arXiv subjects
Publications and source records attributed to Sven Bachmann.
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 consider charge transport for interacting many-body systems with a gapped ground state subspace which is finitely degenerate and topologically ordered. To any locality-preserving, charge-conserving unitary that preserves the ground state space, we associate an index that is an integer multiple of $1/p$, where $p$ is the ground state degeneracy. We prove that the index is additive under composition of unitaries. This formalism gives rise to several applications: fractional quantum Hall conductance, a fractional Lieb-Schultz-Mattis theorem that generalizes the standard LSM to systems where the translation-invariance is broken, and the interacting generalization of the Avron-Dana-Zak relation between Hall conductance and the filling factor.
We propose a many-body index that extends Fredholm index theory to many-body systems. The index is defined for any charge-conserving system with a topologically ordered $p$-dimensional ground state sector. The index is fractional with the denominator given by $p$. In particular, this yields a new short proof of the quantization of the Hall conductance and of Lieb-Schulz-Mattis theorem. In the case that the index is non-integer, the argument provides an explicit construction of Wilson loop operators exhibiting a non-trivial braiding and that can be used to create fractionally charged Abelian anyons.
In general, linear response theory expresses the relation between a driving and a physical system's response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm's law. We show here that, in the case of the quantum Hall effect, all higher order corrections vanish. We prove this in a fully interacting setting and without flux averaging.
We review proofs of a theorem of Bloch on the absence of macroscopic stationary currents in quantum systems. The standard proof shows that the current in 1D vanishes in the large volume limit under rather general conditions. In higher dimension, the total current across a cross-section does not need to vanish in gapless systems but it does vanish in gapped systems. We focus on the latter claim and give a self-contained proof motivated by a recently introduced index for many-body charge transport in quantum lattice systems having a conserved $U(1)$-charge.
In growing plant cells, parallel ordering of microtubules (MTs) along the inner surface of the cell membrane influences the direction of cell expansion and thereby plant morphology. For correct expansion of organs that primarily grow by elongating, such as roots and stems, MTs must bend in the high-curvature direction along the cylindrically shaped cell membrane in order to form the required circumferential arrays. Computational studies, which have recapitulated the self-organization of these arrays, ignored MT mechanics and assumed MTs follow geodesics of the cell surface. Here, we show, through analysis of a derived Euler-Lagrange equation, that an elastic MT constrained to a cylindrical surface will deflect away from geodesics and toward low curvature directions to minimize bending energy. This occurs when the curvature of the cell surface is relatively high for a given anchor density. In the limit of infinite anchor density, MTs always follow geodesics. We compare our analytical predictions to measured curvatures and anchor densities and find that the regime in which cells are forming these cortical arrays straddles the region of parameter space in which arrays must form under the antagonistic influence of this mechanically induced deflection. Although this introduces a potential obstacle to forming circumferentially orientated arrays that needs to be accounted for in the models, it also raises the question of whether plants use this mechanical phenomenon to regulate array orientation. The model also constitutes an elegant generalization of the classical Euler-bucking instability along with an intrinsic unfolding of the associated pitchfork bifurcation.
We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.
In these lecture notes, we review the adiabatic theorem in quantum mechanics, focusing on a recent extension to many-body systems. The role of locality is emphasized and the relation to the quasi-adiabatic flow discussed. An important application of these results to linear response theory is also reviewed.
We provide a short proof of the quantisation of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus. This is not new and should be seen as an adaptation of the proof of [1], simplified by making the stronger assumption that the Hamiltonian remains gapped when threading the torus with fluxes. We argue why this assumption is very plausible. The conductance is given by Berry's curvature and our key auxiliary result is that the curvature is asymptotically constant across the torus of fluxes.
