Hans-Peter Eckle | University of Ulm (original) (raw)
Papers by Hans-Peter Eckle
Oxford University Press eBooks, Jul 25, 2019
The crystal structure of several of the phases of ice shows a peculiarity associated with a speci... more The crystal structure of several of the phases of ice shows a peculiarity associated with a special type of disorder: the one hydrogen atom between the two oxygen atoms is closer to one or the other of the two. This peculiarity depends only on the configuration and is independent of temperature. It gives rise to a finite entropy of ice, even at zero temperature, i.e. the residual entropy. This observation is used as a physical motivation to study a certain type of two-dimensional statistical mechanical models, the so-called vertex models, the exemplary vertex model being the ice model, for which we introduce the ice rule.
We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in... more We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a side-branch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.
Journal of Physics A, Aug 4, 2023
Oxford University Press eBooks, Jul 25, 2019
The connection between statistical mechanics on the one side and quantum mechanics and quantum fi... more The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.
Oxford University Press eBooks, Jul 25, 2019
The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensio... more The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
Oxford University Press eBooks, Jul 25, 2019
This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic ... more This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic Bethe ansatz, for the antiferromagnetic isotropic Heisenberg quantum spin chain, the XXX quantum spin chain. It discusses how the added complications of this model arise from the more complicated structure of excitations of the quantum spin chain, the complex string excitations, which have to be included in the Bethe ansatz thermodynamics. It derives the integral equations of the thermodynamic Bethe ansatz for the XXX quantum spin chain and mentions explicit formulas for the free energy of the quantum spin chain and some interesting physical quantities, especially making contact with predictions of conformal symmetry.
Oxford University Press eBooks, Jul 25, 2019
Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fund... more Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.
Oxford University Press eBooks, Jul 25, 2019
This chapter considers the special case of the six-vertex model on a square lattice using a trigo... more This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.
Physical Review Letters, Mar 13, 2002
Eckle et al. Reply: In the preceding Comment [1], Affleck and Simon argue that the approach adopt... more Eckle et al. Reply: In the preceding Comment [1], Affleck and Simon argue that the approach adopted in our Letter [2] is not appropriate for calculating the persistent current in a mesoscopic ring with a sidebranch quantum dot, when tuned to a Kondo resonance. Specifically, they suggest that our assumption of a linear dispersion relation is the reason why we obtain a result in contradiction to theirs [3,4]. In response to this, we point out that the use of a linear dispersion relation for calculating persistent currents is by now well established and has been extensively exploited for the case of studying, e.g., parity effects [5] or effects from disordering potentials [6] in 1D rings. Our definition of the persistent current in terms of excess numbers [Eq. (7) in [2] ] is conceptually the same as that in [5] where the persistent current is defined via a topological number describing the difference between the number of right-and left-moving particles. In our Letter [2], we argue that since the Kondo resonance of the ring and the sidebranch dot is a Fermi-level property, the use of a linear dispersion and, a fortiori, the definition in Eq. (7) in [2], should remain valid also in the presence of the dot. We admit that this is an assumption that needs to be carefully examined. Although the curvature of the "true" dispersion close to the Fermi level is known to generate only irrelevant operators in 1D (in renormalization-group sense) [7], it is still conceivable that these operators could "tie together" the charge and spin sectors of the integrable theory in such a way as to modify the persistent current. We are presently investigating this possibility [8]. Considering the approach advocated by Affleck and Simon [3,4], we note that it rests on the assumption that a persistent current can be calculated as if it were a transport current, and hence is completely determined by the transmission amplitude of an electron at the Fermi level. This is a nontrivial assertion, which has so far been demonstrated convincingly only for the case of a 1D ring with potential scatterers [9]. However, in an interacting 1D system with spin-charge separation, a finite persistent current will result if collective charge excitations can encircle the ring, even if an electron, which carries both charge and spin, cannot. In our Letter [2], we argued that this is precisely the case for Kondo scattering in a 1D ring. The perturbative renormalization-group argument in [3,4] that the persistent current is a "universal property," and that therefore the analysis for potential scatterers in [9] can be directly transferred to the present case, begs the question, in our opinion. The fundamental issue at stake in our disagreement with Refs. [3,4] is whether or not spin-charge separation manifests itself in boundary effects like the persistent current. We agree with Affleck and Simon that more work is needed, including large-scale numerical simulations, to eventually settle this interesting question.
