### Stripe phases and ground states with inhomogeneous spin, charge and orbital order in strongly correlated transition metal oxides

Strongly correlated transition metal oxides such as cuprates, manganites, nickelates, cobaltates and ruthenates are investigated using microscopic Hamiltonians in order to detect stripe-like ground states and to establish basic physical characteristics of these compounds (in context of their possible applications in quantum solid-state technology). The computations are performed on atomic clusters using Hartree-Fock method – electronic correlations are included on top of it by application of the so-called local-Ansatz (this is a variant of theoretical chemistry *ab-initio* coupled-cluster method which includes only leading types of electron correlations).

The microscopic models we use are as close to reality as possible. Therefore, they are very complicated, difficult to formulate and much more elaborate than vastly-simplified models typically used in the scientific literature. They include such effects as anisotropic electron hoppings between different atoms (dozens of different hopping amplitudes), on site Coulomb interactions parameterized by intraorbital Hubbard repulsion and full Hund's exchange tensor, crystal field terms (coming from atomic embedding of the investigated finite clusters) and Jahn-Teller static distortions of oxygen-metal-ion octahedra.

We also investigate the electronic properties of doped systems, including high-*T*_{c} cuprates, manganites, vanadates and other 3d systems.

### Order, disorder and phase transitions in systems with orbital degrees of freedom

Theoretical modeling of the phase diagram of the vanadium perovskites is very attractive as several ordered phases occur here, depending on the composition and doping. For instance, phase transition from the *G*-AF/*C*-AO phase, with AF and FO order along the **c** axis, to the inverted order in the *C*-AF/*G*-AO phase, with FM and AO order along this direction was first discovered in YVO_{3 } and is accompanied by the puzzling magnetization reversal. The above competition between different types of spin-orbital order in the perovskites has its analog in epitaxially grown films and superlattices of vanadium oxides. We plan to investigate this problem in more detail, and to include the interplay between lattice distortion and spin-orbital superexchange by realistic modeling.

Also systems with *p* orbital degrees are of interest, with some similarities to the "classical" 3*d* orbital systems, and attract a lot of attention recently. We develop spin-orbital models for such systems which might share some of their features both with above mentioned 3*d* systems and with the novel *p*-orbital systems in optical lattices.

We investigate ground states of various orbital and spin-orbital models with frustrated interactions. These models serve to describe realistic systems or exhibit interesting interplay between different types of order or describe spin or spin-orbital liquids. Quantum phase transitions and entanglement in the ground states are particular challenges in these systems.

### Excitations and quantum entanglement in spin-orbital models

The spin-orbital coupling is an intrinsic effect in several strongly correlated electron systems. While spin-orbital entanglement in ground states is to some extent understood, at present we are investigating entanglement in excited states. Here it is of interest to investigate the mechanisms responsible for the coupling between magnons and collective orbital excitations (orbitons), and also the nature and stability of composite collective spin-orbital excitations.

Our research is conducted to determine the orbiton spectral function for certain 2D types of spin-orbital order in a realistic case of finite Hund's exchange. As a preliminary step to understand the effect of the superexchange induced coupling between spin and orbital degrees of freedom on the orbiton dispersion in 2D and 3D systems, we are extending the mapping presented recently to the case of finite Hund's exchange coupling. The obtained results will be compared against the current experimental results, and in particular we are investigating the origin of absent orbiton dispersion observed in many 2D and 3D transition metal oxides and how this phenomenon is influenced by the spin-orbital coupling.

Particular attention is paid recently to systems with strong spin-orbit coupling, where we investigate phase diagrams for the effective models with strongly frustrated interactions and the hole propagation in various situations. A theoretical challenge here is theoretical description of rather peculiar behavior observed in doped iridates.

