93,135 research outputs found
Exact Matrix Product States for Quantum Hall Wave Functions
We show that the model wave functions used to describe the fractional quantum
Hall effect have exact representations as matrix product states (MPS). These
MPS can be implemented numerically in the orbital basis of both finite and
infinite cylinders, which provides an efficient way of calculating arbitrary
observables. We extend this approach to the charged excitations and numerically
compute their Berry phases. Finally, we present an algorithm for numerically
computing the real-space entanglement spectrum starting from an arbitrary
orbital basis MPS, which allows us to study the scaling properties of the
real-space entanglement spectra on infinite cylinders. The real-space
entanglement spectrum obeys a scaling form dictated by the edge conformal field
theory, allowing us to accurately extract the two entanglement velocities of
the Moore-Read state. In contrast, the orbital space spectrum is observed to
scale according to a complex set of power laws that rule out a similar
collapse.Comment: 10 pages and Appendix, v3 published versio
Quartic Gauge Couplings from K3 Geometry
We show how certain F^4 couplings in eight dimensions can be computed using
the mirror map and K3 data. They perfectly match with the corresponding
heterotic one-loop couplings, and therefore this amounts to a successful test
of the conjectured duality between the heterotic string on T^2 and F-theory on
K3. The underlying quantum geometry appears to be a 5-fold, consisting of a
hyperk"ahler 4-fold fibered over a IP^1 base. The natural candidate for this
fiber is the symmetric product Sym^2(K3). We are lead to this structure by
analyzing the implications of higher powers of E_2 in the relevant Borcherds
counting functions, and in particular the appropriate generalizations of the
Picard-Fuchs equations for the K3.Comment: 32 p, harvmac; One footnote on page 11 extended; results unchanged;
Version subm. to ATM
BKM Lie superalgebra for the Z_5 orbifolded CHL string
We study the Z_5-orbifolding of the CHL string theory by explicitly
constructing the modular form tilde{Phi}_2 generating the degeneracies of the
1/4-BPS states in the theory. Since the additive seed for the sum form is a
weak Jacobi form in this case, a mismatch is found between the modular forms
generated from the additive lift and the product form derived from threshold
corrections. We also construct the BKM Lie superalgebra, tilde{G}_5,
corresponding to the modular form tilde{Delta}_1 (Z) = tilde{Phi}_2 (Z)^{1/2}
which happens to be a hyperbolic algebra. This is the first occurrence of a
hyperbolic BKM Lie superalgebra. We also study the walls of marginal stability
of this theory in detail, and extend the arithmetic structure found by Cheng
and Dabholkar for the N=1,2,3 orbifoldings to the N=4,5 and 6 models, all of
which have an infinite number of walls in the fundamental domain. We find that
analogous to the Stern-Brocot tree, which generated the intercepts of the walls
on the real line, the intercepts for the N >3 cases are generated by linear
recurrence relations. Using the correspondence between the walls of marginal
stability and the walls of the Weyl chamber of the corresponding BKM Lie
superalgebra, we propose the Cartan matrices for the BKM Lie superalgebras
corresponding to the N=5 and 6 models.Comment: 30 pages, 2 figure
Bounds on the Power of Constant-Depth Quantum Circuits
We show that if a language is recognized within certain error bounds by
constant-depth quantum circuits over a finite family of gates, then it is
computable in (classical) polynomial time. In particular, our results imply
EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the
class EQP. On the other hand, we adapt and extend ideas of Terhal and
DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates
including Hadamard and CNOT gates, computing the acceptance probabilities of
depth-five circuits over F is just as hard as computing these probabilities for
circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P
where NQNC^0 is the constant-depth analog of the class NQP. This essentially
refutes a conjecture of Green et al. that NQACC is contained in TC^0
(quant-ph/0106017)
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