30 research outputs found
Entanglement Content of Quantum Particle Excitations I. Free Field Theory
We evaluate the entanglement entropy of a single connected region in excited states of one-dimensional massive free theories with finite numbers of particles, in the limit of large volume and region length. For this purpose, we use finite-volume form factor expansions of branch-point twist field two-point functions. We find that the additive contribution to the entanglement due to the presence of particles has a simple "qubit" interpretation, and is largely independent of momenta: it only depends on the numbers of groups of particles with equal momenta. We conjecture that at large momenta, the same result holds for any volume and region lengths, including at small scales. We provide accurate numerical verifications
Entanglement Dynamics after a Quench in Ising Field Theory: A Branch Point Twist Field Approach
We extend the branch point twist field approach for the calculation of entanglement entropies to time-dependent problems in 1+1-dimensional massive quantum field theories. We focus on the simplest example: a mass quench in the Ising field theory from initial mass m0 to final mass m. The main analytical results are obtained from a perturbative expansion of the twist field one-point function in the post-quench quasi-particle basis. The expected linear growth of the Rényi entropies at large times mt ≫ 1 emerges from a perturbative calculation at second order. We also show that the Rényi and von Neumann entropies, in infinite volume, contain subleading oscillatory contributions of frequency 2m and amplitude proportional to (mt)−3/2. The oscillatory terms are correctly predicted by an alternative perturbation series, in the pre-quench quasi-particle basis, which we also discuss. A comparison to lattice numerical calculations carried out on an Ising chain in the scaling limit shows very good agreement with the quantum field theory predictions. We also find evidence of clustering of twist field correlators which implies that the entanglement entropies are proportional to the number of subsystem boundary points
Classical and Quantum Solitons in the Symmetric Space Sine-Gordon Theories
We construct the soliton solutions in the symmetric space sine-Gordon
theories. The latter are a series of integrable field theories in
1+1-dimensions which are associated to a symmetric space F/G, and are related
via the Pohlmeyer reduction to theories of strings moving on symmetric spaces.
We show that the solitons are kinks that carry an internal moduli space that
can be identified with a particular co-adjoint orbit of the unbroken subgroup H
of G. Classically the solitons come in a continuous spectrum which encompasses
the perturbative fluctuations of the theory as the kink charge becomes small.
We show that the solitons can be quantized by allowing the collective
coordinates to be time-dependent to yield a form of quantum mechanics on the
co-adjoint orbit. The quantum states correspond to symmetric tensor
representations of the symmetry group H and have the interpretation of a fuzzy
geometric version of the co-adjoint orbit. The quantized finite tower of
soliton states includes the perturbative modes at the base.Comment: 53 pages, additional comments and small errors corrected, final
journal versio
On the perturbative S-matrix of generalized sine-Gordon models
Motivated by its relation to the Pohlmeyer reduction of AdS_5 x S^5
superstring theory we continue the investigation of the generalized sine-Gordon
model defined by SO(N+1)/SO(N) gauged WZW theory with an integrable potential.
Extending our previous work (arXiv:0912.2958) we compute the one-loop
two-particle S-matrix for the elementary massive excitations. In the N = 2 case
corresponding to the complex sine-Gordon theory it agrees with the charge-one
sector of the quantum soliton S-matrix proposed in hep-th/9410140. In the case
of N > 2 when the gauge group SO(N) is non-abelian we find a curious anomaly in
the Yang-Baxter equation which we interpret as a gauge artifact related to the
fact that the scattered particles are not singlets under the residual global
subgroup of the gauge group
g-Functions and gluon scattering amplitudes at strong coupling
We study gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills
theory at strong coupling by calculating the area of the minimal surfaces in
AdS_3 based on the associated thermodynamic Bethe ansatz system. The remainder
function of the amplitudes is computed by evaluating the free energy, the T-
and Y-functions of the homogeneous sine-Gordon model. Using conformal field
theory (CFT) perturbation, we examine the mass corrections to the free energy
around the CFT point corresponding to the regular polygonal Wilson loop. Based
on the equivalence between the T-functions and the g-functions, which measure
the boundary entropy, we calculate corrections to the T- and Y-functions as
well as express them at the CFT point by the modular S-matrix. We evaluate the
remainder function around the CFT point for 8 and 10-point amplitudes
explicitly and compare these analytic expressions with the 2-loop formulas. The
two rescaled remainder functions show very similar power series structures.Comment: 51 pages, 4 figures, v2: some comments and references added, based on
the published version, v3: minor change
Positivity, entanglement entropy, and minimal surfaces
The path integral representation for the Renyi entanglement entropies of
integer index n implies these information measures define operator correlation
functions in QFT. We analyze whether the limit , corresponding
to the entanglement entropy, can also be represented in terms of a path
integral with insertions on the region's boundary, at first order in .
