## Final Report Summary - COEL (Numerical study of dynamics and magnetic properties of strongly correlated electron systems)

Advanced experimental techniques led to discovery of novel phenomena and fascinating new materials, including unusual heavy-fermion states, unconventional superconductivity, new quantum phases, topological insulators, or artificial cold atomic gases, which keeps the physics of strongly correlated electron systems in the forefront of modern physics. In all of these examples strong correlations due to Coulomb interaction play a central role. Comprehension of the complex phenomena that arise from strong interactions between electrons is, however, one of the greatest challenges in condensed matter physics today. Still, it is driving a revolution in applications such as quantum information technology or spin transport electronics, aiming to replace conventional electronics by using electron spins as information carriers. Treating the strong correlations, however, can only be performed by using powerful numerical techniques.

Within the 'Numerical study of dynamics and magnetic properties of strongly correlated electron systems' (COEL) project, the researcher investigated peculiar magnetic and electronic properties of various systems with strong correlations by means of the continuous-time quantum Monte Carlo simulation method, a powerful method which is being developed in the last decade.

Project context and objectives:

The intent of the study of strongly correlated systems is twofold: one is the theoretical understanding of fundamental phenomena, and the other is characterizing and using of this phenomena for potential future applications. In spite of the intensive study of interacting many-particle systems in the past several decades, a number of problems still remains unsolved until today, and in addition, there is a constant flow of new materials and phenomena being discovered both by the experimentalists and theoreticians.

One of the open issues is the understanding of unusual heavy-fermion states realized in certain rare-earth and actinide compounds. Heavy-fermion behavior, where the electron has an effective mass a thousand times greater than its rest mass, can be found in diverse compounds showing insulating, metallic, magnetic or unconventional superconducting states at low temperatures. As an example, in the samarium compounds SmOs4Sb12, SmPt4Ge12, and SmT2Al20 with T=Ti, V, Cr, the measured specific heat grows linearly with temperature, with a steep slope indicating a heavy-fermion state with a huge effective mass. The temperature dependence of the electrical resistivity shows an emerging energy scale which is much smaller than the energy scale of the valence fluctuations in these materials. While these are typical for a heavy fermion material, in these samarium compounds the heavy-fermion state is insensitive to external magnetic field, ruling out the usual Kondo effect which involves spin-dependent scattering as the origin of the anomalies. It raises a question: Would a novel charge fluctuation mechanism explain the unusual properties of the samarium systems?

For another example we might recall the URu2Si2 compound, which has the most famous hidden order in the literature. The nature and origin of this phase with unknown order parameter has been in center of debates for a long time. Along with the hidden phase, it also possesses an anomalous heavy-fermion state in the dilute case, UxTh(1-x)Ru2Si2, showing decreasing electrical resistivity by decreasing temperature down to low values, which is in contrast to the behavior in usual Kondo insulators. The understanding of the physics of the dilute system, where the uranium ions can be regarded as impurities, might help toward the understanding of the nature of the hidden ordered phase as well.

Another systems where strong correlations lead to intriguing physics are the Mott insulators with integer number of fermions per site. In the simplest case, the fermionic degrees of freedom order antiferromagnetically. However, under particular circumstances the antiferromagnetic order is suppressed and more exotic quantum phases such as spin liquids may appear. Recently, complex systems such as fermions with high spin can be realized with ultracold atoms in optical lattices in a controllable way, serving as a quantum simulators of various interacting fermion models. The comprehensive theoretical study of such interacting fermions having N internal degrees of freedom (or N-flavors) is, however, a challenging problem and can only be performed by using powerful numerical methods due to the increased size of the local Hilbert space. Although the case of N=2, which corresponds to the ordinary electrons with spin one-half, is well-understood, much less is known about the systems with N>2, and in many cases the results are contradictory. Thus, systematic study of the N-flavor classes of the interacting fermion models is timely, and may reveal new quantum states.

An emerging field of modern physics with potential for future applications is spin transport electronics, or spintronics, which aims to replace conventional electronics by using electron spins as information carriers. The central issue in spintronics is the magnitude of spin-relaxation, i.e. the decay of the polarization of a spin-ensemble, which determines the utility of new spin-carrying devices. Therefore, theoretical study of spin-relaxation processes is indispensable for potential application of spintronics in near future.

