Локальная физика фазовых переходов без параметра порядка/ Local Physics of Phase Transitions without an Order Parameter тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Марков Антон Александрович

• Марков Антон Александрович
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Марков Антон Александрович

Contents

1 Introduction

1.1 Phases without an Order Parameter

1.2 Thesis Structure and Content

1.3 Dissertation Goals

1.4 Statements to Defend

1.5 Theoretical and Practical Significance

1.6 List of Publications

1.7 Validity and Methods

1.8 Validation of the Thesis

1.9 Personal contribution of the author

2 Phase Transitions Without an Order Parameter

2.1 Mott-Hubbard physics

2.1.1 HubbardModel

2.1.2 Metallic Phase. Weak Coupling

2.1.3 Mott insulator. Strong coupling

2.1.4 Numerical Approaches to the Hubbard Model

2.1.5 Experimental Signatures of the Mott Transition. The Case of Study:VO2

2.2 Topological Phase Transitions

2.2.1 Topology and Condensed Matter

2.2.2 Quantum Hall Effect and Topology

2.2.3 Twisted Boundary Conditions and Many-Body Chern Number

2.2.4 Single-particle Greens Functions and Topology

2.2.5 Chern-Simons Action and the Streda Formula

2.2.6 Chern Insulators: Materials and Experiments

2.2.7 Local Topological Markers for Chern Insulators

2.3 Dynamical Quantum Phase Transitions

2.3.1 Lieb-Robinson Bounds

2.3.2 Dynamical Quantum Phase Transitions

3 Hofstadter-Hubbard Physics

3.1 Hofstadter-Hubbard Model and DMFT Approximation

3.2 Effect of Finite Interactions and Temperatures on the Lattice Quantum Hall Effect

3.2.1 The Hall Conductivity at Finite Temperatures

3.2.2 Numerical Results

3.2.3 Importance of Correlations

3.3 Orbital Magnetization on the Edge of the Mott Transition

3.3.1 Orbital Magnetization in Insulators

3.3.2 Orbital Magnetization in Hofstadter-Hubbard Model

3.3.3 Numerical Results

3.3.4 Strong-Coupling Expansion

3.3.5 Band and Mott Insulators

3.4 Orbital Magnetic Field Driven Metal-Insulator Transition

3.4.1 Results

3.4.2 Discussion

3.4.3 Relation to Experimental Studies

4 Local Markers for Phase Transitions Without an Order Parameter

4.1 Local Marker for Interacting Topological Insulators

4.2 Numerical Results

4.3 Locality of Topological Dynamics in Chern Insulators

4.3.1 Topological Markers Out of Equilibrium

4.3.2 Chern insulators and their Local Topological Markers

4.3.3 Topological Markers' Dynamics

4.3.4 Local Chern Marker

4.3.5 Local Streda Formula

4.3.6 Quench Dynamics

4.3.7 Translationally Invariant Bulk. QWZ model

4.3.8 Hofstadter-Harper Model

4.3.9 Non-local Transport of the Marker

4.3.10 Slow markers' dynamics

4.3.11 Experimental Relevance of the Results

4.4 Dynamical Quantum Phase Transition Without An Order Parameter

4.4.1 System and Method

4.4.2 Main Results

4.4.3 Low Energy Excitations and Fisher Zeros

4.4.4 Weak Disorder and DQPT

4.4.5 Proof of the Theorem

4.4.6 Application of Theorem 1 to Harmonic Perturbation

4.4.7 Influence of Disorder on the Fermionic Correlators

Appendices

4.A Locality of Topological Dynamics

4.A.1 Local Streda Marker in Equilibrium

4.A.2 Local Streda Marker out of Equilibrium

4.A.3 Spectral Power of the Markers Currents

4.A.4 The Effect of Disorder on the Dynamics

4.A.5 On the Gauge Invariance of the Local Chern Marker Currents

4.A.6 M currents

4.B DQPT without an Order Paramter

4.B.1 Shape of BCS-matrix G without Disorder

4.B.2 Asymptotics of Commutator Series

4.B.3 Bounds on Change of Fermionic Correlators

4.B.4 Fermions in Energy Basis

4.B.5 Fermions in the Position Basis

4.B.6 Finite Disorder

4.B.7 Perturbations of Different Wavelength

4.B.8 Low Energy Part of BCS-matrix, Numerical Test

5 Conclusions

5.1 Hubbard-Hofstadter physics

5.2 Local Markers for Phase Transitions Without an Order Parameter

Введение диссертации (часть автореферата) на тему «Локальная физика фазовых переходов без параметра порядка/ Local Physics of Phase Transitions without an Order Parameter»

Chapter 1 Introduction

1.1 Phases without an Order Parameter

The concept of a phase is natural when the physics of many-body systems, with their enormous number of degrees of freedom, is considered. Most generally, the definition of a phase can be boiled down to: a class of states of a many-body system with qualitatively distinct behavior in their observables and correlators. Fixing a phase, rather than a state, allows one to describe a whole N-body quantum system with a few parameters and a few characteristics, whereas an exponentially large (in N) number of parameters is required to fix a state. For many centuries, three phases of matter were known. These are solid, liquid, and gas, described by only three parameters: temperature, volume, and pressure. In the 20th century, it was realized that there are many more phases of matter. This required a more constructive definition of a phase and classification schemes for the phases themselves. There are three main approaches aimed at defining and classifying the equilibrium phases of matter. The first classifies the phases by their symmetries, the second by the low-energy effective theories, and the last by the topological properties. These three approaches are illustrated schematically in Fig (1.1).

Figure 1.1. Schematic illustration of a phase diagram from the perspectives of (a) Landau theory, (b) Renormalization Group, (c) Topological classification.

Landau [1] proposed the first scheme to classify different phases. According to his theory, phases are uniquely characterized by their symmetries. During a phase transition, the symmetry of a state changes, leading to qualitatively different physics. For example, a solid breaks the

translational invariance of a liquid, and a ferromagnet breaks the spin-rotational symmetry of a paramagnet. This process of symmetry breaking can be described by an order parameter. In the simplest cases, such an order parameter is just one real number, r In a disordered phase, symmetry is preserved on average, and the order parameter is zero, while in an ordered phase, the symmetry is broken, and it acquires a non-zero value. In an infinite homogeneous system, a single global order parameter r suffices. However, to address phenomena in inhomogeneous systems, where different parts may be in different phases, it is not sufficient. In 1950, Ginzburg and Landau [2] introduced a spatially dependent local order parameter r(x) to describe the surface tension at the boundary between a superconductor and a normal (non-superconducting) state, as well as the destruction of the superconducting phase by a magnetic field.

One of the most important counterexamples to this classification scheme is the Kondo problem [3]. The physics of metals with a small number of magnetic impurities is very different at low and high temperatures; however, no phase transition is observed between these states, and no symmetry is broken. The first part of the Kondo problem was revealed experimentally in 1930 [4], predating the inception of Landau theory. W. Meissner and G. Voigt found that metals (presumed pure) exhibited resistance minima at non-zero temperatures, while a decrease in resistivity at low temperatures was expected. In a series of experiments [5,6], this effect was observed in the presence of impurities with partially filled d-levels close to the Fermi energy of a host metal. The second puzzle constituting the Kondo problem was the formation of local moments on magnetic impurities embedded in a metal. Matthias 1 and collaborators [7] observed that a small fraction (« 1%) of Fe immersed in transition metals and their alloys either forms or does not form localized magnetic moments. In a further experiment [8] at Bell Laboratories, these two phenomena—anomalous low-temperature resistance and the formation of local moments—were connected. The modern view of the Kondo problem is that the low- and high-temperature physics of magnetic impurities in metals is described by very different effective low-energy theories. A "poor man's version" of this idea is due to Anderson [9]. A more systematic and rigorous way to construct these low-energy theories came with the development of the Renormalization Group (RG) [10,11].

The main idea of the RG, in its various incarnations [10,12-14], is that the low-energy, longdistance physics should not depend on the microscopic high-energy details. Consider a microscopic theory with a set of couplings {J} describing interactions between the system's constituents. Under RG, the high-energy degrees of freedom are successively eliminated, leading to a change (or flow) in the couplings {J}. At some point (perhaps only in the sense of a limit), the couplings no longer change, and a fixed point is obtained, with a new set of parameters {J}. It turns out that most of the couplings among {J} flow to zero; such couplings are called irrelevant. Thus, often the low-energy theory is much simpler than the original microscopic one. Different phases of matter correspond to basins of attraction around fixed points, which are themselves stable with respect to the RG procedure. Unstable fixed points form critical surfaces separating such basins and correspond to phase transitions. When applied to the Kondo problem, RG shows that the coupling between a magnetic moment and the conducting electrons is an example of a relevant coupling; that is, it flows to infinity at low energies. Thus, at high temperatures, we deal with weakly interacting, high-energy physics. However, lowering the temperature, we instead probe the low-energy physics of the system, which is strongly coupled.

The relevance of topology to the classification of phases beyond the symmetry-breaking paradigm began to be appreciated in the 1960s-70s. Perhaps the first example of a topological phase transition was provided by Lifshitz in 1960 [15]. He noticed that a change in the topology

:The same Matthias as in "Matthias rules". These are famous empirical rules for identifying superconductors. The last rule advises experimentalists to "Stay away from theorists!".

of a Fermi surface drastically alters the density of states and, thus, thermodynamic quantities. He further argued that, in general, no symmetry has to be broken during such a process. The next key insight came from Berezinskii [16] in 1971. He showed that correlators of the two-dimensional XY-model obey different asymptotic forms in the low- and high-temperature limits. According to the Mermin-Wagner theorem [17], however, a continuous symmetry cannot be broken in two-dimensional systems; therefore, no phase transition was expected in the model. Berezinskii connected the behavior of long-range correlators to the possibility of the vector order parameter of the XY model making several full revolutions along the path between two points.

A similar result was independently obtained by Kosterlitz and Thouless [18,19]. They provided a much simpler and more intuitive picture for the phase transition, connecting the revolution of the order parameter with the existence of topological excitations in the system. These are vortices and anti-vortices. At low temperatures, there are no such excitations, but at higher temperatures, the entropy gain allows the formation of vortex-anti-vortex pairs, which can separate and behave as independent excitations. This line of thought was later systematized in a homotopy theory of topological defects by Thouless and Kleman [20], as well as independently by Volovik and Mineev [21] and Rogula [22] in 1976.

However, the field of topological phase transitions truly bloomed with the discovery of the

Quantum Hall Effect [23]. The Hall conductivity was observed to exhibit an astonishingly precise

2 2

quantization in units of , despite 2 the presence of a significant number of impurities. Thouless and collaborators explained [24] that this precision could be understood using topological arguments. They provided the connection between the Hall conductivity and a topological invariant of the corresponding single-particle Hamiltonian. Since then, the idea that different phases can be classified by topological invariants, rather than symmetries, has spread across the entire condensed matter community. Several definitions of a topological phase have been proposed, each slightly different yet intertwined. The key idea is to define a set of continuous deformations of Hamiltonians, field configurations, or wave-functions. Then, phases can be classified by the properties of such objects that remain invariant under the aforementioned set of continuous transformations. Physically, topological stability means robustness against certain external perturbations. Thus, the transverse conductivity in the Quantum Hall case is stable against moderate disorder and external fields.

Topologically inhomogeneous samples require special care from a theoretical perspective. Global topological indices, e.g. Chern number [24], are not applicable directly to such systems, as they characterize the whole system, just like a single order parameter in the Landau theory. Recently, a family of quasi-topological quantities, called local topological markers, has been developed and studied [25-28]. Such markers depend on the exponentially localized density matrix's elements in equilibrium. Thus, one uses local information to estimate a global topological index. Topological markers are not necessarily strictly quantized; rather, the average of the marker over large areas of a system tends towards a quantized value [29]. The requirement that a marker is a local representative of a global index does not produce a unique definition. Indeed, the Chern number has several local counterparts [25-28,30], each coming with its own merits and drawbacks. The topological markers can be seen as a counterpart of Landau-Ginzburg local order parameter.

