Ramis Movassagh | Massachusetts Institute of Technology (MIT) (original) (raw)
Papers by Ramis Movassagh
Physical Review Letters
Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions,... more Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in translationally invariant systems, we show that disorder can be engineered to stabilize the subharmonic response of Majorana modes. A viable observation of this phenomenon can be achieved using modern multiqubit hardware, such as superconducting circuits and cold atomic systems.
Physical review letters, Jan 21, 2018
In recent experiments, time-dependent periodic fields are used to create exotic topological phase... more In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These nonequilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder similar to equilibrium topological phases. However, proving the existence of such topological phases in a general setting is an open problem. We propose a universal effective theory that leverages on modern free probability theory and ideas in random matrices to analytically predict the existence of the topological phase for finite driving frequencies and across a range of disorder. We find that, depending on the strength of disorder, such systems may be topological or trivial and that there is a transition between the two. In particular, the theory predicts the critical point for the transition between the two phases and provides the critical exponents. We corroborate our result...
Physical review letters, 2017
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a c... more We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work...
Chemical reviews, Jan 23, 2018
In this paper, we explore quantum interference (QI) in molecular conductance from the point of vi... more In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's...
We generalize the previous results of [1] by proving unfrustration condition and degeneracy of th... more We generalize the previous results of [1] by proving unfrustration condition and degeneracy of the ground states of qudits (d-dimensional spins) on a k-child tree with generic local interactions. We find that the dimension of the ground space grows doubly exponentially in the region where rk<=(d^2)/4 for k>1. Further, we extend the results in [1] by proving that there are no zero energy ground states when r>(d^2)/4 for k=1 implying that the effective Hamiltonian is invertible.
Bulletin of the American Physical Society, Mar 5, 2015
By Bernoulli's law, an increase in the relative speed of a fluid around a body is accompanies by ... more By Bernoulli's law, an increase in the relative speed of a fluid around a body is accompanies by a decrease in the pressure. Therefore, a rotating body in a fluid stream experiences a force perpendicular to the motion of the fluid because of the unequal relative speed of the fluid across its surface. It is well known that light has a constant speed irrespective of the relative motion. Does a rotating body immersed in a stream of photons experience a Bernoulli-like force? We show that, indeed, a rotating dielectric cylinder experiences such a lateral force from an electromagnetic wave. In fact, the sign of the lateral force is the same as that of the fluid-mechanical analogue as long as the electric susceptibility is positive (ϵ > ϵ 0), but for negative-susceptibility materials (e.g. metals) we show that the lateral force is in the opposite direction. Because these results are derived from a classical electromagnetic scattering problem, Mie-resonance enhancements that occur in other scattering phenomena also enhance the lateral force.
ABSTRACT We approximate the density of states in disordered systems by decomposing the Hamiltonia... more ABSTRACT We approximate the density of states in disordered systems by decomposing the Hamiltonian into two random matrices and constructing their free convolution. The error in this approximation is determined using asymptotic moment expansions. Each moment can be decomposed into contributions from specific joint moments of the random matrices; each of which has a combinatorial interpretation as the weighted sum of returning trajectories. We show how the error, like the free convolution itself, can be calculated without explicit diagonalization of the Hamiltonian. We apply our theory to Hamiltonians for one-dimensional tight binding models with Gaussian and semicircular site disorder. We find that the particular choice of decomposition crucially determines the accuracy of the resultant density of states. From a partitioning of the Hamiltonian into diagonal and off-diagonal components, free convolution produces an approximate density of states which is correct to the eighth moment. This allows us to explain the accuracy of mean field theories such as the coherent potential approximation, as well as the results of isotropic entanglement theory.
This article puts forth a process applicable to central force scatterings. Under certain assumpti... more This article puts forth a process applicable to central force scatterings. Under certain assumptions, we show that in attractive force fields a high speed particle with a small mass speeding through space, statistically loses energy by colliding softly with large masses that move slowly and randomly. Furthermore, we show that the opposite holds in repulsive force fields: the small particle statistically gains energy. This effect is small and is mainly due to asymmetric energy exchange of the transverse (i.e., perpendicular) collisions. We derive a formula that quantifies this effect (Eq.(12)). We then put this work in a broader statistical context and discuss its consistency with established results.
