We show that some gapped quantum many-body systems have a ground state degeneracy that is stable to long-range (e.g., power-law) perturbations, in the sense that any ground state energy splitting induced by such perturbations is exponentially small i
n the system size. More specifically, we consider an Ising symmetry-breaking Hamiltonian with several exactly degenerate ground states and an energy gap, and we then perturb the system with Ising symmetric long-range interactions. For these models we prove (1) the stability of the gap, and (2) that the residual splitting of the low-energy states below the gap is exponentially small in the system size. Our proof relies on a convergent polymer expansion that is adapted to handle the long-range interactions in our model. We also discuss applications of our result to several models of physical interest, including the Kitaev p-wave wire model perturbed by power-law density-density interactions with an exponent greater than 1.
We consider the process of flux insertion for ground states of almost local commuting projector Hamiltonians in two spatial dimensions. In the case of finite dimensional local Hilbert spaces, we prove that this process cannot pump any charge and we conclude that the Hall conductance must vanish.
We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunit
ary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $omega$ of the cohomology group $H^3(G, U_T(1))$. This element $omega in H^3(G, U_T(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$-symbol associated with domain walls in a symmetry broken edge theory; this domain wall $F$-symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.
We revisit the problem of characterizing band topology in dynamically-stable quadratic bosonic Hamiltonians that do not conserve particle number. We show this problem can be rigorously addressed by a smooth and local adiabatic mapping procedure to a
particle number conserving Hamiltonian. In contrast to a generic fermionic pairing Hamiltonian, such a mapping can always be constructed for bosons. Our approach shows that particle non-conserving bosonic Hamiltonians can be classified using known approaches for fermionic models. It also provides a simple means for identifying and calculating appropriate topological invariants. We also explicitly study dynamically stable but non-positive definite Hamiltonians (as arise frequently in driven photonic systems). We show that in this case, each band gap is characterized by two distinct invariants.
Biological information processing is generally assumed to be classical. Measured cellular energy budgets of both prokaryotes and eukaryotes, however, fall orders of magnitude short of the power required to maintain classical states of protein conform
ation and localization at the AA, fs scales predicted by single-molecule decoherence calculations and assumed by classical molecular dynamics models. We suggest that decoherence is limited to the immediate surroundings of the cell membrane and of intercompartmental boundaries within the cell, and that bulk cellular biochemistry implements quantum information processing. Detection of Bell-inequality violations in responses to perturbation of recently-separated sister cells would provide a sensitive test of this prediction. If it is correct, modeling both intra- and intercellular communication requires quantum theory.
We investigate whether there could exist topological invariants of gapped 2D materials related to dissipationless thermoelectric transport at low temperatures. We give both macroscopic and microscopic arguments showing that thermoelectric transport c
oefficients vanish in the limit of zero temperature and thus topological invariants arise only from the electric Hall conductance and the thermal Hall conductance. Our arguments apply to systems with arbitrarily strong interactions. We also show that there is no analog of the Thouless pump for entropy.
We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of 2D topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of
different string-net configurations, where each string-net configuration is a trivalent graph with labeled edges, drawn in the $xy$ plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground state wave function $Phi$ is isotropic: i.e. in the generalized setup, two string-net configurations $X_1, X_2$ that can be continuously deformed into one another can have different ground state amplitudes, $Phi(X_1) eq Phi(X_2)$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground state wave functions on the plane or the sphere -- a property that may be useful in some applications.
An electron-muon collider with an asymmetric collision profile targeting multi-ab$^{-1}$ integrated luminosity is proposed. This novel collider, operating at collisions energies of e.g. 20-200 GeV, 50-1000 GeV and 100-3000 GeV, would be able to probe
charged lepton flavor violation and measure Higgs boson properties precisely. The collision of an electron and muon beam leads to less physics background compared with either an electron-electron or a muon-muon collider, since electron-muon interactions proceed mostly through higher order vector boson fusion and vector boson scattering processes. The asymmetric collision profile results in collision products that are boosted towards the electron beam side, which can be exploited to reduce beam-induced background from the muon beam to a large extent. With this in mind, one can imagine a lepton collider complex, starting from colliding order 10 GeV electron and muon beams for the first time in history and to probe charged lepton flavor violation, then to be upgraded to a collider with 50-100 GeV electron and 1-3 TeV muon beams to measure Higgs properties and search for new physics, and finally to be transformed to a TeV scale muon muon collider. The cost should vary from order 100 millions to a few billion dollars, corresponding to different stages, which make the funding situation more practical.
Most theoretical studies of topological superconductors and Majorana-based quantum computation rely on a mean-field approach to describe superconductivity. A potential problem with this approach is that real superconductors are described by number-co
nserving Hamiltonians with long-range interactions, so their topological properties may not be correctly captured by mean-field models that violate number conservation and have short-range interactions. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the mean-field models, including a finite energy gap in a sector of fixed total particle number, the existence of long range Majorana-like correlations between the ends of an open wire, and a change in the ground state fermion parity for periodic vs. anti-periodic boundary conditions. These results show that many of the remarkable properties of mean-field models of topological superconductivity persist in more realistic models with number-conserving dynamics.
We study anomalies in time-reversal ($mathbb{Z}_2^T$) and $U(1)$ symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-D system but can be realized on the surface of a $(3+1
)$-D symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators --- functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e. $U(1)rtimesmathbb{Z}_2^T$ and $U(1)timesmathbb{Z}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)$-D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)$-D SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.
