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Correlations in one dimensional quantum impurity problems with an external field or a temperature.

Frédéric Lesage, Hubert Saleur

Published Jan 1, 1996• 5 citations
OpenAlex

Abstract

We discuss in more details the theory of low energy excitations in quantum impurity problems with an external field at vanishing temperature, giving further support to results of the previous paper. We then extend these results to the next order at low frequency, obtaining in particular the exact expression, as a function of the bias, of the first two derivatives of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. We also extend our approach to the case of non vanishing temperature and no external field. Fendley et al. had obtained in that case an expression for the Hall conductance with a single impurity using a Landauer-Büttiker type approach. We recover their result in the framework of linear response theory using renormalized form-factors. We also obtain, as a function of the temperature, the first derivative of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. 12/96

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Theory of properties of some one-dimensional systems

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This thesis consists in a theoretical study of some one-dimensional models. Special attention is given to simple but non-trivial soluble models. The study starts with a resume of the Green function method which is the formalism used throughout the work. A discussion of coupled electron-phonon and spin-phonon systems is presented and a decoupling scheme introduced. The approximation gives the transition temperature for one-dimensional systems exactly, namely T = 0, and consequently represents a great improvement on the mean-field approximation. The impure XY-chain in a transverse field is studied and the formal solution presented. The static and time-dependent correlations are discussed, and the effect of the boundary term reviewed in details. The one-impurity case is solved for the isotropic chain, and it is shown how to derive from it the open chain result in the thermodynamic limit. The heat-flux operator is obtained for the impure isotropic chain by solving the continuity equation for energy density. The dynamics of the one-dimensional transverse Ising model is discussed within several approximations. The limitations of each approximation are examined in detail and the time-dependent correlations are calculated using the most successful approximation. It is also shown that this approximation is only valid in the high temperature limit in which case no critical behaviour is to be observed. The one-impurity solution of the isotropic impure XY-chain is extended to many impurities in two different cases. In the first case the specific heat of the dilute chain is calculated in the framework of the average t-matrix approximation, and the results compared with exact numerical calculations for finite chains. In the second case the specific heat and thermal conductivity of the isotropic XY-chain in a random transverse field are calculated using the coherent-potential approximation (CPA). The results are used to explain the low temperature thermal properties of praseodymium ethyl sulphate. Finally, two interacting systems, an electron-phonon system and a spin-phonon system, are studied after a brief discussion of the so-called Peierls instability. The first model is solved exactly in the framework of the approximation discussed at the beginning of the work. It presents a giant Kohn anomaly at zero temperature which drives a Peierls transition. The renormalized modes are discussed, and the real and imaginary parts of the phonon propagator are presented for wave vector π⁄a and several coupling constants and temperatures. The spin-phonon system is also studied in the framework of the approximation discussed initially in the work, and it is chosen in such a way that the relevant properties are given in terms of Green functions calculated in the transverse Ising model. The results obtained for the dynamics of this model are used, and since they are valid only in the high temperature limit, the discussion is restricted to this temperature region. Since this region is far away from the critical one, no critical behaviour is observed, and the study is restricted to a simple discussion of modes. The real and imaginary parts of the renormalized phonon propagator are presented for the wave vector π⁄a, high temperature and various coupling constants.

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Landauer formula for the conductance of interacting electrons

W. Apel

An exact formula for the conductance of a finite sample in which interacting electrons move in one dimension in the presence of random impurity potentials is derived. It has a form similar to the single-channel Landauer formula (1970) for non-interacting particles. The transmission coefficient is given by a temperature-dependent 2k f density response function of the interacting system.

Journal of Physics C Solid State Physics•May 30, 1983•7 citations

Υπολογισμός πολυχρονικών συναρτήσεων συσχέτισης στην αποσβεστική κβαντική φ^4 θεωρία