Geometrization of dynamics consists of representing trajectories by geodesics on a configuration ... more Geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical-and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme. I. STABILITY OF HAMILTONIAN SYSTEMS
The second part is devoted to ground states and gap of QMBS. I first give the necessary backgroun... more The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free (FF) Hamiltonians, real and imaginary time evolution within MPS representation and a numerical implementation. I then prove the degeneracy and FF condition for quantum spin chains with generic local interactions, including corrections to our earlier assertions. I then summarize my efforts in proving lower bounds for the entanglement of the ground states, which includes some new results, with the hope that they inspire future work resulting in solving the conjecture given therein. Next I discuss two interesting measure zero examples where FF Hamiltonians are carefully constructed to give unique ground states with high entanglement. One of the examples (i.e., d=4d=4d=4) has not appeared elsewhere. In particular, we calculate the Schmidt numbers exactly, entanglement entropies and introduce a novel technique for calculating the gap which may be of independent interest. The last chapter elaborates on one of the measure zero examples (i.e., d=3d=3d=3) which is the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits signatures of a critical behavior.
Journal of Mathematical Physics
Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid st... more Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d−dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.
Journal of Mathematical Physics
Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid st... more Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d−dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.
Proceedings of the National Academy of Sciences
Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simulta... more Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially ...
Eprint Arxiv Math Ph 0610084, Oct 1, 2006
Geometrization of dynamics using (non)-affine parametization of arc length with time is investiga... more Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi metric results in positive values of geometrical lyapunov exponent. The non-physical instabilities are shown to be due to a non-affine parametrization. In addition the degree of instability is a monotonically increasing function of the fluctuations in the kinetic energy. We argue that the Jacobi metric gives equivalent results as Eisenhart metric for ergodic systems at equilibrium, where number of degrees of freedom NtoinftyN\to\inftyNtoinfty. We conclude that, in addition to being computationally more expensive, geometrization using the Jacobi metric is meaningful only when the kinetic energy of the system is a positive constant.
Eprint Arxiv 1008 0875, Aug 4, 2010
This article puts forth a process applicable to central force scatterings. Under certain assumpti... more This article puts forth a process applicable to central force scatterings. Under certain assumptions, we show that in attractive force fields a high speed particle with a small mass speeding through space, statistically loses energy by colliding softly with large masses that move slowly and randomly. Furthermore, we show that the opposite holds in repulsive force fields: the small particle statistically gains energy. This effect is small and is mainly due to asymmetric energy exchange of the transverse (i.e., perpendicular) collisions. We derive a formula that quantifies this effect (Eq. 12). We then put this work in a broader statistical context and discuss its consistency with established results.
ACS nano, Jan 29, 2015
An exponential falloff with separation of electron transfer and transport through molecular wires... more An exponential falloff with separation of electron transfer and transport through molecular wires is observed and has attracted theoretical attention. In this study, the attenuation of transmission in linear and cyclic polyenes is related to bond alternation. The explicit form of the zeroth Green's function in a Hückel model for bond-alternated polyenes leads to an analytical expression of the conductance decay factor β. The β values calculated from our model (βCN values, per repeat unit of double and single bond) range from 0.28 to 0.37, based on carotenoid crystal structures. These theoretical β values are slightly smaller than experimental values. The difference can be assigned to the effect of anchoring groups, which are not included in our model. A local transmission analysis for cyclic polyenes, and for [14]annulene in particular, shows that bond alternation affects dramatically not only the falloff behavior but also the choice of a transmission pathway by electrons. Trans...
Geometrization of dynamics consists of representing trajectories by geodesics on a suitably defin... more Geometrization of dynamics consists of representing trajectories by geodesics on a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical- and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme.
The Journal of chemical physics, Jan 14, 2014
The explicit form of the zeroth Green's function in the Hückel model, approximated by the neg... more The explicit form of the zeroth Green's function in the Hückel model, approximated by the negative of the inverse of the Hückel matrix, has direct quantum interference consequences for molecular conductance. We derive a set of rules for transmission between two electrodes attached to a polyene, when the molecule is extended by an even number of carbons at either end (transmission unchanged) or by an odd number of carbons at both ends (transmission turned on or annihilated). These prescriptions for the occurrence of quantum interference lead to an unexpected consequence for switches which realize such extension through electrocyclic reactions: for some specific attachment modes the chemically closed ring will be the ON position of the switch. Normally the signs of the entries of the Green's function matrix are assumed to have no physical significance; however, we show that the signs may have observable consequences. In particular, in the case of multiple probe attachments - i...
2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers, 2010
Physical Review Letters
Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions,... more Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in translationally invariant systems, we show that disorder can be engineered to stabilize the subharmonic response of Majorana modes. A viable observation of this phenomenon can be achieved using modern multiqubit hardware, such as superconducting circuits and cold atomic systems.
