We study the quantum phase transition driven by a point impurity in a chain seamed with ring frustration. With strong coupling and light impurity, the system is in a topological extended-kink (TEK) phase, which exhibits gapless excitations in the bulk. With strong coupling and heavy impurity, the system is in a gapped kink bound state (KBS) phase. Two-point bulk and impurity correlations are defined to characterize the two phases. In the TEK phase, both the bulk and impurity correlations are long range and factorizable so that scaling functions can be parsed. The scaling functions relies on the distance scaled by the system's size. An impurity correlation length can be extracted from the impurity correlation. In the transition from TEK to KBS, the scaling function of the bulk correlation undergoes an abrupt steplike change. Meanwhile, the impurity correlation length decreases from a divergent value to a finite one. The ground state of the TEK phase retains a relatively high value of entanglement entropy due to the absence of symmetry breaking. However, spontaneous symmetry breaking occurs in the KBS phase, which induces antiferromagnetic order in the bulk and entangled spin configuration near the impurity.
We study the quantum phase transition driven by a point impurity in a chain seamed with ring frustration. With strong coupling and light impurity, the system is in a topological extended-kink (TEK) phase, which exhibits gapless excitations in the bulk. With strong coupling and heavy impurity, the system is in a gapped kink bound state (KBS) phase. Two-point bulk and impurity correlations are defined to characterize the two phases. In the TEK phase, both the bulk and impurity correlations are long range and factorizable so that scaling functions can be parsed. The scaling functions relies on the distance scaled by the system's size. An impurity correlation length can be extracted from the impurity correlation. In the transition from TEK to KBS, the scaling function of the bulk correlation undergoes an abrupt steplike change. Meanwhile, the impurity correlation length decreases from a divergent value to a finite one. The ground state of the TEK phase retains a relatively high value of entanglement entropy due to the absence of symmetry breaking. However, spontaneous symmetry breaking occurs in the KBS phase, which induces antiferromagnetic order in the bulk and entangled spin configuration near the impurity.
On a ring, a single Jordan-Wigner transformation between the Kitaev model and the spin model suffers redundant degrees of freedom. However, we can establish an exact quaternary Jordan-Wigner mapping involving two Kitaev rings and two spin rings with periodic or antiperiodic boundary conditions. This mapping allows us to demonstrate exactly how a topological extended-kink (TEK) phase develops in the interacting Kitaev ring with odd number of lattice sites. The emergence of this new phase is attributed to the effect of geometrical ring frustration. Unlike the usual topological phases protected by an energy gap in noninteracting systems, the TEK phase is gapless. And because the spectra of low-energy excitations are quadratic, the specific heat per site approaches half of the Boltzmann constant near absolute zero temperature. More interestingly, the ground state is unique, immune to spontaneous symmetry breaking. It exhibits a long-range correlation function with a nonlocal factor, but no local order parameter can be defined. As a concomitant effect, a special kind of localized kink zero mode (KZM) takes place if we introduce a type of bond defect. We also show that the KZM is robust against moderate disorders.
In this paper, we work on the pure and mixed cluster models with periodic boundary condition. The first purpose is to establish the concept of quantum cluster kink. We clarify that there are two types of cluster kinks since there are two types of ground states depending on the choice of cluster length, of which the first type exhibits symmetry breaking order and the second one string order. Simple pictures are constructed according to the pure cluster model, which facilitates us to introduce the quantity, cluster kink number. As we demonstrate, cluster kinks deriving from different sources can coexist, compete with each other, and lead to quantum phase transition in a mixed cluster model. The second purpose is to elucidate that the effect of ring frustration can be realized in the cluster model with symmetry breaking order, instead of the one with string order. The reason lies in that ring frustration can induce a huge ground-state degeneracy or a special extended-kink phase with gapless excitations in the former case. And, although ring frustration does not change the phase transition point, it can produce unusual ground state in the extended-kink phase, whose special properties are uncovered by the nonlocal scaling factor in the correlation function and the doubled degeneracies of the eigenvalues of the entanglement spectrum.
We investigate quantum scaling phenomena driven by lower-dimensional defects in quantum Ising-like models. We consider quantum Ising rings in the presence of a bond defect. In the ordered phase, the system undergoes a quantum transition driven by the bond defect between a magnet phase, in which the gap decreases exponentially with increasing size, and a kink phase, in which the gap decreases instead with a power of the size. Close to the transition, the system shows a universal scaling behavior, which we characterize by computing, either analytically or numerically, scaling functions for the low-level energy differences and the two-point correlation function. We discuss the implications of these results for the nonequilibrium dynamics in the presence of a slowly varying parallel magnetic field h, when going across the first-order quantum transition at h=0.
Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time ${\ensuremath{\tau}}_{Q}$ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
We study symmetry breaking quantum phase transitions in topological insulators and superconductors where the single electron gap remains open in the bulk. Specifically, we consider spontaneous breaking of the symmetry that protects the gapless boundary modes, so that in the ordered phase these modes are gapped. Here we determine the fate of the topological boundary modes right at the transition where they are coupled to the strongly fluctuating order parameter field. Using a combination of exact solutions and renormalization group techniques, we find that the surface fermionic modes either decouple from the bulk fluctuations, or flow to a strongly coupled fixed point which remains gapless. In addition, we study transitions where the critical fluctuations are confined only to the surface and find that in several cases the critical point is naturally supersymmetric. This allows a determination of critical exponents and points to an underlying connection between band topology and supersymmetry. Finally, we study the fate of gapless Majorana modes localized on point and line defects in topological superconductors at bulk criticality, which is analogous to a quantum impurity problem. Again, an interplay of topology and strong correlations causes these modes to remain gapless but in a strongly coupled state. Experimental candidates for realizing these phenomena are discussed.
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.