Evolution of complexity following a quantum quench in free field theory

We expand on the investigation of the universal scaling properties in the\nearly time behaviour of fast but smooth quantum quenches in a general\n$d$-dimensional conformal field theory deformed by a relevant operator of\ndimension $\\Delta$ with a time-dependent coupling. The quench consists of\nchanging the coupling from an initial constant value $\\lambda_1$ by an amount\nof the order of $\\delta \\lambda$ to some other final value $\\lambda_2$, over a\ntime scale $\\delta t$. In the fast quench limit where $\\delta t$ is smaller\nthan all other length scales in the problem, $ \\delta t \\ll\n\\lambda_1^{1/(\\Delta-d)}, \\lambda_2^{1/(\\Delta-d)}, \\delta\n\\lambda^{1/(\\Delta-d)}$, the energy (density) injected into the system scales\nas $\\delta{\\cal E} \\sim (\\delta \\lambda)^2 (\\delta t)^{d-2\\Delta}$. Similarly,\nthe change in the expectation value of the quenched operator at times earlier\nthan the endpoint of the quench scales as $<{\\cal O}_\\Delta> \\sim \\delta\n\\lambda (\\delta t)^{d-2\\Delta}$, with further logarithmic enhancements in\ncertain cases. While these results were first found in holographic studies, we\nrecently demonstrated that precisely the same scaling appears in fast mass\nquenches of free scalar and free fermionic field theories. As we describe in\ndetail, the universal scaling refers to renormalized quantities, in which the\nUV divergent pieces are consistently renormalized away by subtracting\ncounterterms derived with an adiabatic expansion. We argue that this scaling\nlaw is a property of the conformal field theory at the UV fixed point, valid\nfor arbitrary relevant deformations and insensitive to the details of the\nquench protocol. Our results highlight the difference between smooth fast\nquenches and instantaneous quenches where the Hamiltonian abruptly changes at\nsome time.\n

In this article, we review some illustrative results in the study of two related toy models for quantum gravity, namely cylindrical waves (which are cylindrically symmetric gravitational fields)and parametrized field theory (which is just free scalar field theory on a flat space–time in generally covariant disguise). In the former, we focus on the phenomenon of unexpected large quantum gravity effects in regions of weak classical gravitational fields and on an analysis of causality in a quantum geometry. In the latter, we focus on Dirac quantization, argue that this is related to the unitary implementability of free scalar field evolution along curved foliations of the flat space–time and review the relevant results for unitary implementability.

We consider the algebras generated by observables in quantum field theory localized in regions in the null plane. For a scalar free field theory, we show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres and the modular operator decomposes accordingly. This implies that a certain form of QNEC is valid in free fields involving the causal completions of half-spaces on the null plane (null cuts). We also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states.

We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massless Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the higher dimensional Legendre differential operator. The quadratic form of the massless modular Hamiltonian is expressed in terms of an integral of the energy density with the parabolic distribution. We then get the formula for the local entropy of a wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we provide the passivity characterisation of the modular Hamiltonian within the standard subspace setup.

We describe an analog model for quantum gravity effects in condensed matter physics. The situation discussed is that of phonons propagating in a fluid with a random velocity wave equation. We consider that there are random fluctuations in the reciprocal of the bulk modulus of the system and study free phonons in the presence of Gaussian colored noise with zero mean. We show that, in this model, after performing the random averages over the noise function a free conventional scalar quantum field theory describing free phonons becomes a self-interacting model.

We compute the Krylov Complexity of a light operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where ${S}_{\mathrm{out}}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.

We numerically compute the Rényi entropy for four-dimensional free scalar field theory with a spherical entangling surface. As is well known, the Rényi entropy as a function of the boundary area exhibits linear dependence in the leading order. The coefficient of the subleading logarithmic term from our numerical data, as a function of the Rényi order q, agrees nicely with the general prediction of conformal field theory computation. The motivation of this work is also partly to see how the efficiency of numerical computation changes as a function of q. For q<1 the summation over eigenvalues of reduced density matrix takes longer since the series converges more slowly than for q=1. For q>1 the convergence is faster, but the relative error becomes large as a general trend.