Let us now discuss how these three approaches to the phases of equilibrium matter are related to one another. Naturally, there are symmetry-breaking phase transitions within a single topological phase. On the other hand, a topological phase transition does not require symmetry breaking. Therefore, the Landau and topological classifications can be considered complementary to each other. It is possible to incorporate both schemes into a single classification by considering

2Or rather, due to impurities, in fact

symmetry-protected topological phase transitions. Symmetry-protected topological phases are those that remain stable against continuous deformations as long as a particular symmetry is preserved. Therefore, by imposing this condition on a relevant symmetry, a symmetry-breaking phase transition can also be considered topological [31]. The RG approach can be considered more general than the other two, as it can describe both Landau phase transitions and topological phase transitions [11,32]. By considering the topology or symmetries of a fixed point, the RG phase diagram gains physical content.

Despite significant progress in recent years, out-of-equilibrium collective phenomena are far less understood than equilibrium phenomena [33]. Even the definitions of a "phase" and a "phase transition" are not yet entirely clear. Dynamical Quantum Phase Transitions (DQPT) [34,35] are one of the more established and elaborated attempts to build such an understanding. Equilibrium phase transitions occur when a system substantially and sharply changes its properties as some parameter, for example, temperature or concentration, is varied. A dynamical quantum phase transition is a sharp change in a system's properties, which occurs as a function of time rather than a parameter.

1.2 Thesis Structure and Content

This thesis studies the local physics of phase transitions that lack an order parameter. In particular we concentrate on the three classes of such phase transitions. These are Mott metal-to-insulator phase transitions, topological phase transitions in Chern insulators and dynamical quantum phase transitions.

The thesis is structured as follows. Chapter 2 serves as a literature review for the research part of the thesis. Section 2.1 discusses the physics of the Hubbard model [36] and the Mott transition [37]. We introduce the Hubbard model, discuss the weak coupling and the strong coupling limits where the Hubbard model's behavior can be understood analytically. Thereafter we describe numerical algorithms used to study the model with a focus on the Dynamical Mean Field Theory (DMFT) [38]. The section concludes with a discussion of experimental signatures of the Mott transition. Section 2.2 focuses on the field of topological phases of matter [39]. It examines strongly correlated systems exhibiting the integer quantum Hall effect [23], discussing global topological indices and introducing their localized counterparts, called local topological markers. Section 2.3 addresses Dynamical Quantum Phase Transitions (DQPT) [34]. While ordinary phase transitions occur when some of the system's parameters changes and free energy develops an non-analyticity, DQPT happens at particular points in time, when the return rate becomes nonanalytical as a function of time.

Chapter 3 investigates the interplay between the orbital effects of a magnetic field and strong local correlations. The first section studies the lattice quantum Hall effect under the combined effect of finite interactions and non-zero temperatures. We demonstrate that the integer Hall conductivity values deviate from the integer values at temperatures significantly lower than the renormalized non-interacting band gap as a result of transition from an insulator to a weak metal. In the second section, we explore orbital magnetization oscillations behavior in the Hubbard model in the vicinity of a Mott transition. We observe a sharp transtion from 1/B-periodic de Haas-van Alphen oscillations to B-periodic oscillations. This phenomenon, along with its temperature dependence, is proposed as a method to differentiate between a Mott insulator and a band insulator. Thereafter, in the final section of this chapter we predict an insulator-to-metal phase transition induced solely by the orbital effects of a magnetic field.

Chapter 4 explores whether phases lacking an order parameter can be endowed with an analog

of Landau-Ginzburg local order parameter. In particular we concentrate on topological phase transitions and dynamical quantum phase transitions. In the first section Section 4.1, we introduce a local marker for interacting Chern insulators expressed in terms of the single-particle Green's functions. In the second section Section 4.3, we study the dynamics of topological markers in non-interacting systems and discuss the conditions when the markers obey a local continuity equation. This will allow us to make a connection between the markers out-of-equilibrium and the dynamics of magnetic field-induced charges. Finally in Section 4.4 we shall see that a disorder-induced dynamical quantum phase transitions in the transverse field Ising model, being non-topological, cannot be traced by any reasonably local observable.

1.3 Dissertation Goals

The main goals of the thesis are:

1. To study the interplay between orbital effects of magnetic fields and local many-body correlations. On the one hand, we aim to study the effects of magnetic fields on the Mott transition, and on the other hand, how the topological quantum Hall effect [23] and quantum oscillations are altered in the presence of local Hubbard interactions.

2. To propose an experimentally realistic way to discriminate between a Mott insulator and a band insulator.

3. To explore the possibility of introducing an analog of a Landau-Ginzburg order parameter for phase transitions lacking an order parameter. In particular, we will attempt this task in the case of interacting Chern insulators and systems undergoing a dynamical quantum phase transition.

4. To describe locally the change in a topological phase out of equilibrium.

1.4 Statements to Defend

The main scientifically novel results presented in the thesis are:

1. We have developed an extension of dynamical mean field theory [38] suitable for systems in the presence of orbital magnetic fields.

2. We have demonstrated that the inclusion of local correlations at finite temperatures leads to an instability of the Hofstadter fractal structure of the spectrum. We have found a deviation of the Hall conductivity axy from the topologically protected integer values at much lower temperatures than in a corresponding non-interacting system. We demonstrate that these effects can be related to a correlation-driven filling of the band gaps.

3. We have predicted a Mott insulator-to-metal transition driven solely by the orbital effects of the magnetic field. We found that for sufficiently large magnetic fields, a metallic phase emerges at interaction strengths corresponding to the Mott insulating state in the field-free case. Upon decreasing the field, we observe a metal-insulator transition from this metallic to an insulating state. This effect can be understood as a magnetic-field-induced increase in kinetic energy due to the formation of magnetic mini-bands, which enhances the ratio between the kinetic and potential energy and, thus, triggers a transition between a metallic and insulating phase.

4. We have studied the behavior of orbital magnetization at the verge of the Mott transition and observed features specific to the Mott metal-to-insulator transition on both the metallic and insulating sides of the transition. Therefore, the effect could help shed light on the nature of correlated insulating phases in systems where the Mott transition is under debate, such as in twisted bilayer graphene [40] or in vanadium dioxide (VO2) [41].

5. We propose an intrinsically many-body local topological marker based on the single-particle Green's function. Using this marker, we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to non-equilibrium systems.

6. We have demonstrated that out-of-equilibrium local topological markers obey the local continuity equation for experimentally relevant settings and have connected the out-of-equilibrium marker to the dynamics of magnetic-field-induced charge.

7. We demonstrated that the disorder-induced dynamical quantum phase transition in the Ising model is an example of a non-topological dynamical quantum phase transition without a local order parameter. A series of critical times universally appears for any vanishing perturbation with Fourier components at the lowest momentum of order 1/N3. This could be considered a dynamical counterpart of the Anderson orthogonality catastrophe [42].

1.5 Theoretical and Practical Significance

The first group of results (1-4) is relevant to the physics of strongly interacting systems in large magnetic fields. Magnetic field couples to both spin and orbital magnetic moments. We will concentrate on the orbital coupling. By large magnetic fields, we mean those for which the magnetic length, Ib = eB, is of the order of the lattice spacing, Ib ~ a. This physical regime most directly applies to the physics of moire materials [43]. In these systems, the effective lattice constant can be large enough that moderate magnetic field strengths of several tesla correspond to one flux quantum per unit cell. On the other hand, nearly flat bands emerge in these systems at discrete "magic" twist angles [44]. At these twist angles, strong interactions can be naturally achieved, with U/W ^ 1, where U is the characteristic interaction strength and W is the bandwidth. Moreover, ultra-high magnetic fields of up to 500 T—comprising a noticeable fraction of the flux quantum—have recently been achieved in experiments with more traditional strongly correlated systems, such as VO2 [45] and A-type organic conductors [46].

The results (1-4) might explain magnetic-field-driven insulator-to-metal transitions observed in these systems [40,45,46]. Furthermore, our work may clarify the role of correlations in the formation of insulating correlated states in twisted bilayer graphene [40] and VO2 [45]. Understanding whether an insulating state arises from the Mott mechanism [37] or from symmetry breaking has both practical and fundamental significance. The transition mechanism directly impacts the quantitative characteristics of transistors based on these materials, such as switching times or possible external triggers [41]. Additionally, Mott insulators perhaps doped hosts a variety of exotic quantum phases, including high-temperature superconducting states [47] and spin liquids [48], whereas a doped correlated band insulator typically produces a Fermi liquid state.

The results (5-6) pertain to the physics of spatially inhomogeneous topological systems. A defining property of topological insulators is the formation of robust conducting modes at interfaces between regions with different topological indices [49,50]. This property has numerous

potential applications, including dissipationless power lines [51], new generations of inductors [52], and other electronic devices [53], as well as quantum computation [54]. Therefore, the ability to detect topologically homogeneous regions within a sample and control their locations is of both fundamental and practical interest.

Finally, the last result (7) demonstrates that disorder-induced dynamical quantum phase transitions deviate from the usual equilibrium paradigm. In equilibrium, a phase transition is either topological and characterized by a global "order parameter" or has a local order parameter. We have shown that, out of equilibrium, it is possible for a phase transition to lack both a local order parameter and topological characteristics.

1.6 List of Publications

The work in this thesis is based on the following publications:

1. AA Markov, G Rohringer, and AN Rubtsov. Robustness of the topological quantization of the Hall conductivity for correlated lattice electrons at finite temperatures. Physical Review B, 100(11):115102, 2019

2. AA Markov and AN Rubtsov. Local Marker for Interacting Topological Insulators. Physical Review B, 104(8):L081105, 2021

3. AA Markov, DB Golovanova, AR Yavorsky, and AN Rubtsov. Locality of topological dynamics in Chern insulators. SciPost Physics, 17(6):152, 2024

4. ON Kuliashov, AA Markov, and AN Rubtsov. Dynamical quantum phase transition without an order parameter. Physical Review B, 107(9):094304, 2023

5. G Rohringer and AA Markov. Orbital magnetic field driven metal-insulator transition in strongly correlated electron systems. arXiv preprint arXiv:2406.18729, 2024

6. AA Markov and G Rohringer. Orbital magnetization on the edge of the Mott transition. in preparetion

Four of these are published in peer-reviewed journals.

1.7 Validity and Methods

We have studied the physics of many-body systems using a combination of analytical and numerical approaches. The numerical methods employed in this thesis are either numerically exact (e.g., exact diagonalization) or approximate (dynamical mean field theory [38]). The approximate methods used have been thoroughly tested against other numerical techniques and experimental results [61]. Consequently, all the theoretical results presented in this thesis possess predictive power and can be further validated through experimental testing.