Journal of Physics A, Jun 29, 2017
We consider a generalization of the quantum Rabi model where the twolevel system and the single-m... more We consider a generalization of the quantum Rabi model where the twolevel system and the single-mode cavity oscillator are coupled by an additional Starklike term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. 107, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of twofold near-degenerate levels as the Rabi coupling increases.
APS March Meeting Abstracts, Mar 1, 2001
We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in... more We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a sidebranch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.
Advanced series in mathematical physics, May 1, 1994
Oxford University Press eBooks, Jul 25, 2019
Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of ... more Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.
Oxford University Press eBooks, Jul 25, 2019
The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from ... more The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.
Czechoslovak Journal of Physics, Oct 1, 2006
We consider the problem of a persistent current in a one-dimensional mesoscopic ring with the ele... more We consider the problem of a persistent current in a one-dimensional mesoscopic ring with the electrons coupled by a spin exchange to a magnetic impurity. We show that this problem can be mapped onto an integrable model with a quadratic dispersion (with the latter property allowing for an unambiguous definition of the persistent current). We have solved the model exactly by a Bethe ansatz and found that the current is insensitive to the presence of the impurity. We conjecture that this result holds for any integrable quantum impurity model with an electronic dispersion e(k) that is an even function of k.
Journal of physics, May 7, 1992
Physical review, Nov 13, 2002
The Heisenberg chain with a weak link is studied, as a simple example of a quantum ring with a co... more The Heisenberg chain with a weak link is studied, as a simple example of a quantum ring with a constriction or defect. The Heisenberg chain is equivalent to a spinless electron gas under a Jordan-Wigner transformation. Using density matrix renormalization group and quantum Monte Carlo methods we calculate the spin/charge stiffness of the model, which determines the strength of the 'persistent currents'. The stiffness is found to scale to zero in the weak link case, in agreement with renormalization group arguments of Eggert and Affleck, and Kane and Fisher.
Journal of physics, May 11, 1987
The finite-size corrections to the ground state and the energy of the low magnetisation (S K N) s... more The finite-size corrections to the ground state and the energy of the low magnetisation (S K N) states as a function of the size N are calculated analytically for the onedimensional half-filled Hubbard model with on-site repulsion (U > 0). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N = 512.
Journal of physics, Feb 1, 1987
Oxford University Press eBooks, Jul 25, 2019
Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from... more Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the Bernoulli numbers are needed as a prerequisite for the Euler– Maclaurin summation formula, which allows to transform finite sums into integrals plus, in a systematic way, corrections taking into account the finite size of the system. Applying this mathematical technique to the Bethe ansatz equations results in linear integral equations of the Wiener–Hopf type for the solution of which an elaborate mathematical technique exists, the Wiener–Hopf technique.
Oxford University Press eBooks, Jul 25, 2019
The crystal structure of several of the phases of ice shows a peculiarity associated with a speci... more The crystal structure of several of the phases of ice shows a peculiarity associated with a special type of disorder: the one hydrogen atom between the two oxygen atoms is closer to one or the other of the two. This peculiarity depends only on the configuration and is independent of temperature. It gives rise to a finite entropy of ice, even at zero temperature, i.e. the residual entropy. This observation is used as a physical motivation to study a certain type of two-dimensional statistical mechanical models, the so-called vertex models, the exemplary vertex model being the ice model, for which we introduce the ice rule.
We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in... more We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a side-branch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.
Journal of Physics A, Aug 4, 2023
Oxford University Press eBooks, Jul 25, 2019
The connection between statistical mechanics on the one side and quantum mechanics and quantum fi... more The connection between statistical mechanics on the one side and quantum mechanics and quantum field theory on the other side is based on the analogy between thermal and quantum fluctuations. Formally, the connection is expressed through the mathematical equivalence between the partition function in statistical mechanics and the propagator in quantum field theory. This chapter explores the equivalence between statistical mechanics and quantum mechanics or quantum field theory in general terms using the Feynman path integral and with the example of the equivalence between the classical XY Heisenberg model and the sigma model of quantum field theory. Invoking the concepts of the partition function and the transfer matrix, an example demonstrates the passage from the quantum mechanics of a single degree of freedom, a zero-dimensional system, to the statistical mechanics of a one-dimensional system represented by classical variables.
