Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.
The non-regularizability of free fermion field theories, which is the root of various quantum anomalies, plays a central role in particle physics and modern condensed matter physics. In this paper, we generalize the Nielsen-Ninomiya theorem to all minimal nodal free fermion field theories protected by the time reversal, charge conservation, and charge conjugation symmetries. We prove that these massless field theories cannot be regularized on a lattice.
We introduce exactly solvable gapless quantum systems in $d$ dimensions that support symmetry protected topological (SPT) edge modes. Our construction leads to long-range entangled, critical points or phases that can be interpreted as critical condensates of domain walls "decorated" with dimension $(d-1)$ SPT systems. Using a combination of field theory and exact lattice results, we argue that such gapless SPT systems have symmetry-protected topological edge modes that can be either gapless or symmetry-broken, leading to unusual surface critical properties. Despite the absence of a bulk gap, t…
We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state …
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy…
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical $Z_2$ gauge fields. Requiring $U(1)$ spin conservation, time reversal and certain space group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excita…
We show that the $ν=8$ integer quantum Hall state can support Majorana zero modes at domain walls between its two different stable chiral edge phases without superconductivity. This is due to the existence of an edge phase that does not support gapless fermionic excitations; all gapless excitations are bosonic in this edge phase. Majorana fermion zero modes occur at a domain wall between this edge phase and the more conventional one that does support gapless fermions. Remarkably, due to the chirality of the system, the topological degeneracy of these zero modes has exponential protection, as a…
Gapless Symmetry-Protected Topological States in Measurement-Only Circuits
Fuzhou University（福州大学）；College of Physics and Information Engineering, Fuzhou University（福州大学物理与信息工程学院）；Zhejiang University（浙江大学）；Tsinghua University（清华大学）；Tulane University（杜兰大学）
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single particle excitations in the b…
Topology in quantum matter is typically associated with gapped phases. For example, in symmetry protected topological (SPT) phases, the bulk energy gap localizes edge modes near the boundary. In this work we identify a new mechanism that leads to topological phases which are not only gapless but where the absence of a gap is essential. These `intrinsically gapless SPT phases' have no gapped counterpart and are hence also distinct from recently discovered examples of gapless SPT phases. The essential ingredient of these phases is that on-site symmetries act in an anomalous fashion at low energi…
Recently there has been much effort in understanding topological phases of matter with gapless bulk excitations, which are characterized by topological invariants and protected intrinsic boundary states. Here we show that topological semimetals of Majorana fermions arise in exactly solvable Kitaev spin models on a series of three dimensional lattices. The ground states of these models are quantum spin liquids with gapless nodal spectra of bulk Majorana fermion excitations. It is shown that these phases are topologically stable as long as certain discrete symmetries are protected. The correspon…
We consider quantum many body systems with generalized symmetries, such as the higher form symmetries introduced recently, and the "tensor symmetry". We consider a general form of lattice Hamiltonians which allow a certain level of nonlocality. Based on the assumption of dual generalized symmetries, we explicitly construct low energy excited states. We also derive the 't Hooft anomaly for the general Hamiltonians after "gauging" the dual generalized symmetries. A 3d system with dual anomalous 1-form symmetries can be viewed as the boundary of a 4d generalized symmetry protected topological (SP…
By including a material-relevant off-diagonal interaction called the $Γ$ term into the Kitaev model and introducing spatial anisotropy in the interaction strength on the honeycomb lattice, we obtain a series of nodal Z$_2$ quantum spin liquids (QSLs) from parton approach. These QSLs share the same projective symmetry group and are characterized by certain numbers of symmetry-protected Majorana cones in their low-energy excitation spectrum. We illustrate that the physical properties of the QSLs are dependent on the information of the cones. Using the $\pmb k\cdot\pmb p$ method, we analyze the c…
The theoretical description of non-Fermi liquids is among the most challenging questions in strongly correlated quantum matter. While there are many experimental candidates for such phases, few examples can be studied using controlled theoretical approaches. Here we introduce a novel conceptual perspective to analytically describe the non-Fermi liquid physics of gapless spin liquids arising in a family of two- and three-dimensional Kagome models. We first discuss a symmetry-protection mechanism that is responsible for the stability of the gapless phase. We then re-cast the two-dimensional Kago…
We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by $(\times Z_3)^3\equiv Z_3\times Z_3\times Z_3$ symmetry and smaller $Z_3$ symmetry. We derive microscopic models for the gapless edge, uncover their symmetries, and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary f…
We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadra…
The $S=1/2$ kagome- and triangular-lattice Heisenberg antiferromagnets are investigated using the numerical exact diagonalization and the finite-size scaling analysis. The behaviour of the field derivative at zero magnetization is examined for both systems. The present result indicates that the spin excitation is gapless for each system.