1.8 Validation of the Thesis

The main results were validated during the following international conferences:

1. APS March Meeting, March 7 2019, Boston USA G. Rohringer, A.A. Markov, A.N. Rubtsov "The Hall conductivity in correlated electron systems"

2. Part III Seminar Series, November 30, 2018 Cambridge University, A.A. Markov, G. Rohringer and A.N. Rubtsov "The Quantum Hall Effect in an interacting model"

3. V International Conference on Quantum Technologies, 16th of July 2019 Moscow, A.A. Markov, G. Rohringer and A.N. Rubtsov "Quantum Hall effect in a correlated system at finite temperature"

4. VI International Conference on Quantum Technologies, 14th of July 2021 Moscow, A.A. Markov, G. Rohringer and A.N. Rubtsov "Quantum Hall effect in a correlated system at finite temperature"

5. VI International Conference on Quantum Technologies, 14th of July 2021 Moscow, D.B. Golovanova, A.A. Markov, A.N. Rubtsov "Dynamical control of topological defects in Chern insulator"

6. APS March Meeting March 16, 2022, D.B. Golovanova, A.R. Yavorsky, A.A. Markov, A.N. Rubtsov "Currents of topological marker in dynamically controlled Chern Insulators"

7. Theory Winter School "Modern aspects of quantum condensed matter" January 14th, 2021 Tallahassee, FL , USA "Local marker for topological insulators beyond equilibrium non-interacting systems" A.A. Markov and A.N. Rubtsov

8. ICTP Summer School "New Trends in Modern Quantum Science: from Novel Functional Materials to Quantum Technologies", September 30, 2023 Bukhara, Uzbekiztan A.A. Markov and G. Rohringer "Quantum oscillations of the orbital magnetization in Mott insulators"

Additionally, the results were presented in seminars at the Russian Quantum Center, Moscow State University, Imperial College London and King's College London.

1.9 Personal contribution of the author

All the collaborations are explicitly stated in the corresponding sections. All the statements presented for the defense were initially formulated by the author. The numerical results and analytical proofs presented in the thesis are the product of the author's own work, except for:

• The DMFT calculations of free energy, double occupancy, and magnetization in Section 3.4 and Section 3.3 are due to Georg Rohringer.

• The numerical results and analytical proofs in Section 4.4 were obtained by Oleg Kulyashov. The author co-supervised Oleg's work, proposed the main research directions, and outlined the ideas for the proofs.

Заключение диссертации по теме «Другие cпециальности», Марков Антон Александрович

5.2 Local Markers for Phase Transitions Without an

Order Parameter

In the first section Section 4.1 of the chapter, we proposed a topological marker for interacting systems that does not rely on reference to a non-interacting Hamiltonian. To the best of our knowledge, such a marker has not previously been proposed. We tested our marker in several contexts, calculating it exactly in the presence of the disorder and interactions. From a computational perspective, our invariant is harder to calculate than the local Chern marker obtained from the topological Hamiltonian, as was done in [297,298]. Our marker involves integration over imaginary time (or summation over all the Matsubara frequencies in equilibrium), which imposes substantial numerical requirements.

However, the proposal seems attractive for several reasons. Firstly, the local markers are not exactly topological invariants. They has a quasi-topological nature, so their local value is not invariant under smooth deformation of the system. Only the average of a marker over an infinite number of sites is topologically robust. Thus, an approach based on smoothly removing interactions, as is done in the topological Hamiltonian approach, and then calculating the LCM of the non-interacting system has not been shown to give the local information that exactly corresponds to the interacting system. Secondly, our approach extends the applicability of topological markers to

systems which cannot be adiabatically connected to a non-interacting topological Hamiltonian. An example is systems with a nontrivial frequency-domain winding number [347]. Lastly, our approach allows us to study Local topological markers in non-stationary settings, for example in quenched topological matter. In the case of adiabatic evolution, the use of a topological Hamiltonian can be put on solid grounds, however for fast dynamics it looks inapplicable. On the other hand, our approach seems to admit a non-equilibrium generalization in the Keldysh formalism [348].

Now, let us summarize the main results for the second section Section 4.3. We have demonstrated that out-of-equilibrium topological markers are highly non-local objects due to a very large contribution from long-ranged diagrams presented in Fig (4.7) to the markers' value. Surprisingly, the long-range character of the correlations allows us to approximate the dynamics with a local continuity equation. We have found that the approximate local continuity equation works well for all the times in large translationally invariant patches. In such systems evolution always starts at the boundaries between the patches and then penetrates the bulk with the LiebRobinson velocity [195]. Intuitively it can be understood as follows. In a translationally invariant system, the local Chern marker cannot change. In the case only the bulk is translationally invariant the boundaries are the source of changes. The spread of correlations from boundary to bulk is a local process. Thus, local is the dynamics of the local Chern marker.

The local continuity approximation allowed us to connect the local Chern marker and on-site magnetic field induced charge in systems containing large translationally invariant patches. In such systems the local Streda marker can be used to estimate local Chern marker values. The experimental recipe is to prepare a system in two ground states: one with a small uniform magnetic field and another without. Then both samples should undergo the same evolution, during which the densities should be compared. Let us stress the dynamical Streda marker in general gives the correct topological information about the system only when it is connected to the local Chern marker.

We have found that the markers are able to evolve in the bulk at the very early times in a large patch with a broken translation symmetry. We observe it in a quench dynamics of disordered systems in Section 4.A.4. Remarkably, even a seemingly formal breaking of the translational symmetry, as in Hofstadter-Harper model is enough to change the character of the evolution. In this case, the local continuity equation approximates the dynamics of the marker at the late times only when the correlations are spread across the sample.

While for locality of the marker's dynamics translational invariance plays the key role, our numerical result hints that the connection between the local Chern marker and the magnetic field response should hold more generally than we have proved analytically. In particular, even in a disordered system, the average of the two markers over bulk sites is in noticeable agreement Section 4.A.4.

Let us suggest possible extensions of our work on local topological markers. Fractional Chern insulators - provide a very interesting context in which to apply a Streda-based Chern marker. Its equations of motion can be applied to many-body systems as they do not rely on single-particle projectors. In an interacting system, the projector onto the filled states is not defined, complicating the generalization of such local markers [56]. On the other hand, recent equilibrium calculations indicate that the Streda-based formula may be used as a local marker for fractional phases [349]. In the absence of the Green's functions zeros the local Green marker should correspond to the local Streda formula [165]. However, the fractional Hall systems do have zeros of Green's funcitons [350]. Therefore, it would be interesting to find a Green's function based marker which would account for these. Another important task for out-of-equlibrium setting is to find an optimal method for controlling the distribution of topological properties. This requires further analytical and numerical

studies of non-homogeneous topological systems out of equilibrium.

In the last section Section 4.4 of the chapter we demonstrated that the disorder-induced dynamical quantum phase transition in the Ising model is an example of a non-topological dynamical quantum phase transition without a local order parameter. A series of critical times universally appears for any vanishing perturbation with Fourier components at the lowest momentum of order 1/N3. This could be considered a dynamical counterpart of the Anderson orthogonality catastrophe [42]. That is, a vanishingly small perturbation causes a large deviation in the many-body wave function, while the observables remain intact. In our setting, it is the Loschmidt echo that changes drastically.

Several intriguing questions can be addressed in future studies. As we have an example of the DQPT where no order parameter can be found, a natural question is whether this phase transition belongs to a larger class with the same property. Vice versa: what class of the DQPTs can be endowed with an order parameter?

Список литературы диссертационного исследования кандидат наук Марков Антон Александрович, 2025 год

Bibliography

[1] Lev Davidovich Landau. On the theory of phase transitions. I. Zh. Eksp. Teor. Fiz., 11:19, 1937.

[2] Vitaly L Ginzburg, Vitaly Lazarevich Ginzburg, and LD Landau. On the theory of superconductivity. Springer, 2009.

[3] Jun Kondo. Resistance minimum in dilute magnetic alloys. Progress of theoretical physics, 32(1):37-49, 1964.

[4] W Meissner and B Voigt. Messungen mit Hilfe von flüssigem Helium XI Widerstand der reinen Metalle in tiefen Temperaturen. Annalen der Physik, 399(8):892-936, 1930.

[5] JO Linde. Arkiv. Mat. Astr. Fysik, 36A(10), 1948.

[6] AN Gerritsen and JO Linde. The electrical resistance of silver-manganese alloys at low temperatures: Part I. The resistance from 273° K to 1° K. Physica, 17(6):573-583, 1951.

[7] BT Matthias, M Peter, HJ Williams, AM Clogston, E Corenzwit, and RC Sherwood. Magnetic moment of transition metal atoms in dilute solution and their effect on superconducting transition temperature. Physical Review Letters, 5(12):542, 1960.

[8] MP Sarachik, E Corenzwit, and LD Longinotti. Resistivity of Mo-Nb and Mo-Re alloys containing 1% Fe. Physical Review, 135(4A):A1041, 1964.

[9] PW Anderson. A poor man's derivation of scaling laws for the Kondo problem. In Basic Notions Of Condensed Matter Physics, pages 483-488. CRC Press, 2018.

10] Kenneth G Wilson. Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Physical review B, 4(9):3174, 1971.

11] Kenneth G Wilson. The renormalization group: Critical phenomena and the Kondo problem. Reviews of modern physics, 47(4):773, 1975.

12] Murray Gell-Mann and Francis E Low. Quantum electrodynamics at small distances. Physical Review, 95(5):1300, 1954.

13] Leo P Kadanoff. Scaling laws for Ising models near Tc. Physics Physique Fizika, 2(6):263, 1966.

14] Christof Wetterich. Exact evolution equation for the effective potential. Physics Letters B, 301(1):90-94, 1993.

151 IM Lifshitz et al. Anomalies of electron characteristics of a metal in the high pressure region. Sov. Phys. JETP, 11(5):1130-1135, 1960.

16] VL314399 Berezinskii. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP, 32(3):493-500, 1971.

17] N David Mermin and Herbert Wagner. Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models. Physical Review Letters, 17(22):1133, 1966.

18] J Michael Kosterlitz. Kosterlitz-Thouless physics: a review of key issues. Reports on Progress in Physics, 79(2):026001, 2016.

19] John M Kosterlitz and DJ Thouless. Long range order and metastability in two dimensional solids and superfluids.(application of dislocation theory). Journal of Physics C: Solid State Physics, 5(11):L124, 1972.

20] G Toulouse and M Kleman. Principles of a classification of defects in ordered media. Journal de Physique Lettres, 37(6):149-151, 1976.

21] GE Volovik and VP Mineev. Line and point singularities in superfluid 3He. JETP lett, 24(11):561-563, 1976.

22] D Rogula. Trends in applications of pure mathematics to mechanics. Pitman, London, 1976.

23| K. V. Klitzing, G. Dorda, and M. Pepper. New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance. Phys. Rev. Lett., 45:494-497, Aug 1980.

24] David J Thouless, Mahito Kohmoto, M Peter Nightingale, and Marcel den Nijs. Quantized Hall conductance in a two-dimensional periodic potential. Physical review letters, 49(6):405, 1982.

25] Alexei Kitaev. Anyons in an exactly solved model and beyond. Annals of Physics, 321(1):2-111, 2006.

26] Raffaello Bianco and Raffaele Resta. Mapping topological order in coordinate space. Physical Review] B, 84(24):241106, 2011.

27] Terry A Loring and Matthew B Hastings. Disordered topological insulators via C*-algebras. EPL (Europhysics Letters), 92(6):67004, 2011.

28] Raffaello Bianco and Raffaele Resta. Orbital magnetization as a local property. Physical review letters, 110(8):087202, 2013.

29] Raffaello Bianco. Chern invariant and orbital magnetization as local quantities. 2014.

30] Peru d'Ornellas, Ryan Barnett, and Derek K. K. Lee. Quantized bulk conductivity as a local Chern marker. Phys. Rev. B, 106:155124, Oct 2022.

[31] Kuang-Ting Chen and Patrick A Lee. Unified formalism for calculating polarization, magnetization, and more in a periodic insulator. Physical Review B, 84(20):205137, 2011.

[32] Adrianus MM Pruisken. Field theory, scaling and the localization problem. In The quantum Hall effect, pages 117-173. Springer, 1990.

[33] Jens Eisert, Mathis Friesdorf, and Christian Gogolin. Quantum many-body systems out of equilibrium. Nature Physics, 11(2):124-130, 2015.

