Abstract A central concept in the theory of phase transitions beyond the Landau–Ginzburg–Wilson paradigm is fractionalization: the formation of new quasiparticles that interact via emergent gauge fields. This concept has been extensively explored in the context of continuous quantum phase transitions between distinct orders that break different symmetries. We propose a mechanism for continuous order-to-order quantum phase transitions that operates independently of fractionalization. This mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R 2 Ir2O7. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions.
Abstract A central concept in the theory of phase transitions beyond the Landau–Ginzburg–Wilson paradigm is fractionalization: the formation of new quasiparticles that interact via emergent gauge fields. This concept has been extensively explored in the context of continuous quantum phase transitions between distinct orders that break different symmetries. We propose a mechanism for continuous order-to-order quantum phase transitions that operates independently of fractionalization. This mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R 2 Ir2O7. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions.
Even a century after Heisenberg, Born, Jordan, and others laid the foundations of modern quantum mechanics, quantum phenomena continue to fascinate and puzzle us, with no end to their mysteries in sight. A particularly rich and active field, grounded in a variety of deeply rooted quantum-mechanical concepts, is that of quantum phase transitions, i.e., phase transitions at zero temperature controlled by nonthermal parameters. While a remarkable number of quantum phase transitions are successfully described within the Landau-Ginzburg-Wilson paradigm, particular attention is drawn to cases where a description within the paradigm fails. Prominent examples that evade the Landau-Ginzburg-Wilson paradigm are phase transitions beyond conventional symmetry breaking and phase transitions enriched by additional gapless modes. This dissertation seeks to further advance the theory of phase transitions beyond the Landau-Ginzburg-Wilson paradigm by proposing and analyzing several novel routes beyond the framework and relating them to established concepts in the field. First, we identify an exotic quasiuniversal behavior near the all-in-all-out Weyl quantum critical point in three-dimensional Luttinger semimetals, such as the pyrochlore iridates R₂Ir₂O₇, with R a rare-earth element. The quasiuniversal behavior is characterized by power laws with exponents that vary slowly over several orders of magnitude in energy or length. However, in contrast to the quasiuniversality discussed in the context of deconfined criticality, the present case is characterized by a genuinely universal ultra-low-temperature behavior. In this limit, the pertinent critical exponents can be computed exactly within a renormalization group analysis. This exotic behavior will manifest generically at any quantum critical fixed point possessing a marginally irrelevant operator. Experimental implications for the pyrochlore iridates are outlined and compared with recent experimental observations in Y₂Ir₂O7thinfilms.Second,weproposeanovelmechanismforcontinuousorder−to−orderquantumphasetransitionsthat,unlikedeconfinedcriticality,operatesindependentlyoffractionalization.Theproposedmechanismisbasedonthecollisionandannihilationoftworenormalizationgroupfixedpoints:aquantumcriticalfixedpointandaninfraredstablefixedpoint.Theannihilationofthesefixedpointsrearrangestheflowtopology,eliminatingthedisorderedphaseassociatedwiththeinfraredstablefixedpointandpromotingasecondcriticalfixedpoint,unaffectedbythecollision,toaquantumcriticalpointbetweendistinctorders.Wearguethatthismechanismisrelevanttoabroadspectrumofphysicalsystems.Inparticular,itcanmanifestinLuttingerfermionsystemsinthreespatialdimensions,leadingtoacontinuousquantumphasetransitionbetweenanantiferromagneticWeylsemimetalstate,whichbreakstime−reversalsymmetry,andanematictopologicalinsulator,characterizedbybrokenlatticerotationalsymmetry.Thiscontinuousantiferromagnetic−Weyl−to−nematic−insulatortransitionmightbeobservedinrare−earthpyrochloreiridatesR2Ir2O7.Otherpossiblerealizationsincludekagomequantummagnets,quantumimpuritymodels,andquantumchromodynamicswithsupplementalfour−fermioninteractions.Thisbroadspectrumofpotentialrealizationsprovidesanidealsettingtoexploretheproposedfixed−point−annihilationmechanismanditsinterrelationswithotherconcepts,suchasfractionalization.Third,wepresentmultiplestrandsofongoingresearch,allultimatelyworkingtowardsaputativeSachdev−Ye−Kitaevfixedpointinthreespatialdimensions.Asastartingpoint,weinvestigatethelarge−N$ behavior of a partially bosonized Luttinger fermion theory at a large but finite number of quadratic band touching points N. Building on this, we analyze a random variant of the deterministic Luttinger fermion model, now in the limit of infinitely many fermionic and bosonic degrees of freedom, motivated by low-dimensional Yukawa-Sachdev-Ye-Kitaev theories. In both cases, complications, ultimately resulting from the absence of Lorentz symmetry, arise and motivate the study of a slightly simpler analogue, a relativistic Luttinger fermion theory. Not only does this third study improve our understanding of Luttinger fermion systems at large N in general, but it also reveals a novel fermionic conformal fixed point, which corresponds to an infrared stable fixed point in 8/3 < D < 4 space-time dimensions and to an ultraviolet stable fixed point in 4 < D < 6 space-time dimensions. While the former gives rise to a previously unknown conformal phase, the latter unveils a new asymptotically safe quantum field theory.
