Correlation Functions Of Local Operators Research Articles

Reduced density matrix after a quantum quench

We consider the reduced density matrix (RDM) \rho_A(t) for a finite subsystem A after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of \rho_A(t) is described by the RDM \rho_{GGE,A} of a generalized Gibbs ensemble. Here we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question, how \rho_A(t) approaches \rho_{GGE,A} as a function of time. To that end we introduce a distance on the space of density matrices and show that it approaches zero as a universal power-law t^{-3/2} in time. As the RDM completely determines all local observables within A, this provides information on the relaxation of correlation functions of local operators. We then address the issue, of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.

Perturbative correlation functions of null Wilson loops and local operators

We consider the correlation function of a null Wilson loop with four edges and a local operator in planar MSYM. By applying the insertion procedure, developed for correlation functions of local operators, we give an integral representation for the result at one and two loops. We compute explicitly the one loop result and show that the two loop result is finite.

Invariant-based entanglement monotones as expectation values and their experimental detection

Characterization and quantification of multipartite entanglement is one of the challenges in state-of-the-art experiments in quantum-information processing. According to theory, this is achieved via entanglement monotones, that is, functions that do not increase under stochastic local operations and classical communication (SLOCC). Typically such monotones include the wave function and its time reversal (antilinear-operator formalism) or they are based on not completely positive maps (e.g., partial transpose). Therefore, they are not directly accessible to experimental observations. We show how entanglement monotones derived from polynomial local SL$(2,\phantom{\rule{2.40005pt}{0ex}}\mathrm{width}.04\mathrm{em}\mathrm{height}1.46\mathrm{ex}\mathrm{depth}\ensuremath{-}.07\mathrm{ex}\mathrm{C})$ invariants can be re-written in terms of expectation values of observables. Consequently, the amount of entanglement---of specific SLOCC classes---in a given state can be extracted from the measurement of correlation functions of local operators.

Three-point correlation functions from semiclassical circular strings

The strong-coupling limit of three-point correlation functions of local operators can be analyzed beyond the supergravity regime using vertex operators representing spinning string states. When two of the vertex operators correspond to heavy string states having large quantum numbers, while the third operator corresponds to a light state with fixed charges, the correlator can be computed in the large string tension limit by means of a semiclassical approximation. We study the case when the heavy string states are circular string solutions with one AdS5 spin and three different angular momenta along S5, for several choices of the light string state.

On correlation functions of operators dual to classical spinning string states

We explore how to compute, classically at strong coupling, correlation functions of local operators corresponding to classical spinning string states. The picture we obtain is of ‘fattened’ Witten diagrams, the evaluation of which turns out to be surprisingly subtle and requires a modification of the naive classical action due to a necessary projection onto appropriate wave functions. We examine string solutions which compute the simplest case of a two-point function and reproduce the right scaling with the anomalous dimensions corresponding to the energies of the associated spinning string solutions. We also describe, under some simplifying assumptions, how the spacetime dependence of a conformal three-point correlation function arises in this setup.

Remarks on holography with broken Lorentz invariance

Recently a family of solutions of Einstein equations in backgrounds with broken Lorentz invariance was found. We show that the gravitational solution recently obtained by Kachru et al. is a part of the former solution which was derived earlier in the framework of extra-dimensional theories. We show how the energy-momentum and Einstein tensors are related and establish a correspondence between parameters which govern Lorentz invariance violation. Then we demonstrate that scaling behavior of two point correlation functions of local operators in scalar field theory is reproduced correctly for two cases with critical values of scaling parameters. Therefore, we complete the dictionary of 'tree-level' duality for all known solutions of the bulk theory. In the end we speculate on relations between renormalization group flow of a boundary theory and asymptotic behavior of gravitational solutions in the bulk.

