Zero Modes Research Articles

Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry

We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.

Nonlocal pseudospin dynamics in a quantum Ising chain

The existence of topological zero modes in nontrivial phase of quantum Ising chain results in not only the Kramers-like degeneracy spectrum, but also dynamic response for non-Hermitian perturbation in the ordered phase (2021 Phys. Rev. Lett. 126 116 401). In this work, we investigate the possible response of the degeneracy spectrum for Hermitian perturbations. We provide a single-particle description of the model in the ordered phase, associating with an internal degree of freedom characterized as a pseudospin. The effective magnetic field, arising from both local and nonlocal perturbations in terms of string operators, acts on the pseudospin. We show that the action of string operator can be realized via a quench under the local perturbations. As an application, any ground states and excited states for the Hamiltonian with perturbation can be selected to identify the quantum phase, by adding the other perturbations to trigger a quench and measuring the Loschmidt echo.

System size dependent topological zero modes in coupled topolectrical chains

In this paper, we demonstrate the emergence and disappearance of topological zero modes (TZMs) in a coupled topolectrical (TE) circuit lattice. Specifically, we consider non-Hermitian TE chains in which TZMs do not occur in the individual uncoupled chains, but emerge when these chains are coupled by inter-chain capacitors. The coupled system hosts TZMs which show size-dependent behaviours and vanish beyond a certain critical size. In addition, the emergence or disappearance of the TZMs in the open boundary condition (OBC) spectra for a given size of the coupled system can be controlled by modulating the signs of its inverse decay length. Analytically, trivial and non-trivial phases of the coupled system can be distinguished by the differing ranks of their corresponding Laplacian matrix. The TE circuit framework enables the physical detection of the TZMs via electrical impedance measurements. Our work establishes the conditions for inducing TZMs and modulating their behavior in coupled TE chains.

A 4D IIB flux vacuum and supersymmetry breaking. Part I. Fermionic spectrum

We consider the type-IIB supergravity vacua that include an internal T5, depend on a single coordinate r and respect a four-dimensional Poincaré symmetry, with the aim of highlighting low-energy spectra with broken supersymmetry and a bounded string coupling. These vacua are characterized by the flux Φ of the self-dual five form in the internal torus, the length ℓ of the interval described by the coordinate r, a dilaton profile that is inevitably constant and a strictly positive dimensionless parameter h. As ℓ → ∞ while retaining finite values for Φ and mathrm{ell}{h}^{-frac{5}{4}} , half of the original ten-dimensional supersymmetry is recovered, while finite values of ℓ break it completely. In the large-ℓ limit one boundary disappears but the other is still present, and is felt as a BPS orientifold by a probe brane. In this paper we focus on the fermionic zero modes and show that, although supersymmetry is broken for finite values of ℓ, they are surprisingly those of four-dimensional N = 4 supergravity coupled to five vector multiplets. Still, the gravitini can acquire masses via radiative corrections, absorbing four of the massless spin- frac{1}{2} modes.

Massive Ray-Singer torsion and path integrals

Zero modes are an essential part of topological field theories, but they are frequently also an obstacle to the explicit evaluation of the associated path integrals. In order to address this issue in the case of Ray-Singer Torsion, which appears in various topological gauge theories, we introduce a massive variant of the Ray-Singer Torsion which involves determinants of the twisted Laplacian with mass but without zero modes. This has the advantage of allowing one to explicitly keep track of the zero mode dependence of the theory. We establish a number of general properties of this massive Ray-Singer Torsion. For product manifolds M = N × S1 and mapping tori one is able to interpret the mass term as a flat ℝ+ connection and one can represent the massive Ray-Singer Torsion as the path integral of a Schwarz type topological gauge theory. Using path integral techniques, with a judicious choice of an algebraic gauge fixing condition and a change of variables which leaves one with a free action, we can evaluate the torsion in closed form. We discuss a number of applications, including an explicit calculation of the Ray-Singer Torsion on S1 for G = PSL(2, R) and a path integral derivation of a generalisation of a formula of Fried for the torsion of finite order mapping tori.