Παναγιώτης Αλατάς

In quantum field theories, Feynman diagrams depict visually the interaction of elementary particles and involve closed loops, corresponding to integration over all possible combinations of energy and momentum of the virtual particles traveling in the space-time continuum, leading often to infinities. To treat these infinities, renormalization of the parameters of the theory is required. Renormalization specifies relationships between parameters in the theory whose values at large length scales are different from those at short length scales. Behind renormalization lies the problem of the proper elimination of degrees of freedom in order to bridge the gap between different length and time scales. Nonequilibrium thermodynamics offers an alternative approach to the problem of the time evolution of an open quantum system through a time-evolution master equation for the density matrix of the system, which includes reversible and irreversible contributions. In particular, irreversibility introduces dissipative smoothing mechanisms for the terms responsible for the infinities, providing the proper regularization of the theory. The objective of the present doctoral dissertation is the implementation of the dissipative quantum field theory (DQFT) developed by Prof. Dr. ‪Hans Christian Öttinger to calculate the perturbative expansions up to third order in the interaction parameter of the two- (propagator) and four-point (effective interaction vertex) correlation functions of the scalar theory with quartic interactions (φ^4 theory) in the limit of zero temperature, as well as to extract the respective Feynman diagrams where (in the context of DQFT) collision events have a well-defined arrow of time. To achieve this, we developed a powerful symbolic code which allowed us to organize and simplify the outcome of the commands performing the calculation of terms contained in each function within a given order so that to lead to a compact final expression, thus rendering the identification of the relevant Feynman diagrams more transparent. The use of symbolic computation was motivated by the fact that the perturbative DQFT in d dimensions (d denotes the number of space dimensions) is considerably more complicated than the usual calculations in D = d + 1 dimensions (D the number of space + time dimensions) of the Lagrangian quantum field theory, generating a huge number of terms with increasing order of perturbation. As expected, the resulting expressions for the propagator and the effective interactive vertex contain terms corresponding to zeroth-, first-, second- and third-order contributions. The terms up to second order have already been reported by Prof. Dr. Hans Christian Öttinger. In this Thesis, we computed the third-order contributions to both of them in terms of connected and unconnected Feynman diagrams and studied their topology. In the limit of vanishing dissipation (zero friction coefficient), the final expressions for the two correlation functions analyzed in this Thesis coincide with those obtained from the Lagrangian quantum field theory. This is achieved by integrating out the time or energy component in the D-dimensional integrals of the Lagrangian approach to obtain the d-dimensional integrals of the dissipative approach. The final expression for the propagator up to third order is regularized with two different schemes, one of them being the so-called “on shell” scheme, thus obtaining fully convergent results in d = 3 dimensions. Even more important, by combining our results for the second- and third-order contributions at the long-time or low-energy limit, we confirm that the value of the critical coupling constant λ* as a function of space dimensionality d is correctly reproduced, which demonstrates the power of the new, dissipative approach to quantum field theory and the correct implementation (in its entirety) of the powerful symbolic code developed in this Thesis. The corresponding dissipative result for the effective interaction vertex is expressed in terms of a parameter describing the level of energy conservation violation. A compact and rather friendly expression is obtained by assuming that the external legs are amputated and the mass is equal to zero, demonstrating the finiteness of the terms appearing in the vertex, in the limit of vanishing friction, which is a great outcome of the dissipative approach followed. Moreover, many striking similarities are pointed out and discussed with the corresponding Lagrangian result at the same order of perturbation.

