The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attem... more The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4x4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8x8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.
We extend previous studies of the BCS canonical approach for the attractive Hubbard model. A deri... more We extend previous studies of the BCS canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard and a simpler reduced Hamiltonian. Using direct diagonalization, exact one and two dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one dimensional findings, the BCS is an excellent approximation for the Hubbard ground state energy but not energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wavefunctions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. However for the reduced Hamiltonian model the BCS is an excellent approximation for all observables in both one and two dimensions.
W e introduce a new diagonalization m ethod called quasi-sparse eigenvector diagonalization which... more W e introduce a new diagonalization m ethod called quasi-sparse eigenvector diagonalization which nds the m ost im portant basis vectors ofthe low energy eigenstates ofa quantum Ham iltonian. It can operate using any basis,either orthogonalor non-orthogonal,and any sparse Ham iltonian,either Herm itian,non-Herm itian, nite-dim ensional,or in nite-dim ensional. The m ethod is part ofa new com putationalapproach which com binesboth diagonalization and M onte Carlo techniques.
We introduce a new spectral approach to non-perturbative field theory within the periodic field f... more We introduce a new spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1 + 1 dimensional φ 4 theory, identifying both one-particle and multi-particle contributions. The techniques we discuss have broad applications to the study of particle phenomenology and Minkowski-space dynamics.
We present here a simple proof of the non-existence of a non-periodic invariant point for the qua... more We present here a simple proof of the non-existence of a non-periodic invariant point for the quantum baker's map propagator presented in Rubin and Salwen (Annals of Physics, 1998), for Planck's constant h=1/N and N a positive integer.
Several issues in the modal approach to quantum field theory are discussed. Within the formalism ... more Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hub...
In earlier work, N = (1, 1) super Yang-Mills theory in two dimensions was found to have several i... more In earlier work, N = (1, 1) super Yang-Mills theory in two dimensions was found to have several interesting properties, though these properties could not be investigated in any detail. In this paper we analyze two of these properties. First, we investigate the spectrum of the theory. We calculate the masses of the low-lying states using the supersymmetric discrete light-cone (SDLCQ) approximation and obtain their continuum values. The spectrum exhibits an interesting distribution of masses, which we discuss along with a toy model for this pattern. We also discuss how the average number of partons grows in the bound states. Second, we determine the number of fermions and bosons in the N = (1, 1) and N = (2, 2) theories in each symmetry sector as a function of the resolution. Our finding that the numbers of fermions and bosons in each sector are the same is part of the answer to the question of why the SDLCQ approximation exactly preserves supersymmetry.
We present here a canonical quantization for the baker’s map. The method we use is quite differen... more We present here a canonical quantization for the baker’s map. The method we use is quite different from that used in Balazs and Voros (ref. 1) and Saraceno (ref. 2). We first construct a natural “baker covering map ” on the plane R2. We then use as the quantum algebra of observables the subalgebra of operators on L2 (R) generated by {exp (2πîx),exp (2πîp)}. We construct a unitary propagator such that as � → 0 the classical dynamics is returned. For Planck’s constant h = 1/N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N × N matrix propagator is the same as given in ref. 1 except for a small correction of order h. This correction is shown to preserve the classical symmetry x → 1 − x and p → 1 − p in the quantum dynamics for periodic boundary conditions. 30 pages, 2 figures, 1 table
We propose a method for eliminating the truncation error associated with any subspace diagonaliza... more We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in the initial diagonalization. The method is part of a new approach to computational quantum physics which combines both diagonalization and Monte Carlo techniques.
The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attem... more The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4x4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8x8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.
We present here a complete description of the quantization of the baker's map. The method we use ... more We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators generated by {exp (2πi x) , exp (2πi p)} and construct a unitary propagator such that as → 0, the classical dynamics is returned. For Planck's constant h = 1/N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N × N matrix propagator is the same as given in [BV] except for a small correction of order h. This correction is is shown to preserve the symmetry x → 1 − x and p → 1 − p of the classical map for periodic boundary conditions.
