Dual‐fermion approach to non‐equilibrium strongly correlated problems
C. Jung, A. Lieder, S. Brener, Hartmut Hafermann, Benjamin Baxevanis, Alexander L. Chudnovskiy, Alexey N. Rubtsov, Mikhail I. Katsnelson, Alexander I. Lichtenstein
Abstract We present a generalization of the recently developed dual‐fermion approach introduced for correlated lattices to non‐equilibrium problems. In its local limit, the approach has been used to devise an efficient impurity solver, the superperturbation solver for the Anderson impurity model (AIM). Here we show that the general dual perturbation theory can be formulated on the Keldysh contour. Starting from a reference Hamiltonian system, in which the time‐dependent solution is found by exact diagonalization, we make a dual perturbation expansion in order to account for the relaxation effects from the fermionic bath. Simple test results for closed as well as open quantum systems in a fermionic bath are presented.
Abstract We present a generalization of the recently developed dual‐fermion approach introduced for correlated lattices to non‐equilibrium problems. In its local limit, the approach has been used to devise an efficient impurity solver, the superperturbation solver for the Anderson impurity model (AIM). Here we show that the general dual perturbation theory can be formulated on the Keldysh contour. Starting from a reference Hamiltonian system, in which the time‐dependent solution is found by exact diagonalization, we make a dual perturbation expansion in order to account for the relaxation effects from the fermionic bath. Simple test results for closed as well as open quantum systems in a fermionic bath are presented.
This thesis is devoted to new numerical approaches for the treatment of strongly correlated systems: The dual fermion perturbation theory is generalized to much broader scope, which is not limited to the treatment of correlated lattice models. By considering three different problems, it is shown that the dual fermion scheme is a formalism, which allows to perform a perturbation expansion around an exactly solvable reference system, which has been optimized for the investigation of the particular physical situation. The three examples are:
-The superperturbation impurity solver for quantum impurity problems. Here a finite size system, which can be solved using exact diagonalization, is used to approximate a bath with a continuous energy spectrum. Several exact limits of the approach are discussed and a way to directly compute the density of states on the real axis is introduced. The meaning of the reference system is analyzed in terms of the Kondo problem and a possible application as a solver for the DMFT is discussed. At the end of the chapter a renormalization procedure is formulated, which alleviates the non-causality problem, that can occur at low temperatures.
-The variational lattice approach (VLA). The VLA is a variant of the classical dual fermion approach, which also utilizes exact diagonalization as an impurity solver for the reference system. Using this approach the Mott metal insulator transition is analyzed and the results are compared to a previous CDMFT study. By employing Pade analytic continuation the pseudogap formation in the bad-metal regime is investigated.
-The superperturbation impurity solver for quantum impurity problems out of equilibrium. At the end of the thesis the dual approach is generalized to systems out of equilibrium within the Keldysh framework. A discretization scheme for the numerical treatment is discussed and first results are presented.
Diese Arbeit ist neuen numerischen Ansatzen fur die Behandlung von stark korrelierten Systemen gewidmet. Die Methode der dualen Fermionen wird in ihrem Bedeutungsrahmen erweitert, der nicht auf die Behandlung von korrelierten Gittermodellen begrenzt sein muss. Anhand von drei Problemstellungen wird gezeigt, dass die duale Storungstheorie
ein Formalismus ist, der es erlaubt eine Storungsentwicklung um ein exakt losbares Bezugssystem zu formulieren, welches fur die Untersuchung des vorliegenden physikalischen Problems optimiert wurde. Die drei Beispiele sind:
-Das Superperturbations Verfahren zur Losung von quantenmechanischen Storstellenproblemen. Hierbei wird eine Naherung fur eine Storstelle in einem Bad mit kontinuierlichem Energiespektrum mit Hilfe eines Bades endlicher Grose konstruiert, welches dann durch exakte Diagonalisierung gelost werden kann. Danach werden mehrere exakte Grenzfalle der Theorie erortert und eine Methode zur direkten Berechnung der Zustandsdichte auf der reellen Achse vorgestellt. Die Bedeutung des Referenzsystems wird im Kontext des Kondo Problems analysiert und eine Anwendung der Methode als Losungsverfahren fur die dynamische Molekularfeldtheorie diskutiert. Am Ende des Kapitels wird ein Renormierungsverfahren eingefuhrt, das Nicht-Kausalitatsprobleme, die bei tiefen Temperaturen auftreten
konnen, behebt.
-Die Variations-Gitter Naherung (VLA). VLA ist eine Variante des klassischen Ansatzes dualer Fermionen, welche auch exakte Diagonalisierung zum Losen des Storstellenproblems einsetzt. Mit Hilfe dieser Methode wurde der Mott Ubergang analysiert und mit den Ergebnissen fruherer CDMFT Untersuchungen verglichen. Um die Pseudo-Bandlucke im metallischen Regime zu untersuchen, wurde eine analytische Fortsetzung mit Hilfe von Pade auf die reelle Achse durchgefuhrt.
