We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated fermionic systems. For the 3+1 systems the important universality class is determined by the topologically stable Fermi points in momentum space. In the extreme limit of low energy the condensed matter system of this universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. This introduces the conceptual similarity between such condensed matter and the quantum vacuum of Standard Model, which also contains the Fermi point. For the 2+1 fermionic systems the momentum space topology gives rise to the quantization of different physical parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum phase transitions with the abrupt change of the momentum-space topology. The combined p-space and r-space topology determines in particular the topology of the energy spectrum of fermion zero modes living within the topological defects; fermionic charges of the topologically nontrivial extended objects; axial anomaly; etc.
We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated fermionic systems. For the 3+1 systems the important universality class is determined by the topologically stable Fermi points in momentum space. In the extreme limit of low energy the condensed matter system of this universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. This introduces the conceptual similarity between such condensed matter and the quantum vacuum of Standard Model, which also contains the Fermi point. For the 2+1 fermionic systems the momentum space topology gives rise to the quantization of different physical parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum phase transitions with the abrupt change of the momentum-space topology. The combined p-space and r-space topology determines in particular the topology of the energy spectrum of fermion zero modes living within the topological defects; fermionic charges of the topologically nontrivial extended objects; axial anomaly; etc.
Abstract This chapter discusses effective theories of quantum fermionic liquids. In the low-energy limit, the type of the effective theory depends on the structure of the quasiparticle spectrum, which in turn is determined by topology in momentum space (p-space topology). The p-space topology distinguishes three main generic classes of the stable fermionic spectrum in the quantum vacuum of a 3+1 fermionic system: vacua with Fermi surfaces, vacua with Fermi points, and vacua with a fully gapped fermionic spectrum. Fermi surface is stable because it represents the topological object — the vortex in momentum space. As a result, Fermi liquids with Fermi surface share the properties of their simplest representative: weakly interacting Fermi gas. The low-energy physics of the interacting particles in a Fermi liquid is equivalent to the physics of a gas of quasiparticles moving in collective Bose fields produced by all other particles. Another topological object in p-space is the hedgehog, which is responsible for stability of Fermi points. Example of vacuum of the Fermi-point universality class is provided by 3He-A. Near the Fermi point the effective relativistic field theory emerges with emerging Weyl fermions, gauge fields, and gravity. The chapter discusses p-space and r-space topology, topological invariant for Fermi surface and Fermi points in terms of Green's function, Landau and non-Landau Fermi liquids, collective modes of the Fermi surface, volume of the Fermi surface as invariant of adiabatic deformations, collective modes of vacuum with Fermi points (electromagnetic and gravitational fields), and manifolds of zeros in p-space of higher dimensions.
Many quantum condensed-matter systems, and probably the quantum vacuum of our Universe, are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturbatively. However, physics which emerges in the low-energy does not depend on the complicated details of the system and is relatively simple. It is determined by the nodes in the fermionic spectrum, which are protected by topology in momentum space (in some cases, in combination with the vacuum symmetry). Here we illustrate this universality on some examples of quantum phase transitions, which can occur between the vacua with the same symmetry but with diferent topology in momentum space. The quantum phase transitions between the fully gapped states with different momentum-space topology are also discussed.
Opportunities for topological phase transitions in strongly correlated Fermi systems near a quantum critical point are explored as an alternative to collective scenarios for experimentally observed departures from standard Fermi-liquid behavior. Attention is focused on a quantum critical point at which the effective mass is divergent due to vanishing of the quasiparticle group velocity at the Fermi surface. Working within the original Landau quasiparticle theory, it is demonstrated that the quasiparticle picture can remain meaningful beyond the quantum critical point through rearrangements of the unstable normal Fermi surface and quasiparticle spectrum. Two possibilities emerge at zero temperature, depending on whether the quasiparticle interaction is regular or singular at zero momentum transfer. In the regular case, one type of topological phase transformation leads to a state with a multiconnected Fermi surface. In the singular case, another type of topological phase transition leads to an exceptional state containing a fermion condensate – the Fermi surface swells into a volume in momentum space, within which partial occupation prevails and quasiparticle energies are pinned to the chemical potential. As the temperature increases from zero to a characteristic value Tm, a crossover can occur from the state with multiple Fermi surfaces to that containing a fermion condensate.
In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
The physics of strongly interacting particles is often extraordinarily rich, leading to the \textit{emergence} of fundamentally new phases and phenomena that cannot exist in the absence of correlations. This thesis is devoted to the study of different classes of quantum systems with such strong correlations. Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms to probe quantum many-body phases and their dynamics. In Chapter 2, we start with theories of quantum phase transitions (QPTs) in one-dimensional chains of Rydberg atoms. We demonstrate the existence of a novel QPT in the universality class of the Z3 chiral clock model, calculate its critical exponents, and discuss their measurement with the quantum Kibble-Zurek mechanism. Proceeding to two dimensions, Chapter 3 describes the zero-temperature phase diagram of a square-lattice Rydberg atom array, revealing several experimentally realizable density-wave-ordered phases and QPTs. For a different geometry, namely, the kagome lattice studied in Chapter 4, we identify a regime that constitutes a likely candidate for hosting a phase with long-range quantum entanglement and topological order. Based on mappings to dimer models and gauge theories, we argue for the existence of a quantum spin liquid phase that has proved elusive in solid-state materials. In the next part of this thesis, we focus on similarly strongly correlated phases in the realm of quantum materials. Motivated by recent observations of an enhanced thermal Hall response in the high-temperature cuprate superconductors, in Chapter 5, we study the thermal Hall conductivity, κxy, of chiral spin-liquid ans\"{a}tze on the square lattice, focusing on the contribution from neutral (bosonic or fermionic) spinons. Thereafter, going beyond mean-field theory, we consider the corrections due to emergent gauge fields, which yield a response of opposite sign to that of the fermionic matter. We additionally consider the contribution of phonons to κxy and establish its relation to the spinon Hall viscosity. Thermal Hall measurements also leave intriguing fingerprints in Kitaev materials like α-RuCl3, and in this context, Chapter 6 investigates the unquantized κxy obtained in a gapless spinon Fermi surface state.
