In various topological phases, nontrivial states appear at the boundaries of the system. In this paper, we investigate anomalous dielectric response caused by such states caused by the $\pi$ Zak phase. First, by using the one-dimensional Su-Schrieffer-Heeger model, we show that, when the system is insulating and the Zak phase is $\pi$, the polarization suddenly rises to a large value close to $e/2$, by application of an external electric field. The $\pi$ Zak phase indicates the existence of half-filled edge states, and we attribute this phenomenon to charge transfer between the edge states at the two ends of the system. We extend this idea to two- and three-dimensional insulators with the $\pi$ Zak phase over the Brillouin zone and find similar anomalous dielectric response. We also show that diamond and silicon slabs with (111) surfaces have the $\pi$ Zak phase by ab initio calculations, and show that this anomalous response survives even surface reconstruction involving an odd number of original surface unit cells. Another material example with an anomalous dielectric response is polytetrafluoroethylene (PTFE), showing plateaus of polarization at $\pm e$ by ab initio calculation, in agreement with our theory.

Probing the topological invariants of interacting systems stands as a grand and open challenge. Here we describe a dynamical method to characterize 1D chiral models, based on the direct observation of time-evolving bulk excitations. We present analytical and state-of-the-art numerical calculations on various flavors of interacting Su-Schrieffer-Heeger (SSH) chains, demonstrating how measuring the mean chiral displacement allows us to distinguish between topological insulator, trivial insulator, and symmetry-broken phases. Finally, we provide a readily-feasible experimental blueprint for a model displaying these three phases and we describe how to detect those.

The Dirac mass of a two-dimensional QAH insulator is directly related to the parity anomaly of planar quantum electrodynamics, as shown initially by Niemi and Semenoff [Phys. Rev. Lett. 51, 2077 (1983)]. In this work, we connect the additional momentum-dependent Newtonian mass term of a QAH insulator to the parity anomaly. We reveal that in the calculation of the effective action, before renormalization, the Newtonian mass acts similar to a parity-breaking element of a high-energy regularization scheme. This calculation allows us to derive the finite frequency correction to the DC Hall conductivity of a QAH insulator. We predict that the leading order AC correction contains a term proportional to the Chern number. This term originates from the Newtonian mass and can be measured via electrical or magneto-optical experiments. Moreover, we prove that the Newtonian mass significantly changes the resonance structure of the AC Hall conductivity in comparison to pure Dirac systems like graphene.

We discuss how strongly interacting higher-order symmetry protected topological (HOSPT) phases can be characterized from the entanglement perspective: First, we introduce a topological many-body invariant which reveals the noncommutative algebra between a flux operator and $C_n$ rotations. We argue that this invariant denotes the angular momentum carried by the instanton which is closely related to the discrete Wen-Zee response and the fractional corner charge. Second, we define a new entanglement property, dubbed “higher-order entanglement,” to scrutinize and differentiate various higher-order topological phases from a hierarchical sequence of the entanglement structure. We support our claims by numerically studying a super-lattice Bose-Hubbard model that exhibits different HOSPT phases.

We study topological properties of bound pairs of photons in spatially modulated qubit arrays (arrays of two-level atoms) coupled to a waveguide. While bound pairs behave like Bloch waves, they are topologically nontrivial in the parameter space formed by the center-of-mass momentum and the modulation phase, where the latter plays the role of a synthetic dimension. In a superlattice where each unit cell contains three two-level atoms (qubits), we calculate the Chern numbers for the bound-state photon bands, which are found to be $(1,-2,1)$. For open boundary conditions, we find exotic topological bound-pair edge states with radiative losses. Unlike the conventional case of the bulk-edge correspondence, these novel edge modes not only exist in gaps separating the bound-pair bands but they also may merge with and penetrate into the bands. By joining two structures with different spatial modulations, we find long-lived interface states which may have applications in storage and quantum information processing.