Another hot topic is the theoretical and experimental study of transport effects through artificial single atoms, i.e. quantum dots. The spin-blockade transport effect in double quantum dots can be used, for example, for spin-qubit initialization and read out as well as to probe spin interactions of the electrons in the device. In addition, since hyperfine anisotropy causes pronounced anisotropy of the current, the identification of the chirality of single nanotubes can be facilitated via simple transport measurements.

The aim of the project is the theoretical study and understanding of present-day open issues in condensed matter physics where the strong correlations between electrons play the central role including

- unusual heavy-fermion states in certain samarium and uranium compounds,

- physics of interacting fermions in artificially trapped cold atomic gases,

- spin-relaxation processes in metals and semiconductors,

- transport through carbon-based nanotube double dots.

Project results:

The main results arising from the project are as follows:

- Comprehensive study of the thermodynamical and dynamical properties of the multichannel interacting resonating level model by means of continuous-time quantum Monte Carlo method.

The researcher studied a model, which starts from the Anderson model involving the hybridization of the local charge with one of the conduction electron orbitals, and includes additional Coulomb interaction felt by all conduction orbitals with the local f-electron state. Calculating dynamical one- and two-electron correlation functions for the first time, she proposed a novel charge fluctuation mechanism to explain the peculiar heavy-fermion states in certain samarium compounds such as SmOs4Sb12, SmPt4Ge12 or SmT2Al20 with T=Ti, V, Cr. Furthermore, her calculations revealed that, unlike as the thermodynamics, a single energy scale is not sufficient to describe the observed behavior of the dynamical quantities.

- Investigation of the role of non-spin flip (ordinary) potential scattering beside the usual Kondo exchange for magnetic impurities in metals by means of the continuous-time quantum Monte Carlo numerical method.

By accurate calculation of dynamical properties, the researcher found that under strong potential scattering the quasiparticle density of states obtains an antiresonance in contrast to a resonance of the ordinary Kondo physics. As a result, the electrical resistivity in this reverse Kondo range shows a decrease with decreasing temperature, instead of the logarithmic Kondo increase. Her numerical results demonstrate the importance of the interplay between the Kondo effect and the potential scattering in the formation of the anomalous heavy-fermion state in the dilute system UxTh(1-x)Ru2Si2.

- Study of the static and dynamical properties of the N-flavor Hubbard model for the nontrivial cases N>2.

The researcher extended the conventional, S=1/2 Hubbard model (N=2) to the case of multiple degrees of freedom for the fermions (N>2). She wrote a working code to investigate the case of N=4 within the dynamical mean-field theory by using the continuous-time quantum Monte Carlo method as an impurity solver. Results of this study are expected in the near future.

- Theoretical investigation of spin-relaxation mechanisms due to spin-orbit coupling.

The study includes the most important spin-relaxation mechanisms caused by spin-orbit interaction in metals and semiconductors, which are the so-called Elliott-Yafet and D'yakonov-Perel mechanisms in systems with and without inversion symmetry, respectively. The researcher assisted in providing a unified theory of these two seemingly disjunct mechanisms in the limit of small spin-orbit couplings by calculating a pseudopotential model with spin-orbit interaction, developed for semiconductors with zinc-blende crystal structure. In this model, by changing the antisymmetric component of the lattice potential, the 'degree' of the inversion asymmetry can be tuned, and therefore the competition of the two mechanisms can be studied. In addition, arbitrary values of the spin-orbit coupling can be taken into account.

- Study of the characteristics of the current through carbon-based nanotube double dots where the electrons are subject to anisotropic hyperfine interaction.

The researcher contributed to the extension of the master-equation description of the leakage current in a double quantum dot for the case of anisotropic hyperfine interactions, which was originally developed for isotropic cases. She obtained the average current by numerically averaging over random realizations of nuclear fields in the dots as a function of external magnetic field by using Monte Carlo method. The systematic study allows for the characterization of the anisotropy of the hyperfine interaction as well as the hyperfine energy scale via the spin-blockade current measurement. A possible identification of the chirality of single nanotubes is proposed by using simple transport measurements.

