Parafermion Research Articles

Time-reversal anomalies of QCD3 and QED3

Anomalies of global symmetry provide powerful tool to constrain the dynamics of quantum systems, such as anomaly matching in the renormalization group flow and obstruction to symmetric mass generation. In this note we compute the anomalies in 2+1d time-reversal symmetric gauge theories with massless fermions in the fundamental and rank-two tensor representations, where the gauge groups are SU(N), SO(N), Sp(N), U(1). The fermion parity is part of the gauge group and the theories are bosonic. The time-reversal symmetry satisfies T2 = 1 or T2 = M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document} where M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{M} $$\\end{document} is an internal magnetic symmetry. We show that some of the bosonic gauge theories have time-reversal anomaly with c− ≠ 0 mod 8 that is absent in fermionic systems. The anomalies of the gauge theories can be nontrivial even when the number of Majorana fermions is a multiple of 16 and ν = 0.

Spin-Degeneracy Breaking and Parity Transitions in Three-Terminal Josephson Junctions

Hybrid Josephson junctions (JJs) realized in superconductor-semiconductor heterostructures host fermionic modes known as Andreev bound states (ABSs). In these structures, a promising and yet unexplored avenue for harnessing spin and parity degrees of freedom is offered by JJs with three or more superconducting terminals, where phase-induced spin polarization and transitions of the ground state to an odd parity were predicted to arise. Here we spectroscopically probe the two-dimensional band structure of ABSs in a phase-controlled InAs/Al three-terminal JJ. Andreev bands show signatures of spin-degeneracy breaking, with level splitting in excess of ∼9 GHz, and zero-energy crossings associated to ground state fermion parity transitions. Spin splitting and parity transitions are enabled and controlled by locally applied magnetic fluxes, in the absence of Zeeman effect or Coulomb blockade. Our results underscore the potential of multiterminal hybrid devices for phase engineering ABSs, with significant implications for spin- and parity-based quantum devices. Published by the American Physical Society 2024

Symmetry-Enforced Many-Body Separability Transitions

We study quantum many-body mixed states with a symmetry from the perspective of , i.e., whether a mixed state can be expressed as an ensemble of short-range-entangled symmetric pure states. We provide evidence for “symmetry-enforced separability transitions” in a variety of states, where in one regime the mixed state is expressible as a convex sum of symmetric short-range-entangled pure states, while in the other regime, such a representation is not feasible. We first discuss the Gibbs state of Hamiltonians that exhibit spontaneous breaking of a discrete symmetry, and argue that the associated thermal phase transition can be thought of as a symmetry-enforced separability transition. Next we study cluster states in various dimensions subjected to local decoherence, and identify several distinct mixed-state phases and associated separability phase transitions, which also provides an alternative perspective on recently discussed “average symmetry-protected topological order.” We also study decohered p+ip superconductors, and find that if the decoherence breaks the fermion parity explicitly, then the resulting mixed state can be expressed as a convex sum of nonchiral states, while a fermion parity–preserving decoherence results in a phase transition at a nonzero threshold that corresponds to spontaneous breaking of fermion parity. Finally, we briefly discuss systems that satisfy the no low-energy trivial state property, such as the recently discovered good low-density parity-check codes, and argue that the Gibbs state of such systems exhibits a temperature-tuned separability transition. Published by the American Physical Society 2024

Topological Kondo effect with spinful Majorana fermions

Motivated by the importance of studying topological superconductors beyond the mean-field approximation, we here investigate mesoscopic islands of time-reversal-invariant topological superconductors. We characterize the spectrum in the presence of strong order-parameter fluctuations in the presence of an arbitrary number of Kramers pairs of Majorana edge states and study the effect of coupling the Coulomb blockaded island to external leads. In the case of an odd fermionic parity on the island, we derive an unconventional Kondo Hamiltonian in which metallic leads couple to both topological Majorana degrees of freedom (which keep track of the parity in different leads) and the overall spin 12 in the island. For the simplest case of a single wire (two pairs of Majorana edge states), we demonstrate that anisotropies are irrelevant in the weak coupling renormalization group flow. This permits us to solve the Kondo problem in the vicinity of a Toulouse-type point using Abelian bosonization. We demonstrate a residual ground-state entropy of ln(2), which is protected by spin-rotation symmetry, but reduced to ln(2) (as in the spinless topological Kondo effect) by symmetry-breaking perturbations. In the symmetric case, we further demonstrate the simultaneous presence of both Fermi-liquid and non-Fermi-liquid-like thermodynamics (depending on the observable) and derive charge and spin transport signatures of the Coulomb blockaded island. Published by the American Physical Society 2024