Black hole physics and Liouville theory

We investigate the first law of complexity proposed in Bernamonti et al (2019 Phys. Rev. Lett. 123 081601), i.e., the variation of complexity when the target state is perturbed, in more detail. Based on Nielsen’s geometric approach to quantum circuit complexity, we find the variation only depends on the end of the optimal circuit. We apply the first law to gain new insights into the quantum circuits and complexity models underlying holographic complexity. In particular, we examine the variation of the holographic complexity for both the complexity = action and complexity = volume conjectures in perturbing the AdS vacuum with coherent state excitations of a free scalar field. We also examine the variations of circuit complexity produced by the same excitations for the free scalar field theory in a fixed AdS background. In this case, our work extends the existing treatment of Gaussian coherent states to properly include the time dependence of the complexity variation. We comment on the similarities and differences of the holographic and QFT results.

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

The undisturbed sound field (the sound field that exists without the worker present) is the reference for many damage-risk criterions. The introduction of the worker and a body-mounted microphone perturbs the sound field. The resultant sound field measurement differs from that of the undisturbed sound field. A number of researchers have examined microphone placement effects, however, these have only been conducted at limited azmithual angles in free field. A series of experiments was conducted to explore the interaction of a bodily worn measurement microphone, including the reflections and shielding effects, with the undisturbed field in both reverberant and free sound fields. New data are presented with respect to microphone placement information for 10-deg angle increments for a cohort of various size subjects (male and female). The new microphone placement data were incorporated into a previously published mathematical model to determine the accuracy of a measurement conducted with a body-worn noise dosimeter. Results indicate that body-worn acoustical instruments are accurate to within ±2.0 dB with respect to the undisturbed sound field provided the microphone is located on the shoulder of the most exposed ear.

In order to determine the radiated sound power of an underwater source below the Schroeder cut-off frequency in a non-anechoic tank, a low-frequency extension measuring technique is proposed. This technique is based on a unique relationship between the transmission characteristics of the enclosed field and those of the free field, which can be obtained as a correction term based on previous measurements of a known simple source. The radiated sound power of an unknown underwater source in the free field can thereby be obtained accurately from measurements in a non-anechoic tank. To verify the validity of the proposed technique, a mathematical model of the enclosed field is established using normal-mode theory, and the relationship between the transmission characteristics of the enclosed and free fields is obtained. The radiated sound power of an underwater transducer source is tested in a glass tank using the proposed low-frequency extension measuring technique. Compared with the free field, the radiated sound power level of the narrowband spectrum deviation is found to be less than 3 dB, and the 1/3 octave spectrum deviation is found to be less than 1 dB. The proposed testing technique can be used not only to extend the low-frequency applications of non-anechoic tanks, but also for measurement of radiated sound power from complicated sources in non-anechoic tanks.

On the basis of canonical quantization of free fields, by means of the interaction picture which expresses the evolution of interacting fields in terms of the free field, the scattering matrix is first expressed in Dyson's formula as a perturbation series in a free-field Hamiltonian, then converted to Wick's operator expansion, and finally reduced to matrix elements and the Feynman perturbation series.

This thesis presents various examples of the application of quantum-mechanical methods to the understanding of the structure of space-time. It focuses on noncommutative geometry and the gauge/gravity duality as intermediaries between quantum mechanics and classical geometry. First, we numerically calculate entanglement entropy and mutual information for a massive free scalar field on commutative and noncommutative (fuzzy) spheres. To define a subregion with a well-defined boundary in the noncommutative geometry, we use the symbol map between elements of the noncommutative algebra and functions on the sphere. We show that the UV-divergent part of the entanglement entropy on a fuzzy sphere does not follow an area law. In agreement with holographic predictions, it is extensive for small (but fixed) regions. This is true even in the limit of small noncommutativity. Nonetheless, we find that mutual information (which is UV-finite) is the same in both theories. This suggests that nonlocality at short distances does not affect quantum correlations over large distances in a free field theory. Previous work has shown that a surface embedded in flat ℝ³ can be associated to any three Hermitian matrices. By constructing coherent states corresponding to points in the emergent geometry, we study this emergent surface when the matrices are large. We find that the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes. Finally, we use the gauge/gravity correspondence to translate the positivity of relative entropy on the boundary into constraints on allowable space-time metrics in the bulk. Using the Einstein equations, we interpret these constraints as energy conditions. For certain three-dimensional bulks, we obtain strict constraints coming from the positivity of relative entropy with a thermal reference state which turn out to be equivalent to a version of the weak energy condition near the boundary. In higher dimensions, we use the canonical energy formalism to obtain similar but weaker constraints.