Oxford University Press eBooks, Jul 25, 2019
The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensio... more The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
Oxford University Press eBooks, Jul 25, 2019
This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic ... more This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic Bethe ansatz, for the antiferromagnetic isotropic Heisenberg quantum spin chain, the XXX quantum spin chain. It discusses how the added complications of this model arise from the more complicated structure of excitations of the quantum spin chain, the complex string excitations, which have to be included in the Bethe ansatz thermodynamics. It derives the integral equations of the thermodynamic Bethe ansatz for the XXX quantum spin chain and mentions explicit formulas for the free energy of the quantum spin chain and some interesting physical quantities, especially making contact with predictions of conformal symmetry.
Oxford University Press eBooks, Jul 25, 2019
Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fund... more Chapter 4 reviews the basic notions of equilibrium statistical mechanics and begins with its fundamental postulate and outlines the structure of the theory using the most important of the various statistical ensembles, the microcanonical, the canonical, and the grand canonical ensemble and their corresponding thermodynamic potential, the internal energy, the Helmholtz free energy, and the grand canonical potential. The notions of temperature, pressure, and chemical potential are obtained and it introduces the laws of thermodynamics, the Gibbs entropy, and the concept of the partition function. It also discusses quantum statistical mechanics using the density matrix and as applied to non-interacting Bosonic and Fermionic quantum gases, the former showing Bose–Einstein condensation. The mean-field theory of interacting magnetic moments and the transfer matrix to exactly solve the Ising model in one dimension serve as applications.
Oxford University Press eBooks, Jul 25, 2019
This chapter considers the special case of the six-vertex model on a square lattice using a trigo... more This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.
Physical Review Letters, Mar 13, 2002
Eckle et al. Reply: In the preceding Comment [1], Affleck and Simon argue that the approach adopt... more Eckle et al. Reply: In the preceding Comment [1], Affleck and Simon argue that the approach adopted in our Letter [2] is not appropriate for calculating the persistent current in a mesoscopic ring with a sidebranch quantum dot, when tuned to a Kondo resonance. Specifically, they suggest that our assumption of a linear dispersion relation is the reason why we obtain a result in contradiction to theirs [3,4]. In response to this, we point out that the use of a linear dispersion relation for calculating persistent currents is by now well established and has been extensively exploited for the case of studying, e.g., parity effects [5] or effects from disordering potentials [6] in 1D rings. Our definition of the persistent current in terms of excess numbers [Eq. (7) in [2] ] is conceptually the same as that in [5] where the persistent current is defined via a topological number describing the difference between the number of right-and left-moving particles. In our Letter [2], we argue that since the Kondo resonance of the ring and the sidebranch dot is a Fermi-level property, the use of a linear dispersion and, a fortiori, the definition in Eq. (7) in [2], should remain valid also in the presence of the dot. We admit that this is an assumption that needs to be carefully examined. Although the curvature of the "true" dispersion close to the Fermi level is known to generate only irrelevant operators in 1D (in renormalization-group sense) [7], it is still conceivable that these operators could "tie together" the charge and spin sectors of the integrable theory in such a way as to modify the persistent current. We are presently investigating this possibility [8]. Considering the approach advocated by Affleck and Simon [3,4], we note that it rests on the assumption that a persistent current can be calculated as if it were a transport current, and hence is completely determined by the transmission amplitude of an electron at the Fermi level. This is a nontrivial assertion, which has so far been demonstrated convincingly only for the case of a 1D ring with potential scatterers [9]. However, in an interacting 1D system with spin-charge separation, a finite persistent current will result if collective charge excitations can encircle the ring, even if an electron, which carries both charge and spin, cannot. In our Letter [2], we argued that this is precisely the case for Kondo scattering in a 1D ring. The perturbative renormalization-group argument in [3,4] that the persistent current is a "universal property," and that therefore the analysis for potential scatterers in [9] can be directly transferred to the present case, begs the question, in our opinion. The fundamental issue at stake in our disagreement with Refs. [3,4] is whether or not spin-charge separation manifests itself in boundary effects like the persistent current. We agree with Affleck and Simon that more work is needed, including large-scale numerical simulations, to eventually settle this interesting question.
Journal of Physics A, Jun 29, 2017
We consider a generalization of the quantum Rabi model where the twolevel system and the single-m... more We consider a generalization of the quantum Rabi model where the twolevel system and the single-mode cavity oscillator are coupled by an additional Starklike term. By adapting a method recently introduced by Braak [Phys. Rev. Lett. 107, 100401 (2011)], we solve the model exactly. The low-lying spectrum in the experimentally relevant ultrastrong and deep strong regimes of the Rabi coupling is found to exhibit two striking features absent from the original quantum Rabi model: avoided level crossings for states of the same parity and an anomalously rapid onset of twofold near-degenerate levels as the Rabi coupling increases.