We establish an analytic low-energy theory describing higher-order topological insulator (HOTI) phases in quasicrystalline systems. We apply this to a model consisting of two stacked Haldane models with oppositely propagating edge modes, analogous to the Kane-Mele model, and with a ${30}^{\circ }$ twist. We show that the resulting localized modes at corners, characteristic of a HOTI, are not associated with conventional mass inversions but are instead associated with what we dub “fractional mass kinks.” By generalizing the low-energy theory, we establish a classification for arbitrary $n$-fold rotational symmetries. We also derive a relationship between corner modes in a bilayer and disclination modes in a single layer. By using numerics to go beyond the weak-coupling limit, we show that a hierarchy of additional gaps occurs due to the quasiperiodicity, which also harbor corner-localized modes.

The Mott insulating phase of parent compounds is frequently taken as a starting point for underdoped high-$T_c$ cuprate superconductors. In particular, the pseudogap state is often considered as deriving from the Mott insulator. In this work, we systematically investigate different weakly doped Mott insulators on a square and triangular lattice to clarify the relationship between the pseudogap and Mottness. We show that doping a two-dimensional Mott insulator does not necessarily lead to a pseudogap phase. Despite its inherent strong-coupling nature, we find that the existence or absence of a pseudogap depends sensitively on noninteracting band parameters, and we identify the crucial role played by the van Hove singularities of the system. Motivated by a SU(2) gauge theory for the pseudogap state, we propose and verify numerically a simple equation that governs the evolution of characteristic features in the electronic scattering rate.

A spin-U(1)-symmetry protected momentum-dependent integer-$Z$-valued topological invariant is proposed to time-reversal-invariant superconductivity whose nonzero value will lead to exactly flat surface band(s). The theory is applied to the weakly spin-orbit-coupled quasi-1D ${\mathrm{A}}_2{\mathrm{Cr}}_3{\mathrm{As}}_3$ (A=Na, K, Rb, Cs) superconductors family with highest $T_c$ up to 8.6 K with $p_z$-wave pairing in the $S_z=0$ channel. It's found that up to the leading atomic spin-orbit coupling (SOC), the whole (001) surface Brillouin zone is covered with exactly flat surface bands, with some of the regime hosting three flat bands and the remaining part hosting two. Such exactly flat surface bands will lead to a very sharp zero-bias conductance peak in the scanning tunneling microscopic spectrum. When a tiny subleading spin-flipping SOC is considered, the surface bands will only be slightly split. The application of this theory can be generalized to other unconventional superconductors with weak SOC, particularly to those with mirror-reflection symmetry.

The periodic driving of a quantum system can enable new topological phases without analogs in static systems. This provides a route towards preparing nonequilibrium quantum phases rooted in the nonequilibrium nature by periodic driving engineering. Motivated by the ongoing considerable interest in topological semimetals, we are interested in the novel topological phases in the periodically driven topological semimetals without a static counterpart. We propose to design nonequilibrium topological semimetals in the regime of a weakly driving field where the spectrum width has the same magnitude as the driving frequency. We identify two types of nonequilibrium Weyl semimetals (i.e., Floquet and anomalous Floquet Weyl semimetals) that do not exhibit analogs in equilibrium. The proposed setup is shown to be experimentally feasible using the state-of-the-art techniques used to control ultracold atoms in optical lattices.

