arXiv subjects

Ian Affleck

Publications and source records attributed to Ian Affleck.

At least 37 records · Page 2Linked to original sources

arXiv2020

Comprehensive study of the phase diagram of the spin-1/2 Kitaev-Heisenberg-Gamma chain

A central question on Kitaev materials is the effects of additional couplings on the Kitaev model which is proposed to be a candidate for realizing topological quantum computations. However, two spatial dimension typically suffers the difficulty of lacking controllable approaches. In this work, using a combination of powerful analytical and numerical methods available in one dimension, we perform a comprehensive study on the phase diagram of a one-dimensional version of the spin-1/2 Kitaev-Heisenberg-Gamma model in its full parameter space. A strikingly rich phase diagram is found with nine distinct phases, including four Luttinger liquid phases, a ferromagnetic phase, a Néel ordered phase, an ordered phase of distorted-spiral spin alignments, and two ordered phase which both break a $D_3$ symmetry albeit in different ways, where $D_3$ is the dihedral group of order six. Our work paves the way for studying one-dimensional Kitaev materials and may provide hints to the physics in higher dimensional situations.

cond-mat.str-el↗

arXiv2020

Spin wave theory of one-dimensional generalized Kitaev model

In this work, we perform a combination of classical and spin wave analysis on the one-dimensional spin-$S$ Kitaev-Heisenberg-Gamma model in the region of an antiferromagnetic Kitaev coupling. Four phases are found, including a Néel ordered phase, a phase with $O_h\rightarrow D_3$ symmetry breaking, and "$D_3$-breaking I, II" phases which both break $D_3$ symmetries albeit in different ways, where $O_h$ is the full octahedral group and $D_3$ is the dihedral group of order six. The lowest-lying spin wave mass is calculated perturbatively in the vicinity of the hidden SU(2) symmetric ferromagnetic point.

cond-mat.str-el↗

arXiv2020

Flag manifold sigma models from SU($n$) chains

One dimensional SU($n$) chains with the same irreducible representation $\mathcal{R}$ at each site are considered. We determine which $\mathcal{R}$ admit low-energy mappings to a $\text{SU}(n)/[\text{U}(1)]^{n-1}$ flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each $\mathcal{R}$, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreducible representations that also possess a $\mathbb{Z}_n$ symmetry that acts transitively on the $\text{SU}(n)/[\text{U}(1)]^{n-1}$ fields. Such SU($n$) chains have an 't Hooft anomaly in certain cases, allowing for a generalization of Haldane's conjecture to these novel representations. In particular, for even $n$ and for representations whose Young tableaux have two rows, of lengths $p_1$ and $p_2$ satisfying $p_1\not=p_2$, we predict a gapless ground state when $p_1+p_2$ is coprime with $n$. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if $p_1+p_2$ is not a multiple of $n$.

cond-mat.str-el↗

arXiv2020

Mass generation by fractional instantons in SU($n$) chains

Recently, Haldane's conjecture about SU(2) chains has been generalized to SU($n$) chains in the symmetric representations. For a rank-$p$ representation, a gapless phase is predicted when $p$ and $n$ are coprime; otherwise, a finite energy gap is present above the ground state. In this work, we offer an intuitive explanation of this behavior based on fractional topological excitations, which are able to generate a mass gap except when $p$ and $n$ have no common divisor. This is a generalization of an older work in SU(2), which explains the generation of the Haldane gap in terms of merons in the O(3) nonlinear sigma model.

cond-mat.str-el↗

arXiv2020

Phase diagram of the spin-1/2 Kitaev-Gamma chain and emergent SU(2) symmetry

We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra ``$Γ$-term", using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)$_1$ Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)$_1$ predictions. A modified nonabelian bosonization formula is proposed to capture such exotic emergent ``partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an $O_h\rightarrow D_4$ spontaneous symmetry breaking.