Fermion-parity qubit in a proximitized double quantum dot

Bound states in quantum dots coupled to superconductors can be in a coherent superposition of states with different electron number but with the same fermion parity. Electrostatic gating can tune this superposition to a sweet spot, where the quantum dot has the same mean electric charge independent of its electron-number parity. Here, we propose to encode quantum information in the local fermion parity of two tunnel-coupled quantum dots embedded in a Josephson junction. At the sweet spot, the qubit states have zero charge dipole moment. This protects the qubit from dephasing due to charge noise acting on the potential of each dot, as well as fluctuations of the (weak) interdot tunneling. At weak interdot tunneling, relaxation is suppressed because of disjoint qubit states. However, for strong interdot tunneling the system is protected against noise affecting each quantum dot separately (energy-level noise, dot-superconductor tunneling fluctuations, and hyperfine interactions). Finally, we describe initialization and readout as well as single-qubit and two-qubit gates by pulsing gate voltages. Published by the American Physical Society 2024

Entanglement negativity between separated regions in quantum critical systems

We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show nonperturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the one-dimensional result, and indicates that quantum critical ground states do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix. Published by the American Physical Society 2024

Charge Sensing the Parity of an Andreev Molecule

The proximity effect of superconductivity on confined states in semiconductors gives rise to various bound states such as Andreev bound states, Andreev molecules, and Majorana zero modes. While such bound states do not conserve charge, their fermion parity is a good quantum number. One way to measure parity is to convert it to charge first, which is then sensed. In this work, we sense the charge of Andreev bound states and Andreev molecules in an InSb-Al hybrid nanowire using an integrated quantum dot operated as a charge sensor. We show how charge sensing measurements can resolve the even and odd states of an Andreev molecule, without affecting the parity. Such an approach can be further used for parity measurements of Majorana zero modes in Kitaev chains based on quantum dots. Published by the American Physical Society 2024

Majorana chain and Ising model - (non-invertible) translations, anomalies, and emanant symmetries

We study the symmetries of closed Majorana chains in 1+1d, including the translation, fermion parity, spatial parity, and time-reversal symmetries. The algebra of the symmetry operators is realized projectively on the Hilbert space, signaling anomalies on the lattice, and constraining the long-distance behavior. In the special case of the free Hamiltonian (and small deformations thereof), the continuum limit is the 1+1d free Majorana CFT. Its continuum chiral fermion parity (-1)^{F_L}(−1)FL emanates from the lattice translation symmetry. We find a lattice precursor of its mod 8 ’t Hooft anomaly. Using a Jordan-Wigner transformation, we sum over the spin structures of the lattice model (a procedure known as the GSO projection), while carefully tracking the global symmetries. In the resulting bosonic model of Ising spins, the Majorana translation operator leads to a non-invertible lattice translation symmetry at the critical point. The non-invertible Kramers-Wannier duality operator of the continuum Ising CFT emanates from this non-invertible lattice translation of the transverse-field Ising model.

On the conclusive detection of Majorana zero modes: conductance spectroscopy, disconnected entanglement entropy and the fermion parity noise

Semiconducting nanowires with strong Rashba spin–orbit coupling in the proximity with a superconductor and under a strong Zeeman field can potentially manifest Majorana zero modes (MZMs) at their edges and are a topical candidate for topological superconductivity. However, protocols for their detection based on the local and the non-local conductance spectroscopy have been subject to intense scrutiny. In this work, by taking current experimental setups into account, we detail mathematical ideas related to the entanglement entropy and the fermion parity fluctuations to faithfully distinguish between true MZMs and trivial quasi-MZMs. We demonstrate that the disconnected entanglement entropy, derived from the von Neumann entanglement entropy, provides a distinct and robust signature of the topological phase transition which is immune to system parameters, size and disorders. In order to understand the entanglement entropy of the Rashba nanowire system, we establish its connection to a model of interacting spinfull Kitaev chains. Moreover, we relate the entanglement entropy to the fermionic parity fluctuation, and show that it behaves concordantly with entanglement entropy, hence making it a suitable metric for the detection of MZMs. In connection with the topological gap protocol that is based on the conductance spectra, the aforesaid metrics can reliably point toward the topological transitions even in realistic setups.