We derive the low-energy theory of semi-quantized quantum Hall states, a recently observed class of gapless bilayer fractional quantum Hall states. Our theory shows these states to feature gapless quasiparticles of fractional charge coupled to an emergent Chern-Simons gauge field. These gapless quasiparticles can be understood as composites of electrons and Laughlin-like quasiparticles. We show that semi-quantized quantum Hall states exhibit perfect interlayer drag, host non-Fermi liquid physics, and serve as versatile parent states for fully gapped topological phases hosting anyonic excitatio…
We show that line defects in a non-Abelian Kitaev spin liquid harbor gapless one-dimensional Majorana modes if the interaction across the defect falls below a critical value. Treating the weak interaction at the line defect within a mean-field approximation, we determine the critical interaction strength as a function of the external magnetic field. In the gapless regime, we use the low-energy effective field theory to calculate the spin-lattice relaxation rate for a nuclear spin near the defect and find a cubic temperature dependence that agrees with experiments in the Kitaev material $α\text…
Crystalline topological insulators owe their topological character to the protection that certain boundary states acquire because of certain point-group symmetries. We first show that a Hermitian operator obeying supersymmetric quantum mechanisms (SUSY QM) delivers the entanglement spectrum. We then show that such an entanglement spectrum that is compatible with a certain point-group symmetry obeys a certain local spectral symmetry. The latter result is applied to the stability analysis of four fermionic non-interacting Hamiltonians, the last of which describes graphene with a Kekule distortio…
To reveal the nature of elementary excitations in a quantum spin liquid (QSL), we measured low temperature thermal conductivity and specific heat of 1T-TaS$_2$, a QSL candidate material with frustrated triangular lattice of spin-1/2. The nonzero temperature linear specific heat coefficient $γ$ and the finite residual linear term of the thermal conductivity in the zero temperature limit $κ_0/T=κ/T(T\rightarrow 0)$ are clearly resolved. This demonstrates the presence of highly mobile gapless excitations, which is consistent with fractionalized spinon excitations that form a Fermi surface. Remark…
We introduce a systematic construction of a gapless symmetry protected topological phase in one dimension by "decorating" the domain walls of Luttinger liquids. The resulting strongly interacting phases provide a concrete example of a gapless symmetry protected topological (gSPT) phase with robust symmetry-protected edge modes. Using boundary conformal field theory arguments, we show that while the bulks of such gSPT phases are identical to conventional Luttinger liquids, their boundary critical behavior is controlled by a different, strongly-coupled renormalization group fixed point. Our resu…
We report on the 63/65Cu-nuclear magnetic resonance and nuclear quadrupole resonance (NQR) studies of trivalent Yb zigzag chain compound YbCuS2. Sharp NQR signals were observed in the paramagnetic (PM) state. Below TO = 0.95 K, the multi peaks induced by the internal magnetic fields arising from the antferromagnetic (AFM) ordered moments appear and coexist with the PM signal down to 0.85 K, evidencing the first-order AFM phase transition at TO. In addition, the nuclear spin-lattice relaxation rate 1/T1 abruptly decreases below TO and shows the T-linear behavior below 0.5 K. The significant lar…
Topological quantum matter is typically associated with gapped phases and edge modes protected by the bulk gap. In contrast, recent work (Phys. Rev. B 104, 075132) proposed intrinsically gapless topological phases that, in one dimension, carry protected edge modes only when the bulk is a gapless Luttinger liquid. The edge modes of such a topological Luttinger liquid (TLL) descend from a nonlocal string order that is forbidden in gapped phases and whose precise form depends on the symmetry class of the system. In this work, we propose a powerful and unbiased entanglement-based smoking gun signa…