•May 4, 2023•0 citations
We discuss in more details the theory of low energy excitations in quantum impurity problems with an external field at vanishing temperature, giving further support to results of the previous paper. We then extend these results to the next order at low frequency, obtaining in particular the exact expression, as a function of the bias, of the first two derivatives of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. We also extend our approach to the case of non vanishing temperature and no external field. Fendley et al. had obtained in that case an expression for the Hall conductance with a single impurity using a Landauer-Büttiker type approach. We recover their result in the framework of linear response theory using renormalized form-factors. We also obtain, as a function of the temperature, the first derivative of the response function χ ′ ′ (ω) at ω = 0 in the double well problem of dissipative quantum mechanics. 12/96
We develop an analytical approach using form factors to compute time dependent correlations in integrable quantum impurity problems. As an example, we obtain for the first time the frequency dependent conductivity $G(\ensuremath{\omega})$ for the tunneling between edges in the $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall effect and the spectrum $S(\ensuremath{\omega})$ of the spin-spin correlation in the anisotropic Kondo model and equivalently in the double well system of dissipative quantum mechanics, both at vanishing temperature.
This thesis consists in a theoretical study of some one-dimensional models. Special attention is given to simple but non-trivial soluble models. The study starts with a resume of the Green function method which is the formalism used throughout the work. A discussion of coupled electron-phonon and spin-phonon systems is presented and a decoupling scheme introduced. The approximation gives the transition temperature for one-dimensional systems exactly, namely T = 0, and consequently represents a great improvement on the mean-field approximation. The impure XY-chain in a transverse field is studied and the formal solution presented. The static and time-dependent correlations are discussed, and the effect of the boundary term reviewed in details. The one-impurity case is solved for the isotropic chain, and it is shown how to derive from it the open chain result in the thermodynamic limit. The heat-flux operator is obtained for the impure isotropic chain by solving the continuity equation for energy density. The dynamics of the one-dimensional transverse Ising model is discussed within several approximations. The limitations of each approximation are examined in detail and the time-dependent correlations are calculated using the most successful approximation. It is also shown that this approximation is only valid in the high temperature limit in which case no critical behaviour is to be observed. The one-impurity solution of the isotropic impure XY-chain is extended to many impurities in two different cases. In the first case the specific heat of the dilute chain is calculated in the framework of the average t-matrix approximation, and the results compared with exact numerical calculations for finite chains. In the second case the specific heat and thermal conductivity of the isotropic XY-chain in a random transverse field are calculated using the coherent-potential approximation (CPA). The results are used to explain the low temperature thermal properties of praseodymium ethyl sulphate. Finally, two interacting systems, an electron-phonon system and a spin-phonon system, are studied after a brief discussion of the so-called Peierls instability. The first model is solved exactly in the framework of the approximation discussed at the beginning of the work. It presents a giant Kohn anomaly at zero temperature which drives a Peierls transition. The renormalized modes are discussed, and the real and imaginary parts of the phonon propagator are presented for wave vector π⁄a and several coupling constants and temperatures. The spin-phonon system is also studied in the framework of the approximation discussed initially in the work, and it is chosen in such a way that the relevant properties are given in terms of Green functions calculated in the transverse Ising model. The results obtained for the dynamics of this model are used, and since they are valid only in the high temperature limit, the discussion is restricted to this temperature region. Since this region is far away from the critical one, no critical behaviour is observed, and the study is restricted to a simple discussion of modes. The real and imaginary parts of the renormalized phonon propagator are presented for the wave vector π⁄a, high temperature and various coupling constants.
Recent developments have made possible the computation of equilibrium dynamical correlators in quantum impurity problems. In many situations, however, one is interested in correlators subject to a nonequilibrium initial preparation. This is the case, for instance, for the occupation probability P(t)P(t)P(t) in the double well problem of dissipative quantum mechanics. We show here how to handle this situation in the framework of integrable quantum field theories. This allows us to obtain new exact results for P(t)P(t)P(t): for instance, we find that at large times (or small ggg), the leading behavior for g<12g<\frac{1}{2}g<21​ is $P(t)\ensuremath{\propto}{e}^{\ensuremath{-}\ensuremath{\Gamma}t}\mathrm{cos}\ensuremath{\Omega}t$, with the universal ratio $\frac{\ensuremath{\Omega}}{\ensuremath{\Gamma}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{cot}\frac{\ensuremath{\pi}g}{2(1\ensuremath{-}g)}$.
We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.
In quantum field theories, Feynman diagrams depict visually the interaction of elementary particles and involve closed loops, corresponding to integration over all possible combinations of energy and momentum of the virtual particles traveling in the space-time continuum, leading often to infinities. To treat these infinities, renormalization of the parameters of the theory is required. Renormalization specifies relationships between parameters in the theory whose values at large length scales are different from those at short length scales. Behind renormalization lies the problem of the proper elimination of degrees of freedom in order to bridge the gap between different length and time scales. Nonequilibrium thermodynamics offers an alternative approach to the problem of the time evolution of an open quantum system through a time-evolution master equation for the density matrix of the system, which includes reversible and irreversible contributions. In particular, irreversibility introduces dissipative smoothing mechanisms for the terms responsible for the infinities, providing the proper regularization of the theory. The objective of the present doctoral dissertation is the implementation of the dissipative quantum field theory (DQFT) developed by Prof. Dr. ‪Hans Christian Öttinger to calculate the perturbative expansions up to third order in the interaction parameter of the two- (propagator) and four-point (effective interaction vertex) correlation functions of the scalar theory with quartic interactions (φ^4 theory) in the limit of zero temperature, as well as to extract the respective Feynman diagrams where (in the context of DQFT) collision events have a well-defined arrow of time. To achieve this, we developed a powerful symbolic code which allowed us to organize and simplify the outcome of the commands performing the calculation of terms contained in each function within a given order so that to lead to a compact final expression, thus rendering the identification of the relevant Feynman diagrams more transparent. The use of symbolic computation was motivated by the fact that the perturbative DQFT in d dimensions (d denotes the number of space dimensions) is considerably more complicated than the usual calculations in D = d + 1 dimensions (D the number of space + time dimensions) of the Lagrangian quantum field theory, generating a huge number of terms with increasing order of perturbation. As expected, the resulting expressions for the propagator and the effective interactive vertex contain terms corresponding to zeroth-, first-, second- and third-order contributions. The terms up to second order have already been reported by Prof. Dr. Hans Christian Öttinger. In this Thesis, we computed the third-order contributions to both of them in terms of connected and unconnected Feynman diagrams and studied their topology. In the limit of vanishing dissipation (zero friction coefficient), the final expressions for the two correlation functions analyzed in this Thesis coincide with those obtained from the Lagrangian quantum field theory. This is achieved by integrating out the time or energy component in the D-dimensional integrals of the Lagrangian approach to obtain the d-dimensional integrals of the dissipative approach. The final expression for the propagator up to third order is regularized with two different schemes, one of them being the so-called “on shell” scheme, thus obtaining fully convergent results in d = 3 dimensions. Even more important, by combining our results for the second- and third-order contributions at the long-time or low-energy limit, we confirm that the value of the critical coupling constant λ* as a function of space dimensionality d is correctly reproduced, which demonstrates the power of the new, dissipative approach to quantum field theory and the correct implementation (in its entirety) of the powerful symbolic code developed in this Thesis. The corresponding dissipative result for the effective interaction vertex is expressed in terms of a parameter describing the level of energy conservation violation. A compact and rather friendly expression is obtained by assuming that the external legs are amputated and the mass is equal to zero, demonstrating the finiteness of the terms appearing in the vertex, in the limit of vanishing friction, which is a great outcome of the dissipative approach followed. Moreover, many striking similarities are pointed out and discussed with the corresponding Lagrangian result at the same order of perturbation.