Papers by Bruno Nachtergaele
Communications in Mathematical Physics, 2009
We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and desc... more We prove Lieb-Robinson bounds for systems defined on infinite dimensional Hilbert spaces and described by unbounded Hamiltonians. In particular, we consider harmonic and certain anharmonic lattice systems.
Physical Review B
We introduce a two-parameter family of perturbations of Kitaev's Toric Code Model in which the an... more We introduce a two-parameter family of perturbations of Kitaev's Toric Code Model in which the anyonic excitations acquire an interesting dynamics. We study the dynamics of this model in the space of states with electric and magnetic charge both equal to 1 and find that the model exhibits both bound states and scattering states in a suitable region of the parameters. The bound state is a Majorana fermion with a dispersion relation of Dirac cone type. For a certain range of model parameters, we find that these bound states disappear in a continuum of scattering states at a critical value of the total momentum. The scattering states describe separate electric and magnetic anyons, which in this model each have a sin k dispersion relation.
Journal of Mathematical Physics
Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynam... more Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustration-free models satisfying a Local Topological Quantum Order condition, which we present in a sequel to this paper.
Journal of Mathematical Physics
We investigate the persistence of spectral gaps of one-dimensional frustration free quantum latti... more 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.
Physical Review B
Renormalization-group equations for the uniaxial commensurate-incommensurate (C-Ic) transition in... more Renormalization-group equations for the uniaxial commensurate-incommensurate (C-Ic) transition in two dimensions are derived. The soliton density p is a nonanalytic function of the misfit parameter JM, even at high temperatures where only a floating phase (i.e., algebraic correlations with exponent g) is possible.
Journal of Mathematical Physics
Understanding many-body dynamics lies at the heart of many fundamental problems of mathematical p... more Understanding many-body dynamics lies at the heart of many fundamental problems of mathematical physics. Even when one is not directly concerned with time-dependent phenomena, such as in the study of equilibrium and non-equilibrium stationary states, one is essentially investigating properties of the dynamics. It is also possible to approach spectral questions about the Hamiltonian of a quantum system starting from an analysis of the dynamics it generates.
J Phys a Math Gen, 1988
... Take as subalgebra the von Neumann algebra generated by the spectral family {e( B) I B Borel ... more ... Take as subalgebra the von Neumann algebra generated by the spectral family {e( B) I B Borel subset of R} of the element W(f) for some fixed O#ff X-, ie e“ de(( -CO, t]). W(f) = i ... Leggett AJ, Chakravarty S, Dorsey AT, Fisher MPA, Garg Anupam and Zwerger W 1987 Rec. Mod. ...
日本物理学会講演概要集 年会, Mar 15, 1996
Advances in Theoretical and Mathematical Physics, 1998
We show that the well-known translation invariant ground states and the recently discovered kink ... more We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-S ferromagnetic XXZ chains with Hamiltonian H = − x [S
Two discrete path integral formulations for the ground state of a spin-pinned quantum anisotropic... more Two discrete path integral formulations for the ground state of a spin-pinned quantum anisotropic XXZ Heisenberg chain are introduced. Their properties are discussed and two recursion relations are proved.
A continuum approximation for the excitations of the (1, 1, . . . , 1) interface in the quantum H... more A continuum approximation for the excitations of the (1, 1, . . . , 1) interface in the quantum Heisenberg model *
We supply the mathematical arguments required to complete the proofs of two previously published ... more We supply the mathematical arguments required to complete the proofs of two previously published results: Lieb-Robinson bounds for the dynamics of quantum lattice systems with unbounded on-site terms in the Hamiltonian and the existence of the thermodynamic limit of the dynamics of such systems.
Some recent developments in the theory of quantum spin systems are reviewed.
Lett Math Phys, 1992
ABSTRACT
We prove that the spectral gap of the spin-1/2 ferromagnetic XXZ chain with Hamiltonian $H=-\sum_... more We prove that the spectral gap of the spin-1/2 ferromagnetic XXZ chain with Hamiltonian $H=-\sum_x S^{(1)}_xS^{(1)}_{x+1}+S^{(2)}_xS^{(2)}_{x+1} +\Delta S^{(3)}_xS^{(3)}_{x+1}$, is given by $\Delta-1$ for all $\Delta\geq 1$. This is the gap in the spectrum of the infinite chain in any of its ground states, the translation invariant ones as well as the kink ground states, which contain an interface between an
QP-PQ: Quantum Probability and White Noise Analysis, 1992
ABSTRACT
Journal of Statistical Physics, 2016
We analyze a class of quantum spin models defined on half-spaces in the d-dimensional hypercubic ... more We analyze a class of quantum spin models defined on half-spaces in the d-dimensional hypercubic lattice bounded by a hyperplane with inward unit normal vector m ∈ R d . The family of models was previously introduced as the single species Product Vacua with Boundary States (PVBS) model, which is a spin-1/2 model with a XXZ-type nearest neighbor interactions depending on parameters λ j ∈ (0, ∞), one for each coordinate direction. For any given values of the parameters, we prove an upper bound for the spectral gap above the unique ground state of these models, which vanishes for exactly one direction of the normal vector m. For all other choices of m we derive a positive lower bound of the spectral gap, except for the case λ 1 = · · · = λ d = 1, which is known to have gapless excitations in the bulk.
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Papers by Bruno Nachtergaele