductivity. We also argue that, under certain conditions, this model describes a variant of the periodic Anderson model for heavy fermions. Using a path integral formulation we construct mean field equations, which we study in some detail. We evaluate the superconducting critical temperature and show that it is strongly enhanced by antiferromagnetic order. We also evaluate the superconducting gap, the superconducting density of states, and the tunneling conductivity, and show that the most stable channel usually has a $d_{x^2-y^2}$-wave gap.