88 research outputs found
The infinite XXZ quantum spin chain revisited: Structure of low lying spectral bands and gaps
We study the structure of the spectrum of the infinite XXZ quantum spin
chain, an anisotropic version of the Heisenberg model. The XXZ chain
Hamiltonian preserves the number of down spins (or particle number), allowing
to represent it as a direct sum of N-particle interacting discrete
Schr\"odinger-type operators restricted to the fermionic subspace. In the Ising
phase of the model we use this representation to give a detailed determination
of the band and gap structure of the spectrum at low energy. In particular, we
show that at sufficiently strong anisotropy the so-called droplet bands are
separated from higher spectral bands uniformly in the particle number. Our
presentation of all necessary background is self-contained and can serve as an
introduction to the mathematical theory of the Heisenberg and XXZ quantum spin
chains.Comment: 32 pages, 3 figure
Extensions of dissipative operators with closable imaginary part
Given a dissipative operator on a complex Hilbert space
such that the quadratic form f\mapsto \mbox{Im}\langle f,Af\rangle is
closable, we give a necessary and sufficient condition for an extension of
to still be dissipative. As applications, we describe all maximally accretive
extensions of strictly positive symmetric operators and all maximally
dissipative extensions of a highly singular first-order operator on the
interval.Comment: 11 page
Entanglement bounds in the XXZ quantum spin chain
We consider the XXZ spin chain, characterized by an anisotropy parameter
, and normalized such that the ground state energy is and the
ground state given by the all spins up state. The energies , , can be interpreted as -cluster break-up
thresholds for down spin configurations. We show that, for every , the
bipartite entanglement of all states with energy below the -cluster
break-up satisfies a logarithmically corrected (or enhanced) area law. This
generalizes a result by Beaud and Warzel, who considered energies in the
droplet spectrum (i.e., below the 2-cluster break-up).
For general , we find an upper logarithmic bound with pre-factor .
We show that this constant is optimal in the Ising limit . Beaud
and Warzel also showed that after introducing a random field and disorder
averaging the enhanced area law becomes a strict area law, again for states in
the droplet regime. For the Ising limit with random field, we show that this
result does not extend beyond the droplet regime. Instead, we find states with
energies arbitrarily close to the -cluster break-up whose entanglement
satisfies a logarithmically growing lower bound with pre-factor .Comment: 35 pages, v2: small errors and typos correcte
On the Theory of Dissipative Extensions
We consider the problem of constructing dissipative extensions of given dissipative
operators.
Firstly, we discuss the dissipative extensions of symmetric operators and give a suffcient condition for when these extensions are completely non-selfadjoint. Moreover,
given a closed and densely defined operator A, we construct its closed extensions which we parametrize by suitable subspaces of D(A^*).
Then, we consider operators A and \widetilde{A} that form a dual pair, which means that A\subset \widetilde{A}^*, respectively \widetilde{A}\subset A^* Assuming that A and (-\widetilde{A}) are dissipative, we present a method of determining the proper dissipative extensions \widehat{A} of this dual pair, i.e. we determine all dissipative operators \widehat{A} such that A\subset \subset\widehat{A}\subset\widetilde{A}^* provided that D(A)\cap D(\widetilde{A}) is dense in H. We discuss applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators. Also, we investigate the stability of the numerical ranges of the various proper dissipative extensions of the dual pair (A,\widetilde{A}).
Assuming that zero is in the field of regularity of a given dissipative operator A, we then construct its Krein-von Neumann extension A_K, which we show to be maximally
dissipative. If there exists a dissipative operator (-\widetilde{A}) such that A and \widetilde{A} form a dual
pair, we discuss when A_K is a proper extension of the dual pair (A,\widetilde{A}) and if this is not
the case, we propose a construction of a dual pair (A_0,\widetilde{A}_0), where A_0\subset A and \widetilde{A}_0\subset\widetilde{A} such that A_K is a proper extension of (A_0,\widetilde{A}_0).
After this, we consider dual pairs (A, \widetilde{A}) of sectorial operators and construct proper sectorial extensions that satisfy certain conditions on their numerical range. We apply this result to positive symmetric operators, where we recover the theory of non-negative
selfadjoint and sectorial extensions of positive symmetric operators as described by Birman, Krein, Vishik and Grubb.
Moreover, for the case of proper extensions of a dual pair (A_0,\widetilde{A}_0)of sectorial operators, we develop a theory along the lines of the Birman-Krein-Vishik theory and define an order in the imaginary parts of the various proper dissipative extensions of (A,\widetilde{A}).
We finish with a discussion of non-proper extensions: Given a dual pair (A,\widetilde{A}) that satisfies certain assumptions, we construct all dissipative extensions of A that have domain contained in D(\widetilde{A}^*). Applying this result, we recover Crandall and Phillip's description
of all dissipative extensions of a symmetric operator perturbed by a bounded dissipative operator. Lastly, given a dissipative operator A whose imaginary part induces
a strictly positive closable quadratic form, we find a criterion for an arbitrary extension of A to be dissipative
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