Generalized Supersymmetric Quantum Mechanics and Reflectionless Fermion Bags in 1+1 Dimensions

We study static fermion bags in the 1+1 dimensional Gross-Neveu and Nambu-Jona-Lasinio models. It has been known, from the work of Dashen, Hasslacher and Neveu (DHN), followed by Shei's work, in the 1970's, that the self-consistent static fermion bags in these models are reflectionless. The works of DHN and of Shei were based on inverse scattering theory. Several years ago, we offered an alternative argument to establish the reflectionless nature of these fermion bags, which was based on analysis of the spatial asymptotic behavior of the resolvent of the Dirac operator in the background of a static bag, subjected to the appropriate boundary conditions. We also calculated the masses of fermion bags based on the resolvent and the Gelfand-Dikii identity. Based on arguments taken from a certain generalized one dimensional supersymmetric quantum mechanics, which underlies the spectral theory of these Dirac operators, we now realize that our analysis of the asymptotic behavior of the resolvent was incomplete. We offer here a critique of our asymptotic argument.Comment: 33 pages, 2 figure

Dynamical Generation of Extended Objects in a 1+11+1 Dimensional Chiral Field Theory: Non-Perturbative Dirac Operator Resolvent Analysis

We analyze the $1+1$ dimensional Nambu-Jona-Lasinio model non-perturbatively. In addition to its simple ground state saddle points, the effective action of this model has a rich collection of non-trivial saddle points in which the composite fields \sigx=\lag\bar\psi\psi\rag and \pix=\lag\bar\psi i\gam_5\psi\rag form static space dependent configurations because of non-trivial dynamics. These configurations may be viewed as one dimensional chiral bags that trap the original fermions (``quarks") into stable extended entities (``hadrons"). We provide explicit expressions for the profiles of these objects and calculate their masses. Our analysis of these saddle points is based on an explicit representation we find for the diagonal resolvent of the Dirac operator in a \{\sigx, \pix\} background which produces a prescribed number of bound states. We analyse in detail the cases of a single as well as two bound states. We find that bags that trap $N$ fermions are the most stable ones, because they release all the fermion rest mass as binding energy and become massless. Our explicit construction of the diagonal resolvent is based on elementary Sturm-Liouville theory and simple dimensional analysis and does not depend on the large $N$ approximation. These facts make it, in our view, simpler and more direct than the calculations previously done by Shei, using the inverse scattering method following Dashen, Hasslacher, and Neveu. Our method of finding such non-trivial static configurations may be applied to other $1+1$ dimensional field theories

Does the complex deformation of the Riemann equation exhibit shocks?

The Riemann equation $u_t+uu_x=0$, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is \cP\cT symmetric. A one-parameter \cP\cT-invariant complex deformation of this equation, $u_t-iu(iu_x)^\epsilon= 0$ ($\epsilon$ real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless $\epsilon$ is an odd integer.Comment: latex, 8 page

On Kinks and Bound States in the Gross-Neveu Model

We investigate static space dependent \sigx=\lag\bar\psi\psi\rag saddle point configurations in the two dimensional Gross-Neveu model in the large N limit. We solve the saddle point condition for \sigx explicitly by employing supersymmetric quantum mechanics and using simple properties of the diagonal resolvent of one dimensional Schr\"odinger operators rather than inverse scattering techniques. The resulting solutions in the sector of unbroken supersymmetry are the Callan-Coleman-Gross-Zee kink configurations. We thus provide a direct and clean construction of these kinks. In the sector of broken supersymmetry we derive the DHN saddle point configurations. Our method of finding such non-trivial static configurations may be applied also in other two dimensional field theories.Comment: Revised version. A new section added with derivation of the DHN static configurations in the sector of broken supersymmetry. Some references added as well. 25 pp, latex, e-mail [email protected]

Renormalizing Rectangles and Other Topics in Random Matrix Theory

We consider random Hermitian matrices made of complex or real $M\times N$ rectangular blocks, where the blocks are drawn from various ensembles. These matrices have $N$ pairs of opposite real nonvanishing eigenvalues, as well as $M-N$ zero eigenvalues (for $M>N$.) These zero eigenvalues are ``kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large $N,M$ limit, in which the ``rectangularity" $r={M\over N}$ is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large $N$ renormalization techniques. In addition to the kinematical $\delta$-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if $|r-1|$ is held fixed as $N\rightarrow\infty$, the $N$ non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As $r\rightarrow 1$ the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that $r\rightarrow 1$ drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.Comment: LateX, 34 pages, 3 ps figure

"Single Ring Theorem" and the Disk-Annulus Phase Transition

Recently, an analytic method was developed to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the existing Gaussian non-hermitean literature. One obtains an explicit algebraic equation for the integrated density of eigenvalues from which the Green's function and averaged density of eigenvalues could be calculated in a simple manner. Thus, that formalism may be thought of as the non-hermitean analog of the method due to Br\'ezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. A somewhat surprising result is the so called "Single Ring" theorem, namely, that the domain of the eigenvalue distribution in the complex plane is either a disk or an annulus. In this paper we extend previous results and provide simple new explicit expressions for the radii of the eigenvalue distiobution and for the value of the eigenvalue density at the edges of the eigenvalue distribution of the non-hermitean matrix in terms of moments of the eigenvalue distribution of the associated hermitean matrix. We then present several numerical verifications of the previously obtained analytic results for the quartic ensemble and its phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we demonstrate numerically the "Single Ring" theorem for the sextic potential, namely, the potential of lowest degree for which the "Single Ring" theorem has non-trivial consequences.Comment: latex, 5 eps figures, 41 page

Non-Hermitean Random Matrix Theory: method of hermitization

We consider random non-hermitean matrices in the large $N$ limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitean random matrices, in contrast to hermitean random matrices. To overcome this difficulty, we show that associated to each ensemble of non-hermitean matrices there is an auxiliary ensemble of random hermitean matrices which can be analyzed by the usual methods. We then extract the Green's function and the density of eigenvalues of the non-hermitean ensemble from those of the auxiliary ensemble. We apply this "method of hermitization" to several examples, and discuss a number of related issues.Comment: 46 pages, 3 ps figures, LaTe

Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism

We apply the recently introduced method of hermitization to study in the large $N$ limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the recent Gaussian non-hermitean literature. We develop the general formalism for calculating the Green's function and averaged density of eigenvalues, which may be thought of as the non-hermitean analog of the method due to Br\`ezin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. We obtain an explicit algebraic equation for the integrated density of eigenvalues. A somewhat surprising result of that equation is that the shape of the eigenvalue distribution in the complex plane is either a disk or an annulus. As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitean random variables. We use this formalism to state and prove a non-abelian non-hermitean version of the central limit theorem.Comment: 40 pages, no figures, LaTex. Section 5 has been correcte

The Response to a Perturbation in the Reflection Amplitude

We apply inverse scattering theory to calculate the functional derivative of the potential $V(x)$ and wave function $\psi(x,k)$ of a one-dimensional Schr\"odinger operator with respect to the reflection amplitude $r(k)$.Comment: 16 pages, no figure