Furthermore, we develop methods to calculate the orbital resonant inelastic X-ray scattering (RIXS) response of Mott insulators with orbital degrees of freedom which can then be used as a guideline for future RIXS experiments. We investigate the spectral function for spin excitations in the spin-orbital model which consists of the above mentioned spin-orbital superexchange term but supplemented by the purely orbital Jahn-Teller effect. Particularly attractive in this context are the studies of hole propagation in spin-orbital models using an effective t-J-like model which we investigate either using exact diagonalization (for 1D case) or the self-consistent Born approximation. These studies serve also to make predictions for the inelastic neutron scattering experiments and magnetic RIXS spectra which we compare against those measured in various transition metal oxides with orbital degrees of freedom, including the well-known LaMnO_{3} and KCuF_{3} systems.

### Quantum tensor network algorithms

Computational complexity makes the interacting quantum many-body systems hard to simulate efficiently on a classical computer. The main obstacle stems from their huge Hilbert space which is growing exponentially with the number of particles. The most remarkable example is the Hubbard model of strongly interacting electrons on a two-dimensional square lattice. The model, relevant for high-*T*_{c} superconducting materials, was formulated more than 40 years ago. Still, in spite of an enormous analytical and numerical effort, some of the most fundamental questions, including the existence of stripe phases or the very superconductivity itself, have not been settled yet.

One of the most promising approaches to efficiently simulate quantum many-body systems on a classical computer are quantum tensor network algorithms, a prototypical example of which is the density matrix renormalization group (DMRG). This approach is efficient for a special class of states in the exponentially large Hilbert space that satisfy an area law for entropy of entanglement, i.e., the entropy of any subsystem is proportional to the area separating the subsystem from the rest. Nowadays, DMRG is understood as a variational algorithm in the class of matrix product states (MPS). Its generalization to two dimensions was formulated as PEPS. A qualitatively new class of quantum tensor networks, that goes slightly beyond the area law, is the multi-scale entanglement renormalization Ansatz (MERA) algorithm. It is a real-space renormalization group, where the quantum entanglement is partially disentangled by local unitary transformations before each coarse graining transformation. This is a method tailored to study quantum critical points and it was also successfully applied to a number of problems in two dimensional systems. Being a variational method, the quantum tensor networks do not suffer form the notorious fermionic sign problem, and thus they can be applied to interacting fermions.

### Dynamics of quantum critical phenomena

When simulations on a classical computer fail, the alternative method to use would be quantum emulation using ultracold atoms or trapped ions. The idea of quantum emulation can be traced back to Richard Feynman who pointed out that classical simulation of a quantum system generally scales exponentially with the system size. In case of, say, dynamical response of a quantum system to a change of its Hamiltonian the entanglement quickly violates the area law as a result of e.g. excitation of correlated pairs of quasiparticles. Because of that, the application of quantum tensor networks to such problems is very limited. Feynman's observation suggests both the possibility of exponential speed-up of quantum computers and, for the temporary lack of universal quantum computers on the market, the necessity for quantum emulation.

The quantum emulation is basically a substitution of a quantum system by an equivalent one which is more easily tractable experimentally. The most promising universal quantum emulators at the moment are ultracold quantum gases and trapped ions. For instance, there is a detailed proposal how to emulate the quantum Ising model with arbitrary inter-spin coupling with a chain of trapped ions. Yet another idea, which is closely related to quantum emulation, is adiabatic quantum computation where a solution of a hard computational problem is encoded into a ground state of a suitable Hamiltonian. In order to reach this ground state, the system is initially prepared in a simple ground state of a simple initial Hamiltonian, and then it is driven adiabatically to the desired ground state of the final Hamiltonian. Unfortunately, the interesting final ground state often differs enough from the simple initial one to be separated by a quantum phase transition, where, due to vanishing energy gap, it is hard to keep the evolution adiabatic. A similar problem is encountered in the present day quantum emulators which often relay on the adiabatic protocol. The non-adiabaticity of the transition across a quantum critical point is expected to be described by a quantum version of the Kibble-Zurek (KZ) mechanism. This prediction is partly confirmed by dedicated experiments with ultracold atoms and ion traps.