Impact:

The efficient continuous-time quantum Monte Carlo simulation technique combined with advanced many-body physics opens a new route for studying complex systems with strong correlations in models where the dimension of the local Hilbert space is large. Phenomena including Kondo physics can be studied this way, and quantum impurity description of many-body phenomena realized in single artificial atoms can be provided. Previously, these problems could have been investigated only by limited number of analytic tools. The researcher introduced the continuous-time quantum Monte Carlo technique in her host country during the project by way of various dissemination activities, where even the class of conventional quantum Monte Carlo methods were unfamiliar to the Hungarian scientific society. She applied successfully the method for a diversity of problems, in collaborations with other research groups.

Within the 'Numerical study of dynamics and magnetic properties of strongly correlated electron systems' (COEL) project, the researcher investigated peculiar magnetic and electronic properties of various systems with strong correlations by means of the continuous-time quantum Monte Carlo simulation method, a powerful method which is being developed in the last decade.

Project context and objectives:

The intent of the study of strongly correlated systems is twofold: one is the theoretical understanding of fundamental phenomena, and the other is characterizing and using of this phenomena for potential future applications. In spite of the intensive study of interacting many-particle systems in the past several decades, a number of problems still remains unsolved until today, and in addition, there is a constant flow of new materials and phenomena being discovered both by the experimentalists and theoreticians.

One of the open issues is the understanding of unusual heavy-fermion states realized in certain rare-earth and actinide compounds. Heavy-fermion behavior, where the electron has an effective mass a thousand times greater than its rest mass, can be found in diverse compounds showing insulating, metallic, magnetic or unconventional superconducting states at low temperatures. As an example, in the samarium compounds SmOs4Sb12, SmPt4Ge12, and SmT2Al20 with T=Ti, V, Cr, the measured specific heat grows linearly with temperature, with a steep slope indicating a heavy-fermion state with a huge effective mass. The temperature dependence of the electrical resistivity shows an emerging energy scale which is much smaller than the energy scale of the valence fluctuations in these materials. While these are typical for a heavy fermion material, in these samarium compounds the heavy-fermion state is insensitive to external magnetic field, ruling out the usual Kondo effect which involves spin-dependent scattering as the origin of the anomalies. It raises a question: Would a novel charge fluctuation mechanism explain the unusual properties of the samarium systems?

For another example we might recall the URu2Si2 compound, which has the most famous hidden order in the literature. The nature and origin of this phase with unknown order parameter has been in center of debates for a long time. Along with the hidden phase, it also possesses an anomalous heavy-fermion state in the dilute case, UxTh(1-x)Ru2Si2, showing decreasing electrical resistivity by decreasing temperature down to low values, which is in contrast to the behavior in usual Kondo insulators. The understanding of the physics of the dilute system, where the uranium ions can be regarded as impurities, might help toward the understanding of the nature of the hidden ordered phase as well.

Another systems where strong correlations lead to intriguing physics are the Mott insulators with integer number of fermions per site. In the simplest case, the fermionic degrees of freedom order antiferromagnetically. However, under particular circumstances the antiferromagnetic order is suppressed and more exotic quantum phases such as spin liquids may appear. Recently, complex systems such as fermions with high spin can be realized with ultracold atoms in optical lattices in a controllable way, serving as a quantum simulators of various interacting fermion models. The comprehensive theoretical study of such interacting fermions having N internal degrees of freedom (or N-flavors) is, however, a challenging problem and can only be performed by using powerful numerical methods due to the increased size of the local Hilbert space. Although the case of N=2, which corresponds to the ordinary electrons with spin one-half, is well-understood, much less is known about the systems with N>2, and in many cases the results are contradictory. Thus, systematic study of the N-flavor classes of the interacting fermion models is timely, and may reveal new quantum states.

An emerging field of modern physics with potential for future applications is spin transport electronics, or spintronics, which aims to replace conventional electronics by using electron spins as information carriers. The central issue in spintronics is the magnitude of spin-relaxation, i.e. the decay of the polarization of a spin-ensemble, which determines the utility of new spin-carrying devices. Therefore, theoretical study of spin-relaxation processes is indispensable for potential application of spintronics in near future.

Another hot topic is the theoretical and experimental study of transport effects through artificial single atoms, i.e. quantum dots. The spin-blockade transport effect in double quantum dots can be used, for example, for spin-qubit initialization and read out as well as to probe spin interactions of the electrons in the device. In addition, since hyperfine anisotropy causes pronounced anisotropy of the current, the identification of the chirality of single nanotubes can be facilitated via simple transport measurements.