We consider domain walls in nematic quantum Hall ferromagnets predicted to form in multivalley semiconductors, recently probed by scanning tunnelling microscopy experiments on Bi(111) surfaces. We show that the domain wall properties depend sensitively on the filling factor $ν$ of the underlying (integer) quantum Hall states. For $ν=1$ and in the absence of impurity scattering we argue that the wall hosts a single-channel Luttinger liquid whose gaplessness is a consequence of valley and charge conservation. For $ν=2$, it supports a two-channel Luttinger liquid, which for sufficiently strong in…
$α$-RuCl$_3$ is a leading candidate material for theobservation of physics related to the Kitaev quantum spin liquid (QSL). By combined susceptibility, specific-heat, and nuclear-magnetic-resonance measurements, we demonstrate that $α$-RuCl$_3$ undergoes a quantum phase transition to a QSL in a magnetic field of 7.5 T applied in the $ab$ plane. We show further that this high-field QSL phase has gapless spin excitations over a field range up to 16 T. This highly unconventional result, unknown in either Heisenberg or Kitaev magnets, offers insight essential to establishing the physics of $α$-RuC…
Gutzwiller projection is a way to construct many-body wave functions that could carry topological order or symmetry protected topological (SPT) order. However, an important issue is to determine whether or not a given Gutzwiller-projected wave functions (GWF) carries a non-trivial SPT order, and which SPT order is carried by the wavefunction. In this paper, we numerically study the SPT order in a spin $S = 1$ GWF on the Kagome lattice. Using the standard Monte Carlo method, we directly confirm that the GWF has (1) gapped bulk with short-range correlations, (2) a trivial topological order via n…
Symmetry protected topological (SPT) phases in free fermion and interacting bosonic systems have been classified, but the physical phenomena of interacting fermionic SPT phases have not been fully explored. Here, employing large-scale quantum Monte Carlo simulation, we investigate the edge physics of a bilayer Kane-Mele-Hubbard model with zigzag ribbon geometry. Our unbiased numerical results show that the fermion edge modes are gapped out by interaction, while the bosonic edge modes remain gapless at the $(1+1)d$ boundary, before the bulk quantum phase transition to a topologically trivial ph…
Motivated by recent experiments probing anomalous surface states of Dirac semimetals (DSMs) Na$_3$Bi and Cd$_3$As$_2$, we raise the question posed in the title. We find that, in marked contrast to Weyl semimetals, the gapless surface states of DSMs are not topologically protected in general, except on time-reversal-invariant planes of surface Brillouin zone. We first demonstrate this in a minimal $4$-band model with a pair of Dirac nodes at ${\bf k}=(0,0,\pm Q)$, where gapless states on the side surfaces are protected only near $k_z=0$. We then validate our conclusions about the absence of a t…
The search for Majorana bound states in solid-state physics has been limited to materials which display a gap in their bulk spectrum. We show that such unpaired states appear in certain quasi-one-dimensional Josephson junctions arrays with gapless bulk excitations. The bulk modes mediate a coupling between Majorana bound states via the Ruderman-Kittel-Yosida-Kasuya mechanism. As a consequence, the lowest energy doublet acquires a finite energy difference. For realistic set of parameters this energy splitting remains much smaller than the energy of the bulk eigenstates even for short chains of …
It is well-known that a stable algebraic spin liquid state (or equivalently an algebraic Bose liquid (ABL) state) with emergent gapless photon excitations can exist in quantum spin ice systems, or in a quantum dimer model on a bipartite $3d$ lattice. This photon phase is stable against any weak perturbation without assuming any symmetry. Further works concluded that certain lattice models give rise to more exotic stable algebraic Bose liquid phases with graviton-like excitations. In this paper we will show how these algebraic Bose liquid states can be generalized to stable phases with even mor…