-Das Superperturbations Verfahren zur Losung von quantenmechanischen Storstellenproblemen im Nichtgleichgewicht. Am Ende der Arbeit wird der Ansatz der dualen Fermionen auf Nichtgleichgewichts Systeme verallgemeinert. Dies wird mit Hilfe der Keldysh Theorie bewerkstelligt. Ein Schema zur numerischen Behandlung der auftretenden Grosen wird eingefuhrt und erste Ergebnisse werden prasentiert.
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization, using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate-coupling regime is well reproduced for a single bath site and the lowest-order correction. The method is particularly suited for low temperatures, alleviates analytical continuation of imaginary time data due to the absence of statistical noise and can be generalized to obtain dynamical quantities directly on the real axis.
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization, using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate-coupling regime is well reproduced for a single bath site and the lowest-order correction. The method is particularly suited for low temperatures, alleviates analytical continuation of imaginary time data due to the absence of statistical noise and can be generalized to obtain dynamical quantities directly on the real axis.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. \nIf electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, \ntreating both interactions and disorder on equal footing. \nThe approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. \nTerletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. \nWe extended the approach from Terletska et al. to treat disordered interacting systems. \n \nDual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field \ntheory. \nThe perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, \nbut experience shows that oftentimes good results are obtained for this region. \n \nAn advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. \nAdditionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. \nThus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. \nCluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. \n \nThe new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. \nFor both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. \nThe Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. \n \nFor the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are \nqualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. \nWe found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result \nshows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. \nThis behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. \nWe found this behavior for all values of the disorder that we considered.
Disorder is ubiquitous in real materials and influences the physical properties like the conductivity to varying degrees. If electron-electron interactions are strong, theoretical and numerical treatment of these systems becomes challenging. In this thesis a numerical approach is developed to address these systems, treating both interactions and disorder on equal footing. The approach is based on the dual fermion approach for interacting systems developed by Rubtsov et al. Terletska et al. applied the ideas of the dual fermion approach to disordered non-interacting systems. In this approach, the replica trick is used to integrate out the disorder in favor of an effective electron-electron interaction. We extended the approach from Terletska et al. to treat disordered interacting systems. Dual Fermions allow to take into account non-local fluctuations by means of a perturbative expansion around an impurity problem. The impurity reference system is determined self-consistently, analogously to the dynamical mean-field theory. The perturbative expansion is expected to yield good results for small and large values of interaction strength and disorder. A priori, it is not clear what to expect for intermediate values, but experience shows that oftentimes good results are obtained for this region. An advantage of the dual fermion approach is that there is no sign-problem for a single orbital model if quantum Monte Carlo is used to solve the interacting reference system. Additionally, perturbation theory is usually numerically much cheaper than fully solving an interacting lattice or cluster problem. Thus, the dual fermion approach allows to address regions of parameter space that are not accessible to lattice quantum Monte Carlo calculations or cluster extension of dynamical mean-field theory. Cluster extensions of the dynamical mean-field theory are for example the dynamical cluster approximation or the cellular dynamical mean-field theory. The new approach is benchmarked with respect to the Anderson-Falicov-Kimball model. The Anderson-Hubbard model is the main application. For both models, the algorithm is tested in one dimension for the single particle Green function and compared to the dynamical cluster approximation to estimate the quality of the results. The Anderson-Falicov-Kimball model is studied further in two dimensions and the phase diagram on the disorder-interaction plane is calculated from both the single particle Green function and the dc conductivity. For the Anderson-Hubbard model, the antiferromagnetic phase transition, the Mott transition and the Anderson transition are investigated in three dimensions. The dynamical mean-field theory results are qualitatively confirmed for the most part. Quantitative corrections are in agreement with dynamical cluster approximation results, thus confirming the validity of the results. We found a qualitative difference between the dual fermion approach and dynamical mean-field theory for the temperature dependence of the hysteresis of the double occupancy. The dual fermion result shows a decreasing lower critical interaction strength for decreasing temperature, which is in contrast with the dynamical mean-field result that shows an increase of the lower critical interaction strength. This behavior happens for small temperatures that are not accessible using for example the dynamical cluster approximation, thus highlighting the usefulness of the dual fermion approach. We found this behavior for all values of the disorder that we considered.
Quantum impurity models with strong electron correlations, such as the paradigmatic Anderson Impurity Model (AIM), are central to our understanding of a range of physical phenomena including local moment formation, Coulomb blockade and Kondo screening. They describe magnetic atoms and molecules on surfaces, quantum dot circuits, and correlated materials through dynamical mean field theory. The physics of such systems in strongly non-equilibrium conditions is particularly complex and challenging to capture, whereas Gaussian models of free fermions can be easily solved. Here we show that the time-evolving dynamics of the AIM after a quench can be described by a completely non-interacting version of the model, at the expense of coupling to additional static auxiliary degrees of freedom. Starting from the full solution of the quenched AIM using ED and DMRG, we study the properties of this mapping using numerical optimization, and uncover intriguing structure in the auxiliary system. The method allows us to understand interacting non-equilibrium dynamics through the simpler lens of an effective non-interacting system of larger dimension.