Quantum quenches in driven-dissipative quadratic fermionic systems with parity-time symmetry

We study the quench dynamics of noninteracting fermionic quantum many-body systems that are subjected to Markovian drive and dissipation and are described by a quadratic Liouvillian which has parity-time (PT) symmetry. In recent work, we have shown that such systems relax locally to a maximum entropy ensemble that we have dubbed the PT-symmetric generalized Gibbs ensemble (PTGGE), in analogy to the generalized Gibbs ensemble that describes the steady state of isolated integrable quantum many-body systems after a quench. Here, using driven-dissipative versions of the Su-Schrieffer-Heeger (SSH) model and the Kitaev chain as paradigmatic model systems, we corroborate and substantially expand upon our previous results. In particular, we confirm the validity of a dissipative quasiparticle picture at finite dissipation by demonstrating light cone spreading of correlations and the linear growth and saturation to the PTGGE prediction of the quasiparticle-pair contribution to the subsystem entropy in the PT-symmetric phase. Further, we introduce the concept of directional pumping phases, which is related to the non-Hermitian topology of the Liouvillian and based upon qualitatively different dynamics of the dual string order parameter and the subsystem fermion parity in the SSH model and the Kitaev chain, respectively: depending on the postquench parameters, there can be pumping of string order and fermion parity through both ends of a subsystem corresponding to a finite segment of the one-dimensional lattice, through only one end, or there can be no pumping at all. We show that transitions between dynamical pumping phases give rise to a new and independent type of dynamical critical behavior of the rates of directional pumping, which are determined by the soft modes of the PTGGE. Published by the American Physical Society 2024

Dynamics of parafermionic states in transport measurements

Advances in hybrid fractional quantum Hall (FQH)-superconductor platforms pave the way for realisation of parafermionic modes. We analyse signatures of these non-abelian anyons in transport measurements across devices with \mathbb{Z}_6ℤ6 parafermions (PFs) coupled to an external electrode. Simulating the dynamics of these open systems by a stochastic quantum jump method, we show that a current readout over sufficiently long times constitutes a projective measurement of the fractional charge shared by two PFs. Interaction of these topological modes with the FQH environment, however, may cause poisoning events affecting this degree of freedom which we model by jump operators that describe incoherent coupling of PFs with FQH edge modes. We analyse how this gives rise to a characteristic three-level telegraph noise in the current, constituting a very strong signature of PFs. We discuss also other forms of poisoning and noise caused by interaction with fractional quasiparticles in the bulk of the Hall system. We conclude our work with an analysis of four-PF devices, in particular on how the PF fusion algebra can be observed in electrical transport experiments.

Phase-engineering the Andreev band structure of a three-terminal Josephson junction

In hybrid Josephson junctions with three or more superconducting terminals coupled to a semiconducting region, Andreev bound states may form unconventional energy band structures, or Andreev matter, which are engineered by controlling superconducting phase differences. Here we report tunnelling spectroscopy measurements of three-terminal Josephson junctions realised in an InAs/Al heterostructure. The three terminals are connected to form two loops, enabling independent control over two phase differences and access to a synthetic Andreev band structure in the two-dimensional phase space. Our results demonstrate a phase-controlled Andreev molecule, originating from two discrete Andreev levels that spatially overlap and hybridise. Signatures of hybridisation are observed in the form of avoided crossings in the spectrum and band structure anisotropies in the phase space, all explained by a numerical model. Future extensions of this work could focus on addressing spin-resolved energy levels, ground state fermion parity transitions and Weyl bands in multiterminal geometries.

Nonequilibrium Fractional Josephson Effect.

Josephson tunnel junctions exhibit a supercurrent typically proportional to the sine of the superconducting phase difference ϕ. In general, a term proportional to cos(ϕ) is also present, alongside microscopic electronic retardation effects. We show that voltage pulses sharply varying in time prompt a significant impact of the cos(ϕ) term. Its interplay with the sin(ϕ) term results in a nonequilibrium fractional Josephson effect (NFJE) ∼sin(ϕ/2) in the presence of bound states close to zero frequency. Our microscopic analysis reveals that the interference of nonequilibrium virtual quasiparticle excitations is responsible for this phenomenon. We also analyze this phenomenon for topological Josephson junctions with Majorana bound states. Remarkably, the NFJE is independent of the ground state fermion parity unlike its equilibrium counterpart.

Dynamical Polarization of the Fermion Parity in a Nanowire Josephson Junction.

Josephson junctions in InAs nanowires proximitized with an Al shell can host gate-tunable Andreev bound states. Depending on the bound state occupation, the fermion parity of the junction can be even or odd. Coherent control of Andreev bound states has recently been achieved within each parity sector, but it is impeded by incoherent parity switches due to excess quasiparticles in the superconducting environment. Here, we show that we can polarize the fermion parity dynamically using microwave pulses by embedding the junction in a superconducting LC resonator. We demonstrate polarization up to 94%±1% (89%±1%) for the even (odd) parity as verified by single shot parity readout. Finally, we apply this scheme to probe the flux-dependent transition spectrum of the even or odd parity sector selectively, without any postprocessing or heralding.