We solve N = (8, 8) super Yang-Mills theory in 1+1 dimensions at strong coupling to directly conf... more We solve N = (8, 8) super Yang-Mills theory in 1+1 dimensions at strong coupling to directly confirm the predictions of supergravity at weak coupling. We do our calculations in the large-N c approximation using Supersymmetric Discrete Light-Cone Quantization with up to 3 × 10 12 basis states. We calculate the stress-energy correlator T ++ (r)T ++ (0) as a function of the separation r and find that at intermediate values of r the correlator behaves as r −5 to within errors as predicted by weak-coupling supergravity. We also present an extension to significantly higher resolution of our earlier results for the same correlator in the N = (2, 2) theory and see that in this theory the correlator has very different behavior at intermediate values of r.
Renormalization in spherical field theory 1 Support provided by the NSF under Grant 5-22968 and PHY-9802709. 1
We derive several results concerning non-perturbative renormalization in the spherical field form... more We derive several results concerning non-perturbative renormalization in the spherical field formalism. Using a small set of local counterterms, we are able to remove all ultraviolet divergences in a manner such that the renormalized theory is finite and translationally invariant. As an explicit example we consider massless φ4 theory in four dimensions.
We present here a simple proof of the non-existence of a non-periodic invariant point for the qua... more We present here a simple proof of the non-existence of a non-periodic invariant point for the quantum baker's map propagator presented in Rubin and Salwen (Annals of Physics, 1998), for Planck's constant h=1/N and N a positive integer.
We present a formulation of N = (1, 1) super Yang-Mills theory in 1+1 dimensions at finite temper... more We present a formulation of N = (1, 1) super Yang-Mills theory in 1+1 dimensions at finite temperature. The partition function is constructed by finding a numerical approximation to the entire spectrum. We solve numerically for the spectrum using Supersymmetric Discrete Light-Cone Quantization (SDLCQ) in the large-N c approximation and calculate the density of states. We find that the density of states grows exponentially and the theory has a Hagedorn temperature, which we extract. We find that the Hagedorn temperature at infinite resolution is slightly less than one in units of g 2 N c /π. We use the density of states to also calculate a standard set of thermodynamic functions below the Hagedorn temperature. In this temperature range, we find that the thermodynamics is dominated by the massless states of the theory.
We consider N = (1, 1) super Yang-Mills theory in 1+1 dimensions with fundamentals at large-N c .... more We consider N = (1, 1) super Yang-Mills theory in 1+1 dimensions with fundamentals at large-N c . A Chern-Simons term is included to give mass to the adjoint partons. Using the spectrum of the theory, we calculate thermodynamic properties of the system as a function of the temperature and the Yang-Mills coupling. In the large-N c limit there are two non-communicating sectors, the glueball sector, which we presented previously, and the meson-like sector that we present here. We find that the meson-like sector dominates the thermodynamics. Like the glueball sector, the meson sector has a Hagedorn temperature T H , and we show that the Hagedorn temperature grows with the coupling. We calculate the temperature and coupling dependence of the free energy for temperatures below T H . As expected, the free energy for weak coupling and low temperature grows quadratically with the temperature. Also the ratio of the free energies at strong coupling compared to weak coupling, r s−w , for low temperatures grows quadratically with T. In addition, our data suggest that r s−w tends to zero in the continuum limit at low temperatures.
ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obt... more ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obtained by dimensionally reducing ${\cal N}=1$ super Yang--Mills theory in four dimension to two dimensions. We do our calculations in the large-$N_c$ approximation using Supersymmetric Discrete Light Cone Quantization. The objective is to calculate quantities that might be investigated by researchers using other numerical methods. We present a precision study of the low-mass spectrum and the stress-energy correlator $<T^{++}(r) T^{++}(0)>$. We find that the mass gap of this theory closes as the numerical resolution goes to infinity and that the correlator in the intermediate $r$ region behaves like $r^{-4.75}$. Comment: 18 pages, 8 figures
ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obt... more ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obtained by dimensionally reducing ${\cal N}=1$ super Yang--Mills theory in four dimension to two dimensions. We do our calculations in the large-$N_c$ approximation using Supersymmetric Discrete Light Cone Quantization. The objective is to calculate quantities that might be investigated by researchers using other numerical methods. We present a precision study of the low-mass spectrum and the stress-energy correlator $<T^{++}(r) T^{++}(0)>$. We find that the mass gap of this theory closes as the numerical resolution goes to infinity and that the correlator in the intermediate $r$ region behaves like $r^{-4.75}$. Comment: 18 pages, 8 figures
We solve N = (8, 8) super-Yang-Mills theory in 1 + 1 dimensions at strong coupling to directly co... more We solve N = (8, 8) super-Yang-Mills theory in 1 + 1 dimensions at strong coupling to directly confirm the predictions of supergravity at weak coupling. We do our calculations in the large-N c approximation using Supersymmetric Discrete Light-Cone Quantization with up to 3 × 10 12 basis states. We calculate the stress-energy correlator T ++ (r)T ++ (0) as a function of the separation r and find that at intermediate values of r the correlator behaves as r −5 to within errors as predicted by weak-coupling supergravity. We also present an extension to significantly higher resolution of our earlier results for the same correlator in the N = (2, 2) theory and see that in this theory the correlator has very different behavior at intermediate values of r.
Introduction to stochastic error correction methods
Physics Letters B, 2001
We propose a method for eliminating the truncation error associated with any subspace diagonaliza... more We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in ...
The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attem... more The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4x4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8x8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.
We extend previous studies of the BCS canonical approach for the attractive Hubbard model. A deri... more We extend previous studies of the BCS canonical approach for the attractive Hubbard model. A derivation of the BCS formulation is presented for both the Hubbard and a simpler reduced Hamiltonian. Using direct diagonalization, exact one and two dimensional solutions for both Hamiltonians are compared to BCS variational calculations. Approximate and exact ground state energies and energy gaps are computed for different electron number systems as well as correlation observables not previously predicted. Reproducing published one dimensional findings, the BCS is an excellent approximation for the Hubbard ground state energy but not energy gap, a finding that remains true in two dimensions. Propagators and correlators are found more sensitive to wavefunctions and appreciable differences are computed with the Hubbard model exhibiting a weaker degree of superconductivity than the BCS. However for the reduced Hamiltonian model the BCS is an excellent approximation for all observables in both one and two dimensions.
W e introduce a new diagonalization m ethod called quasi-sparse eigenvector diagonalization which... more W e introduce a new diagonalization m ethod called quasi-sparse eigenvector diagonalization which nds the m ost im portant basis vectors ofthe low energy eigenstates ofa quantum Ham iltonian. It can operate using any basis,either orthogonalor non-orthogonal,and any sparse Ham iltonian,either Herm itian,non-Herm itian, nite-dim ensional,or in nite-dim ensional. The m ethod is part ofa new com putationalapproach which com binesboth diagonalization and M onte Carlo techniques.
We introduce a new spectral approach to non-perturbative field theory within the periodic field f... more We introduce a new spectral approach to non-perturbative field theory within the periodic field formalism. As an example we calculate the real and imaginary parts of the propagator in 1 + 1 dimensional φ 4 theory, identifying both one-particle and multi-particle contributions. The techniques we discuss have broad applications to the study of particle phenomenology and Minkowski-space dynamics.
We present here a simple proof of the non-existence of a non-periodic invariant point for the qua... more We present here a simple proof of the non-existence of a non-periodic invariant point for the quantum baker's map propagator presented in Rubin and Salwen (Annals of Physics, 1998), for Planck's constant h=1/N and N a positive integer.
Several issues in the modal approach to quantum field theory are discussed. Within the formalism ... more Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hub...