[34] Markus Heyl, Anatoli Polkovnikov, and Stefan Kehrein. Dynamical quantum phase transitions in the transverse-field Ising model. Physical review letters, 110(13):135704, 2013.

[35] Markus Heyl. Dynamical quantum phase transitions: a review. Reports on Progress in Physics, 81(5):054001, 2018.

[36] J. Hubbard. Electron Correlations in Narrow Energy Bands. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 276:238-257, 1963.

[37] NF Mott and R Peierls. Discussion of the paper by de Boer and Verwey. Proceedings of the Physical Society, 49(4S):72, 1937.

[38] Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Reviews of Modern Physics, 68(1):13, 1996.

[39] Roderich Moessner and Joel E Moore. Topological phases of matter. Cambridge University Press, 2021.

[40] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L Tomarken, Jason Y Luo, Javier D Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature, 556(7699):80-84, 2018.

[41] Zewei Shao, Xun Cao, Hongjie Luo, and Ping Jin. Recent progress in the phase-transition mechanism and modulation of vanadium dioxide materials. NPG Asia Materials, 10(7):581-605, 2018.

[42] Philip W Anderson. Infrared catastrophe in Fermi gases with local scattering potentials. Physical Review Letters, 18(24):1049, 1967.

[43] Eva Y Andrei, Dmitri K Efetov, Pablo Jarillo-Herrero, Allan H MacDonald, Kin Fai Mak, T Senthil, Emanuel Tutuc, Ali Yazdani, and Andrea F Young. The marvels of moire materials. Nature Reviews Materials, 6(3):201-206, 2021.

[44] Rafi Bistritzer and Allan H MacDonald. Moire bands in twisted double-layer graphene. Proceedings of the National Academy of Sciences, 108(30):12233-12237, 2011.

[45] Yasuhiro H Matsuda, Daisuke Nakamura, Akihiko Ikeda, Shojiro Takeyama, Yuki Suga, Hayato Nakahara, and Yuji Muraoka. Magnetic-field-induced insulator-metal transition in W-doped VO2 at 500 T. Nature communications, 11(1):3591, 2020.

[46] S. Fukuoka, T. Oka, Y. Ihara, A. Kawamoto, S. Imajo, and K. Kindo. Magnetic field induced insulator-to-metal Mott transition in A-type organic conductors. Phys. Rev. B, 109:195142, May 2024.

[47] J George Bednorz and K Alex Muller. Possible high Tc superconductivity in the Ba-La-Cu-O system. Zeitschrift fur Physik B Condensed Matter, 64(2):189-193, 1986.

[48] Lucile Savary and Leon Balents. Quantum spin liquids: a review. Reports on Progress in Physics, 80(1):016502, 2016.

[49] Bertrand I Halperin. Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Physical Review B, 25(4):2185, 1982.

[50] Jeffrey CY Teo and Charles L Kane. Topological defects and gapless modes in insulators and superconductors. Physical Review B, 82(11):115120, 2010.

[51] Markus Buttiker. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Physical Review B, 38(14):9375, 1988.

[52] Timothy M Philip and Matthew J Gilbert. High-performance nanoscale topological energy transduction. Scientific reports, 7(1):1-10, 2017.

[53] Matthew J Gilbert. Topological electronics. Communications Physics, 4(1):1-12, 2021.

[54] Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconductors. Reviews of Modern Physics, 83(4):1057, 2011.

[55] AA Markov, G Rohringer, and AN Rubtsov. Robustness of the topological quantization of the Hall conductivity for correlated lattice electrons at finite temperatures. Physical Review B, 100(11):115102, 2019.

[56] AA Markov and AN Rubtsov. Local Marker for Interacting Topological Insulators. Physical Review B, 104(8):L081105, 2021.

[57] AA Markov, DB Golovanova, AR Yavorsky, and AN Rubtsov. Locality of topological dynamics in Chern insulators. SciPost Physics, 17(6):152, 2024.

[58] ON Kuliashov, AA Markov, and AN Rubtsov. Dynamical quantum phase transition without an order parameter. Physical Review B, 107(9):094304, 2023.

[59] G Rohringer and AA Markov. Orbital magnetic field driven metal-insulator transition in strongly correlated electron systems. arXiv preprint arXiv:2406.18729, 2024.

[60] AA Markov and G Rohringer. Orbital magnetization on the edge of the Mott transition. in preparetion.

[61] Mingpu Qin, Thomas Schafer, Sabine Andergassen, Philippe Corboz, and Emanuel Gull. The Hubbard model: A computational perspective. Annual Review of Condensed Matter Physics, 13(1):275-302, 2022.

[62] Subir Sachdev. Quantum phases of matter. Cambridge University Press, 2023.

[63] Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Quantum mechanics: non-relativistic theory, volume 3. Elsevier, 2013.

[64] Walter Kohn and Lu Jeu Sham. Self-consistent equations including exchange and correlation effects. Physical review, 140(4A):A1133, 1965.

[65] Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura. Metal-insulator transitions. Reviews of modern physics, 70(4):1039, 1998.

[66] MS Bahovadinov, RO Sharipov, BL Altshuler, and GV Shlyapnikov. Enhanced Drude weight in a one-dimensional system of fermions with pair-hopping events. Physical Review B, 109(19):195159, 2024.

[67] JE Hirsch. Metallic ferromagnetism in a single-band model. IV. Effect of pair hopping. Physical Review B, 43(1):705, 1991.

[68] Google LLC, 2021.

[69] Kyounghwan Kim, Ashley DaSilva, Shengqiang Huang, Babak Fallahazad, Stefano Larentis, Takashi Taniguchi, Kenji Watanabe, Brian J LeRoy, Allan H MacDonald, and Emanuel Tutuc. Tunable moire bands and strong correlations in small-twist-angle bilayer graphene. Proceedings of the National Academy of Sciences, 114(13):3364-3369, 2017.

[70] Anton Mazurenko, Christie S Chiu, Geoffrey Ji, Maxwell F Parsons, Marton Kanasz-Nagy, Richard Schmidt, Fabian Grusdt, Eugene Demler, Daniel Greif, and Markus Greiner. A cold-atom Fermi-Hubbard antiferromagnet. Nature, 545(7655):462-466, 2017.

[71] Eugene Demler and Takuya Kitagawa. Strongly correlated systems in atomic and condensed matter physics. Lecture notes for Physics, 284:28, 2011.

[72] Stefano Giorgini, Lev P Pitaevskii, and Sandro Stringari. Theory of ultracold atomic Fermi gases. Reviews of Modern Physics, 80(4):1215-1274, 2008.

[73] Robert Jordens, Niels Strohmaier, Kenneth Günter, Henning Moritz, and Tilman Esslinger. A Mott insulator of fermionic atoms in an optical lattice. Nature, 455(7210):204-207, 2008.

[74] Piers Coleman. Introduction to many-body physics. Cambridge University Press, 2015.

[75] S Raghu, SA Kivelson, and DJ Scalapino. Superconductivity in the repulsive Hubbard model: An asymptotically exact weak-coupling solution. Physical Review B—Condensed Matter and Materials Physics, 81(22):224505, 2010.

[76] Daniel P Arovas, Erez Berg, Steven A Kivelson, and Srinivas Raghu. The hubbard model. Annual review of condensed matter physics, 13(1):239-274, 2022.

[77] JPF LeBlanc, Andrey E Antipov, Federico Becca, Ireneusz W Bulik, Garnet Kin-Lic Chan, Chia-Min Chung, Youjin Deng, Michel Ferrero, Thomas M Henderson, Carlos A Jimenez-Hoyos, et al. Solutions of the two-dimensional hubbard model: benchmarks and results from a wide range of numerical algorithms. Physical Review X, 5(4):041041, 2015.

[78] L. D. Landau. The Theory of a Fermi Liquid. Zh. Eksp. Teor. Fiz., 30(6):1058, 1956.

[79] Aleksei Alekseevich Abrikosov, Lev Petrovich Gorkov, and Igor Ekhielevich Dzyaloshinski. Methods of quantum field theory in statistical physics. Courier Corporation, 2012.

[80] Rev Shankar. Renormalization-group approach to interacting fermions. Reviews of Modern Physics, 66(1):129, 1994.

[81] K Andres, JE Graebner, and HR Ott. 4 f-virtual-bound-state Formation in Ce Al 3 at Low Temperatures. Physical Review Letters, 35(26):1779, 1975.

[82] John R Schrieffer and Peter A Wolff. Relation between the anderson and kondo hamiltonians. Physical Review, 149(2):491, 1966.

[83] Tosio Kato. On the convergence of the perturbation method. I. Progress of Theoretical Physics, 4(4):514-523, 1949.

[84] Minoru Takahashi. Half-filled Hubbard model at low temperature. Journal of Physics C: Solid State Physics, 10(8):1289, 1977.

[85] Allan H MacDonald, SM Girvin, and DT Yoshioka. t/U expansion for the Hubbard model. Physical Review B, 37(16):9753, 1988.

[86] Efstratios Manousakis. The spin-1/2 Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides. Reviews of Modern Physics, 63(1):1, 1991.

[87] Ryogo Kubo. The spin-wave theory of antiferromagnetics. Physical Review, 87(4):568, 1952.

[88] Philip W Anderson. An approximate quantum theory of the antiferromagnetic ground state. Physical Review, 86(5):694, 1952.

[89] Freeman J Dyson, Elliott H Lieb, and Barry Simon. Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. Statistical Mechanics: Selecta of Elliott H. Lieb, pages 163-211, 2004.

[90] Tom Kennedy, Elliott H Lieb, and B Sriram Shastry. Existence of neel order in some spin-1/2 Heisenberg antiferromagnets. Journal of statistical physics, 53:1019-1030, 1988.

[91] E Jordao Neves and J Fernando Perez. Long range order in the ground state of two-dimensional antiferromagnets. Physics Letters A, 114(6):331-333, 1986.

[92] Yi Zhou, Kazushi Kanoda, and Tai-Kai Ng. Quantum spin liquid states. Reviews of Modern Physics, 89(2):025003, 2017.

[93] Stephane Pairault, David Senechal, and A-MS Tremblay. Strong-coupling expansion for the hubbard model. Physical review letters, 80(24):5389, 1998.

[94] Stephane Pairault, David Senechal, and A-MS Tremblay. Strong-coupling perturbation theory of the Hubbard model. The European Physical Journal B-Condensed Matter and Complex Systems, 16:85-105, 2000.

[95] Walter Metzner. Linked-cluster expansion around the atomic limit of the Hubbard model. Physical Review B, 43(10):8549, 1991.

[96] Gabriel Kotliar and Andrei E Ruckenstein. New functional integral approach to strongly correlated fermi systems: The gutzwiller approximation as a saddle point. Physical review letters, 57(11):1362, 1986.

[97] Martin C Gutzwiller. Effect of correlation on the ferromagnetism of transition metals. Physical Review Letters, 10(5):159, 1963.

[98] Rubem Mondaini, Sabyasachi Tarat, and Richard T Scalettar. Quantum critical points and the sign problem. Science, 375(6579):418-424, 2022.

[99] J. E. Hirsch and R. M. Fye. Monte Carlo Method for Magnetic Impurities in Metals. Phys. Rev. Lett., 56:2521, June 1986.

100] Steven R White. Density matrix formulation for quantum renormalization groups. Physical review letters, 69(19):2863, 1992.

101] Pierre Hohenberg and Walter Kohn. Inhomogeneous electron gas. Physical review, 136(3B):B864, 1964.

102] Antoine Georges and Gabriel Kotliar. Hubbard model in infinite dimensions. Physical Review B, 45(12):6479, 1992.

103] David Jacob, Gianluca Stefanucci, and Stefan Kurth. Mott metal-insulator transition from steady-state density functional theory. Physical Review Letters, 125(21):216401, 2020.