Fermionic defects of topological phases and logical gates

We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called “gauged Gu-Wen SPT defects” that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D \mathbb{Z}_2ℤ2 toric code stacked with a (2+1)D ancilla trivial atomic insulator. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the \mathbb{Z}_8ℤ8 classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.

Enforced symmetry breaking by invertible topological order

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time-reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking (ESB) by topological orders. In this work, we systematically study ESB of a finite symmetry group ${G}_{f}$ by fermionic invertible topological orders. Mathematically, the group ${G}_{f}$ is a central extension over a bosonic symmetry group $G$ by the fermion parity group ${\mathbb{Z}}_{2}^{f}$, characterized by a 2-cocycle $\ensuremath{\lambda}\ensuremath{\in}{\mathcal{H}}^{2}(G,{\mathbb{Z}}_{2})$. For given $G$ and $\ensuremath{\lambda}$, we are able to obtain a series of criteria on the existence or nonexistence of ESB by the corresponding fermionic invertible topological orders. For 2D systems, we define a set of physical quantities to describe symmetry-enriched invertible topological orders and derive obstruction functions using both fermionic and bosonic languages. The study in the bosonic language is performed after gauging the fermion parity, and we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. The obstruction functions are crucial components of the ESB criteria that we derive. With these criteria, we discover many new ESB examples, e.g., we find that the quaternion group ${Q}_{8}$ is incompatible with two copies of $p+ip$ superconductors. We also obtain explicit results on the ESB phenomena of the continuous group ${\mathrm{SU}}_{f}(N)$ by 2D invertible topological orders through a different argument.

Enhanced 2π-periodic Aharonov–Bohm Effect as a Signature of Majorana Zero Modes Probed by Nonlocal Measurements

We propose the $2\pi$-periodic Aharonov-Bohm (AB) effect as a nonlocal probe of Majorana zero modes (MZMs) without the restriction of fermion parity. We demonstrate the enhancement of the AB effect, where the topological protection of MZMs yields amplified and robust Andreev reflection mediated by MZMs at multiple superconductor-normal metal junctions. We investigate the influence of trivial bound states and show that a nonlocal index enables a more explicit distinction between the trivial and topological bound states than local probes.

Full counting statistics in a Majorana single-charge transistor

We study full counting statistics of electron transport through a Majorana single-charge transistor. At low bias voltage, transport is dominated by the so-called Josephson-Majorana cycle, a sequence of normal and anomalous single-charge and Josephson tunneling. Factorial cumulants characterizing the full counting statistics elucidate the correlated nature of the charge transfers in this cycle. Moreover, we predict a topological transition in the full counting statistics from a perfect Poissonian transfer of Cooper pairs to a correlated switching between two distinct fermion parity states with increasing Josephson coupling.

Nonperturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1) dimensions

In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group $G_f$ of the ground state splits as $G_f = G_b \times \mathbb{Z}_2^f$, where $\mathbb{Z}_2^f$ is generated by fermion parity, $(-1)^F$. In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when $G_f$ is not of the form $G_b \times \mathbb{Z}_2^f$. In this paper we study how $G_f$ together with the chiral central charge $c_-$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}(1)^f\rtimes H,\mathrm{SU}(2)^f \times H, \mathrm{U}(2)^f\rtimes H $, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr$_2$RuO$_4$, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.

Cooper Pair Excitation Mediated by a Molecular Quantum Spin on a Superconducting Proximitized Gold Film.

Breaking a correlated pair in a superconductor requires an even number of fermions providing at least twice the pairing energy Δ. Here, we show that a single tunneling electron can also excite a pair breaking excitation in a proximitized gold film in the presence of magnetic impurities. Combining scanning tunneling spectroscopy with theoretical modeling, we map the excitation spectrum of an Fe-porphyrin molecule on the Au/V(100) proximitized surface into a manifold of entangled Yu-Shiba-Rusinov and spin excitations. Pair excitations emerge in the tunneling spectra as peaks outside the spectral gap only in the strong coupling regime, where the presence of a bound quasiparticle in the ground state ensures the even fermion parity of the excitation. Our results unravel the quantum nature of magnetic impurities on superconductors and demonstrate that pair excitations unequivocally reveal the parity of the ground state.