The aim of the project is the theoretical study and understanding of present-day open issues in condensed matter physics where the strong correlations between electrons play the central role including

- unusual heavy-fermion states in certain samarium and uranium compounds,

- physics of interacting fermions in artificially trapped cold atomic gases,

- spin-relaxation processes in metals and semiconductors,

- transport through carbon-based nanotube double dots.

Project results:

The main results arising from the project are as follows:

- Comprehensive study of the thermodynamical and dynamical properties of the multichannel interacting resonating level model by means of continuous-time quantum Monte Carlo method.

The researcher studied a model, which starts from the Anderson model involving the hybridization of the local charge with one of the conduction electron orbitals, and includes additional Coulomb interaction felt by all conduction orbitals with the local f-electron state. Calculating dynamical one- and two-electron correlation functions for the first time, she proposed a novel charge fluctuation mechanism to explain the peculiar heavy-fermion states in certain samarium compounds such as SmOs4Sb12, SmPt4Ge12 or SmT2Al20 with T=Ti, V, Cr. Furthermore, her calculations revealed that, unlike as the thermodynamics, a single energy scale is not sufficient to describe the observed behavior of the dynamical quantities.

- Investigation of the role of non-spin flip (ordinary) potential scattering beside the usual Kondo exchange for magnetic impurities in metals by means of the continuous-time quantum Monte Carlo numerical method.

By accurate calculation of dynamical properties, the researcher found that under strong potential scattering the quasiparticle density of states obtains an antiresonance in contrast to a resonance of the ordinary Kondo physics. As a result, the electrical resistivity in this reverse Kondo range shows a decrease with decreasing temperature, instead of the logarithmic Kondo increase. Her numerical results demonstrate the importance of the interplay between the Kondo effect and the potential scattering in the formation of the anomalous heavy-fermion state in the dilute system UxTh(1-x)Ru2Si2.

- Study of the static and dynamical properties of the N-flavor Hubbard model for the nontrivial cases N>2.

The researcher extended the conventional, S=1/2 Hubbard model (N=2) to the case of multiple degrees of freedom for the fermions (N>2). She wrote a working code to investigate the case of N=4 within the dynamical mean-field theory by using the continuous-time quantum Monte Carlo method as an impurity solver. Results of this study are expected in the near future.

- Theoretical investigation of spin-relaxation mechanisms due to spin-orbit coupling.

The study includes the most important spin-relaxation mechanisms caused by spin-orbit interaction in metals and semiconductors, which are the so-called Elliott-Yafet and D'yakonov-Perel mechanisms in systems with and without inversion symmetry, respectively. The researcher assisted in providing a unified theory of these two seemingly disjunct mechanisms in the limit of small spin-orbit couplings by calculating a pseudopotential model with spin-orbit interaction, developed for semiconductors with zinc-blende crystal structure. In this model, by changing the antisymmetric component of the lattice potential, the 'degree' of the inversion asymmetry can be tuned, and therefore the competition of the two mechanisms can be studied. In addition, arbitrary values of the spin-orbit coupling can be taken into account.

- Study of the characteristics of the current through carbon-based nanotube double dots where the electrons are subject to anisotropic hyperfine interaction.

The researcher contributed to the extension of the master-equation description of the leakage current in a double quantum dot for the case of anisotropic hyperfine interactions, which was originally developed for isotropic cases. She obtained the average current by numerically averaging over random realizations of nuclear fields in the dots as a function of external magnetic field by using Monte Carlo method. The systematic study allows for the characterization of the anisotropy of the hyperfine interaction as well as the hyperfine energy scale via the spin-blockade current measurement. A possible identification of the chirality of single nanotubes is proposed by using simple transport measurements.

Impact:

The efficient continuous-time quantum Monte Carlo simulation technique combined with advanced many-body physics opens a new route for studying complex systems with strong correlations in models where the dimension of the local Hilbert space is large. Phenomena including Kondo physics can be studied this way, and quantum impurity description of many-body phenomena realized in single artificial atoms can be provided. Previously, these problems could have been investigated only by limited number of analytic tools. The researcher introduced the continuous-time quantum Monte Carlo technique in her host country during the project by way of various dissemination activities, where even the class of conventional quantum Monte Carlo methods were unfamiliar to the Hungarian scientific society. She applied successfully the method for a diversity of problems, in collaborations with other research groups.