Scattering of fermionic isodoublets on the sine-Gordon kink

The scattering of Dirac fermions on the sine-Gordon kink is studied both analytically and numerically. To achieve invariance with respect to a discrete symmetry, the sine-Gordon model is treated as a nonlinear sigma -model with a circular target space that interacts with fermionic isodublets through the Yukawa interaction. It is shown that the diagonal and antidiagonal parts of the fermionic wave function interact independently with the external field of the sine-Gordon kink. The wave functions of the fermionic scattering states are expressed in terms of the Heun functions. General expressions for the transmission and reflection coefficients are derived, and their dependences on the fermion momentum and mass are studied numerically. The existence condition is found for two fermionic zero modes, and their analytical expressions are obtained. It is shown that the zero modes do not lead to fragmentation of the fermionic charge, but can lead to polarization of the fermionic vacuum. The scattering of the diagonal and antidiagonal fermionic states is found to be significantly different; this difference is shown to be due to the different dependences of the energy levels of these bound states on the fermion mass, and is in accordance with Levinson’s theorem.

Topological order in random interacting Ising-Majorana chains stabilized by many-body localization

We numerically explore $\mathbb Z_2$-symmetric random interacting Ising-Majorana chains at high energy. A very rich phase diagram emerges with two topologically distinct many-body localization (MBL) regimes separated by a much broader thermal phase than previously found. This is a striking consequence of the avalanche theory. We further find MBL spin-glass order always associated to a many-body spectral pairing, presumably signaling a strong zero mode operator which opens fascinating perspectives for MBL-protected topological qubits.

Two-dimensional microtwist modeling of topological polarization in hinged Kagome lattices and its experimental validation

Kagome lattices have recently attracted a great attention because of the unique mechanical properties including their topological polarization and localized zero modes at certain edges, which challenge the standard effective continuum theories. The previous study of these systems has been predominantly focused on the ideal Kagome lattice with the spring–mass models. In this study, we stretch this paradigm by exploring the hinged Kagome lattices towards practical application to understand the topological polarization under the framework of the microtwist continuum. The hinges are modeled by ligaments capable of supporting stretching, shear and bending deformations. The microtwist elasticity is then formulated thanks to leading order two-scale asymptotics and its constitutive and balance equations are derived. Performance of the proposed theory is validated by the exact solution for predicting dispersion relations and periodic zero modes. We further demonstrate the effectiveness of this theory through numerical simulations as well as experimental testing. Finally, nonuniform deformation under complex loadings and parity asymmetric surface waves in microtwist media are explored. Our study provides a great potential of using the microtwist medium to design, control and program hinge-based metamaterials.

Linear zero mode spectra for quasicrystals

A converse is given to the well-known fact that a hyperplane localised zero mode of a crystallographic bar-joint framework gives rise to a line or lines in the zero mode (RUM) spectrum. These connections motivate definitions of linear zero mode spectra for an aperiodic bar-joint framework G that are based on relatively dense sets of linearly localised flexes. For a Delone framework in the plane the limit spectrumLlim(G,a_) is defined in this way, as a subset of the reciprocal space for a reference basis a_ of the ambient space. A smaller spectrum, the slippage spectrumLslip(G,a_), is also defined. For quasicrystal parallelogram frameworks associated with regular multi-grids, in the sense of de Bruijn and Beenker, these spectra coincide and are determined in terms of the geometry of G.

Probing Majorana modes via local spin dynamics

We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here, the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for p-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap whilst retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the inter-dimer coupling to generate a finite size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.

D-instanton induced superpotential

We use string field theory to fix the normalization of the D-instanton corrections to the superpotential involving the moduli fields of type II string theory compactified on an orientifold of a Calabi-Yau threefold in the absence of fluxes. We focus on O(1) instantons whose only zero modes are the four bosonic modes associated with translation of the instanton in non-compact directions and a pair of fermionic zero modes associated with the two supercharges broken by the instanton. We work with a generic superconformal field theory and express our answer in terms of the spectrum of open strings on the instanton. We analyse the contribution of multi-instantons of this kind to the superpotential and argue that it vanishes when background fluxes are absent.