Physical review letters, Jan 21, 2018
In recent experiments, time-dependent periodic fields are used to create exotic topological phase... more In recent experiments, time-dependent periodic fields are used to create exotic topological phases of matter with potential applications ranging from quantum transport to quantum computing. These nonequilibrium states, at high driving frequencies, exhibit the quintessential robustness against local disorder similar to equilibrium topological phases. However, proving the existence of such topological phases in a general setting is an open problem. We propose a universal effective theory that leverages on modern free probability theory and ideas in random matrices to analytically predict the existence of the topological phase for finite driving frequencies and across a range of disorder. We find that, depending on the strength of disorder, such systems may be topological or trivial and that there is a transition between the two. In particular, the theory predicts the critical point for the transition between the two phases and provides the critical exponents. We corroborate our result...
Physical review letters, 2017
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a c... more We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work...
Chemical reviews, Jan 23, 2018
In this paper, we explore quantum interference (QI) in molecular conductance from the point of vi... more In this paper, we explore quantum interference (QI) in molecular conductance from the point of view of graph theory and walks on lattices. By virtue of the Cayley-Hamilton theorem for characteristic polynomials and the Coulson-Rushbrooke pairing theorem for alternant hydrocarbons, it is possible to derive a finite series expansion of the Green's function for electron transmission in terms of the odd powers of the vertex adjacency matrix or Hückel matrix. This means that only odd-length walks on a molecular graph contribute to the conductivity through a molecule. Thus, if there are only even-length walks between two atoms, quantum interference is expected to occur in the electron transport between them. However, even if there are only odd-length walks between two atoms, a situation may come about where the contributions to the QI of some odd-length walks are canceled by others, leading to another class of quantum interference. For nonalternant hydrocarbons, the finite Green's...
We generalize the previous results of [1] by proving unfrustration condition and degeneracy of th... more We generalize the previous results of [1] by proving unfrustration condition and degeneracy of the ground states of qudits (d-dimensional spins) on a k-child tree with generic local interactions. We find that the dimension of the ground space grows doubly exponentially in the region where rk<=(d^2)/4 for k>1. Further, we extend the results in [1] by proving that there are no zero energy ground states when r>(d^2)/4 for k=1 implying that the effective Hamiltonian is invertible.
Bulletin of the American Physical Society, Mar 5, 2015
By Bernoulli's law, an increase in the relative speed of a fluid around a body is accompanies by ... more By Bernoulli's law, an increase in the relative speed of a fluid around a body is accompanies by a decrease in the pressure. Therefore, a rotating body in a fluid stream experiences a force perpendicular to the motion of the fluid because of the unequal relative speed of the fluid across its surface. It is well known that light has a constant speed irrespective of the relative motion. Does a rotating body immersed in a stream of photons experience a Bernoulli-like force? We show that, indeed, a rotating dielectric cylinder experiences such a lateral force from an electromagnetic wave. In fact, the sign of the lateral force is the same as that of the fluid-mechanical analogue as long as the electric susceptibility is positive (ϵ > ϵ 0), but for negative-susceptibility materials (e.g. metals) we show that the lateral force is in the opposite direction. Because these results are derived from a classical electromagnetic scattering problem, Mie-resonance enhancements that occur in other scattering phenomena also enhance the lateral force.
ABSTRACT We approximate the density of states in disordered systems by decomposing the Hamiltonia... more ABSTRACT We approximate the density of states in disordered systems by decomposing the Hamiltonian into two random matrices and constructing their free convolution. The error in this approximation is determined using asymptotic moment expansions. Each moment can be decomposed into contributions from specific joint moments of the random matrices; each of which has a combinatorial interpretation as the weighted sum of returning trajectories. We show how the error, like the free convolution itself, can be calculated without explicit diagonalization of the Hamiltonian. We apply our theory to Hamiltonians for one-dimensional tight binding models with Gaussian and semicircular site disorder. We find that the particular choice of decomposition crucially determines the accuracy of the resultant density of states. From a partitioning of the Hamiltonian into diagonal and off-diagonal components, free convolution produces an approximate density of states which is correct to the eighth moment. This allows us to explain the accuracy of mean field theories such as the coherent potential approximation, as well as the results of isotropic entanglement theory.
This article puts forth a process applicable to central force scatterings. Under certain assumpti... more This article puts forth a process applicable to central force scatterings. Under certain assumptions, we show that in attractive force fields a high speed particle with a small mass speeding through space, statistically loses energy by colliding softly with large masses that move slowly and randomly. Furthermore, we show that the opposite holds in repulsive force fields: the small particle statistically gains energy. This effect is small and is mainly due to asymmetric energy exchange of the transverse (i.e., perpendicular) collisions. We derive a formula that quantifies this effect (Eq.(12)). We then put this work in a broader statistical context and discuss its consistency with established results.