Continuous quantum phase transitions beyond the conventional paradigm of fluctuations of a symmetry-breaking order parameter are challenging for theory.These phase transitions often involve emergent deconfined gauge fields at the critical points as demonstrated in phase transitions between different broken-symmetry states of 2 þ 1-dimensional quantum magnets, as well as those between symmetry-protected topological (SPT) phases.In this paper, we present several examples of deconfined quantum critical points between SPT phases in 3 þ 1-D for both bosonic and fermionic systems.These critical theories can be formulated as non-Abelian gauge theories either in the infrared-free regime or in the conformal window when they flow to the Banks-Zaks fixed points.We explicitly demonstrate several interesting quantum critical phenomena.We describe situations in which the same phase transition allows for multiple universality classes controlled by distinct fixed points.We exhibit the possibility-which we dub "unnecessary quantum critical points"-of stable generic continuous phase transitions within the same phase.We present examples of interaction-driven, band-theory-forbidden, continuous phase transitions between two distinct band insulators.The understanding we develop leads us to suggest an interesting possible 3 þ 1-D field theory duality between SUð2Þ gauge theory coupled to one massless adjoint Dirac fermion and the theory of a single massless Dirac fermion augmented by a decoupled topological field theory.
We present the critical theory of a number of zero-temperature phase transitions of quantum antiferromagnets and interacting boson systems in two dimensions. The most important example is the transition of the S=1∕2 square lattice antiferromagnet between the N\'eel state (which breaks spin rotation invariance) and the paramagnetic valence bond solid (which preserves spin rotation invariance but breaks lattice symmetries). We show that these two states are separated by a second-order quantum phase transition. This conflicts with Landau-Ginzburg-Wilson theory, which predicts that such states with distinct broken symmetries are generically separated either by a first-order transition, or by a phase with co-existing orders. The critical theory of the second-order transition is not expressed in terms of the order parameters characterizing either state, but involves fractionalized degrees of freedom and an emergent, topological, global conservation law. A closely related theory describes the superfluid-insulator transition of bosons at half filling on a square lattice, in which the insulator has a bond density wave order. Similar considerations are shown to apply to transitions of antiferromagnets between the valence bond solid and the Z2 spin liquid: the critical theory has deconfined excitations interacting with an emergent U(1) gauge force. We comment on the broader implications of our results for the study of quantum criticality in correlated electron systems.
The Landau-Ginzburg-Wilson (LGW) paradigm is the backbone of the modern understanding of critical phenoma. It rests on the assumption that a continuous phase transition can be described solely in terms of a fluctuating order parameter. In this sandwich thesis, we analyze several quantum phase transitions in two-dimensional systems where this basic assumption is violated. Our analysis is structured in three parts: In the first part, two ordering transitions in metals are studied, where the presence of gapless fermionic modes invalidates the pure order parameter description. Instead, we apply a renormalization group approach which retains both fermionic and bosonic (order paramater) degrees of freedom, using the deviation from an upper critical dimension of d = 5/2 as a control parameter. We first apply this technique to describe the quantum phase transition between a normal metal and an inhomogeneous (FFLO) superconductor. Our analysis rigorously confirms the mean field expectation that this transition is continuous, and shows that interesting non-Fermi liquid physics can arise at the critical point, manifesting itself in unusual scaling of various observables. The second case study is the onset of incommensurate 2kF charge density wave order. While non-Fermi liquid features are less pronounced at this transition, we find a strong dynamical nesting of the Fermi surface, which stabilizes the density wave formation and results in a continuous transition, opposed to early theoretical claims. In the second part, we present a novel case study of “deconfined criticality” for a quantum magnet: Here, symmetries are broken on both sides of the phase transition, which would be first order within a conventional two-order-parameter LGW description. By contrast, the scenario of deconfined criticality predicts a continuous transition, driven by condensation of topological defects which carry quantum numbers of the opposite phase. Its paradigmatic application is the transition from a SU (2)-Neél state to a valence-bond solid (a singlet covering configuration that breaks spatial symmetries) on the square lattice. Here, we propose a novel extension and study a SU (3) antiferromagnet on a triangular lattice, which supports a transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU (3) singlet phase. We provide a critical theory in terms of fractional bosonic degrees of freedom (and with a topological defect interpretation), and study its fixed point properties with functional renormalization group. This yields a critical fixed point in a suitable large-N -limit, implying a continuous transition in accordance with the deconfined criticality framework. In the third and final part, we study the “polaron”-problem of a single impurity coupled to a majority Fermi sea, which is particularly interesting when formation of a bound state (“molecule”) between impurity and majority is allowed for. Starting from the exactly solvable limit of infinite impurity mass, we present a controlled computation of impurity spectra (both single- and two-particle) for heavy impurities based on Feynman diagram techniques, with an eye for experiments on doped semiconductors and ultracold gases. Furthermore, we discuss various aspects of the “molecule-to-polaron” transition that occurs in these systems, which also defies a simple LGW description in the single-impurity limit.
We investigate nematic quantum phase transitions in two different Dirac fermion models. The models feature twofold and fourfold, respectively, lattice rotational symmetries that are spontaneously broken in the ordered phase. Using negative-sign-free quantum Monte Carlo simulations and an ε-expansion renormalization group analysis, we show that both models exhibit continuous phase transitions. In contrast to generic Gross-Neveu dynamical mass generation, the quantum critical regime is characterized by large velocity anisotropies, with fixed-point values being approached very slowly. Both experimental and numerical investigations will not be representative of the infrared fixed point, but of a quasiuniversal regime where the drift of the exponents tracks the velocity anisotropy.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.