104] Frank Verstraete and J Ignacio Cirac. Renormalization algorithms for quantum-many body systems in two and higher dimensions. arXiv preprint cond-mat/04 07066, 2004.

105] Jens Eisert, Marcus Cramer, and Martin B Plenio. Colloquium: Area laws for the entanglement entropy. Reviews of modern physics, 82(1):277-306, 2010.

106] AN Rubtsov, MI Katsnelson, and AI Lichtenstein. Dual fermion approach to nonlocal correlations in the Hubbard model. Physical Review B, 77(3):033101, 2008.

107] Georg Rohringer. New routes towards a theoretical treatment of nonlocal electronic correlations. PhD thesis, Technische Universitat Wien, 2013.

108] Michel Caffarel and Werner Krauth. Exact diagonalization approach to correlated fermions in infinite dimensions: Mott transition and superconductivity. Physical review letters, 72(10):1545, 1994.

109] Philipp Werner, Armin Comanac, Luca De Medici, Matthias Troyer, and Andrew J Millis. Continuous-time solver for quantum impurity models. Physical Review Letters, 97(7):076405, 2006.

110] Ralf Bulla, Theo A Costi, and Thomas Pruschke. Numerical renormalization group method for quantum impurity systems. Reviews of Modern Physics, 80(2):395, 2008.

111] G Rohringer, H Hafermann, A Toschi, AA Katanin, AE Antipov, MI Katsnelson, AI Lichtenstein, AN Rubtsov, and K Held. Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. Reviews of Modern Physics, 90(2):025003, 2018.

113

114

115

116

117

118

119

120

121

122

123

124

125

FJ Morin. Oxides which show a metal-to-insulator transition at the Neel temperature. Physical review letters, 3(1):34, 1959.

W Rüdorff, G Walter, and J Stadler. Magnetismus, leitfähigkeit und Reflexionsspektren von Vanadindioxyd und Vanadindioxyd-Titandioxyd-Mischkristallen. Zeitschrift für anorganische und allgemeine Chemie, 297(1-2):1-13, 1958.

Jean-Paul Pouget. Basic aspects of the metal-insulator transition in vanadium dioxide VO _{2}: a critical review. Comptes Rendus. Physique, 22(1):37-87, 2021.

J Jaffray and AJ Dumas. Recherches Centre Natl. Recherches Sci. Labs.. Bellevue (Paris), 5:360, 1954.

John B Goodenough. Direct cation-cation interactions in several oxides. Physical Review, 117(6):1442, 1960.

Nevill F Mott. The transition to the metallic state. Philosophical Magazine, 6(62):287-309, 1961.

Jun-ichi Umeda, Hazime Kusumoto, Koziro Narita, and Eizaburo Yamada. Nuclear magnetic resonance in polycrystalline VO2. The Journal of Chemical Physics, 42(4):1458-1459, 1965.

Koji Kosuge. The phase transition in VO2. Journal of the Physical Society of Japan, 22(2):551-557, 1967.

Walter D Knight. Nuclear magnetic resonance shift in metals. Physical Review, 76(8):1259, 1949.

AS Barker Jr, HW Verleur, and HJ Guggenheim. Infrared optical properties of vanadium dioxide above and below the transition temperature. Physical Review Letters, 17(26):1286, 1966.

Daniel Wegkamp, Marc Herzog, Lede Xian, Matteo Gatti, Pierluigi Cudazzo, Christina L McGahan, Robert E Marvel, Richard F Haglund Jr, Angel Rubio, Martin Wolf, et al. Instantaneous band gap collapse in photoexcited monoclinic VO 2 due to photocarrier doping. Physical review letters, 113(21):216401, 2014.

AMNF Zylbersztejn and Nevill Francis Mott. Metal-insulator transition in vanadium dioxide. Physical Review B, 11(11):4383, 1975.

John B Goodenough. The two components of the crystallographic transition in VO2. Journal of Solid State Chemistry, 3(4):490-500, 1971.

M Marezio, Do B McWhan, JP Remeika, and PD Dernier. Structural Aspects of the Metal-Insulator Transitions in Cr-Doped V O 2. Physical Review B, 5(7):2541, 1972.

J Laverock, S Kittiwatanakul, AA Zakharov, YR Niu, B Chen, SA Wolf, JW Lu, and KE Smith. Direct observation of decoupled structural and electronic transitions and an ambient pressure monocliniclike metallic phase of VO 2. Physical review letters, 113(21):216402, 2014.

[127] Renata M Wentzcovitch, Werner W Schulz, and Philip B Allen. VO 2: Peierls or Mott-Hubbard? A view from band theory. Physical review letters, 72(21):3389, 1994.

[128] Volker Eyert. VO 2: A novel view from band theory. Physical Review Letters, 107(1):016401, 2011.

[129] Ricardo Grau-Crespo, Hao Wang, and Udo Schwingenschlögl. Why the Heyd-Scuseria-Ernzerhof hybrid functional description of VO 2 phases is not correct. Physical Review B—Condensed Matter and Materials Physics, 86(8):081101, 2012.

[130] Silke Biermann, A Poteryaev, AI Lichtenstein, and A Georges. Dynamical singlets and correlation-assisted Peierls transition in VO 2. Physical review letters, 94(2):026404, 2005.

[131] Cedric Weber, David D O'Regan, Nicholas DM Hine, Mike C Payne, Gabriel Kotliar, and Peter B Littlewood. Vanadium dioxide: A Peierls-Mott insulator stable against disorder. Physical review letters, 108(25):256402, 2012.

[132] Philip B Allen, Renata M Wentzcovitch, Werner W Schulz, and Paul C Canfield. Resistivity of the high-temperature metallic phase of VO 2. Physical Review B, 48(7):4359, 1993.

[133] Sangwook Lee, Kedar Hippalgaonkar, Fan Yang, Jiawang Hong, Changhyun Ko, Joonki Suh, Kai Liu, Kevin Wang, Jeffrey J Urban, Xiang Zhang, et al. Anomalously low electronic thermal conductivity in metallic vanadium dioxide. Science, 355(6323):371-374, 2017.

[134] Charles Kittel. Introduction to solid state physics Eighth edition. 2021.

[135] John D Budai, Jiawang Hong, Michael E Manley, Eliot D Specht, Chen W Li, Jonathan Z Tischler, Douglas L Abernathy, Ayman H Said, Bogdan M Leu, Lynn A Boatner, et al. Metallization of vanadium dioxide driven by large phonon entropy. Nature, 515(7528):535-539, 2014.

[136] Henri Poincare. Analysis situs. Gauthier-Villars Paris, France, 1895.

[137] Andreas P Schnyder, Shinsei Ryu, Akira Furusaki, and Andreas WW Ludwig. Classification of topological insulators and superconductors in three spatial dimensions. Physical Review B—Condensed Matter and Materials Physics, 78(19):195125, 2008.

[138] Alexander Altland and Martin R Zirnbauer. Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures. Physical Review B, 55(2):1142, 1997.

[139] David J Thouless, Mahito Kohmoto, M Peter Nightingale, and Md den Nijs. Quantized Hall conductance in a two-dimensional periodic potential. Physical Review Letters, 49(6):405, 1982.

[140] Joseph E Avron and Ruedi Seiler. Quantization of the Hall conductance for general, multiparticle Schrüdinger Hamiltonians. Physical review letters, 54(4):259, 1985.

[141] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order. Physical Review B—Condensed Matter and Materials Physics, 82(15):155138, 2010.

[142] GE Volovik. Quantum phase transitions from topology in momentum space. In Quantum analogues: from phase transitions to black holes and cosmology, pages 31-73. Springer, 2007.

143] Grigory E Volovik. The universe in a helium droplet, volume 117. OUP Oxford, 2003.

144] Klaus von Klitzing. Quantum Hall effect: Discovery and application. Annual Review of Condensed Matter Physics, 8(1):13-30, 2017.

145] LD Landau. Diamagnetismus der metalle. Zeitschrift fur Physik, 64:629-637, 1930.

146] Tsuneya Ando, Yukio Matsumoto, and Yasutada Uemura. Theory of Hall effect in a two-dimensional electron system. Journal of the Physical Society of Japan, 39(2):279-288, 1975.

147] F Duncan M Haldane. Nobel lecture: Topological quantum matter. Reviews of Modern Physics, 89(4):040502, 2017.

148] Ryogo Kubo. Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan, 12(6):570-586, 1957.

149] Barry Simon. Holonomy, the quantum adiabatic theorem, and Berry's phase. Physical Review Letters, 51(24):2167, 1983.

150] Michael Victor Berry. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences., 392, 1984.

151] Daniel C Tsui, Horst L Stormer, and Arthur C Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters, 48(22):1559, 1982.

152] Lukasz Fidkowski and Alexei Kitaev. Effects of interactions on the topological classification of free fermion systems. Physical Review B, 81(13):134509, 2010.

153] Yuan-Ming Lu and Ashvin Vishwanath. Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach. Physical Review B—Condensed Matter and Materials Physics, 86(12):125119, 2012.

154] Andreas Bauer, Jens Eisert, and Carolin Wille. A unified diagrammatic approach to topological fixed point models. SciPost Physics Core, 5(3):038, 2022.

155] Xiao-Liang Qi, Yong-Shi Wu, and Shou-Cheng Zhang. General theorem relating the bulk topological number to edge states in two-dimensional insulators. Physical Review B—Condensed Matter and Materials Physics, 74(4):045125, 2006.

156] Qian Niu and DJ Thouless. Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction. Journal of Physics A: Mathematical and General, 17(12):2453, 1984.

157] Qian Niu, Ds J Thouless, and Yong-Shi Wu. Quantized Hall conduct ance as a topological invariant. Physical Review B, 31(6):3372, 1985.

158] Robert B Laughlin. Quantized Hall conductivity in two dimensions. Physical Review B, 23(10):5632, 1981.

159] Matthew B Hastings and Tohru Koma. Spectral gap and exponential decay of correlations. Communications in mathematical physics, 265:781-804, 2006.

160] Koji Kudo, Haruki Watanabe, Toshikaze Kariyado, and Yasuhiro Hatsugai. Many-body chern number without integration. Physical Review Letters, 122(14):146601, 2019.

161] Byungmin Kang, Wonjun Lee, and Gil Young Cho. Many-Body Invariants for Chern and Chiral Hinge Insulators. arXiv preprint arXiv:2003.13706, 2020.

162] Raffaele Resta. Quantum-mechanical position operator in extended systems. Physical Review Letters, 80(9):1800, 1998.

163] Matthew B Hastings and Spyridon Michalakis. Quantization of Hall conductance for interacting electrons on a torus. Communications in Mathematical Physics, 334(1):433-471, 2015.

164] Kenzo Ishikawa and Toyoki Matsuyama. Magnetic field induced multi-component QED3 and quantum Hall effect. Zeitschrift fur Physik C Particles and Fields, 33(1):41-45, 1986.

165] Lucila Peralta Gavensky, Subir Sachdev, and Nathan Goldman. Connecting the many-body Chern number to luttinger's theorem through streda's formula. Physical review letters, 131(23):236601, 2023.

166] Jinchao Zhao, Peizhi Mai, Barry Bradlyn, and Philip Phillips. Failure of topological invariants in strongly correlated matter. Physical review letters, 131(10):106601, 2023.

167] Xiao-Liang Qi, Taylor L Hughes, and Shou-Cheng Zhang. Topological field theory of time-reversal invariant insulators. Physical Review B, 78(19):195424, 2008.

168] Zhong Wang and Shou-Cheng Zhang. Simplified topological invariants for interacting insulators. Physical Review X, 2(3):031008, 2012.

169] Jun-Hui Zheng, Tao Qin, and Walter Hofstetter. Interaction-enhanced integer quantum Hall effect in disordered systems. Physical Review B, 99(12):125138, 2019.

