conferred him PhD degree after successful defense of PhD thesis entitled "Statistical theory of partially excited systems", which Oleg elaborated under the guidance of Professor Ihor Yukhnovskii and Professor Roman Levitskii.Over the past few decades Oleg Derzhko has been scientifically working at the Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine in Lviv, at first, as a research associate at the former Lviv Division of the Institute for Theoretical Physics of the Academy of Sciences of the UkrSSR (since 1982), and finally as head of Department for Theory of Model Spin Systems (2003-2016) and head of Department for Quantum Statistics (since 2016 up till now).During this period, he has also performed a long-term lecturing of several courses of theoretical physics at Ivan Franko National University in Lviv and Lviv Polytechnic National University.Oleg's scientific interests cover various modern problems of condensed matter physics and statistical physics.At the Institute for Condensed Matter Physics in Lviv OlegDerzhko has successfully developed a research direction, which is primarily focused on the state-of-theart problems of quantum spin models with a particular emphasis on the exactly solvable models and is already recognized worldwide.In addition to this challenging research topic in its own right, Oleg has a lot of other scientific interests including, but not limited to, unsolved problems of phase transitions of atomic liquids.It is impossible to summarize here all Oleg's achievements, so let us briefly review at least a few most important milestones of his scientific career.Early-career scientific contributions of Oleg Derzhko were devoted to equilibrium properties of partially excited gas and an influence of resonance irradiation upon a liquid-gas phase transition [1].He did not stop taking interest in this topic across his entire scientific career.The further research in this direction has been developing mainly in collaboration with Dr. Vasyl Myhal where their joint efforts are focused on the properties of a two-phase fluid of two-level atoms, the part of which is excited.Among other matters, Oleg and Vasyl brought a deeper understanding into the vapor-to-liquid transition, wetting under electromagnetic resonance irradiation, a microscopic theory of photonucleation with the special emphasis laid on nucleation phenomena in an electric field, surface tension, cavitation and so on [2][3][4].Exactly solvable problems in general and particularly one-dimensional quantum XY models and quantum Ising models in a transverse magnetic field have apparently become Oleg's greatest scientific passion and delight, where he substantially extended the rigorous results with the help of Jordan-Wigner fermionization technique [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].To this end, Oleg Derzhko and his collaborators, which were predominantly from the Institute for Condensed Matter Physics in Lviv, have nontrivially adapted the recipé originally envisaged in a seminal paper by Lieb, Schultz, and Mattis [20].A big step forward at that time was the formulation of the exact numerical method for the calculation of time-dependent spin correlation functions and the corresponding structure factors [5][6][7].Exact solutions for numerous random and periodic non-uniform versions of quantum XY and Ising models achieved by Oleg and by his research group brought a considerable progress in theoretical understanding of a lot of exciting phenomena emergent in one-dimensional quantum spin chains.For instance, in this field of study Oleg has significantly contributed to a clarification of the effect of regular nonuniformity, randomness and correlated disorder on static and dynamical properties of several exactly solved quantum XY and Ising chains [8][9][10][11][12][13][14].Oleg achieved another important breakthrough in this research area by a detailed examination of spin-Peierls instabilities of a certain class of quantum XY chains [15][16][17], as well as, quantum phase transitions and the dynamics of exactly solved XY chains with three-site interactions [16,18,19].