Dirac materials are of great interest as condensed matter realizations of the Dirac and Weyl equations. In particular, they serve as a starting point for the study of topological phases. This physics has been extensively studied in electronic systems such as graphene, Weyl, and Dirac semimetals. In contrast, recent studies have highlighted several examples of Dirac-like cones in collective excitation spectra, viz. in phonon, magnon, and triplon bands. These cannot be directly related to the Dirac or Weyl equations as they are bosonic in nature with pseudounitary band bases. We address this issue by constructing a generic deformation scheme that maps any fermionic Hamiltonian and its spectrum to that of a bosonic problem. In particular, we show that any Dirac-like equation can be deformed into a suitable bosonic form. The resulting bosonic spectra bear a two-to-one relation to that of the parent Dirac system. Their dispersions inherit several interesting properties including conical band touching points and a gap-opening role for ‘mass’ terms. The relationship also extends to the band eigenvectors with the bosonic states carrying the same Berry connections as the parent fermionic states. The bosonic bands thus inherit topological character as well. If the parent fermionic system has nontrivial topology that leads to midgap surface states, the bosonic analog also hosts surface states that lie within the corresponding band gap. The proposed bosonic Dirac structure appears in several known models. In materials, it is realized in ${\mathrm{Ba}}_2{\mathrm{CuSi}}_2{\mathrm{O}}_6{\mathrm{Cl}}_2$ and possibly in ${\mathrm{CoTiO}}_3$ as well as in paramagnetic honeycomb ruthenates. Our results allow for a rigorous understanding of Dirac phononic and magnonic systems and enable concrete predictions, e.g., of surface states in magnonic topological insulators and Weyl semimetals.

Quantum many-body systems described by Lagrangian path integrals can realize many different topological phases of matter. One of the most important problems in condensed-matter physics is to extract topological invariants from the Lagrangian and to determine the topological order in the systems. In this paper, we suggest a general solution to this problem. Given a path integral on space-time lattice $C^{d+1}$ that describes a short-range correlated (i.e., gapped) system, we design systematic ways to extract topological invariants. For example, we show how to use nonuniversal partition functions $Z\left(C^{2+1}\right)$ on several space-time lattices with related topologies to extract ${\left(M_f\right)}_{11}$ and $\text{Tr}\left(M_f\right)$, where $M_f$ is a representation of the modular group $\text{SL}(2,\mathbf{Z})$, a topological invariant that characterizes ($2+1$)-dimensional topological orders. Our approach is guided by a notion of quantum volume. A path integral gives rise to a wave function $|\mathrm{\Psi }\rangle$ on the boundary of ($d+1$)-dimensional space-time $C^{d+1}$. We show that $V=\text{ln}\sqrt{\langle \mathrm{\Psi }|\mathrm{\Psi }\rangle }$ satisfies the inclusion-exclusion property $\frac{V(A\cup B)+V(A\cap B)}{V\left(A\right)+V\left(B\right)}=1$ and behaves like a volume of the space-time $C^{d+1}$ in the thermodynamics limit if the system is short range correlated. This leads to a proposal that the vector $|\mathrm{\Psi }\rangle$ itself is the quantum volume of the space-time $C^{d+1}$. The subleading term of thermodynamics limit gives rise to the topological invariants.

In the study of $\mathcal{P}\mathcal{T}$-symmetric quantum systems with non-Hermitian perturbations, one of the most important questions is whether eigenvalues stay real or whether $\mathcal{P}\mathcal{T}$ symmetry is spontaneously broken when eigenvalues meet. A particularly interesting set of eigenstates is provided by the degenerate ground-state subspace of systems with topological order. In this paper, we present simple criteria that guarantee the protection of $\mathcal{P}\mathcal{T}$ symmetry and, thus, the reality of the eigenvalues in topological many-body systems. We formulate these criteria in both geometric and algebraic form and demonstrate them using the toric code and several different fracton models as examples. Our analysis reveals that $\mathcal{P}\mathcal{T}$ symmetry is robust against a remarkably large class of non-Hermitian perturbations in these models; this is particularly striking in the case of fracton models due to the exponentially large number of degenerate states.