170] Zhong Wang, Xiao-Liang Qi, and Shou-Cheng Zhang. Topological order parameters for interacting topological insulators. Physical review letters, 105(25):256803, 2010.

171] Edward Witten. Topological quantum field theory. Communications in Mathematical Physics, 117(3):353-386, 1988.

172] SM Girvin and Allan H MacDonald. Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect. Physical review letters, 58(12):1252, 1987.

173] S C_ Zhang, Th H Hansson, and S Kivelson. Effective-field-theory model for the fractional quantum Hall effect. Physical review letters, 62(1):82, 1989.

174] B Andrei Bernevig and Taylor L Hughes. Topological insulators and topological superconductors. Princeton university press, 2013.

[175] P Streda. Theory of quantised Hall conductivity in two dimensions. Journal of Physics C: Solid State Physics, 15(22):L717, 1982.

[176] Daniel Arovas, John R Schrieffer, and Frank Wilczek. Fractional statistics and the quantum Hall effect. Physical review letters, 53(7):722, 1984.

[177] James Nakamura, Shuang Liang, Geoffrey C Gardner, and Michael J Manfra. Direct observation of anyonic braiding statistics. Nature Physics, 16(9):931-936, 2020.

[178] F Duncan M Haldane. Model for a quantum Hall effect without Landau levels: Condensedmatter realization of the" parity anomaly". Physical review letters, 61(18):2015, 1988.

[179] Charles L Kane and Eugene J Mele. Z 2 topological order and the quantum spin Hall effect. Physical review letters, 95(14):146802, 2005.

[180] Charles L Kane and Eugene J Mele. Quantum spin Hall effect in graphene. Physical review letters, 95(22):226801, 2005.

[181] K Hatsuda, H Mine, T Nakamura, J Li, R Wu, S Katsumoto, and J Haruyama. Evidence for a quantum spin Hall phase in graphene decorated with Bi2Te3 nanoparticles. Science advances, 4(11):eaau6915, 2018.

[182] B Andrei Bernevig, Taylor L Hughes, and Shou-Cheng Zhang. Quantum spin Hall effect and topological phase transition in hgte quantum wells. science, 314(5806):1757-1761, 2006.

[183] Markus Konig, Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut Buhmann, Laurens W Molenkamp, Xiao-Liang Qi, and Shou-Cheng Zhang. Quantum spin Hall insulator state in hgte quantum wells. Science, 318(5851):766-770, 2007.

[184] Joel E Moore and Leon Balents. Topological invariants of time-reversal-invariant band structures. Physical Review B—Condensed Matter and Materials Physics, 75(12):121306, 2007.

[185] David Hsieh, Dong Qian, Lewis Wray, Yiman Xia, Yew San Hor, Robert Joseph Cava, and M Zahid Hasan. A topological Dirac insulator in a quantum spin Hall phase. Nature, 452(7190):970-974, 2008.

[186] Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science, 340(6129):167-170, 2013.

[187] Cui-Zu Chang, Chao-Xing Liu, and Allan H MacDonald. Colloquium: Quantum anomalous hall effect. Reviews of Modern Physics, 95(1):011002, 2023.

[188] M Serlin, CL Tschirhart, H Polshyn, Y Zhang, J Zhu, K Watanabe, T Taniguchi, L Balents, and AF Young. Intrinsic quantized anomalous Hall effect in a moire heterostructure. Science, 367(6480):900-903, 2020.

[189] Si-Yu Li, Yu Zhang, Ya-Ning Ren, Jianpeng Liu, Xi Dai, and Lin He. Experimental evidence for orbital magnetic moments generated by moire-scale current loops in twisted bilayer graphene. Physical Review B, 102(12):121406, 2020.

[190] Sameer Grover, Matan Bocarsly, Aviram Uri, Petr Stepanov, Giorgio Di Battista, Indranil Roy, Jiewen Xiao, Alexander Y Meltzer, Yuri Myasoedov, Keshav Pareek, et al. Chern mosaic and berry-curvature magnetism in magic-angle graphene. Nature Physics, 18(8):885-892, 2022.

191

192

193

194

195

196

197

198

199

200

201

202

203

Emil Prodan, Taylor L Hughes, and B Andrei Bernevig. Entanglement spectrum of a disordered topological chern insulator. Physical review letters, 105(11):115501, 2010.

RO Umucalilar, Hui Zhai, and MO Oktel. Trapped Fermi gases in rotating optical lattices: Realization and detection of the topological Hofstadter insulator. Physical review letters, 100(7):070402, 2008.

Gerhard C Hegerfeldt. Violation of causality in relativistic quantum theory? Physical review letters, 54(22):2395, 1985.

Phillippe H Eberhard and Ronald R Ross. Quantum field theory cannot provide faster-than-light communication. Foundations of Physics Letters, 2(2):127-149, 1989.

Elliott H Lieb and Derek W Robinson. The finite group velocity of quantum spin systems. In Statistical mechanics, pages 425-431. Springer, 1972.

Matthew B Hastings. Lieb-Schultz-Mattis in higher dimensions. Physical review b, 69(10):104431, 2004.

Bruno Nachtergaele and Robert Sims. Lieb-Robinson bounds and the exponential clustering theorem. Communications in mathematical physics, 265:119-130, 2006.

Sergey Bravyi, Matthew B Hastings, and Frank Verstraete. Lieb-Robinson bounds and the generation of correlations and topological quantum order. Physical review letters, 97(5):050401, 2006.

Marc Cheneau, Peter Barmettler, Dario Poletti, Manuel Endres, Peter Schauß, Takeshi Fukuhara, Christian Gross, Immanuel Bloch, Corinna Kollath, and Stefan Kuhr. Light-conelike spreading of correlations in a quantum many-body system. Nature, 481(7382):484-487, 2012.

Szabolcs Vajna and Balazs Dora. Disentangling dynamical phase transitions from equilibrium phase transitions. Physical Review B, 89(16):161105, 2014.

C Karrasch and D Schuricht. Dynamical phase transitions after quenches in nonintegrable models. Physical Review B, 87(19):195104, 2013.

Markus Schmitt and Stefan Kehrein. Dynamical quantum phase transitions in the Kitaev honeycomb model. Physical Review B, 92(7):075114, 2015.

Szabolcs Vajna and Balazs Dora. Topological classification of dynamical phase transitions. Physical Review B, 91(15):155127, 2015.

Elena Canovi, Philipp Werner, and Martin Eckstein. First-order dynamical phase transitions. Physical Review Letters, 113(26):265702, 2014.

[205] Jan Carl Budich and Markus Heyl. Dynamical topological order parameters far from equilibrium. Physical Review B, 93(8):085416, 2016.

[206] Markus Heyl. Scaling and universality at dynamical quantum phase transitions. Physical Review Letters, 115(14):140602, 2015.

[207] Daniele Trapin, Jad C Halimeh, and Markus Heyl. Unconventional critical exponents at dynamical quantum phase transitions in a random Ising chain. Physical Review B, 104(11):115159, 2021.

[208] P Jurcevic, H Shen, P Hauke, C Maier, T Brydges, C Hempel, BP Lanyon, Markus Heyl, R Blatt, and CF Roos. Direct observation of dynamical quantum phase transitions in an interacting many-body system. Physical review letters, 119(8):080501, 2017.

[209] Simon A Weidinger, Markus Heyl, Alessandro Silva, and Michael Knap. Dynamical quantum phase transitions in systems with continuous symmetry breaking. Physical Review B, 96(13):134313, 2017.

[210] Jad C Halimeh, Daniele Trapin, Maarten Van Damme, and Markus Heyl. Local measures of dynamical quantum phase transitions. Physical Review B, 104(7):075130, 2021.

[211] Fengcheng Wu, Timothy Lovorn, Emanuel Tutuc, and Allan H MacDonald. Hubbard model physics in transition metal dichalcogenide moire bands. Physical review letters, 121(2):026402, 2018.

[212] Le Liu, Yanbang Chu, Guang Yang, Yalong Yuan, Fanfan Wu, Yiru Ji, Jinpeng Tian, Rong Yang, Kenji Watanabe, Takashi Taniguchi, et al. Quantum oscillations in correlated insulators of a moir\'e superlattice. arXiv preprint arXiv:2205.10025, 2022.

[213] Yasuhiro H. Matsuda, Yuji Muraoka, Daisuke Nakamura, Akihiko Ikeda, Yuto Ishii, Xu-Guang Zhou, Hironobu Sawabe, and Shojiro Takeyama. Magnetic-Field-Induced Insulator Metal Transition of w-doped VO2 Observed by Electromagnetic Flux Compression at ISSP. Journal of the Physical Society of Japan, 91(10):101008, 2022.

[214] R. Peierls. Zur Theorie des Diamagnetismus von Leitungselektronen. Zeitschrift für Physik, 80(11):763-791, Nov 1933.

[215] Jan M Tomczak. Spectral and optical properties of correlated materials. PhD thesis, PhD thesis, Ecole Polytechnique, 2007.

[216] Mark Ya Azbel. Energy spectrum of a conduction electron in a magnetic field. Soviet Physics Jetp-Ussr, 19(3):634-645, 1964.

[217] J Zak. Magnetic translation group. Physical Review, 134(6A):A1602, 1964.

[218] Philip George Harper. Single band motion of conduction electrons in a uniform magnetic field. Proceedings of the Physical Society. Section A, 68(10):874, 1955.

[219] Douglas R Hofstadter. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Physical review B, 14(6):2239, 1976.

[220] XG Wen and A Zee. Winding number, family index theorem, and electron hopping in a magnetic field. Nuclear Physics B, 316(3):641-662, 1989.

[221] Mahito Kohmoto. Zero modes and the quantized Hall conductance of the two-dimensional lattice in a magnetic field. Physical Review B, 39(16):11943, 1989.

[222] Indubala I Satija. Butterfly in the Quantum World. Phys. Rev, 97:869-83, 2016.

[223] Jan M. Tomczak and Silke Biermann. Optical properties of correlated materials: Generalized Peierls approach and its application to vo2. Phys. Rev. B, 80:085117, Aug 2009.

[224] Georg Rohringer, Angelo Valli, and Alessandro Toschi. Local electronic correlation at the two-particle level. Physical Review B, 86(12):125114, 2012.

[225] AG Rojo, Gabriel Kotliar, and GS Canright. Sign of equilibrium Hall conductivity in strongly correlated systems. Physical Review B, 47(14):9140, 1993.

[226] FF Assaad and M Imada. Hall coefficient for the two-dimensional Hubbard model. Physical review letters, 74(19):3868, 1995.

[227] Massimo Capone, Luca de' Medici, and Antoine Georges. Solving the dynamical mean-field theory at very low temperatures using the Lanczos exact diagonalization. Phys. Rev. B, 76:245116, Dec 2007.

[228] WJ De Haas and PM Van Alphen. The dependence of the susceptibility of diamagnetic metals upon the field. In Proc. Netherlands Roy. Acad. Sci, volume 33, page 170, 1930.

[229] David Shoenberg. Magnetic oscillations in metals. Cambridge university press, 2009.

[230] L Onsager. Interpretation of the de haas-van Alphen effect. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 43(344):1006-1008, 1952.

[231] IM Lifshitz and AM Kosevich. Theory of magnetic susceptibility in metals at low temperatures. Sov. Phys. JETP, 2(4):636-645, 1956.

[232] Mark V Kartsovnik. High magnetic fields: a tool for studying electronic properties of layered organic metals. Chemical reviews, 104(11):5737-5782, 2004.

[233] Baptiste Vignolle, A Carrington, RA Cooper, MMJ French, AP Mackenzie, C Jaudet, D Vignolles, Cyril Proust, and NE Hussey. Quantum oscillations in an overdoped high-t c superconductor. Nature, 455(7215):952-955, 2008.