Geometrization of dynamics consists of representing trajectories by geodesics on a configuration ... more Geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical-and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme. I. STABILITY OF HAMILTONIAN SYSTEMS
The second part is devoted to ground states and gap of QMBS. I first give the necessary backgroun... more The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free (FF) Hamiltonians, real and imaginary time evolution within MPS representation and a numerical implementation. I then prove the degeneracy and FF condition for quantum spin chains with generic local interactions, including corrections to our earlier assertions. I then summarize my efforts in proving lower bounds for the entanglement of the ground states, which includes some new results, with the hope that they inspire future work resulting in solving the conjecture given therein. Next I discuss two interesting measure zero examples where FF Hamiltonians are carefully constructed to give unique ground states with high entanglement. One of the examples (i.e., d=4d=4d=4) has not appeared elsewhere. In particular, we calculate the Schmidt numbers exactly, entanglement entropies and introduce a novel technique for calculating the gap which may be of independent interest. The last chapter elaborates on one of the measure zero examples (i.e., d=3d=3d=3) which is the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits signatures of a critical behavior.
Journal of Mathematical Physics
Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid st... more Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d−dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.
Journal of Mathematical Physics
Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid st... more Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, G, of the N × N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d−dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green's function and discuss applications related to transport and conductivity.
Proceedings of the National Academy of Sciences
Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simulta... more Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an “area law”: The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system’s size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system’s size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially ...
Eprint Arxiv Math Ph 0610084, Oct 1, 2006
Geometrization of dynamics using (non)-affine parametization of arc length with time is investiga... more Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi metric results in positive values of geometrical lyapunov exponent. The non-physical instabilities are shown to be due to a non-affine parametrization. In addition the degree of instability is a monotonically increasing function of the fluctuations in the kinetic energy. We argue that the Jacobi metric gives equivalent results as Eisenhart metric for ergodic systems at equilibrium, where number of degrees of freedom NtoinftyN\to\inftyNtoinfty. We conclude that, in addition to being computationally more expensive, geometrization using the Jacobi metric is meaningful only when the kinetic energy of the system is a positive constant.
Eprint Arxiv 1008 0875, Aug 4, 2010
This article puts forth a process applicable to central force scatterings. Under certain assumpti... more This article puts forth a process applicable to central force scatterings. Under certain assumptions, we show that in attractive force fields a high speed particle with a small mass speeding through space, statistically loses energy by colliding softly with large masses that move slowly and randomly. Furthermore, we show that the opposite holds in repulsive force fields: the small particle statistically gains energy. This effect is small and is mainly due to asymmetric energy exchange of the transverse (i.e., perpendicular) collisions. We derive a formula that quantifies this effect (Eq. 12). We then put this work in a broader statistical context and discuss its consistency with established results.
ACS nano, Jan 29, 2015
An exponential falloff with separation of electron transfer and transport through molecular wires... more An exponential falloff with separation of electron transfer and transport through molecular wires is observed and has attracted theoretical attention. In this study, the attenuation of transmission in linear and cyclic polyenes is related to bond alternation. The explicit form of the zeroth Green's function in a Hückel model for bond-alternated polyenes leads to an analytical expression of the conductance decay factor β. The β values calculated from our model (βCN values, per repeat unit of double and single bond) range from 0.28 to 0.37, based on carotenoid crystal structures. These theoretical β values are slightly smaller than experimental values. The difference can be assigned to the effect of anchoring groups, which are not included in our model. A local transmission analysis for cyclic polyenes, and for [14]annulene in particular, shows that bond alternation affects dramatically not only the falloff behavior but also the choice of a transmission pathway by electrons. Trans...
Geometrization of dynamics consists of representing trajectories by geodesics on a suitably defin... more Geometrization of dynamics consists of representing trajectories by geodesics on a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical- and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme.
The Journal of chemical physics, Jan 14, 2014
The explicit form of the zeroth Green's function in the Hückel model, approximated by the neg... more The explicit form of the zeroth Green's function in the Hückel model, approximated by the negative of the inverse of the Hückel matrix, has direct quantum interference consequences for molecular conductance. We derive a set of rules for transmission between two electrodes attached to a polyene, when the molecule is extended by an even number of carbons at either end (transmission unchanged) or by an odd number of carbons at both ends (transmission turned on or annihilated). These prescriptions for the occurrence of quantum interference lead to an unexpected consequence for switches which realize such extension through electrocyclic reactions: for some specific attachment modes the chemically closed ring will be the ON position of the switch. Normally the signs of the entries of the Green's function matrix are assumed to have no physical significance; however, we show that the signs may have observable consequences. In particular, in the case of multiple probe attachments - i...
2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers, 2010