cond-mat.str-el↗

arXiv2020

Floating, critical and dimerized phases in a frustrated spin-3/2 chain

We study spontaneous dimerization and emergent criticality in a spin-3/2 chain with antiferromagnetic nearest-neighbor $J_1$, next-nearest-neighbor $J_2$ and three-site $J_3$ interactions. In the absence of three-site interaction $J_3$, we provide evidence that the model undergoes a remarkable sequence of three phase transitions as a function of $J_2/J_1$, going successively through a critical commesurate phase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommensurate order, to end up in a fully dimerized phase at very large $J_2/J_1$. In the field theory language, this implies that the coupling constant of the marginal operator responsible for dimerization changes sign three times. For large enough $J_3$, the fully dimerized phase is stabilized for all $J_2$, and the phase transitions between the critical phases and this phase are both Wess-Zumino-Witten (WZW) SU(2)$_3$ along part of the boundary and turn first order at some point due to the presence of a marginal operator in the WZW SU(2)$_3$ model. By contrast, the transition between the two dimerized phase is always first order, and the phase transitions between the partially dimerized phase and the critical phases are Kosterlitz-Thouless. Finally, we discuss the intriguing spin-1/2 edge states that emerge in the partially dimerized phase for even chains. Unlike their counterparts in the spin-1 chain, they are not confined and disappear upon increasing $J_2$ in favour of a reorganization of the dimerization pattern.

cond-mat.str-el↗

arXiv2019

Anyons in Multichannel Kondo Systems

Fractionalized quasiparticles - anyons - bear a special role in present-day physics. At the same time, they display properties of interest both foundational, with quantum numbers that transcend the spin-statistics laws, and applied, providing a cornerstone for decoherence-free quantum computation. The development of platforms for realization and manipulation of these objects, however, remains a challenge. Typically these entail the zero-temperature ground-state of incompressible, gapped fluids. Here, we establish a strikingly different approach: the development and probing of anyon physics in a gapless fluid. The platform of choice is a chiral, multichannel, multi-impurity realization of the Kondo effect. We discuss how, in the proper limit, anyons appear at magnetic impurities, protected by an asymptotic decoupling from the fluid and by the emerging Kondo length scale. We discuss possible experimental realization schemes using integer quantum Hall edges. The gapless and charged degrees of freedom coexistent with the anyons suggest the possibility of extracting quantum information data by transport and simple correlation functions. To show that this is the case, we generalize the fusion ansatz of Cardy's boundary conformal field theory, now in the presence of multiple localized perturbations. The generalized fusion ansatz captures the idea that multiple impurities share quantum information non-locally, in a way formally identical to anyonic zero modes. We display several examples supporting and illustrating this generalization and the extraction of quantum information data out of two-point correlation functions. With the recent advances in mesoscopic realizations of multichannel Kondo devices, our results imply that exotic anyon physics may be closer to reach than presently imagined.

cond-mat.str-el↗

arXiv2019

Generalization of the Haldane conjecture to SU($n$) chains

Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-$p$ symmetric ones with $p$ not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU($n$) chains in the symmetric representation. For a rank-$p$ symmetric representation with $n$ and $p$ coprime, we predict gapless excitations above the ground state. If $p$ is a multiple of $n$, we predict a unique ground state with a finite energy gap. Finally, if $p$ and $n$ have a greatest common divisor $1<q<n$, we predict a ground state degeneracy of $n/q$, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU($n$) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Shultz-Mattis-Affleck theorem, and extend it to SU($n$) models with longer range interactions.

cond-mat.str-el↗

arXiv2019

Generalization of the Haldane conjecture to SU(3) chains

We apply field theory methods to $\mbox{SU}(3)$ chains in the symmetric representation, with $p$ boxes in the Young tableau, mapping them into a flag manifold non-linear $σ$-model with a topological angle $θ=2πp/3$. Generalizing the Haldane conjecture, we argue that the models are gapped for $p=3m$ but gapless for $p=3m\pm 1$ (for integer $m$), corresponding to a massless phase of the $σ$-model at $θ=\pm 2π/3$. We confirm this with Monte Carlo calculations on the $σ$-model.