Lieb-Schultz-Mattis theorem for quasitopological systems

In this paper we address the question of the existence of a spectral gap in a class of local Hamiltonians. These Hamiltonians have the following properties: their ground states are known exactly; all equal-time correlation functions of local operators are short-ranged; and correlation functions of certain nonlocal operators are critical. A variational argument shows gaplessness with $\ensuremath{\omega}\ensuremath{\propto}{k}^{2}$ at critical points defined by the absence of certain terms in the Hamiltonian, which is remarkable because equal-time correlation functions of local operators remain short ranged. We call such critical points, in which spatial and temporal scaling are radically different, quasitopological. When these terms are present in the Hamiltonian, the models are in gapped topological phases which are of special interest in the context of topological quantum computation.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program — II

We study the two-dimensional twisted (0, 2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence—at the level of the holomorphic chiral algebra—between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical -algebras and a level relation which underlie a geometric Langlands correspondence for G = SL(N, C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G = SL(N, C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G = SL(N, C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

Gauging spacetime symmetries on the worldsheet and the geometric Langlands program

We study the two-dimensional twisted (0,2) sigma-model on various smooth complex flag manifolds G/B, and explore its relevance to the geometric Langlands program. We find that an equivalence - at the level of the holomorphic chiral algebra - between a bosonic string on G/B and a B-gauged version of itself on G, will imply an isomorphism of classical W-algebras and a level relation which underlie a geometric Langlands correspondence for G=SL(N,C). This furnishes an alternative physical interpretation of the geometric Langlands correspondence for G=SL(N,C), to that demonstrated earlier by Kapustin and Witten via an electric-magnetic duality of four-dimensional gauge theory. Likewise, the Hecke operators and Hecke eigensheaves will have an alternative physical interpretation in terms of the correlation functions of local operators in the holomorphic chiral algebra of a quasi-topological sigma-model without boundaries. A forthcoming paper will investigate the interpretation of a ``quantum'' geometric Langlands correspondence for G=SL(N,C) in a similar setting, albeit with fluxes of the sigma-model moduli which induce a ``quantum'' deformation of the relevant classical algebras turned on.

Cluster Property and Robustness of Ground States of Interacting Many Bosons

We study spatial correlation functions of local operators of interacting many bosons confined in a box of a large, but volume V, for various `ground states' whose energy densities are almost degenerate. The ground states include the coherent state of interacting bosons (CSIB), the number state of interacting bosons (NSIB), and the number-phase squeezed state of interacting bosons, which interpolates between the CSIB and NSIB. It was shown previously that only the CSIB is robust (i.e., does not decohere for a macroscopically long time) against the leakage of bosons into an environment. We show that for the CSIB the spatial correlation of any local operators A(r) and B(r') (which are localized around r and r', respectively) vanishes as |r - r' | \sim V^{1/3} \to \infty, i.e., the CSIB has the `cluster property.' In contrast, the other ground states do not possess the cluster property. Therefore, we have successfully shown that the robust state has the cluster property. This ensures the consistency of the field theory of bosons with macroscopic theories.

Quantum groups, correlation functions and infrared divergences

We show in two simple examples that for one-dimensional quantum chains with quantum group symmetries, the hamiltonian being nonhermitian in the massless regime, the correlation functions of local operators are, in general, infrared divergent. If one considers, however, correlation functions invariant under the quantum group, the divergences cancel out.

THREE-POINT FUNCTIONS OF NON-UNITARY MINIMAL MATTER COUPLED TO GRAVITY

The tree-level three-point correlation functions of local operators in the general (p, q) minimal models coupled to gravity are calculated in the continuum approach. On one hand, the result agrees with the unitary series (q=p+1); and on the other hand, for p=2, q=2k−1, we find agreement with the one-matrix model results.

Double-scaled field theory at c = 1

We investigate the double-scaled free fermion theory of the c = 1 matrix model. We compute correlation functions of the eigenvalue density field and compare with the predictions of a relativistic boson theory. The c = 1 theory behaves as a relativistic theory at long distances, but has softer behavior at short distances. The soft short-distance behavior is closely related to the breakdown of the topological expansion at high energies. We also compute macroscopic loop amplitudes at c = 1, finding an integral representation for n-loop amplitudes to all orders of perturbation theory. We evaluate the integrals explicitly for two, three, and four macroscopic loops. The small loop length asymptotic expansion then gives correlation functions of local operators in the theory. The two-macroscopic-loop formula gives information on the spectrum and wave functions in the theory. The three- and four-loop amplitudes give scattering amplitudes for tachyon operators to all orders of perturbation theory. Again, the topological expansion breaks down at high energies. We compare our amplitudes with predictions from the Liouville theory.