Dulmage-Mendelsohn Percolation: Geometry of Maximally Packed Dimer Models and Topologically Protected Zero Modes on Site-Diluted Bipartite Lattices

A new kind of quantum percolation phenomenon---of particles hopping on a lattice with missing sites---has implications for the effect of nonmagnetic impurities in quantum antiferromagnets and could aid the search for Majorana excitations in solid-state devices.

Linear mode analysis and spin relaxation

In this paper, a detailed analysis on normal modes of the linearized Hermite collision operator is presented, which follows from linearizing spin Boltzmann equation for massive fermions proposed in [Phys. Rev. D 104, 016022 (2021)] with the nondiagonal part of the transition rate neglected and approximating what we got with a mutilated operator. With the assumption of total angular momentum conservation, the collision term is proven to well describe the equilibrium state and gives proper interpretation for collisional invariants, thus is relevant for the research on local spin polarization. Following the familiar fashion as used in quantum mechanics, we treat the problem of solving normal modes as a degenerate perturbation problem and calculate the dispersion relations for intriguing eleven zero modes, which form one-to-one correspondence to all collisional invariants. We find that the results of spinless modes appearing in ordinary hydrodynamics are consistent with available conclusions in textbooks. As for spin-related modes, we obtain the frequencies up to second order in a wave vector and relate them with the dissipation of spin density fluctuation. In addition, the ratio of two relaxation timescales for spin and momentum is shown as a function of reduced mass, which reads that based on present framework spin equilibration is almost as slow as momentum equilibration as far as the strange quark spins in quark gluon plasma (QGP) are concerned.

Edge states in a non-Hermitian chiral lattice

Non-Hermiticity can alter the topological properties of energy bands and drive topologically trivial systems with Hermitian limits to generate nontrivial topological edge states. Nevertheless, most efforts were devoted to the study of isotropic systems containing multiple pairs of non-Hermitian components, while the influence of distribution of non-Hermitian components was seldom explored. Here, we first investigated a two-dimensional square-lattice non-Hermitian model in which the losses increase in either clockwise or counterclockwise direction, leading to distinct non-Hermitian chiral structures. The effective Hamiltonian satisfied non-Hermitian particle-hole symmetry, which can induce non-Hermitian zero modes. By combing two chiral structures, various edge states emerged from the interface, whose bands strongly depend on the distribution of non-Hermitian components. And the band properties of edge states were analyzed and predicted basing on the non-Hermitian particle-hole symmetry and hypothetical parity-time symmetry. Furthermore, to validate our theory, we proposed a piezoelectric phononic crystal with an external circuit. The band structures were coincident with our theoretical results. And robust wave propagation was also identified. Our study may provide an alternative method to actively engineer propagating waves in non-Hermitian systems.

Scars from protected zero modes and beyond in $U(1)$ quantum link and quantum dimer models

We demonstrate the presence of anomalous high-energy eigenstates, or many-body scars, in U(1)U(1) quantum link and quantum dimer models on square and rectangular lattices. In particular, we consider the paradigmatic Rokhsar-Kivelson Hamiltonian H=\mathcal{O}_{\mathrm{kin}} + \lambda \mathcal{O}_{\mathrm{pot}}H=𝒪kin+λ𝒪pot where \mathcal{O}_{\mathrm{pot}}𝒪pot (\mathcal{O}_{\mathrm{kin}}𝒪kin) is defined as a sum of terms on elementary plaquettes that are diagonal (off-diagonal) in the computational basis. Both these interacting models possess an exponentially large number of mid-spectrum zero modes in system size at \lambda=0λ=0 that are protected by an index theorem preventing any mixing with the nonzero modes at this coupling. We classify different types of scars for |\lambda| \lesssim \mathcal{O}(1)|λ|≲𝒪(1) both at zero and finite winding number sectors complementing and significantly generalizing our previous work [Banerjee and Sen, Phys. Rev. Lett. 126, 220601 (2021)]. The scars at finite \lambdaλ show a rich variety with those that are composed solely from the zero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin, those that contain an admixture of both the zero and the nonzero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin, and finally those composed solely from the nonzero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin. These scars have tell-tale energies such as (non-zero) integers and irrationals like \pm \sqrt{2}±2 at \lambda=0λ=0 or n_1 \lambda \pm n_2n1λ±n2 at \lambda \neq 0λ≠0 where both n_1, n_2n1,n2 are integers. We give analytic expressions for certain ``lego scars’’ for the quantum dimer model on rectangular lattices where one of the linear dimensions can be made arbitrarily large, with the building blocks (legos) being composed of emergent singlets and other more complicated entangled structures.