In earlier work, N = (1, 1) super Yang-Mills theory in two dimensions was found to have several i... more In earlier work, N = (1, 1) super Yang-Mills theory in two dimensions was found to have several interesting properties, though these properties could not be investigated in any detail. In this paper we analyze two of these properties. First, we investigate the spectrum of the theory. We calculate the masses of the low-lying states using the supersymmetric discrete light-cone (SDLCQ) approximation and obtain their continuum values. The spectrum exhibits an interesting distribution of masses, which we discuss along with a toy model for this pattern. We also discuss how the average number of partons grows in the bound states. Second, we determine the number of fermions and bosons in the N = (1, 1) and N = (2, 2) theories in each symmetry sector as a function of the resolution. Our finding that the numbers of fermions and bosons in each sector are the same is part of the answer to the question of why the SDLCQ approximation exactly preserves supersymmetry.
We present here a canonical quantization for the baker’s map. The method we use is quite differen... more We present here a canonical quantization for the baker’s map. The method we use is quite different from that used in Balazs and Voros (ref. 1) and Saraceno (ref. 2). We first construct a natural “baker covering map ” on the plane R2. We then use as the quantum algebra of observables the subalgebra of operators on L2 (R) generated by {exp (2πîx),exp (2πîp)}. We construct a unitary propagator such that as � → 0 the classical dynamics is returned. For Planck’s constant h = 1/N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N × N matrix propagator is the same as given in ref. 1 except for a small correction of order h. This correction is shown to preserve the classical symmetry x → 1 − x and p → 1 − p in the quantum dynamics for periodic boundary conditions. 30 pages, 2 figures, 1 table
We propose a method for eliminating the truncation error associated with any subspace diagonaliza... more We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in the initial diagonalization. The method is part of a new approach to computational quantum physics which combines both diagonalization and Monte Carlo techniques.
The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attem... more The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4x4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8x8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.
We present here a complete description of the quantization of the baker's map. The method we use ... more We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators generated by {exp (2πi x) , exp (2πi p)} and construct a unitary propagator such that as → 0, the classical dynamics is returned. For Planck's constant h = 1/N, we show that the dynamics can be reduced to the dynamics on an N-dimensional Hilbert space, and the unitary N × N matrix propagator is the same as given in [BV] except for a small correction of order h. This correction is is shown to preserve the symmetry x → 1 − x and p → 1 − p of the classical map for periodic boundary conditions.
We solve N = (8, 8) super Yang-Mills theory in 1+1 dimensions at strong coupling to directly conf... more We solve N = (8, 8) super Yang-Mills theory in 1+1 dimensions at strong coupling to directly confirm the predictions of supergravity at weak coupling. We do our calculations in the large-N c approximation using Supersymmetric Discrete Light-Cone Quantization with up to 3 × 10 12 basis states. We calculate the stress-energy correlator T ++ (r)T ++ (0) as a function of the separation r and find that at intermediate values of r the correlator behaves as r −5 to within errors as predicted by weak-coupling supergravity. We also present an extension to significantly higher resolution of our earlier results for the same correlator in the N = (2, 2) theory and see that in this theory the correlator has very different behavior at intermediate values of r.
Renormalization in spherical field theory 1 Support provided by the NSF under Grant 5-22968 and PHY-9802709. 1
We derive several results concerning non-perturbative renormalization in the spherical field form... more We derive several results concerning non-perturbative renormalization in the spherical field formalism. Using a small set of local counterterms, we are able to remove all ultraviolet divergences in a manner such that the renormalized theory is finite and translationally invariant. As an explicit example we consider massless φ4 theory in four dimensions.
We present here a simple proof of the non-existence of a non-periodic invariant point for the qua... more We present here a simple proof of the non-existence of a non-periodic invariant point for the quantum baker's map propagator presented in Rubin and Salwen (Annals of Physics, 1998), for Planck's constant h=1/N and N a positive integer.