conferred him PhD degree after successful defense of PhD thesis entitled "Statistical theory of partially excited systems", which Oleg elaborated under the guidance of Professor Ihor Yukhnovskii and Professor Roman Levitskii.Over the past few decades Oleg Derzhko has been scientifically working at the Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine in Lviv, at first, as a research associate at the former Lviv Division of the Institute for Theoretical Physics of the Academy of Sciences of the UkrSSR (since 1982), and finally as head of Department for Theory of Model Spin Systems (2003-2016) and head of Department for Quantum Statistics (since 2016 up till now).During this period, he has also performed a long-term lecturing of several courses of theoretical physics at Ivan Franko National University in Lviv and Lviv Polytechnic National University.Oleg's scientific interests cover various modern problems of condensed matter physics and statistical physics.At the Institute for Condensed Matter Physics in Lviv OlegDerzhko has successfully developed a research direction, which is primarily focused on the state-of-theart problems of quantum spin models with a particular emphasis on the exactly solvable models and is already recognized worldwide.In addition to this challenging research topic in its own right, Oleg has a lot of other scientific interests including, but not limited to, unsolved problems of phase transitions of atomic liquids.It is impossible to summarize here all Oleg's achievements, so let us briefly review at least a few most important milestones of his scientific career.Early-career scientific contributions of Oleg Derzhko were devoted to equilibrium properties of partially excited gas and an influence of resonance irradiation upon a liquid-gas phase transition [1].He did not stop taking interest in this topic across his entire scientific career.The further research in this direction has been developing mainly in collaboration with Dr. Vasyl Myhal where their joint efforts are focused on the properties of a two-phase fluid of two-level atoms, the part of which is excited.Among other matters, Oleg and Vasyl brought a deeper understanding into the vapor-to-liquid transition, wetting under electromagnetic resonance irradiation, a microscopic theory of photonucleation with the special emphasis laid on nucleation phenomena in an electric field, surface tension, cavitation and so on [2][3][4].Exactly solvable problems in general and particularly one-dimensional quantum XY models and quantum Ising models in a transverse magnetic field have apparently become Oleg's greatest scientific passion and delight, where he substantially extended the rigorous results with the help of Jordan-Wigner fermionization technique [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].To this end, Oleg Derzhko and his collaborators, which were predominantly from the Institute for Condensed Matter Physics in Lviv, have nontrivially adapted the recipé originally envisaged in a seminal paper by Lieb, Schultz, and Mattis [20].A big step forward at that time was the formulation of the exact numerical method for the calculation of time-dependent spin correlation functions and the corresponding structure factors [5][6][7].Exact solutions for numerous random and periodic non-uniform versions of quantum XY and Ising models achieved by Oleg and by his research group brought a considerable progress in theoretical understanding of a lot of exciting phenomena emergent in one-dimensional quantum spin chains.For instance, in this field of study Oleg has significantly contributed to a clarification of the effect of regular nonuniformity, randomness and correlated disorder on static and dynamical properties of several exactly solved quantum XY and Ising chains [8][9][10][11][12][13][14].Oleg achieved another important breakthrough in this research area by a detailed examination of spin-Peierls instabilities of a certain class of quantum XY chains [15][16][17], as well as, quantum phase transitions and the dynamics of exactly solved XY chains with three-site interactions [16,18,19].
The study of non-equilibrium behaviors of quantum matter is one of the research directions at the frontiers of condensed matter physics and, in general, of theoretical and experimental physics. Understanding the mechanisms which prevent many quantum particles from reaching thermal equilibrium can ultimately shed light on the foundations of statistical physics and provide insight into the anomalous long-time collective behaviors observed in experiments. In this thesis, we present new theoretical developments in this direction. In the first part of the thesis we discuss the non-equilibrium dynamics of the two-dimensional quantum Ising model, the prototypical model of magnetism. In particular, we investigate the dynamics of interfaces separating different spin domains. In some regimes, we can provide an exact description of their unitary time evolution, but we also discuss the robustness of our predictions away from the exactly-solvable limit. Moreover, we address the same problem in the presence of disorder and provide a hydrodynamic interpretation of our findings. In the second part of the thesis, we investigate the localization properties of disordered quantum systems. In particular, we develop a renormalization-group picture of the Anderson localization transition in generic geometries. This provides a new framework to understand the phase diagram of such systems and their critical behavior beyond the usual finite-size scaling analysis. We also provide further insights on this topic both by studying resonances and their connection with observables and by employing unsupervised non-parametric learning techniques to probe disordered systems in an experimentally accessible form. The results presented in this thesis constitute a step forward in understanding the non-equilibrium phases and regimes of quantum matter and they pave the way to interesting future developments.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.
