The concepts of deconfinement and topological order are of great current interest for quantum information science and for our understanding of quantum materials. Here, we introduce a simple model of three antiferromagnetically coupled Kondo impurities, a ``Kondo triangle'', which can be used to further extend the application of these concepts to electronic systems. We show that by tuning the magnetic frustration, the Kondo triangle undergoes a quantum phase transition between two phases of unbroken symmetry, signaling a phase transition beyond the Landau paradigm. We demonstrate that the frustrated ``spin liquid'' phase is described by a three-channel Kondo fixed point and thus displays an irrational ground state degeneracy. Using an Abrikosov pseudofermion representation this quantum state is categorized by an emergent U(1) gauge field and its projective symmetry group. The gauge theory is deconfining in the sense that a miniature Wilson loop orders and that topological defects (instantons in the gauge field) are expelled. This phase persists in presence of moderate Kondo screening until the proliferation of topological defects leads to a quantum phase transition to an unfrustrated Fermi liquid phase. Based on this evidence, we propose that the three-channel Kondo phase displays topological order in a similar sense as gapless spin liquids.
Multichannel Kondo lattice models are examples of strongly correlated electronic systems that exhibit non-Fermi-liquid behavior due to the presence of a continuous channel symmetry. Mean-field analyses have predicted that these systems undergo channel symmetry breaking at low temperature. We use the dynamical large- N technique to study temporal and spatial fluctuations of the multichannel Kondo model on a honeycomb lattice and find that this prediction is not generally true. Rather, we find a ( 2+1 )-dimensional conformally invariant fixed point, governed by critical exponents that are found numerically. When we break time-reversal symmetry by adding a Haldane mass to the conduction electrons, three phases, separated by continuous transitions, are discernible: one characterized by dynamic mass generation and spontaneous breaking of the channel symmetry, one where topological defects restore channel symmetry but preserve the gap, and one with a Kondo-coupled chiral spin liquid. We argue that the last phase is a fractional Chern insulator with anyonic excitations. Published by the American Physical Society 2024
The two-channel Kondo (2CK) effect with its exotic ground state properties has remained difficult to realize in physical systems. At low energies, a quantum impurity with orbital degree of freedom, like a proton bound in an interstitial lattice space, comprises a 3-level system with a unique ground state and (at least) doubly degenerate rotational excitations with excitation energy
Delta0. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings
J), while the electron spin is conserved. We show by extensive numerical renormalization group (NRG) calculations that without fine-tuning of parameters this system exhibits a 2CK fixed point, due to Kondo correlations in the excited-state doublet whose degeneracy is stabilized by the host lattice parity, while the channel symmetry (electron spin) is guaranteed by time reversal symmetry. We find a pronounced plateau in the entropy at
S(TK<T<Delta0)=kBln2 between the high-
T value,
S(TggDelta0)=kBln3, and the 2CK ground state value,
S(0)=kBlnsqrt2. This indicates a downward renormalization of the doublet below the non-interacting ground state, thus realizing the 2CK fixed point, in agreement with earlier conjectures. We mapped out the phase diagram of the model in the
J−Delta0 plane. The Kondo temperature
TK shows non-monotonic
J-dependence, characteristic for 2CK systems.newline indent Beside the two-channel Kondo effect of the model, we also study the single-channel version, which is realized by applying a strong magnetic field to the conduction band electrons so that their degeneracy is lifted and consequently having only one kind of electrons scattering off the impurity. This single-channel case is easier to analyze since the Hilbert space is not as large as that of the 2CK. We equally find a downward renormalization of the excited state energy by the Kondo correlations in the SU(2) doublet. In a wide range of parameter values this stabilizes the single-channel Kondo fixed point and a phase diagram is also mapped out for the model. In the single-channel version a plateau is found in the entropy at
S(TK<T<Delta0)=kBln2 between high-
T value,
S(TggDelta0)=kBln3, and the single-channel Kondo ground state value,
S(0)=kBln1.
Insulating
Z2 spin liquids are a phase of matter with bulk anyonic quasiparticle excitations and ground-state degeneracies on manifolds with nontrivial topology. We construct a time-reversal symmetric
Z2 spin liquid in two spatial dimensions using an array of quantum wires. We identify the anyons as kinks in the appropriate Luttinger-liquid description, compute their mutual statistics, and construct local operators that transport these quasiparticles. We also present a construction of a fractionalized Fermi liquid (FL*) by coupling the spin sector of the
Z2 spin liquid to a Fermi liquid via a Kondo-like coupling.
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.\nFor the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.\nFor the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic
r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For
r>rmax, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions
r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.