This conjecture has been used in the literature in several occasions, and
specially in an attempt to prove the Ryu-Takayanagi holographic entanglement
entropy formula. We show it leads to conditional positivity of the entropy
correlation matrices, which is equivalent to an infinite series of polynomial
inequalities for the entropies in QFT or the areas of minimal surfaces
representing the entanglement entropy in the AdS-CFT context. We check these
inequalities in several examples. No counterexample is found in the few known
exact results for the entanglement entropy in QFT. The inequalities are also
remarkable satisfied for several classes of minimal surfaces but we find
counterexamples corresponding to more complicated geometries. We develop some
analytic tools to test the inequalities, and as a byproduct, we show that
positivity for the correlation functions is a local property when supplemented
with analyticity. We also review general aspects of positivity for large N
theories and Wilson loops in AdS-CFT.Comment: 36 pages, 10 figures. Changes in presentation and discussion of
Wilson loops. Conclusions regarding entanglement entropy unchange
TBA for non-perturbative moduli spaces
Recently, an exact description of instanton corrections to the moduli spaces
of 4d N=2 supersymmetric gauge theories compactified on a circle and Calabi-Yau
compactifications of Type II superstring theories was found. The equations
determining the instanton contributions turn out to have the form of
Thermodynamic Bethe Ansatz. We explore further this relation and, in
particular, we identify the contact potential of quaternionic string moduli
space with the free energy of the integrable system and the Kahler potential of
the gauge theory moduli space with the Yang-Yang functional. We also show that
the corresponding S-matrix satisfies all usual constraints of 2d integrable
models, including crossing and bootstrap, and derive the associated Y-system.
Surprisingly, in the simplest case the Y-system is described by the MacMahon
function relevant for crystal melting and topological strings.Comment: 25 pages, 1 figur
Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS_3
We study classical open string solutions with a null polygonal boundary in
AdS_3 in relation to gluon scattering amplitudes in N=4 super Yang-Mills at
strong coupling. We derive in full detail the set of integral equations
governing the decagonal and the dodecagonal solutions and identify them with
the thermodynamic Bethe ansatz equations of the homogeneous sine-Gordon models.
By evaluating the free energy in the conformal limit we compute the central
charges, from which we observe general correspondence between the polygonal
solutions in AdS_n and generalized parafermions.Comment: 25 pages, 4 figures, v2: a figure and references added, minor
corrections, v3: references added, minor corrections, to appear in JHE
One-point functions in massive integrable QFT with boundaries
We consider the expectation value of a local operator on a strip with
non-trivial boundaries in 1+1 dimensional massive integrable QFT. Using finite
volume regularisation in the crossed channel and extending the boundary state
formalism to the finite volume case we give a series expansion for the
one-point function in terms of the exact form factors of the theory. The
truncated series is compared with the numerical results of the truncated
conformal space approach in the scaling Lee-Yang model. We discuss the
relevance of our results to quantum quench problems.Comment: 43 pages, 20 figures, v2: typos correcte
The Relativistic Avatars of Giant Magnons and their S-Matrix
The motion of strings on symmetric space target spaces underlies the
integrability of the AdS/CFT correspondence. Although these theories, whose
excitations are giant magnons, are non-relativistic they are classically
equivalent, via the Polhmeyer reduction, to a relativistic integrable field
theory known as a symmetric space sine-Gordon theory. These theories can be
formulated as integrable deformations of gauged WZW models. In this work we
consider the class of symmetric spaces CP^{n+1} and solve the corresponding
generalized sine-Gordon theories at the quantum level by finding the exact
spectrum of topological solitons, or kinks, and their S-matrix. The latter
involves a trignometric solution of the Yang-Baxer equation which exhibits a
quantum group symmetry with a tower of states that is bounded, unlike for
magnons, as a result of the quantum group deformation parameter q being a root
of unity. We test the S-matrix by taking the semi-classical limit and comparing
with the time delays for the scattering of classical solitons. We argue that
the internal CP^{n-1} moduli space of collective coordinates of the solitons in
the classical theory can be interpreted as a q-deformed fuzzy space in the
quantum theory. We analyse the n=1 case separately and provide a further test
of the S-matrix conjecture in this case by calculating the central charge of
the UV CFT using the thermodynamic Bethe Ansatz.Comment: 33 pages, important correction to S-matrix to ensure crossing
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