The study of quantum dynamics is an increasingly important aspect of atomic, molecular and optical physics, as well as condensed-matter physics and other disciplines. It is essential in order to understand non-equilibrium phenomena which do not fit within the framework of equilibrium thermodynamics, such as many-body localization, dynamical quantum phase transitions, and non-equilibrium steady states in driven-dissipative systems. However, the exact calculation of dynamical observables is extremely difficult in general for several reasons, which makes it challenging for theoretical predictions and analyses to keep pace with rapidly advancing experiments. In this thesis I use and develop several computational methods to investigate the dynamics of quantum many-body systems, with the two goals of better understanding the evolution of local observables and correlation functions in spin and Fermion systems as well as creating and testing more effective numerical methods for solving quantum dynamics problems. I explore quantum dynamics by using Hamiltonian parameter quenches in several commonly-used models in AMO physics and elsewhere: the Fermi-Hubbard model, the XXZ model, and the transverse-Ising model. The first represents a system of short-range interacting Fermions, while the latter two represent spin systems with different spin-spin couplings and magnetic field terms. All of these models can be realized in current experiments. First, I derive an exact solution for an extreme interaction quench in the Fermi-Hubbard model, from a strongly-interacting initial state to a noninteracting final Hamiltonian. From this result I show that there are nontrivial transient connected correlations, despite the fact that the system has very high temperature and a noninteracting Hamiltonian. Those features would suppress such correlations in equilibrium, but the dynamical correlations are still significant. Then, I develop and examine two numerical methods for solving dynamics of the spin expectation values and correlation functions in the transverse-Ising and XXZ models. Both methods rely on approximating a thermodynamically large system with a much smaller and more computationally tractable cluster of lattice sites, or a series of clusters. The first method is a dynamical numerical linked cluster expansion (NLCE). NLCEs have already been used successfully on systems in thermodynamic equilibrium, and I extended the technique to directly compute the dynamics of an observable after a quantum quench. I review the general NLCE method, describe my dynamical variant of it, and compute the quench dynamics of spin and correlation observables in the transverse-Ising and XXZ models. The second method, which I term the adaptive-boundary cluster method, relies on recently refined bounds on information propagation in quantum systems to choose a cluster of lattice sites that minimizes the finite size error bound of a dynamical computation. I briefly review these bounds, and describe a new algorithm which uses them to construct a cluster whose boundary is chosen according to a related error estimate. Such an adaptive-boundary cluster has the property that there exists a computable upper bound on the finite size error of observables measured at the cluster center, and this error bound is nearly minimal when compared to that of other clusters of the same size. I then compute the quench dynamics of spin and correlation observables in the transverse-Ising model using the adaptive boundary clusters. I apply this method to both uniform and strongly disordered systems. To evaluate the effectiveness of both dynamical NLCE and adaptive-boundary clusters, I compare the results obtained using them to results from a standard and general purpose numerical method: exact solvers applied a rectangular cluster with periodic boundary conditions. Both of the new methods usually converge faster and are more accurate than the standard method for comparable cluster sizes. This thesis illustrates that choosing numerical methods which take advantage of the geometric and entanglement structure of the initial state and Hamiltonian can substantially enhance our ability to study quantum dynamics, by reducing the computational cost of numerically exact solutions. I conclude with a discussion of various avenues for future research to build on this work.
The book reviews several theoretical, mostly exactly solvable, models for selected systems in condensed states of matter, including the solid, liquid, and disordered states, and for systems of few or many bodies, both with boson, fermion, or anyon statistics. Some attention is devoted to models for quantum liquids, including superconductors and superfluids. Open problems in relativistic fields and quantum gravity are also briefly reviewed.The book ranges almost comprehensively, but concisely, across several fields of theoretical physics of matter at various degrees of correlation and at different energy scales, with relevance to molecular, solid-state, and liquid-state physics, as well as to phase transitions, particularly for quantum liquids. Mostly exactly solvable models are presented, with attention also to their numerical approximation and, of course, to their relevance for experiments.
This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.