Papers by Marvin Weinstein
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Dec 16, 2009
Bulletin of the American Physical Society, Feb 29, 2012
Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition MARVIN WEINSTEIN... more Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition MARVIN WEINSTEIN, SLAC National Accelerator Laboratory, RAVI CHANDRA, ASSA AUERBACH, Technion, Haifa -We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements. In contrast to variational approaches and most implementations of DMRG, Lanczos rotations towards the ground state do not involve incremental minimizations, (e.g. sweeping procedures) which may get stuck in false local minima. The lattice of size N is partitioned into two subclusters. At each iteration the rotating Lanczos vector is compressed into two sets of n svd small subcluster vectors using singular value decomposition. For low entanglement entropy S ee , (satisfied by short range Hamiltonians), the truncation error is bounded by exp(-n 1/S ee svd ). Convergence is tested for the Heisenberg model on Kagomé clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm to multiple partitioning is discussed, and comparisons to other techniques are given. Reference: arXiv:1105.
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information), Oct 19, 2005
Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstate... more Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that are widely believed not to be solvable by such methods. The novel feature of adaptive perturbation theory is that it decomposes a given Hamiltonian, H, into an unperturbed part and a perturbation in a way which extracts the leading non-perturbative behavior of the problem exactly. In this talk I will introduce the method in the context of the pure anharmonic oscillator and then apply it to the case of tunneling between symmetric minima. After that, I will show how this method can be applied to field theory. In that discussion I will show how one can non-perturbatively extract the structure of mass, wavefunction and coupling constant renormalization.
In this talk I show how to canonically quantize a massless scalar field in the background of a Sc... more In this talk I show how to canonically quantize a massless scalar field in the background of a Schwarzschild black hole in Lemaître coordinates and then present a simplified derivation of Hawking radiation based upon this procedure. The key result of quantization procedure is that the Hamiltonian of the system is explicitly time dependent and so problem is intrinsically non-static. From this it follows that, although a unitary time-development operator exists, it is not useful to talk about vacuum states; rather, one should focus attention on steady state phenomena such as the Hawking radiation. In order to clarify the approximations used to study this problem I begin by discussing the related problem of the massless scalar field theory calculated in the presence of a moving mirror.
Nuclear physics, Feb 1, 2010
Physical Review E, Nov 9, 2011
We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically red... more We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements, without restricting to variational ansatzes. The lattice of size N is partitioned into two subclusters. At each iteration the Lanczos vector is projected into two sets of n svd smaller subcluster vectors using singular value decomposition. For low entanglement entropy See, (satisfied by short range Hamiltonians), the truncation error is expected to vanish as exp(−n 1/See svd). Convergence is tested for the Heisenberg model on Kagomé clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm to multiple partitioning is discussed, and comparisons to other techniques are given.
Physical review, Jan 11, 2000
The Contractor Renormalization Group method (CORE) is used to establish the equivalence of variou... more The Contractor Renormalization Group method (CORE) is used to establish the equivalence of various Hamiltonian free fermion theories and a class of generalized frustrated antiferromagnets. In particular, after a detailed discussion of a simple example, it is argued that a generalized frustrated SU(3) antiferromagnet whose single-site states have the quantum numbers of mesons and baryons is equivalent to a theory of free massless quarks. Furthermore, it is argued that for slight modification of the couplings which define the frustrated antiferromagnet Hamiltonian, the theory becomes a theory of quarks interacting with color gauge-fields.
I discuss the problem of inflation in the context of Friedmann-Robertson-Walker Cosmology and sho... more I discuss the problem of inflation in the context of Friedmann-Robertson-Walker Cosmology and show how, after a simple change of variables, to quantize the problem in a way which parallels the classical discussion. The result is that two of the Einstein equations arise as exact equations of motion and one of the usual Einstein equations (suitably quantized) survives as a constraint equation to be imposed on the space of physical states. However, the Friedmann equation, which is also a constraint equation and which is the basis of the Wheeler-deWitt equation, acquires a welcome quantum correction that becomes significant for small scale factors. To clarify how things work in this formalism I briefly outline the way in which our formalism works for the exactly solvable case of de-Sitter space.
Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstate... more Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that heretofore were not believed to be treatable by such methods. The novel feature of adaptive perturbation theory is that it decomposes a given Hamiltonian, H, into an unperturbed part and a perturbation in a way which extracts the leading non-perturbative behavior of the problem exactly. This paper introduces the method in the context of the pure anharmonic oscillator and then goes on to apply it to the case of tunneling between both symmetric and asymmetric minima. It concludes with an introduction to the extension of these methods to the discussion of a quantum field theory. A more complete discussion of this issue will be given in the second paper in this series, and it will show how to use the method of adaptive perturbation theory to non-perturbatively extract the structure of mass, wavefunction and coupling constant renormalization.
The COntractor REnormalization group (CORE) approximation, a new method for solving Hamiltonian l... more The COntractor REnormalization group (CORE) approximation, a new method for solving Hamiltonian lattice systems, is introduced. The approach c o m bines variational and contraction techniques with the real-space renormalization group approach and is systematically improvable. Since it applies to lattice systems of in nite extent, the method is suitable for studying critical phenomena and phase structure systems with dynamical fermions can also be treated. The method is tested using the 1+1-dimensional Ising model.
Physical review, Sep 15, 1974
A new method of renormalizing Abelian gauge theories without regulators is developed in this seri... more A new method of renormalizing Abelian gauge theories without regulators is developed in this series of papers. In the present part infrared problems are avoided by modifying the Lagrangian such that all masses are non-vanishing.