APS March Meeting Abstracts, Mar 1, 2001
We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in... more We study the persistent currents induced by both the Aharonov-Bohm and Aharonov-Casher effects in a one-dimensional mesoscopic ring coupled to a sidebranch quantum dot at Kondo resonance. For privileged values of the Aharonov-Bohm-Casher fluxes, the problem can be mapped onto an integrable model, exactly solvable by a Bethe ansatz. In the case of a pure magnetic Aharonov-Bohm flux, we find that the presence of the quantum dot has no effect on the persistent current. In contrast, the Kondo resonance interferes with the spin-dependent Aharonov-Casher effect to induce a current which, in the strong-coupling limit, is independent of the number of electrons in the ring.
Advanced series in mathematical physics, May 1, 1994
Oxford University Press eBooks, Jul 25, 2019
Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of ... more Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle Hilbert spaces to describe large assemblies of interacting systems composed of Bosons or Fermions, which lead to the versatile formalism of second quantization as a convenient and eminently practical language ubiquitous in the mathematical formulation of the theory of many-particle systems of quantum matter. The main objects in which the formalism of second quantization is expressed are the Bosonic or Fermionic creation and annihilation operators that become, in the position basis, the quantum field operators.
Oxford University Press eBooks, Jul 25, 2019
The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from ... more The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.
Czechoslovak Journal of Physics, Oct 1, 2006
We consider the problem of a persistent current in a one-dimensional mesoscopic ring with the ele... more We consider the problem of a persistent current in a one-dimensional mesoscopic ring with the electrons coupled by a spin exchange to a magnetic impurity. We show that this problem can be mapped onto an integrable model with a quadratic dispersion (with the latter property allowing for an unambiguous definition of the persistent current). We have solved the model exactly by a Bethe ansatz and found that the current is insensitive to the presence of the impurity. We conjecture that this result holds for any integrable quantum impurity model with an electronic dispersion e(k) that is an even function of k.
Journal of physics, May 7, 1992
Physical review, Nov 13, 2002
The Heisenberg chain with a weak link is studied, as a simple example of a quantum ring with a co... more The Heisenberg chain with a weak link is studied, as a simple example of a quantum ring with a constriction or defect. The Heisenberg chain is equivalent to a spinless electron gas under a Jordan-Wigner transformation. Using density matrix renormalization group and quantum Monte Carlo methods we calculate the spin/charge stiffness of the model, which determines the strength of the 'persistent currents'. The stiffness is found to scale to zero in the weak link case, in agreement with renormalization group arguments of Eggert and Affleck, and Kane and Fisher.
Journal of physics, May 11, 1987
The finite-size corrections to the ground state and the energy of the low magnetisation (S K N) s... more The finite-size corrections to the ground state and the energy of the low magnetisation (S K N) states as a function of the size N are calculated analytically for the onedimensional half-filled Hubbard model with on-site repulsion (U > 0). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N = 512.
Journal of physics, Feb 1, 1987
Oxford University Press eBooks, Jul 25, 2019
Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from... more Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the Bernoulli numbers are needed as a prerequisite for the Euler– Maclaurin summation formula, which allows to transform finite sums into integrals plus, in a systematic way, corrections taking into account the finite size of the system. Applying this mathematical technique to the Bethe ansatz equations results in linear integral equations of the Wiener–Hopf type for the solution of which an elaborate mathematical technique exists, the Wiener–Hopf technique.
The quantum Rabi model Quantum Rabi model simplest model to describe interaction light ⇔ matter t... more The quantum Rabi model Quantum Rabi model simplest model to describe interaction light ⇔ matter two-level atom (qubit) interacting with single bosonic mode nuclear magnetic resonance (Isidor Isaac Rabi, 1936) cavity and circuit quantum electrodynamics quantum dots, trapped ions superconducting qubits nanoelectromechanical devices (photons ⇒ phonons) candidate for physical realization of quantum information processing (quantum computing) Generic Hamiltonian H Rabi = ωa † a + ∆σ z + g a + a † σ x No complete exact solution until recently: Braak (2011)
Philosophie in Zeiten der Quantentechnologie, 2020
Public Lecture