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arXiv:1310.7434

UV-IR transmutation for hybrid realizations of Z_k parafermion systems

Abolhassan Vaezi, Eun-Ah Kim

Published Oct 28, 2013• 1 citation
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Abstract

Recent experiments brought Majorana particles closer to reality and raised interest in realizing $\mathbb{Z}_k$ parafermions for general value of $k$ in hybrid structures. Of particular interest is the prospect of realizing a two dimensional system with the edge states described by $\mathbb{Z}_k$ parafermion conformal field theory (CFT) by coupling one dimensional hybrid systems. However in order to understand the effective perturbation due to local inter-wire coupling, it is crucial to relate short distance ultra-violet (UV) degrees of freedom to primary fields of the CFT describing long distance infra-red (IR) physics. Here we consider two recent proposals for realizing a one dimensional system described by $\mathbb{Z}_k$ parafermion CFT upon tuning to quantum critical points: a parafermion chain and a pair of $ν=2/k$ quantum Hall chiral edge states under uniform backscattering and pairing, where the latter is represented by self-dual sine-Gordon model. We explicitly derive the relation between distinct UV degrees of freedom relevant for each proposals and the primary fields of $\mathbb{Z}_3$ parafermion CFT. Our results point to a marked difference in the effect of local coupling between wires represented by each proposals, regarding robustness of the edge states in the resulting two dimensional systems.

DOI
10.48550/arxiv.1310.7434

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Recent experiments brought Majorana particles closer to reality and raised interest in realizing Zk\mathbb{Z}_kZk​ parafermions for general value of kkk in hybrid structures. Of particular interest is the prospect of realizing a two dimensional system with the edge states described by Zk\mathbb{Z}_kZk​ parafermion conformal field theory (CFT) by coupling one dimensional hybrid systems. However in order to understand the effective perturbation due to local inter-wire coupling, it is crucial to relate short distance ultra-violet (UV) degrees of freedom to primary fields of the CFT describing long distance infra-red (IR) physics. Here we consider two recent proposals for realizing a one dimensional system described by Zk\mathbb{Z}_kZk​ parafermion CFT upon tuning to quantum critical points: a parafermion chain and a pair of ν=2/kν=2/kν=2/k quantum Hall chiral edge states under uniform backscattering and pairing, where the latter is represented by self-dual sine-Gordon model. We explicitly derive the relation between distinct UV degrees of freedom relevant for each proposals and the primary fields of Z3\mathbb{Z}_3Z3​ parafermion CFT. Our results point to a marked difference in the effect of local coupling between wires represented by each proposals, regarding robustness of the edge states in the resulting two dimensional systems.
On the classification of UV completions of integrable TT‾ T\overline{T} TT deformations of CFT
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The construction of fractional quantum Hall (FQH) states from the two-dimensional array of quantum wires provides a useful way to control strong interactions in microscopic models and has been successfully applied to the Laughlin, Moore-Read, and Read-Rezayi states. We extend this construction to the Abelian and non-Abelian $SU(N\ensuremath{-}1)$-singlet FQH states at filling fraction $\ensuremath{\nu}=k(N\ensuremath{-}1)/[N+k(N\ensuremath{-}1)m]$ labeled by integers kkk and mmm, which are potentially realized in multicomponent quantum Hall systems or SU(N)SU(N)SU(N) spin systems. Utilizing the bosonization approach and conformal field theory (CFT), we show that their bulk quasiparticles and gapless edge excitations are both described by an $(N\ensuremath{-}1)$-component free-boson CFT and the $SU{(N)}_{k}/{[U(1)]}^{N\ensuremath{-}1}$ CFT known as the Gepner parafermion. Their generalization to different filling fractions is also proposed. In addition, we argue possible applications of these results to two kinds of lattice systems: bosons interacting via occupation-dependent correlated hoppings and an SU(N)SU(N)SU(N) Heisenberg model.
A bstract We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski’s equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the su2 {\mathfrak{su}}_2 su2​ su 2 Wess-Zumino-Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz-Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.
We introduce a family of commuting-projector Hamiltonians whose degrees of freedom involve Z3{\mathbb{Z}}_{3}Z3​ parafermion zero modes residing in a parent fractional-quantum-Hall fluid. These commuting-projector models inherit nontrivial Hall conductance from the parent quantum-Hall states in which they are defined, and thus can describe chiral topological phases. The two simplest models in this family emerge from dressing Ising-paramagnet and toric-code spin models with parafermions; we study their edge properties, anyonic excitations, and ground-state degeneracy. We show that the first model realizes a symmetry-enriched topological phase (SET) for which Z2{\mathbb{Z}}_{2}Z2​ spin-flip symmetry from the Ising paramagnet permutes the anyons. Interestingly, the interface between this SET and the parent quantum-Hall phase realizes symmetry-enforced Z3{\mathbb{Z}}_{3}Z3​ parafermion criticality with no fine-tuning required. The second model exhibits a non-Abelian phase that is consistent with SU(2)4\text{SU}{(2)}_{4}SU(2)4​ topological order, and can be accessed by gauging the Z2{\mathbb{Z}}_{2}Z2​ symmetry in the SET. Employing Levin-Wen string-net models with Z2{\mathbb{Z}}_{2}Z2​-graded structure, we generalize this picture to construct a large class of commuting-projector models for Z2{\mathbb{Z}}_{2}Z2​ SETs and non-Abelian topological orders exhibiting the same relation. Our construction provides the first commuting-projector-Hamiltonian realization of chiral bosonic non-Abelian topological order.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.
We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski's equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the su2\mathfrak{su}_2su2​ Wess--Zumino--Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz--Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.