Physical review, Aug 15, 1982
Physical review, Oct 9, 2000
In order to better understand what to expect from numerical CORE computations for two-dimensional... more In order to better understand what to expect from numerical CORE computations for two-dimensional massless QED (the Schwinger model) we wish to obtain some analytic control over the approach to the continuum limit for various choices of fermion derivative. To this end we study the Hamiltonian formulation of the lattice Schwinger model (i.e., the theory defined on the spatial lattice with continuous time) in A 0 = 0 gauge. We begin with a discussion of the solution of the Hamilton equations of motion in the continuum, we then parallel the derivation of the continuum solution within the lattice framework for a range of fermion derivatives. The equations of motion for the Fourier transform of the lattice charge density operator show explicitly why it is a regulated version of this operator which corresponds to the pointsplit operator of the continuum theory and the sense in which the regulated lattice operator can be treated as a Bose field. The same formulas explicitly exhibit operators whose matrix elements measure the lack of approach to the continuum physics. We show that both chirality violating Wilson-type and chirality preserving SLAC-type derivatives correctly reproduce the continuum theory and show that there is a clear connection between the strong and weak coupling limits of a theory based upon a generalized SLAC-type derivative.
Physical Review Letters, Nov 24, 1986
Physical review, Oct 15, 1973
This paper discusses the role played by conserved currents in fixing the structure of currently p... more This paper discusses the role played by conserved currents in fixing the structure of currently popular renormalizable theories of strong electromagnetic and weak interactions. The major objective of this work is to show that these theories correspond to another kind of symmetry-which we call a Higgs-type symmetry-and to clarify the relation of this scheme to the already familiar normal and Goldstone symmetries. In order to do this, we introduce a language * which makes no reference to any specific Lagrangian formalism and so avoids questions of whether or not hadrons are composite and whether or not the Goldstone bosons (massless particles of these theories) necessarily-have massive partners, For pedagogical reasons, we discuss the original Weinberg model of leptons and a model coupling leptons and hadrons from the current algebra point of view.
Physical review, Apr 15, 1985
This paper develops a framework which allows us to treat the topology and dimension of the space-... more This paper develops a framework which allows us to treat the topology and dimension of the space-time continuum as dynamically generated. We present examples of quantum systems which are defined without a notion of space, but which nevertheless undergo a transition to a space-time phase. The dimension of the space is an integer valued order parameter which characterizes distinct phases of a single system. We also show the interactions between the low energy particles of the system are gauge-like. Finally, we discuss the computability of Newton's constant in this class of theories.
International Journal of Modern Physics D, Sep 1, 2004
We develop a Hamiltonian formalism which can be used to discuss the physics of a massless scalar ... more We develop a Hamiltonian formalism which can be used to discuss the physics of a massless scalar field in a gravitational background of a Schwarzschild black hole. Using this formalism we show that the time evolution of the system is unitary and yet all known results such as the existence of Hawking radiation can be readily understood. We then point out that the Hamiltonian formalism leads to interesting observations about black hole entropy and the information paradox.
arXiv (Cornell University), Jan 11, 2011
This paper makes the simple observation that a fundamental length, or cutoff, in the context of F... more This paper makes the simple observation that a fundamental length, or cutoff, in the context of Friedmann-Lemaître-Robertson-Walker (FRW) cosmology implies very different things than for a static universe. It is argued that it is reasonable to assume that this cutoff is implemented by fixing the number of quantum degrees of freedom per co-moving volume (as opposed to a Planck volume) and the relationship of the vacuum-energy of all of the fields in the theory to the cosmological constant (or dark energy) is reexamined. The restrictions that need to be satisfied by a generic theory to avoid conflicts with current experiments are discussed, and it is shown that in any theory satisfying these constraints knowing the difference between w and minus one allows one to predictẇ. It is argued that this is a robust result and if this prediction fails the idea of a fundamental cutoff of the type being discussed can be ruled out. Finally, it is observed that, within the context of a specific theory, a co-moving cutoff implies a predictable time variation of fundamental constants. This is accompanied by a general discussion of why this is so, what are the strongest phenomenological limits upon this predicted variation, and which limits are in tension with the idea of a co-moving cutoff. It is pointed out, however, that a careful comparison of the predicted time variation of fundamental constants is not possible without restricting to a particular model field-theory and that is not done in this paper.
This paper discusses the problem of inflation in the context of Friedmann-Robertson-Walker Cosmol... more This paper discusses the problem of inflation in the context of Friedmann-Robertson-Walker Cosmology. We show how, after a simple change of variables, one can quantize the problem in a way which parallels the classical discussion. The result is that two of the Einstein equations arise as exact equations of motion and one of the usual Einstein equations (suitably quantized) survives as a constraint equation to be imposed on the space of physical states. However, the Friedmann equation, which is also a constraint equation and which is the basis of the Wheeler-DeWitt equation, acquires a welcome quantum correction that becomes significant for small scale factors. We discuss the extension of this result to a full quantum mechanical derivation of the anisotropy (δρ/ρ) in the cosmic microwave background radiation, and the possibility that the extra term in the Friedmann equation could have observable consequences. Finally, we suggest interesting ways in which these techniques can be generalized to cast light on the question of chaotic or eternal inflation. In particular, we suggest one can put an experimental lower bound on the distance to a universe with a scale factor very different from our own, by looking at its effects on our CMB radiation.
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Papers by Marvin Weinstein