We review low- and high-field magnetotransport in 80-nm-thick strained HgTe, a material that belongs to the class of strong three-dimensional topological insulators. Utilizing a top gate, the Fermi level can be tuned from the valence band via the Dirac surface states into the conduction band and allows studying Landau quantization in situations where different species of charge carriers contribute to magnetotransport. Landau fan charts, mapping the conductivity ${\sigma }_{xx}\left(V_g,B\right)$ in the whole magnetic field–gate voltage range, can be divided into six areas, depending on the state of the participating carrier species. Key findings are (i) the interplay of bulk holes (spin degenerate) and Dirac surface electrons (nondegenerate), coexisting for $E_F$ in the valence band, leads to a periodic switching between odd and even filling factors and thus odd and even quantized Hall voltage values. (ii) We found a similar though less pronounced behavior for coexisting Dirac surface and conduction band electrons. (iii) In the bulk gap, quantized Dirac electrons on the top surface coexist at lower B with nonquantized ones on the bottom side, giving rise to quantum Hall plateau values depending—for a given filling factor—on the magnetic field strength. In stronger $B$ fields, Landau level separation increases; charge transfer between different carrier species becomes energetically favorable and leads to the formation of a global (i.e., involving top and bottom surfaces) quantum Hall state. Simulations using the simplest possible theoretical approach are in line with the basic experimental findings, describing correctly the central features of the transitions from classical to quantum transport in the respective areas of our multicomponent charge carrier system.

The protected surface states of topological insulators (TIs) form gapless Dirac cones corresponding nondegenerate eigenstates with helical spin polarization. The presence of a warping term deforms the isotropic cone of the most simple model into snowflake Fermi surfaces as in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Their features have been identified in STM quasiparticle interference (QPI) experiments on isolated surfaces. Here we investigate the QPI spectrum for the TI thin film geometry with a finite tunneling between the surface states. This leads to a dramatic change of spectrum due to gapping and a change in spin texture that should leave distinct signatures in the QPI pattern. We consider both normal and magnetic exchange scattering from the surface impurities and obtain the scattering $t$ matrix in Born approximation as well as the general closed solution. We show the expected systematic variation of QPI snowflake intensity features by varying film thickness and study, in particular, the influence on backscattering processes. We predict the variation of QPI spectrum for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ thin films using the observed gap dependence from ARPES results.

Majorana zero modes (MZMs), fundamental building blocks for realizing topological quantum computers, can appear at the interface between a superconductor and a topological material. One of the experimental signatures that has been widely pursued to confirm the existence of MZMs is the observation of a large, quantized zero-bias conductance peak (ZBCP) in the differential conductance measurements. In this Letter, we report observation of such a large ZBCP in junction structures of normal metal (titanium/gold Ti/Au)—Dirac semimetal (cadmium-arsenide $\mathrm{C}{\mathrm{d}}_3\mathrm{A}{\mathrm{s}}_2$)—conventional superconductor (aluminum Al), with a value close to four times that of the normal state conductance. Our detailed analyses suggest that this large ZBCP is most likely not caused by MZMs. We attribute the ZBCP, instead, to the existence of a supercurrent between two far-separated superconducting Al electrodes, which shows up as a zero-bias peak because of the circuitry and thermal fluctuations of the supercurrent phase, a mechanism conceived by Ivanchenko and Zil'berman more than 50 years ago [Ivanchenko and Zil’berman, JETP 28, 1272 (1969)]. Our results thus call for extreme caution when assigning the origin of a large ZBCP to MZMs in a multiterminal semiconductor or topological insulator/semimetal setup. We thus provide criteria for identifying when the ZBCP is definitely not caused by an MZM. Furthermore, we present several remarkable experimental results of a supercurrent effect occurring over unusually long distances and clean perfect Andreev reflection features.

Quasiparticle interference (QPI) of the electronic states has been widely applied in scanning tunneling microscopy to analyze the electronic band structure of materials. Single-defect-induced QPI reveals defect-dependent interaction between a single atomic defect and electronic states, which deserves special attention. Due to the weak signal of single-defect-induced QPI, the signal-to-noise ratio is relatively low in a standard two-dimensional QPI measurement. In this paper, we introduce a projective quasiparticle interference (PQPI) method in which a one-dimensional measurement is taken along high-symmetry directions centered on a specified defect. We apply the PQPI method to the topological nodal-line semimetal ZrSiS. We focus on two special types of atomic defects that scatter the surface and bulk electronic bands. With an enhanced signal-to-noise ratio in PQPI, the energy dispersions are clearly resolved along high-symmetry directions. We discuss the defect-dependent scattering of bulk bands with the nonsymmorphic symmetry-enforced selection rules. Furthermore, an energy shift of the surface floating band is observed, and a branch of energy dispersion (${\mathit{q}}_6$) is resolved. This PQPI method can be applied to other complex materials to explore defect-dependent interactions in the future.