cond-mat.str-el↗

arXiv2019

Kondo effect due to a hydrogen impurity in graphene: a multichannel Kondo problem with diverging hybridization

We consider the Kondo effect arising from a hydrogen impurity in graphene. As a first approximation, the strong covalent bond to a carbon atom removes that carbon atom without breaking the $C_{3}$ rotation symmetry, and we only retain the Hubbard interaction on the three nearest neighbors of the removed carbon atom which then behave as magnetic impurities. These three impurity spins are coupled to three conduction channels with definite helicity, two of which support a diverging local density of states (LDOS) $\propto 1/ [ | ω\ | \ln ^{2}( Λ/| ω\ | \ ) \ ] $ near the Dirac point $ω\rightarrow 0$ even though the bulk density of states vanishes linearly. We study the resulting 3-impurity multi-channel Kondo model using the numerical renormalization group method. For weak potential scattering, the ground state of the Kondo model is a particle-hole symmetric spin-$1/2$ doublet, with ferromagnetic coupling between the three impurity spins; for moderate potential scattering, the ground state becomes a particle-hole asymmetric spin singlet, with antiferromagnetic coupling between the three impurity spins. This behavior is inherited by the Anderson model containing the hydrogen impurity and all four carbon atoms in its vicinity.

cond-mat.str-el↗

arXiv2019

Self-conjugate representation SU(3) chains

It was recently argued that SU(3) chains in the $p$-box symmetric irreducible representation (irrep) exhibit a "Haldane gap" when $p$ is a multiple of 3 and are otherwise gapless [Nucl. Phys. B $\boldsymbol{924}$, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with $p$ columns of length 2 and $p$ columns of length 1 in the Young tableau ($p=1$ corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for $p$ odd but not even.

cond-mat.str-el↗

arXiv2019

Phase diagrams of Majorana-Hubbard ladders

Models of interacting Majorana modes may be realized in vortex lattices in superconducting films in contact with topological insulators and may be tuned to the strong-interaction regime by adjusting the chemical potential. Extending the results on one- and two-dimensional Majorana-Hubbard models, here we determine the phase diagrams of two- and four-leg ladders using both field theory and the density-matrix renormalization group (DMRG) methods.

cond-mat.str-el↗

arXiv2019

Novel families of ${\rm SU}(N)$ AKLT states with arbitrary self-conjugate edge states

Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color ${\rm SU}(N)$ symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of ${\rm SU}(N)$. In particular, ${\rm SU}(2)$ spin-$1$ AKLT states with edge states of arbitrary spin $s=1/2,1,3/2,\cdots$ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are unique ground states. This enables us to study the stability of the edge states by interpolating between exact AKLT Hamiltonians. As an example, in the case of spin-$1$ physical degrees of freedom, it is shown that a quantum phase transition of central charge $c = 1$ separates the Symmetry Protected Topological (SPT) phase with spin-$1/2$ edge states from a topologically trivial phase with spin-$1$ edge states. We also address some specificities of the generalization to ${\rm SU}(N)$ with $N>2$, in particular regarding the construction of parent Hamiltonians. For the AKLT state of the ${\rm SU}(3)$ model with the $3$-box symmetric representation, we prove that the edge states are in the $8$-dimensional adjoint irrep, and for the ${\rm SU}(3)$ model with adjoint irrep at each site, we are able to construct two different reflection-symmetric AKLT Hamiltonians, each with a unique ground state which is either even or odd under reflection symmetry and with edge states in the adjoint irrep. Finally, examples of two-column and adjoint physical irreps for ${\rm SU}(N)$ with $N$ even and with edge states living in the antisymmetric irrep with $N/2$ boxes are given, with a conjecture about the general formula for their correlation lengths.