A wideband mmWave microstrip antenna based on zero‐mode and TM ‐mode resonances

Millimeter-Wave (mmWave) antennas for 5G communication require wide bandwidth, directional radiation patterns, low-profile design and multilayer compatibility for module-level integration. In this paper, we introduce a method of loading shorting pins to a patch antenna to generate extra zero modes. By merging the 2nd zero-mode, TM01 mode, 3rd zero-mode, and TM20 mode in the frequency spectrum, a wide bandwidth varying from 23 to 34 GHz and a low-profile as 0.762 mm can be obtained. Based on this wideband patch antenna, a 4 × 2 antenna array is obtained with the ±40° scanning performance. Theoretical analysis, full-wave simulations, and experimental performances are presented, validating the effectiveness of this method to achieve a wideband performance in mmWave band. It can be applied to 5G communication systems in mmWave band.

Emerging Majorana Zero Modes in Topologically Trivial Dirac Nodal Loop Semimetal Superconductors with Even Parity

Emerging Majorana Zero Modes in Topologically Trivial Dirac Nodal Loop Semimetal Superconductors with Even Parity

Correlating confinement to topological fluctuations near the crossover transition in QCD

We show the existence of strong (anti)correlations between the topological hot spots and the local values of the trace of the Polyakov loop in $2+1$ flavor QCD with physical quark mass, in the vicinity of the crossover transition corresponding to the simultaneous restoration of chiral symmetry and deconfinement. Using sophisticated lattice techniques, we have carefully identified the topological hot spots using quark zero modes and measured the short-distance fluctuations of the Polyakov loop about them, showing how the latter is repelled quite strongly around the peak of the zero modes. Though we could explain some aspects of these correlations within the instanton-dyon picture, our work sets the stage for a larger goal towards a systematic study of the role of different topological species that interact with the Polyakov loop, establishing the strong connection between topology and confinement.

Parafermions in a multilegged geometry: Towards a scalable parafermionic network

Parafermionic zero modes are non-Abelian excitations which have been predicted to emerge at the boundary of topological phases of matter. Contrary to earlier proposals, here we show that such zero modes may also exist in multilegged star junctions of quantum Hall states. We demonstrate that the quantum states spanning the degenerate parafermionic Hilbert space may be detected and manipulated through protocols employing quantum antidots and fractional edge modes. Such star-shaped setups may be the building blocks of two-dimensional parafermionic networks.

Consecutive level spacings in the chiral Gaussian unitary ensemble: from the hard and soft edge to the bulk

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the kth and (k + 1)st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large-n. The result depends on the number of the generic zero modes ν and the number of flavours N f, which are given in terms of characteristic polynomials, as motivated by quantum chromodynamics (QCD). We find that the convergence in n is very rapid. The same can be said separately about the limit k → ∞ (limit to the bulk) and ν → ∞ (limit to the soft edge). Interestingly, the Wigner surmise is a very good approximation for all these cases and, apart from k = 1, shows a deviation below one percent. These findings are corroborated with Monte-Carlo simulations. We finally compare for k = 1 with data from QCD on the lattice, being in this symmetry class.