We consider a Josephson bijunction consisting of a thin $SIS\pi$-Josephson junction sandwiched between two-dimensional semiconducting layers with strong Rashba spin-orbit interaction. Each of these layers forms an $SNS$ junction due to proximity-induced superconductivity. The $SIS$ junction is assumed to be thin enough such that the two Rashba layers are tunnel coupled. We show that, by tuning external gates, this system can be controllably brought into a time-reversal invariant topological superconducting phase with a Kramers pair of Majorana bound states being localized at the end of the normal region for a large parameter phase space. In particular, in the strong spin-orbit interaction limit, the topological phase can be accessed already in the regime of small tunneling amplitudes.

The fractional quantum Hall effect is a paradigm of topological order and has been studied thoroughly in two dimensions. Here, we construct a different type of fractional quantum Hall system, which has the special property that it lives in fractal dimensions. We provide analytical wave functions and exact few-body parent Hamiltonians, and we show numerically for several different Hausdorff dimensions between 1 and 2 that the systems host anyons. We also find examples of fractional quantum Hall physics in fractals with Hausdorff dimension 1 and $ln\left(4\right)/ln\left(5\right)$. Our results suggest that the local structure of the investigated fractals is more important than the Hausdorff dimension to determine whether the systems are in the desired topological phase. The study paves the way for further investigations of strongly correlated topological systems in fractal dimensions.

Correlations in topological states of matter provide a rich phenomenology, including a reduction in the topological classification of the interacting system compared to its noninteracting counterpart. This happens when two phases that are topologically distinct on the noninteracting level become adiabatically connected once interactions are included. Here, we use a numerically exact quantum Monte Carlo method to study such a topological reduction. We consider a two-dimensional charge-conserving analog of the Levin-Gu superconductor whose classification is reduced from $\mathbb{Z}$ to ${\mathbb{Z}}_4$. We may expect any symmetry-preserving interaction that leads to a symmetric gapped ground state at strong coupling, and consequently a gapped symmetric surface, to be sufficient for such reduction. Here, we provide a counterexample by considering an interaction that (i) leads to a symmetric gapped ground state at sufficient strength, and (ii) does not allow for any adiabatic path connecting the trivial phase to the topological phase with $w=4$. The latter statement is established by numerically mapping the phase diagram as a function of the interaction strength and a parameter tuning the topological invariant. Instead of the adiabatic connection, the system exhibits an extended region of spontaneous symmetry breaking separating the topological sectors. Upon the inclusion of frustration, the size of this long-range-ordered region is reduced until it gives way to a first-order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free-fermion states into each other. We conclude that an interaction that trivializes the surface of a gapped topological phase is a necessary but not sufficient condition to establish an adiabatic path within the interaction-reduced classification. In other words, the class of interactions that trivializes the surface is different (likely larger) from the class of interactions that establishes an adiabatic connection in the bulk.

In topological systems, a modulation in the gap onset near interfaces can lead to the appearance of massive edge states, as were first described by Volkov and Pankratov. In this work, we study graphene nanoribbons in the presence of intrinsic spin-orbit coupling smoothly modulated near the system edges. We show that this space modulation leads to the appearance of Volkov-Pankratov states, in addition to the topologically protected ones. We obtain this result by means of two complementary methods, one based on the effective low-energy Dirac equation description and the other on a fully numerical tight-binding approach, finding excellent agreement between the two. We then show how transport measurements might reveal the presence of Volkov-Pankratov states, and discuss possible graphenelike structures in which such states might be observed.