cond-mat.str-el↗

arXiv2019

Asymptotic Freedom and Large Spin Antiferromagnetic Chains

Building on the mapping of large-$S$ spin chains onto the O($3$) nonlinear $σ$ model with coupling constant $2/S$, and on general properties of that model (asymptotic freedom, implying that perturbation theory is valid at high energy, and Elitzur's conjecture that rotationally invariant quantities are infrared finite in perturbation theory), we use the Holstein-Primakoff representation to derive analytic expressions for the equal-time and dynamical spin-spin correlations valid at distances smaller than $S^{-1} \exp(πS)$ or at energies larger than $J S^2 \exp(-πS)$, where $J$ is the Heisenberg exchange coupling. This is supported by comparing the static correlations with quantum Monte Carlo simulations for $S = 5/2$.

cond-mat.str-el↗

arXiv2019

Real fermion modes, impurity entropy, and nontrivial fixed points in the phase diagram of junctions of interacting quantum wires and topological superconductors

We discuss how to extend the impurity entropy to systems with boundary interactions depending on zero-mode real fermion operators (Majorana modes as well as Klein factors). As specific applications of our method, we consider a junction between N interacting quantum wires and a topological superconductor, as well as a Y-junction of three spinless interacting quantum wires. In addition we find a remarkable correspondence between the N=2 topological superconductor junction and the Y-junction. On one hand, this allows us to determine the range of the system parameters in which a stable phase of the N=2 junction is realized as a nontrivial, finite-coupling fixed point corresponding to the M-fixed point in the phase diagram of the Y-junction. On the other hand, it enables us to show the occurrence of a novel "planar" finite-coupling fixed point in the phase diagram of the Y-junction. Eventually, we discuss how to set the system's parameters to realize the correspondence.

cond-mat.str-el↗

arXiv2018

Topological Superconductor-Luttinger Liquid Junctions

Experimental evidence was recently obtained for topological superconductivity in spin-orbit coupled nano wires in a magnetic field, proximate to an s-wave superconductor. When only part of the wire contacts the superconductor, a localized Majorana mode exists at the junction between superconducting and normal parts of the nanowire. We consider here the case of a T-junction between the superconductor and two normal nanowires and also the case of a single wire with two (or more) partially filled bands in the normal part. We find that coupling this 2-channel Luttinger liquid to the single Majorana mode at the junction produces frustration, leading to a critical point separating phases with perfect Andreev scattering in one channel and perfect normal scattering in the other.

cond-mat.str-el↗

arXiv2018

Renormalization group analysis of phase transitions in the two dimensional Majorana-Hubbard model

A lattice of interacting Majorana modes can occur in a superconducting film on a topological insulator in a magnetic field. The phase diagram as a function of interaction strength for the square lattice was analyzed recently using a combination of mean field theory and renormalization group methods, and was found to include second order phase transitions. One of these corresponds to spontaneous breaking of an emergent U(1) symmetry, for attractive interactions. Despite the fact that the U(1) symmetry is not exact, this transition was claimed to be in a supersymmetric universality class when time reversal symmetry is present and in the conventional XY universality class otherwise. Another second order transition was predicted for repulsive interactions with time reversal symmetry to be in the same universality class as the transition occurring in the Gross-Neveu model, despite the fact that the U(1) symmetry is not exact in the Majorana model. We analyze these phase transitions using a modified $ε$-expansion, confirming the previous conclusions.

cond-mat.str-el↗

arXiv2018

Topological qubits from valence bond solids

Topological qubits based on $SU(N)$-symmetric valence-bond solid models are constructed. A logical topological qubit is the ground subspace with two-fold degeneracy, which is due to the spontaneous breaking of a global parity symmetry. A logical $Z$-rotation by angle $\frac{2π}{N}$, for any integer $N > 2$, is provided by a global twist operation, which is of topological nature and protected by the energy gap. A general concatenation scheme with standard quantum error-correction codes is also proposed, which can lead to better codes. Generic error-correction properties of symmetry-protected topological order are also demonstrated.

quant-ph↗