We present a formulation of N = (1, 1) super Yang-Mills theory in 1+1 dimensions at finite temper... more We present a formulation of N = (1, 1) super Yang-Mills theory in 1+1 dimensions at finite temperature. The partition function is constructed by finding a numerical approximation to the entire spectrum. We solve numerically for the spectrum using Supersymmetric Discrete Light-Cone Quantization (SDLCQ) in the large-N c approximation and calculate the density of states. We find that the density of states grows exponentially and the theory has a Hagedorn temperature, which we extract. We find that the Hagedorn temperature at infinite resolution is slightly less than one in units of g 2 N c /π. We use the density of states to also calculate a standard set of thermodynamic functions below the Hagedorn temperature. In this temperature range, we find that the thermodynamics is dominated by the massless states of the theory.
We consider N = (1, 1) super Yang-Mills theory in 1+1 dimensions with fundamentals at large-N c .... more We consider N = (1, 1) super Yang-Mills theory in 1+1 dimensions with fundamentals at large-N c . A Chern-Simons term is included to give mass to the adjoint partons. Using the spectrum of the theory, we calculate thermodynamic properties of the system as a function of the temperature and the Yang-Mills coupling. In the large-N c limit there are two non-communicating sectors, the glueball sector, which we presented previously, and the meson-like sector that we present here. We find that the meson-like sector dominates the thermodynamics. Like the glueball sector, the meson sector has a Hagedorn temperature T H , and we show that the Hagedorn temperature grows with the coupling. We calculate the temperature and coupling dependence of the free energy for temperatures below T H . As expected, the free energy for weak coupling and low temperature grows quadratically with the temperature. Also the ratio of the free energies at strong coupling compared to weak coupling, r s−w , for low temperatures grows quadratically with T. In addition, our data suggest that r s−w tends to zero in the continuum limit at low temperatures.
ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obt... more ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obtained by dimensionally reducing ${\cal N}=1$ super Yang--Mills theory in four dimension to two dimensions. We do our calculations in the large-$N_c$ approximation using Supersymmetric Discrete Light Cone Quantization. The objective is to calculate quantities that might be investigated by researchers using other numerical methods. We present a precision study of the low-mass spectrum and the stress-energy correlator $<T^{++}(r) T^{++}(0)>$. We find that the mass gap of this theory closes as the numerical resolution goes to infinity and that the correlator in the intermediate $r$ region behaves like $r^{-4.75}$. Comment: 18 pages, 8 figures
ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obt... more ABSTRACT We consider the (1+1)-dimensional ${\cal N}=(2,2)$ super Yang--Mills theory which is obtained by dimensionally reducing ${\cal N}=1$ super Yang--Mills theory in four dimension to two dimensions. We do our calculations in the large-$N_c$ approximation using Supersymmetric Discrete Light Cone Quantization. The objective is to calculate quantities that might be investigated by researchers using other numerical methods. We present a precision study of the low-mass spectrum and the stress-energy correlator $<T^{++}(r) T^{++}(0)>$. We find that the mass gap of this theory closes as the numerical resolution goes to infinity and that the correlator in the intermediate $r$ region behaves like $r^{-4.75}$. Comment: 18 pages, 8 figures
We solve N = (8, 8) super-Yang-Mills theory in 1 + 1 dimensions at strong coupling to directly co... more We solve N = (8, 8) super-Yang-Mills theory in 1 + 1 dimensions at strong coupling to directly confirm the predictions of supergravity at weak coupling. We do our calculations in the large-N c approximation using Supersymmetric Discrete Light-Cone Quantization with up to 3 × 10 12 basis states. We calculate the stress-energy correlator T ++ (r)T ++ (0) as a function of the separation r and find that at intermediate values of r the correlator behaves as r −5 to within errors as predicted by weak-coupling supergravity. We also present an extension to significantly higher resolution of our earlier results for the same correlator in the N = (2, 2) theory and see that in this theory the correlator has very different behavior at intermediate values of r.
Introduction to stochastic error correction methods
Physics Letters B, 2001
We propose a method for eliminating the truncation error associated with any subspace diagonaliza... more We propose a method for eliminating the truncation error associated with any subspace diagonalization calculation. The new method, called stochastic error correction, uses Monte Carlo sampling to compute the contribution of the remaining basis vectors not included in ...
Uploads
Papers by Nathan Salwen