We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let $ψ$ be an arbitrary two-qubit state. We consider a chain of $n$ qubits with open boundary conditions and Hamiltonian $H_n(ψ)$ which is defined as the sum of rank-1 projectors onto $ψ$ applied to consecutive pairs of qubits. We show that the spectral gap of $H_n(ψ)$ is upper bounded by $1/(n-1)$ if the eigenvalues of a certain two-by-two matrix simply related to $ψ$ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of $n$ (depending only on $ψ$). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
Key properties of a physical system depend on whether it is gapped, i.e. whether its spectral gap has a positive lower bound that is independent of system size. In quantum information theory, the question of whether a system is gapped has essential computational significance as well. Here, we introduce a rigorous renormalization method to prove that a spin chain is gapped. This approach exploits the fact that ground states of gapped systems exhibit decaying correlations. We apply the method to show that two interesting models are gapped, successfully completing proofs even where the previously…
We propose an efficient algorithm for the ground state of frustration-free one-dimensional gapped Hamiltonians. This algorithm is much simpler than the original one by Landau et al., and thus may be easily accessible to a general audience in the community. We present all the details in two pages.
For every Matrix Product State (MPS) one can always construct a so-called parent Hamiltonian. This is a local, frustration free, Hamiltonian which has the MPS as ground state and is gapped. Whenever that parent Hamiltonian has a degenerate ground state (the so-called non-injective case), we construct another 'uncle' Hamiltonian which is local and frustration free but gapless, and its spectrum is $\R^+$. The construction is obtained by linearly perturbing the matrices building up the state in a random direction, and then taking the limit where the perturbation goes to zero. For MPS where the pa…
We study the stability with respect to a broad class of perturbations of gapped ground state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi-Hastings-Michalakis (BHM) strategy that under a condition of Local Topological Quantum Order, the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models …
We investigate the behavior of genuine multiparticle entanglement, as quantified by the generalized geometric measure, in gapless-to-gapped quantum transitions of one- and two-dimensional quantum spin models. The investigations are performed in the exactly solvable one-dimensional $XY$ models, as well as two-dimensional frustrated $J_{1}-J_{2}$ models, including the Shastry-Sutherland model. The generalized geometric measure shows non-monotonic features near such transitions in the frustrated quantum systems. We also compare the features of the generalized geometric measure near the quantum cr…
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…
We investigate the persistence of spectral gaps of one-dimensional frustration free quantum lattice systems under weak perturbations and with open boundary conditions. Assuming the interactions of the system satisfy a form of local topological quantum order, we prove explicit lower bounds on the ground state spectral gap and higher gaps for spin and fermion chains. By adapting previous methods using the spectral flow, we analyze the bulk and edge dependence of lower bounds on spectral gaps.
We improve Knabe's spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-$m$ chain with periodic boundary conditions, while the local gap is that of a subchain of size $n<m$ with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value $1/(n-1)$ for some $n>2$, then the global gap is lower bounded by a positive constant in the thermodynamic limit $m\rightarrow \infty$. Here we improve the threshold to $\frac{6}…
A frustration-free local Hamiltonian has the property that its ground state minimises the energy of all local terms simultaneously. In general, even deciding whether a Hamiltonian is frustration-free is a hard task, as it is closely related to the QMA1-complete quantum satisfiability problem (QSAT) -- the quantum analogue of SAT, which is the archetypal NP-complete problem in classical computer science. This connection shows that the frustration-free property is not only relevant to physics but also to computer science. The Quantum Lovász Local Lemma (QLLL) provides a sufficient condition for …
In this paper, we consider classification problem of asymmetric gapped Hamiltonians, which are given as the non-degenerate part of the Hamiltonians introduced in [O1]. We consider the $C^1$-classification, which takes into account the effect of boundaries. We show that the left and right degeneracies of edge ground states are the complete invariant. As a corollary, we consider the bulk-classification problem. We study Hamiltonians that1.are given by translation invariant finite range interactions, 2.are gapped in the bulk, 3.are frustration-free, 4.have uniformly bounded ground state degenerac…
Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap $ε$ has correlation length $ξ$ upper bounded as $ξ=O(1/ε)$. In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound $ξ=O(1/\sqrtε)$ in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained …
We prove that the entanglement entropy of the ground state of a locally gapped frustration-free 2D lattice spin system satisfies an area law with respect to a vertical bipartition of the lattice into left and right regions. We first establish that the ground state projector of any locally gapped frustration-free 1D spin system can be approximated to within error $ε$ by a degree $O(\sqrt{n\log(ε^{-1})})$ multivariate polynomial in the interaction terms of the Hamiltonian. This generalizes the optimal bound on the approximate degree of the boolean AND function, which corresponds to the special c…
We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on …
The existence of a spectral gap above the ground state has far-reaching consequences for the low-energy physics of a quantum many-body system. A recent work of Movassagh [R. Movassagh, PRL 119 (2017), 220504] shows that a spatially random local quantum Hamiltonian is generically gapless. Here we observe that a gap is more common for translation-invariant quantum spin chains, more specifically, that these are gapped with a positive probability if the interaction is of small rank. This is in line with a previous analysis of the spin-$1/2$ case by Bravyi and Gosset. The Hamiltonians are construct…
We study the dynamics of discrete-time quantum walk using quantum coin operations, $\hat{C}(θ_1)$ and $\hat{C}(θ_2)$ in time-dependent periodic sequence. For the two-period quantum walk with the parameters $θ_1$ and $θ_2$ in the coin operations we show that the standard deviation [$σ_{θ_1, θ_2} (t)$] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations $θ_1$ or $θ_2$, $σ_{θ_1, θ_2}(t) = \min \{σ_{θ_1}(t), σ_{θ_2}(t) \}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum …
We investigate the von Neumann entanglement entropy and Schmidt gap in the ground state of the Kitaev model on the honeycomb lattice for square and cylindrical subsystems. We find that, for both the subsystems, the free-fermionic contribution to the entanglement entropy $S_E$ exhibits signatures of the phase transitions between the gapless and gapped phases. However, we find that $S_E$ does not show an expected monotonic behaviour as a function of the coupling $J_z$ between the suitably defined one-dimensional chains within the gapless phase for either geometry; moreover the system generically…
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory. In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions. We find that both gapped and gapless phases with fractonic excitations can emerge from the models.…
Topological phases in spinless non-Hermitian models have been widely studied both theoretically and experimentally in some artificial materials using photonics, phononics and magnon. In this work, we investigate the interplay between non-Hermitian loss and gain and non-Abelian gauge potential realized in a two-component superconducting circuit. In our model, the non-Hermiticity along only gives rise to trivial gain and loss to the states; while the non-Abelian gauge along gives rise to flying butterfly spectra and associated edge modes, which in photonics can be directly measured by the intens…
Thermal robustness of sensing quantum phases of matter via qubit probes
FZU - Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）；Department of Physics, King’s College London（伦敦国王学院物理系）；School of Physics, Trinity College Dublin（都柏林三一学院物理学院）
The geometric properties of a lattice can have profound consequences on its band spectrum. For example, symmetry constraints and geometric frustration can give rise to topologicially nontrivial and dispersionless bands, respectively. Line-graph lattices are a perfect example of both of these features: their lowest energy bands are perfectly flat, and here we develop a formalism to connect some of their geometric properties with the presence or absence of fragile topology in their flat bands. This theoretical work will enable experimental studies of fragile topology in several types of line-gra…