[234] Valentin Leeb and Johannes Knolle. Quantum oscillations in a doped Mott insulator beyond onsager's relation. arXiv preprint arXiv:2301.08685, 2023.

[235] Valentin Leeb, K Polyudov, S Mashhadi, S Biswas, Roser Valenti, M Burghard, and J Knolle. Anomalous quantum oscillations in a heterostructure of graphene on a proximate quantum spin liquid. Physical Review) Letters, 126(9):097201, 2021.

[236] Ming-Che Chang and Qian Niu. Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands. Physical Review B, 53(11):7010, 1996.

[237] Davide Ceresoli, T Thonhauser, David Vanderbilt, and Raffaele Resta. Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals. Physical Review B—Condensed Matter and Materials Physics, 74(2):024408, 2006.

[238] Emil Prodan and Hermann Schulz-Baldes. Bulk and boundary invariants for complex topological insulators. K, 2016.

[239] Gang Li, Ziji Xiang, Fan Yu, Tomoya Asaba, Benjamin Lawson, Peng Cai, Colin Tinsman, Adam Berkley, Steven Wolgast, Yun Suk Eo, et al. Two-dimensional Fermi surfaces in Kondo insulator smb6. Science, 346(6214):1208-1212, 2014.

[240] Johannes Knolle and Nigel R Cooper. Quantum oscillations without a Fermi surface and the anomalous de haas-van Alphen effect. Physical review letters, 115(14):146401, 2015.

[241] Long Zhang, Xue-Yang Song, and Fa Wang. Quantum oscillation in narrow-gap topological insulators. Physical review letters, 116(4):046404, 2016.

[242] G Baskaran. Majorana Fermi sea in insulating smb6: a proposal and a theory of quantum oscillations in Kondo insulators. arXiv preprint arXiv:1507.034 77, 2015.

[243] Onur Erten, Pouyan Ghaemi, and Piers Coleman. Kondo breakdown and quantum oscillations in smb 6. Physical review letters, 116(4):046403, 2016.

[244] Andrew Allocca and Nigel R Cooper. Quantum oscillations in interaction-driven insulators. SciPost Physics, 12(4):123, 2022.

[245] Z Xiang, Y Kasahara, Tomoya Asaba, B Lawson, Colin Tinsman, Lu Chen, K Sugimoto, S Kawaguchi, Y Sato, G Li, et al. Quantum oscillations of electrical resistivity in an insulator. Science, 362(6410):65-69, 2018.

[246] Hsu Liu, Mate Hartstein, Gregory J Wallace, Alexander J Davies, Monica Ciomaga Hatnean, Michelle D Johannes, Natalya Shitsevalova, Geetha Balakrishnan, and Suchitra E Sebastian. Fermi surfaces in Kondo insulators. Journal of Physics: Condensed Matter, 30(16):16LT01, 2018.

[247] Pengjie Wang, Guo Yu, Yanyu Jia, Michael Onyszczak, F Alexandre Cevallos, Shiming Lei, Sebastian Klemenz, Kenji Watanabe, Takashi Taniguchi, Robert J Cava, et al. Landau quantization and highly mobile fermions in an insulator. Nature, 589(7841):225-229, 2021.

[248] Harris Pirie, Eric Mascot, Christian E Matt, Yu Liu, Pengcheng Chen, MH Hamidian, Shanta Saha, Xiangfeng Wang, Johnpierre Paglione, Graeme Luke, et al. Visualizing the atomic-scale origin of metallic behavior in Kondo insulators. Science, 379(6638):1214-1218, 2023.

[249] Nigel R Cooper and Jack Kelsall. Quantum oscillations in an impurity-band Anderson insulator. SciPost Physics, 15(3):118, 2023.

[250] Gurpreet Singh and Hridis K Pal. Effect of many-body interaction on de haas-van Alphen oscillations in insulators. Physical Review B, 108(20):L201103, 2023.

[251] Michele Fabrizio. Spin-liquid insulators can be landau's Fermi liquids. Physical Review Letters, 130(15):156702, 2023.

[252] Inti Sodemann, Debanjan Chowdhury, and T Senthil. Quantum oscillations in insulators with neutral Fermi surfaces. Physical Review B, 97(4):045152, 2018.

[253] Olexei I Motrunich. Orbital magnetic field effects in spin liquid with spinon Fermi sea: Possible application to K-(et) 2 Cu 2 (C N) 3. Physical Review B, 73(15):155115, 2006.

[254] Diptiman Sen and R Chitra. Large-U limit of a Hubbard model in a magnetic field: Chiral spin interactions and paramagnetism. Physical Review B, 51(3):1922, 1995.

[255] LL Hirst. The microscopic magnetization: concept andapplication. Reviews of Modern Physics, 69(2):607, 1997.

[256] R Nourafkan, G Kotliar, and A-MS Tremblay. Orbital magnetization of correlated electrons with arbitrary band topology. Physical Review B, 90(12):125132, 2014.

[257] Perry T Mahon, Rodrigo A Muniz, and JE Sipe. Microscopic polarization and magnetization fields in extended systems. Physical Review B, 99(23):235140, 2019.

[258] WH Xu, LP Yang, MP Qin, and T Xiang. Probing the Hofstadter butterfly with the quantum oscillation of magnetization. Physical Review B, 78(24):241102, 2008.

[259] Joaquin Mazdak Luttinger and John Clive Ward. Ground-state energy of a many-fermion system. II. Physical Review, 118(5):1417, 1960.

[260] Michael Potthoff. Non-perturbative construction of the Luttinger-Ward functional. arXiv preprint cond-mat/04 06671, 2004.

[261] J Vucicevic and R Zitko. Electrical conductivity in the Hubbard model: orbital effects of magnetic field. Physical Review B, 104(20):205101, 2021.

[262] JC Slater. Magnetic effects and the Hartree-Fock equation. Physical Review, 82(4):538, 1951.

[263] K. L. Gurunatha, S. Sathasivam, J. Li, M. Portnoi, I. P. Parkin, and Papakonstantinou I. Combined effect of temperature induced strain and oxygen vacancy on metal-insulator transition of VO2 colloidal particles. Adv. Fund. Mater., 30:2005311, 2020.

[264] Chen Ling, Zhengjing Zhao, Xinyuan Hu, Jingbo Li, Xushan Zhao, Zongguo Wang, Yongjie Zhao, and Haibo Jin. W Doping and Voltage Driven Metal-Insulator Transition in VO2 Nano-Films for Smart Switching Devices. ACS Appl. Nano Mater., 2(10):6738-6746, 2019.

[265] G. Mazza, A. Amaricci, M. Capone, and M. Fabrizio. Field-Driven Mott Gap Collapse and Resistive Switch in Correlated Insulators. Phys. Rev. Lett., 117:176401, Oct 2016.

[266] F. Kagawa, T. Itou, K. Miyagawa, and K. Kanoda. Magnetic-Field-Induced Mott Transition in a Quasi-Two-Dimensional Organic Conductor. Phys. Rev. Lett., 93:127001, Sep 2004.

[267] Z. L. Sun, A. F. Wang, H. M. Mu, H. H. Wang, Z. F. Wang, T. Wu, Z. Y. Wang, X. Y. Zhou, and X. H. Chen. Field-induced metal-to-insulator transition and colossal anisotropic magnetoresistance in a nearly Dirac material eumnsb2. npj Quantum Materials, 6(1):94, 2021.

[268] A P Ramirez. Colossal magnetoresistance. Journal of Physics: Condensed Matter, 9(39):8171, sep 1997.

[269] Elbio Dagotto, Takashi Hotta, and Adriana Moreo. Colossal magnetoresistant materials: the key role of phase separation. Physics Reports, 344(1):1-153, 2001.

[270] Myron B. Salamon and Marcelo Jaime. The physics of manganites: Structure and transport. Rev. Mod. Phys, 73:583-628, Aug 2001.

[271] Katarzyna Czajka, Anna Gorczyca, Maciej M. Maska, and Marcin Mierzejewski. Hofstadter butterfly for a finite correlated system. Phys. Rev. B, 74:125116, Sep 2006.

[272] S. Acheche, L.-F. Arsenault, and A.-M. S. Tremblay. Orbital effect of the magnetic field in dynamical mean-field theory. Phys. Rev. B, 96:235135, Dec 2017.

[273] A. A. Markov, G. Rohringer, and A. N. Rubtsov. Robustness of the topological quantization of the Hall conductivity for correlated lattice electrons at finite temperatures. Phys. Rev. B, 100:115102, Sep 2019.

[274] Jak sa Vucicevic and Rok Zitko. Universal Magnetic Oscillations of dc Conductivity in the Incoherent Regime of Correlated Systems. Phys. Rev. Lett., 127:196601, Nov 2021.

[275] J. Vucicevic and R. Zitko. Electrical conductivity in the Hubbard model: Orbital effects of magnetic field. Phys. Rev. B, 104:205101, Nov 2021.

[276] Jixun K. Ding, Wen O. Wang, Brian Moritz, Yoni Schattner, Edwin W. Huang, and Thomas P. Devereaux. Thermodynamics of correlated electrons in a magnetic field. Communications Physics, 5(1):204, 2022.

[277] D. Vollhardt and P. Wolfe. Electronic Phase Transitions, chapter 1, pages 1-78. Elsevier Science Publishers B.V., 1992.

[278] Antoine Georges and Werner Krauth. Numerical solution of the d = ro Hubbard model: Evidence for a Mott transition. Phys. Rev. Lett., 69(8):1240-1243, August 1992.

[279] Antoine Georges and Gabriel Kotliar. Hubbard model in infinite dimensions. Phys. Rev. B, 45:6479-6483, March 1992.

[280] Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys., 68:13, January 1996.

[281] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L. Tomarken, Jason Y. Luo, Javier D. Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, Ray C. Ashoori, and Pablo Jarillo-Herrero. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature, 556(7699):80-84, 2018.

[282] J M Pizarro, M J Calderon, and E Bascones. The nature of correlations in the insulating states of twisted bilayer graphene. Journal of Physics Communications, 3(3):035024, mar 2019.

[283] Yves H. Kwan, S.A. Parameswaran, and S. L. Sondhi. Twisted bilayer graphene in a parallel magnetic field. Phys. Rev. B, 101:205116, May 2020.

[284] Laurent Laloux, Antoine Georges, and Werner Krauth. Effect of a magnetic field on Mott-Hubbard systems. Physical Review B, 50(5):3092, 1994.

[285] J. Bauer and A. C. Hewson. Field-dependent quasiparticles in the infinite-dimensional Hubbard model. Phys. Rev. B, 76:035118, Jul 2007.

[286] Peter GJ van Dongen and C Leinung. Mott-Hubbard transition in a magnetic field. Annalen der Physik, 509(1):45-67, 1997.

[287] Markus Wallerberger, Andreas Hausoel, Patrik Gunacker, Alexander Kowalski, Nicolaus Parragh, Florian Goth, Karsten Held, and Giorgio Sangiovanni. w2dynamics: Local one- and two-particle quantities from dynamical mean field theory. Computer Physics Communications, 235:388 - 399, 2019.

[288] G. Rohringer and A Markov. Magnetic field driven metal-insulator transition in strongly correlated electron systems, 2024.

[289] Haichang Lu and John Robertson. Density Functional Theory Studies of the Metal-Insulator Transition in Vanadium Dioxide Alloys. physica status solidi (b), 256(12):1900210, 2019.

[290] Songhee Choi, Gihyeon Ahn, Soon Jae Moon, and Shinbuhm Lee. Tunable resistivity of correlated VO2(A) and VO2(B) via tungsten doping. Scientific Reports, 10(1):9721, 2020.

