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 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.
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.
This thesis is devoted to the study of strongly correlated electron systems in two spatial dimensions. The main perspective used in this work is that of scaling, implemented by the renormalization group formalism. This thesis can be divided into three main parts. The first part, comprised of chapters 2-5, investigates various aspects of two dimensional insulating quantum antiferromagnets. In particular, chapters 2-4 deal with the phase transition between an ordered antiferromagnet and a valence bond solid on a square lattice. This transition falls outside the conventional Landau-Ginzburg-Wilson paradigm and is described by a so-called “deconfined ” quantum critical point. In chapter 2, we present an elegant calculation of the scaling dimension of the valence bond solid order parameter at this quantum critical point. Chapters 3 and 4 investigate signatures of deconfined criticality in the response of an antiferromagnet to non-magnetic impurities. Chapter 5 describes the edge response of a conventional ordered square-lattice antiferromagnet. The second part of this thesis, comprised of chapters 6 and 7, studies the subject
We perform a numerical study of a spin-1/2 model with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry in one dimension which demonstrates an interesting similarity to the physics of two-dimensional deconfined quantum critical points (DQCP). Specifically, we investigate the quantum phase transition between Ising ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working directly in the thermodynamic limit using uniform matrix product states, we find evidence for a direct continuous phase transition that lies outside of the Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is found everywhere on the phase boundary. We find that the magnetic and VBS correlations show very close power-law exponents, which is expected from the self-duality of the parton description of this DQCP. Critical exponents vary continuously along the phase boundary in a manner consistent with the predictions of the field theory for this transition. We also find a regime where the phase boundary splits, as suggested by the theory, introducing an intermediate phase of coexisting ferromagnetic and VBS order parameters. Interestingly, we discover a transition involving this coexistence phase which is similar to the DQCP, being also disallowed by the Landau-Ginzburg-Wilson symmetry-breaking theory.
We study a number of quantum phase transitions, which are exotic in their nature and separates non-trivial phases of matter. Since quantum fluctuations, which drive these phase transitions, are stronger in low-dimensions, we concentrate on low-dimensional systems. We consider two different two-dimensional systems in this thesis and study their phase transition. \n\nFirst, we investigate a phase transition in graphene, one of the most famous two-dimensional systems in condensed matter. For a suspended bilayer graphene in ν = 0 quantum Hall regime, the conductivity data and mean-field analysis suggests a phase transition from an antiferromagnetic (AF) state to a valence bond solid (VBS) state, when perpendicular electric field is increased. This AF to VBS phase transition is reminiscent of deconfined criticality, which is a novel phase transition that cannot be explained by Landau’s theory of symmetry breaking. We show that in the strong coupling regime of bilayer graphene, the AF state is destabilized by the transverse electric field, likely resulting in a VBS state. We also consider monolayer and bilayer graphene in the large cyclotron gap limit and show that the effective action for the AF and VBS order parameters have a topological Wess-Zumino-Witten term, supporting that the phase transition observed in experiments is in the deconfined criticality class. \n\nSecond, we study the model systems of cuprate superconductor, which is effectively a two-dimensionalal system in the CuO_2 plane. The proposal that the pseudogap metal is a fractionalized Fermi liquid described by a quantum dimer model is extended using the density matrix renormalization group. Measuring the Friedel oscillations in the open boundaries reveals that the fermionic dimers have dispersion minima near (π/2,π/2), which is compatible with the Fermi arcs in photoemission. Moreover, investigating the entanglement entropy suggests that the dimer model with low fermion density is similar to the free fermion system above the Lifshitz transition. We also study the phase transition from a metal with SU(2) spin symmetry to an AF metal. By applying the functional renormalization group to the two-band spin-fermion model, we establish the existence of a strongly coupled fixed point and calculate critical exponents of the fixed point.
Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N ¼ 1, 2, and 3.The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm.In particular, one of the prominent phases for N > 1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z 2 gauge field in its deconfined phase.However, unlike a conventional Z 2 gauge theory, we do not impose "Gauss's Law" by hand; instead, it emerges because of spontaneous symmetry breaking.Correspondingly, unlike a conventional Z 2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss's law constraint at zero temperature.By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N ¼ 2 and a valence-bond solid for N ¼ 3. Furthermore, for N ¼ 3, the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.
Topology plays a cardinal role in explaining phases and quantum phase transitions beyond the Landau-Ginzburg-Wilson paradigm. In this study, we formulate a set of models of Dirac fermions in 2+1 dimensions with SU(N)×SU(2)×U(1) symmetry that have the potential to host critical points described by field theories with topological terms. For N=2 it shows a rich phase diagram containing semimetallic, quantum spin Hall insulating, Kekulé valence bond solid and s-wave superconducting phases and features multiple Landau-Ginzburg-Wilson phase transitions driven by interaction strength. At N=1 a deconfined quantum critical point is observed. At N=2 one expects the critical theory to correspond to a level 2 Wess-Zumino-Witten theory in 2+1 dimensions. Here the numerical results however show a strong first order transition. Another transition can be governed by a topological θ-term which is rendered irrelevant for even values of N thus leading to Landau-Ginzburg-Wilson behaviour.