We investigate conformal field theories with gauge group U($\textit{N}$) at arbitrary rank $\textit{N}$, focusing on the role of trace relations in determining the structure of the Hilbert space. Working in the free trace algebra without imposing relations, we identify a class of evanescent states that vanish at finite $\textit{N}$. Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge $\textit{Q}$ . This framework allows us to define and compute transition amplitudes between evanescent and physical states, which we show correspond precisely to ordinary CFT amplitudes analytically continued in $\textit{N}$. Our results provide a direct algebraic realization of the proposals which realize trace relations in the bulk as over-maximal giant gravitons [1–3] and establish analytic continuation in $\textit{N}$ as a powerful tool for understanding finite-$\textit{N}$ effects.

Japan

Poland

Sweden

Pawel Caputa

China

South Africa

Robert Mello Koch

Gauge invariants at arbitrary $\textit{N}$ and trace relations

In this work, we investigate local quench dynamics in two-dimensional conformal field theories using Krylov space methods. We derive Lanczos coefficients, spread complexity, and Krylov entropies for local joining and splitting quenches in theories on an infinite line, a circle, a finite interval, and at finite temperature. We examine how these quantities depend on the central charge of the underlying conformal field theory and find that both spread complexity and Krylov entropy are proportional to it. Interestingly, Krylov entropies evolve logarithmically with time, mirroring standard entanglement entropies, making them useful for extracting the central charge. In the large central charge limit, using holography, we establish a connection between the rate of spread complexity and the proper momentum of the tip of the end-of-the world brane, which probes the bulk analogously to a point particle. Our results further demonstrate that spread complexity and Krylov entropies are powerful tools for probing non-equilibrium dynamics of interacting quantum systems.

Giuseppe Giulio

Local quenches from a Krylov perspective

In this work, we develop a generalisation of the thermal entropy to complex inverse temperatures, which we call the thermal pseudo-entropy. We show that this quantity represents the pseudo-entropy of the transition matrix between Thermofield Double states at different times. We have studied its properties in various quantum mechanical setups, Schwarzian theory, Random Matrix Theories, and 2D CFTs, including symmetric orbifolds. Our findings indicate a close relationship between the averaged thermal pseudo-entropy and the spectral form factor, which is instrumental in distinguishing chaotic and integrable models. Moreover, we have observed a logarithmic scaling of this quantity in models with a continuous spectrum, with a universal coefficient that is sensitive to the scaling of the density of states near the edge of the spectrum. Lastly, we found the connection between the real and imaginary parts of the thermal pseudo-entropy through the Kramers-Kronig relations.

Bowen Chen

Tadashi Takayanagi

Takashi Tsuda

Thermal pseudo-entropy

We point out an interesting connection between the mathematical framework of the Krylov basis, which is used to quantify quantum complexity, and the entanglement entropy in high energy QCD. In particular, we observe that the cascade equation of the dipole model is equivalent to the <math><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></math> Schrödinger equation in the Krylov basis. Consequently, the Krylov complexity corresponds to the average distribution of partons and the Krylov entropy is the counterpart of the entanglement entropy computations of [D. E. Kharzeev and E. M. Levin, Phys. Rev. D 95, 114008 (2017).]. Our work not only brings new tools for exploring quantum information and complexity in QCD, but also gives hope for experimental tests of some of the recent, physical probes of quantum complexity.

Krzysztof Kutak

Krylov complexity and gluon cascades in the high energy limit

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity (C  K ) and Spread complexity (C  S ) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between C  K  and 2C  S . Furthermore, for maximally entangled pure states, we find that the moment-generating function of C  K  becomes the Spectral Form Factor and, at late-times, C  K  is simply related to NC  S  for N ≥ 2 within the N-dimensional Hilbert space. Notably, we confirm that C  K  = 2C  S  holds across all times when N = 2. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

Spain

Hyun-Sik Jeong

Sinong Liu

Juan Pedraza

Le-Chen Qu

Krylov complexity of density matrix operators

We investigate the complexity of states and operators evolved with the modular Hamiltonian by using the Krylov basis. In the first part, we formulate the problem for states and analyze different examples, including quantum mechanics, two-dimensional conformal field theories and random modular Hamiltonians, focusing on relations with the entanglement spectrum. We find that the modular Lanczos spectrum provides a different approach to quantum entanglement, opening new avenues in many-body systems and holography. In the second part, we focus on the modular evolution of operators and states excited by local operators in two-dimensional conformal field theories. We find that, at late modular time, the spread complexity is universally governed by the modular Lyapunov exponent <math><msubsup><mi>λ</mi><mi>L</mi><mtext>mod</mtext></msubsup><mo>=</mo><mn>2</mn><mi>π</mi></math> and is proportional to the local temperature of the modular Hamiltonian. Our analysis provides explicit examples where entanglement entropy is indeed not enough; however the entanglement spectrum is, and encodes the same information as complexity.

Argentina

Javier M. Magan

Dimitrios Patramanis

Italy

Erik Tonni

Krylov complexity of modular Hamiltonian evolution

A correction to this paper has been published: https://doi.org/10.1007/JHEP04(2022)081

Germany

Hugo Camargo

India

Pratik Nandy

Erratum to: Q-curvature and path integral complexity

In this work we study families of generalised coherent states constructed from SL(2,R) subalgebras of the Virasoro algebra in two-dimensional conformal field theories. We derive the energy density and entanglement entropy and discuss their equivalence with analogous quantities computed in locally excited states. Moreover, we analyze their dual, holographic geometries and reproduce entanglement entropies from the Ryu-Takayanagi prescription. Finally, we outline possible applications of this universal class of states to operator growth and inhomogeneous quenches.

Dongsheng Ge

Entanglement and geometry from subalgebras of the Virasoro algebra

A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor — the Kitaev chain — as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.

Nitin Gupta

S. Haque

Jeff Murugan

Hendrik Zyl

Spread complexity and topological transitions in the Kitaev chain

A correction to this paper has been published: https://doi.org/10.1007/JHEP07(2021)016

Jan Boruch

Erratum to: Holographic path-integral optimization