[291] T. J. Huffman, C. Hendriks, E. J. Walter, Joonseok Yoon, Honglyoul Ju, R. Smith, G. L. Carr, H. Krakauer, and M. M. Qazilbash. Insulating phases of vanadium dioxide are Mott-Hubbard insulators. Phys. Rev. B, 95:075125, Feb 2017.

[292] Urs Gebert, Bernhard Irsigler, and Walter Hofstetter. Local Chern marker of smoothly confined Hofstadter fermions. Physical Review A, 101(6):063606, 2020.

[293] Huaqing Huang and Feng Liu. Quantum spin Hall effect and spin Bott index in a quasicrystal lattice. Physical review letters, 121(12):126401, 2018.

[294] Duc-Thanh Tran, Alexandre Dauphin, Nathan Goldman, and Pierre Gaspard. Topological Hofstadter insulators in a two-dimensional quasicrystal. Physical Review B, 91(8):085125, 2015.

[295] Adhip Agarwala. Topological insulators in amorphous systems. In Excursions in Ill-Condensed Quantum Matter, pages 61-79. Springer, 2019.

[296] Marcello Davide Caio, Gunnar Möller, Nigel R Cooper, and MJ Bhaseen. Topological marker currents in Chern insulators. Nature Physics, 15(3):257-261, 2019.

[297] A Amaricci, L Privitera, F Petocchi, M Capone, G Sangiovanni, and B Trauzettel. Edge state reconstruction from strong correlations in quantum spin Hall insulators. Physical Review B, 95(20):205120, 2017.

[298] Bernhard Irsigler, Jun-Hui Zheng, and Walter Hofstetter. Interacting Hofstadter Interface. Physical review letters, 122(1):010406, 2019.

[299] Haim Brezis and Louis Nirenberg. Degree theory and BMO; part I: Compact manifolds without boundaries. Sel. Math. New Ser., 1(2):197-264, 1995.

[300] Zhiyuan Wang and Kaden RA Hazzard. Tightening the Lieb-Robinson Bound in Locally Interacting Systems. PRX Quantum, 1(1):010303, 2020.

[301] Stephane Jaffard. Proprietes des matrices bien localisees pres de leur diagonale et quelques applications. In Annales de l'Institut Henri Poincare (C) Non Linear Analysis, volume 7, pages 461-476. Elsevier, 1990.

[302] Michele Benzi. Localization in matrix computations: Theory and applications. In Exploiting Hidden Structure in Matrix Computations: Algorithms and Applications, pages 211-317. Springer, 2016.

[303] Xiao-Liang Qi, Yong-Shi Wu, and Shou-Cheng Zhang. Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors. Physical Review B, 74(8):085308, 2006.

[304] Jean Bellissard, Andreas van Elst, and Hermann Schulz-Baldes. The noncommutative geometry of the quantum Hall effect. Journal of Mathematical Physics, 35(10):5373-5451, 1994.

[305] Masaru Onoda, Yshai Avishai, and Naoto Nagaosa. Localization in a quantum spin Hall system. Phys. Rev. Lett., 98(7):076802, 2007.

[306] Eduardo V Castro, M Pilar Lopez-Sancho, and Maria AH Vozmediano. Anderson localization and topological transition in Chern insulators. Phys. Rev. B, 92(8):085410, 2015.

[307] Elbio Dagotto. Correlated electrons in high-temperature superconductors. Rev. Mod. Phys., 66(3):763, 1994.

[308] Mitsuaki Kawamura, Kazuyoshi Yoshimi, Takahiro Misawa, Youhei Yamaji, Synge Todo, and Naoki Kawashima. Quantum lattice model solver H$. Comput. Phys. Commun., 217:180-192, 2017.

[309] Christopher N Varney, Kai Sun, Marcos Rigol, and Victor Galitski. Interaction effects and quantum phase transitions in topological insulators. Physical Review B, 82(11):115125, 2010.

[310] Marcelo J Rozenberg, Goetz Moeller, and Gabriel Kotliar. The metal-insulator transition in the Hubbard model at zero temperature ii. Modern Physics Letters B, 8(08n09):535-543, 1994.

[311] Bernhard Irsigler, Jun-Hui Zheng, and Walter Hofstetter. Microscopic characteristics and tomography scheme of the local Chern marker. Physical Review A, 100(2):023610, 2019.

[312] Luca D'Alessio and Marcos Rigol. Dynamical preparation of Floquet Chern insulators. Nature communications, 6(1):1-8, 2015.

[313] Lorenzo Privitera and Giuseppe E Santoro. Quantum annealing and nonequilibrium dynamics of Floquet Chern insulators. Physical Review B, 93(24):241406, 2016.

[314] Daniele Toniolo. Time-dependent topological systems: A study of the Bott index. Physical Review B, 98(23):235425, 2018.

[315] MD Caio, Nigel R Cooper, and MJ Bhaseen. Quantum quenches in Chern insulators. Physical review letters, 115(23):236403, 2015.

[316] Michele Benzi, Paola Boito, and Nader Razouk. Decay properties of spectral projectors with applications to electronic structure. SIAM review, 55(1):3-64, 2013.

[317] Noah P Mitchell, Lisa M Nash, Daniel Hexner, Ari M Turner, and William TM Irvine. Amorphous topological insulators constructed from random point sets. Nature Physics, 14(4):380-385, 2018.

[318] Pasquale Calabrese and John Cardy. Evolution of entanglement entropy in one-dimensional systems. Journal of Statistical Mechanics: Theory and Experiment, 2005(04):P04010, 2005.

[319] Rudolph Peierls. Zur theorie des diamagnetismus von leitungselektronen. Zeitschrift für Physik, 80(11-12):763-791, 1933.

[320] Tim Langen, Remi Geiger, and Jorg Schmiedmayer. Ultracold atoms out of equilibrium. Annu. Rev. Condens. Matter Phys., 6(1):201-217, 2015.

[321] Waseem S Bakr, Jonathon I Gillen, Amy Peng, Simon Fölling, and Markus Greiner. A quantum gas microscope for detecting single atoms in a hubbard-regime optical lattice. Nature, 462(7269):74-77, 2009.

[322] Ahmed Omran, Martin Boll, Timon A Hilker, Katharina Kleinlein, Guillaume Salomon, Immanuel Bloch, and Christian Gross. Microscopic observation of Pauli blocking in degenerate fermionic lattice gases. Physical review letters, 115(26):263001, 2015.

[323] Wenjin Zhao, Kaifei Kang, Yichi Zhang, Patrick Knöppel, Zui Tao, Lizhong Li, Charles L Tschirhart, Evgeny Redekop, Kenji Watanabe, Takashi Taniguchi, et al. Realization of the Haldane Chern insulator in a moire lattice. Nature Physics, 20(2):275-280, 2024.

[324] Guorui Chen, Aaron L Sharpe, Eli J Fox, Ya-Hui Zhang, Shaoxin Wang, Lili Jiang, Bosai Lyu, Hongyuan Li, Kenji Watanabe, Takashi Taniguchi, et al. Tunable correlated Chern insulator and ferromagnetism in a moire superlattice. Nature, 579(7797):56-61, 2020.

[325] D Krökel, D Grischkowsky, and MB Ketchen. Subpicosecond electrical pulse generation using photoconductive switches with long carrier lifetimes. Applied Physics Letters, 54(11):1046-1047, 1989.

[326] Kadir Vahaplar, AM Kalashnikova, AV Kimel, Stefan Gerlach, Denise Hinzke, Ulrich Nowak, Roy Chantrell, Arata Tsukamoto, A Itoh, Andrei Kirilyuk, et al. All-optical magnetization reversal by circularly polarized laser pulses: Experiment and multiscale modeling. Physical Review B—Condensed Matter and Materials Physics, 85(10):104402, 2012.

[327] Monika Aidelsburger, Sylvain Nascimbene, and Nathan Goldman. Artificial gauge fields in materials and engineered systems. Comptes Rendus Physique, 19(6):394-432, 2018.

[328] Kangkai Liang, Liya Bi, Qingyi Zhu, Hao Zhou, and Shaowei Li. Ultrafast dynamics revealed with time-resolved scanning tunneling microscopy: A review. ACS Applied Optical Materials, 1(5):924-938, 2023.

[329] Luis A Pena Ardila, Markus Heyl, and Andre Eckardt. Measuring the single-particle density matrix for fermions and hard-core bosons in an optical lattice. Physical review letters, 121(26):260401, 2018.

[330] Thomas Chalopin, Tanish Satoor, Alexandre Evrard, Vasiliy Makhalov, Jean Dalibard, Raphael Lopes, and Sylvain Nascimbene. Probing chiral edge dynamics and bulk topology of a synthetic Hall system. Nature Physics, 16(10):1017-1021, 2020.

[331] Kaiyuan Cao, Wenwen Li, Ming Zhong, and Peiqing Tong. Influence of weak disorder on the dynamical quantum phase transitions in the anisotropic XY chain. Physical Review B, 102(1):014207, 2020.

[332] Pascual Jordan and Eugene Paul Wigner. Uber das paulische aquivalenzverbot. In The Collected Works of Eugene Paul Wigner, pages 109-129. Springer, 1993.

[333] N Bogoliubov. On the theory of superfluidity. J. Phys, 11(1):23, 1947.

[334] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407-466, 1961.

[335] David J Thouless. Stability conditions and nuclear rotations in the Hartree-Fock theory. Nuclear Physics, 21:225-232, 1960.

[336] Peter Ring and Peter Schuck. The nuclear many-body problem. Springer Science & Business Media, 2004.

[337] Ming Zhong and Peiqing Tong. Loschmidt echo of a two-level qubit coupled to nonuniform anisotropic X Y chains in a transverse field. Physical Review A, 84(5):052105, 2011.

[338] Michael E Fisher. The nature of critical points. University of Colorado Press, 1965.

[339] Eytan Barouch and Barry M McCoy. Statistical mechanics of the X y model. II. spin-correlation functions. Physical Review A, 3(2):786, 1971.

[340] Subir Sachdev. Quantum phase transitions. Physics world, 12(4):33, 1999.

[341] A Yu Kitaev. Unpaired Majorana fermions in quantum wires. Physics-uspekhi, 44(10S):131, 2001.

[342] Pasquale Calabrese, Fabian HL Essler, and Maurizio Fagotti. Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators. Journal of Statistical Mechanics: Theory and Experiment, 2012(07):P07016, 2012.

[343] A Porro and T Duguet. On the off-diagonal wick's theorem and Onishi formula: Alternative and consistent approach to off-diagonal operator and norm kernels. The European Physical Journal A, 58(10):197, 2022.

[344] Rudolf Ernst Peierls. Quantum theory of solids. Clarendon Press, 1996.

[345] Thomas Schafer, Faruk Geles, Daniel Rost, Georg Rohringer, Enrico Arrigoni, Karsten Held, Nils Blömer, Markus Aichhorn, and Alessandro Toschi. Fate of the false Mott-Hubbard transition in two dimensions. Physical Review B, 91(12):125109, 2015.

[346] G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held. Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. Rev. Mod. Phys., 90:025003, May 2018.

[347] Lei Wang, Xi Dai, and XC Xie. Frequency domain winding number and interaction effect on topological insulators. Physical Review B, 84(20):205116, 2011.

[348] Leonid V Keldysh et al. Diagram technique for nonequilibrium processes. Sov. Phys. JETP, 20(4):1018-1026, 1965.

[349] Botao Wang, Xiao-Yu Dong, and Andre Eckardt. Measurable signatures of bosonic fractional Chern insulator states and their fractional excitations in a quantum-gas microscope. arXiv preprint arXiv:2111.01110, 2021.

[350] Victor Gurarie and Andrew M Essin. Topological invariants for fractional quantum hall states. JETP letters, 97:233-238, 2013.