We relate explicitly the adiabatic curvature -- in flux space -- of an interacting Hall insulator with nondegenerate ground state to various linear response coefficients, in particular the Kubo response and the adiabatic response. The flexibility of the setup, allowing for various driving terms and currents, reflects the topological nature of the adiabatic curvature. We also outline an abstract connection between Kubo response and adiabatic response, corresponding to the fact that electric fields can be generated both by electrostatic potentials and time-dependent magnetic fields. Our treatment fits in the framework of rigorous many-body theory, thanks to the gap assumption.
The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter $ε$. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g.~for the integer quantum Hall effect.
The first proof of the quantum adiabatic theorem was given as early as 1928. Today, this theorem is increasingly applied in a many-body context, e.g. in quantum annealing and in studies of topological properties of matter. In this setup, the rate of variation $\varepsilon$ of local terms is indeed small compared to the gap, but the rate of variation of the total, extensive Hamiltonian, is not. Therefore, applications to many-body systems are not covered by the proofs and arguments in the literature. In this letter, we prove a version of the adiabatic theorem for gapped ground states of quantum spin systems, under assumptions that remain valid in the thermodynamic limit. As an application, we give a mathematical proof of Kubo linear response formula for a broad class of gapped interacting systems.
We discuss thermotropic nematic liquid crystals in the mean-field regime. In the first part of this article, we rigorously carry out the mean-field limit of a system of $N$ rod-like particles as $N\to\infty$, which yields an effective `one-body' free energy functional. In the second part, we focus on spatially homogeneous systems, for which we study the associated Euler-Lagrange equation, with a focus on phase transitions for general axisymmetric potentials. We prove that the system is isotropic at high temperature, while anisotropic distributions appear through a transcritical bifurcation as the temperature is lowered. Finally, as the temperature goes to zero we also prove, in the concrete case of the Maier-Saupe potential, that the system converges to perfect nematic order.
In this comprehensive study of Kitaev's abelian models defined on a graph embedded on a closed orientable surface, we provide complete proofs of the topological ground state degeneracy, the absence of local order parameters, compute the entanglement entropy exactly and characterise the elementary anyonic excitations. The homology and cohomolgy groups of the cell complex play a central role and allow for a rigorous understanding of the relations between the above characterisations of topological order.
We study the evolution of a driven harmonic oscillator with a time-dependent frequency $ω_t \propto |t|$. At time $t=0$ the Hamiltonian undergoes a point of infinite spectral degeneracy. If the system is initialized in the instantaneous vacuum in the distant past then the asymptotic future state is a squeezed state whose parameters are explicitly determined. We show that the squeezing is independent on the sweeping rate. This manifests the failure of the adiabatic approximation at points where infinitely many eigenvalues collide. We extend our analysis to the situation where the gap at $t=0$ remains finite. We also discuss the natural geometry of the manifold of squeezed states. We show that it is realized by the Poincaré disk model viewed as a Kähler manifold.
Motivated by the study of area laws for the entanglement entropy of gapped ground states of quantum spin systems and their stability, we prove that the unitary cocycle generated by a local time-dependent Hamiltonian can be approximated, for any finite set $X$, by a tensor product of the corresponding unitaries in $X$ and its complement, multiplied by a dynamics strictly supported in the neighbourhood of the surface $\partial X$. The error decays almost exponentially in the size of the neighbourhood and grows with the square of the area~$\vert \partial X\vert^2$.
We consider translation invariant gapped quantum spin systems satisfying the Lieb-Robinson bound and containing single-particle states in a ground state representation. Following the Haag-Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding $S$-matrix. We also obtain some general restrictions on the shape of the energy-momentum spectrum. For the purpose of our analysis we adapt the concepts of almost local observables and energy-momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb-Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields.
We consider the $C^1$-classification of gapped Hamiltonians introduced in [Fannes-Nachtergaele-Werner, Nachtergaele] as parents Hamiltonians of translation invariant finitely correlated states. Within this family, we show that the number of edge modes, which is equal at the left and right edge, is the complete invariant. The construction proves that translation invariance of the `bulk' ground state does not need to be broken to establish $C^1$-equivalence, namely that the spin chain does not need to be blocked.