SciPost Phys. 10, 007 (2021) ·
published 13 January 2021
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We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.
SciPost Phys. 8, 034 (2020) ·
published 3 March 2020
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The non-equilibrium steady states of integrable models are believed to be
described by the Generalized Gibbs Ensemble (GGE), which involves all local and
quasi-local conserved charges of the model. In this work we investigate
integrable lattice models solvable by the nested Bethe Ansatz, with group
symmetry $SU(N)$, $N\ge 3$. In these models the Bethe Ansatz involves various
types of Bethe rapidities corresponding to the "nesting" procedure, describing
the internal degrees of freedom for the excitations. We show that a complete
set of charges for the GGE can be obtained from the known fusion hierarchy of
transfer matrices. The resulting charges are quasi-local in a certain regime in
rapidity space, and they completely fix the rapidity distributions of each
string type from each nesting level.
SciPost Phys. 6, 002 (2019) ·
published 8 January 2019
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We construct gauge theories with a vector-multiplet and hypermultiplets of
$(1,0)$ supersymmetry on the six-sphere. The gauge coupling on the sphere
depends on the polar angle. This has a natural explanation in terms of the
tensor branch of $(1,0)$ theories on the six-sphere. For the vector-multiplet
we give an off-shell formulation for all supersymmetries. For hypermultiplets
we give an off-shell formulation for one supersymmetry. We show that the path
integral for the vector-multiplet localizes to solutions of the
Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self
duality condition to higher dimensions. For the hypermultiplet, the path
integral localizes to configurations where the field strengths of two complex
scalars are related by an almost complex structure.
SciPost Phys. 5, 032 (2018) ·
published 12 October 2018
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We investigate the short-time regime of the KPZ equation in $1+1$ dimensions
and develop a unifying method to obtain the height distribution in this regime,
valid whenever an exact solution exists in the form of a Fredholm Pfaffian or
determinant. These include the droplet and stationary initial conditions in
full space, previously obtained by a different method. The novel results
concern the droplet initial condition in a half space for several Neumann
boundary conditions: hard wall, symmetric, and critical. In all cases, the
height probability distribution takes the large deviation form $P(H,t) \sim
\exp( - \Phi(H)/\sqrt{t})$ for small time. We obtain the rate function
$\Phi(H)$ analytically for the above cases. It has a Gaussian form in the
center with asymmetric tails, $|H|^{5/2}$ on the negative side, and $H^{3/2}$
on the positive side. The amplitude of the left tail for the half-space is
found to be half the one of the full space. As in the full space case, we find
that these left tails remain valid at all times. In addition, we present here
(i) a new Fredholm Pfaffian formula for the solution of the hard wall boundary
condition and (ii) two Fredholm determinant representations for the solutions
of the hard wall and the symmetric boundary respectively.
SciPost Phys. 4, 037 (2018) ·
published 25 June 2018
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Motivated by the calculation of correlation functions in inhomogeneous
one-dimensional (1d) quantum systems, the 2d Inhomogeneous Gaussian Free Field
(IGFF) is studied and solved. The IGFF is defined in a domain $\Omega \subset
\mathbb{R}^2$ equipped with a conformal class of metrics $[{\rm g}]$ and with a
real positive coupling constant $K: \Omega \rightarrow \mathbb{R}_{>0}$ by the
action $\mathcal{S}[h] = \frac{1}{8\pi } \int_\Omega \frac{\sqrt{{\rm g}} d^2
{\rm x}}{K({\rm x})} \, {\rm g}^{i j} (\partial_i h)(\partial_j h)$. All
correlations functions of the IGFF are expressible in terms of the Green's
functions of generalized Poisson operators that are familiar from 2d
electrostatics in media with spatially varying dielectric constants.
This formalism is then applied to the study of ground state correlations of
the Lieb-Liniger gas trapped in an external potential $V(x)$. Relations with
previous works on inhomogeneous Luttinger liquids are discussed. The main
innovation here is in the identification of local observables $\hat{O} (x)$ in
the microscopic model with their field theory counterparts $\partial_x h, e^{i
h(x)}, e^{-i h(x)}$, etc., which involve non-universal coefficients that
themselves depend on position --- a fact that, to the best of our knowledge,
was overlooked in previous works on correlation functions of inhomogeneous
Luttinger liquids ---, and that can be calculated thanks to Bethe Ansatz form
factors formulae available for the homogeneous Lieb-Liniger model. Combining
those position-dependent coefficients with the correlation functions of the
IGFF, ground state correlation functions of the trapped gas are obtained.
Numerical checks from DMRG are provided for density-density correlations and
for the one-particle density matrix, showing excellent agreement.
