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 Y₂Ir₂$O₇ thin films. Second, we propose a novel mechanism for continuous order-to-order quantum phase transitions that, unlike deconfined criticality, operates independently of fractionalization. The proposed mechanism is based on the collision and annihilation of two renormalization group fixed points: a quantum critical fixed point and an infrared stable fixed point. The annihilation of these fixed points rearranges the flow topology, eliminating the disordered phase associated with the infrared stable fixed point and promoting a second critical fixed point, unaffected by the collision, to a quantum critical point between distinct orders. We argue that this mechanism is relevant to a broad spectrum of physical systems. In particular, it can manifest in Luttinger fermion systems in three spatial dimensions, leading to a continuous quantum phase transition between an antiferromagnetic Weyl semimetal state, which breaks time-reversal symmetry, and a nematic topological insulator, characterized by broken lattice rotational symmetry. This continuous antiferromagnetic-Weyl-to-nematic-insulator transition might be observed in rare-earth pyrochlore iridates R₂Ir₂O₇. Other possible realizations include kagome quantum magnets, quantum impurity models, and quantum chromodynamics with supplemental four-fermion interactions. This broad spectrum of potential realizations provides an ideal setting to explore the proposed fixed-point-annihilation mechanism and its interrelations with other concepts, such as fractionalization. Third, we present multiple strands of ongoing research, all ultimately working towards a putative Sachdev-Ye-Kitaev fixed point in three spatial dimensions. As a starting point, we investigate the large-$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.
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.
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.
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.
The Mott critical point between a metal and a correlated insulator has usually been studied via density or spin density bosonic mode fluctuations according to the standard Ginzburg-Landau-Wilson phase transition paradigm. A moment's reflection leads to increasing doubts that such an approach should work as the transition is nonmagnetic, voiding the relevance of spin density modes. Charge density modes are irrelevelant since the long range Coulomb interaction leads to a large plasmon gap and their incompressibility. In solidarity with these doubts, recent measurements of the Mott critical point in low dimensional organic materials yield critical exponents in violent diasagreement with the bosonic mode criticality lore. We propose that fermionic fluctuations control the behavior of the Mott transition. The transition thus has an intrinsic quantum aspect despite being a finite temperature phase transition. We develop this hitherto unexplored physics, obtain experimental predictions and find agreement with one of the novel unexplained experimental exponents. We conclude that this Mott transition corresponds to a new universality class of finite temperature critical points that contains quantum effects and cannot be accounted for by conventional Ginzburg-Landau-Wilson wisdom.
In this PhD thesis, the physics of quantum critical matter and exotic quantum state close to quantum phase transitions is investigated. We will focus on three different examples that highlight some of the interesting phenomena related to quantum phase transitions. Firstly, we discuss the physics of quantum phase transitions in quantum wires as a function of an external gate voltage when new subbands are activated. We find that at these transitions, strong correlations lead to the formation of an impenetrable gas of polarons, and identify criteria for possible instabilities in the spin- and charge sectors of the model. Our analysis is based on the combination of exact resummations, renormalization group techniques and Luttinger liquid approaches. \n \nSecondly, we turn to the physics of multiple divergent time scales close to a quantum critical point. Using an appropriately generalized renormalization group approach, we identify that the presence of multiple dynamics at a quantum phase transition can lead to the emergence of new critical scaling exponents and thus to the breakdown of the ususal scaling schemes. We calculate the critical behavior of various thermodynamic properties and detail how unusual physics can arise. It is hoped that these results might be helpful for the interpretation of experimental scaling puzzles close to quantum critical points. \n \nThirdly, we turn to the physics of topological transitions, and more precisely the physics of Weyl superconductors. The latter are the superconducting variant of the topologically non-trivial Weyl semimetals, and emerge at the quantum phase transition between a topological superconductor and a normal insulator upon perturbing the transition with a time reversal symmetry breaking perturbation, such as magnetism. We characterize the topological properties of Weyl superconductors and establish a topological phase diagram for a particular realization in heterostructures. We discuss the physics of vortices in Weyl superconductors, and establish under which conditions they can trap zero energy Majorana modes. Our disucssion ends with some remarks on possible experimental signatures.
A unified theory of quantum critical points beyond the conventional Landau-Ginzburg-Wilson paradigm remains unknown. According to Landau cubic criterion, phase transitions should be first-order when cubic terms of order parameters are allowed by symmetry in the Landau-Ginzburg free energy. Here, from renormalization group analysis, we show that second-order quantum phase transitions can occur at such putatively first-order transitions in interacting two-dimensional Dirac semimetals. As such type of Landau-forbidden quantum critical points are induced by gapless fermions, we call them fermion-induced quantum critical points. We further introduce a microscopic model of SU(N) fermions on the honeycomb lattice featuring a transition between Dirac semimetals and Kekule valence bond solids. Remarkably, our large-scale sign-problem-free Majorana quantum Monte Carlo simulations show convincing evidences of a fermion-induced quantum critical points for N = 2, 3, 4, 5 and 6, consistent with the renormalization group analysis. We finally discuss possible experimental realizations of the fermion-induced quantum critical points in graphene and graphene-like materials.Quantum phase transitions are governed by Landau-Ginzburg theory and the exceptions are rare. Here, Li et al. propose a type of Landau-forbidden quantum critical points induced by gapless fermions in two-dimensional Dirac semimetals.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].