"This thesis is devoted to the investigation of the quantum liquid crystal phases in strongly correlated electronic systems. Such phases are characterized by their partially broken spatial symmetries and are observed in various strongly correlated systems as being summarized in Chapter 1. In quantum liquid crystal phases formed by fermions, behaviors beyond Landau's theory of the Fermi liquids appear naturally, signed by the absence of well-defined quasiparticle excitations. Although quantum liquid crystal phases often involve strong-coupling physics, the associated symmetry analysis can often be carried out also in a relatively weak coupling regime and this is the main approach taken in this thesis. Two different quantum liquid crystal phases, the nematic state and the smectic state, are studied in Chapters 2 and 3 respectively, by investigating various instabilities of a Landau Fermi liquid. A nematic phase is a state that breaks the rotational symmetry spontaneously but preserves the translational symmetries. It can be reached by the Pomeranchuk instability of a Landau Fermi liquid. The Pomeranchuk instability in the charge (spin-singlet) sector of a Fermi liquid was studied previously within a random phase approximation and by a high-dimensional bosonization method. In Chapter 2, we concentrate on the Pomeranchuk instability in the spin-triplet sector, and two different classes of ordered phases are discovered, dubbed the a and ,8-phases by analogy to the superfluid 3He-A and B-phases. The Fermi surfaces in the a-phases exhibit spontaneous anisotropic distortions, while in the ,8-phases the Fermi surfaces remain circular or spherical with topologically non-trivial spin configurations in momentum space. The behaviors of the low energy bosonic excitations are studied both in the order phases (a and .B) and at the quantum critical point separating the ordered phases from the Fermi liquid phase. In Chapter 3, the smectic phase and the quantum phase transition between a metallic nematic state and a metallic smectic phase are investigated using the random phase approximation, by considering the formation of a unidirectional charge density wave in an electronic nematic state. The associated quantum critical point is investigated within the Hertz-Millis approach. Non-Fermi liquid behaviors are discovered both at the quantum phase transition and in the smectic ordered phase. The theory we studied here provides a simple description for the so-called ""fluctuating stripe"" phase in cuprates. Similar to its classical counterpart, the phase transition between the nematic and smectic phases are described by a ""gauge-like"" theory as required by the rotational invariance in close analogy with the McMillan-deGennes theory of the smectic-nematic phase transition in classical liquid crystals. However, unlike the classical McMillandeGennes theory, in which the coupling between the ""gauge-like"" nematic fluctuations and the ""matter-like"" smectic fields is relevant, in the sense of renormalization group, this coupling is irrelevant at the quantum phase transition. This property enables the study of low energy fluctuations and the fate of fermions near this quantum critical point. In Chapter 4, quantum liquid crystal phases with a broken time-reversal symmetry are investigated. We start by studying the general scenario of timereversal symmetry breaking phases in the absence of magnetic ordering in systems with long-lived low-energy fermionic quasiparticle excitations in two dimensions. Using a Berry phase approach, we classified possible time-reversal symmetry breaking phases by the accompanying spatial symmetry breaking patterns. Two of the simplest situations are referred to as the type I state, which breaks simultaneously the space-inversion symmetry and the time-reversal symmetry, and the type I I state, in which the chiral symmetry is broken in addition to the breaking of the time-reversal symmetry. Although the starting point is general and has little relation with the quantum liquid crystal phases, both type I and type II states defined above are realized by (generalized) Pomeranchuk instabilities. In fact , a type I I state is usually a nematic state which also breaks the time-reversal symmetry. Interestingly, the type II states also shows spontaneously generated Hall effect even in the absence of a magnetic field, which is known as the spontaneous anomalous Hall effect. In Chapter 5, we investigate the stability of a quadratic band-crossing point in two-dimensional fermionic systems that provide an explicit example, within well controlled approximations, for the nematic ordering discussed in Chapter 2 and spontaneous anomalous Hall effect discussed in Chapter 4. At the non-interacting level, we show that a quadratic band-crossing point exists and is topologically stable when the Berry flux of this crossing point is ±27r and the point symmetry group at this point has either fourfold or sixfold rotational symmetries. Strikingly, we find that this would-be topologically stable quadratic band-crossing is marginally unstable against short-range interactions: a quadratic band-crossing point in two dimensions is unstable against shortrange repulsion, no matter how weak! Four ordered phases are found in this system, the quantum anomalous Hall state, the quantum spin Hall phase, the nematic phase, and the nematic-spin-nematic phase."