In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL(2$\textit{,}$ ℝ) Hamiltonians. In the Vacuum state, we find that the K-Complexity exhibits a universal phase structure. The phase structure involves the K-Complexity exhibiting an oscillatory behaviour in the non-heating phase, which contrasts with the exponential growth observed in the heating phase, while it displays polynomial growth at the phase boundary. Furthermore, we extend our analysis to compute the K-Complexity of a light operator in excited states, considering both large-c CFT and free field theory. In the free field theory, we find a state-independent phase structure of K-Complexity. However, in the large-c CFT, the behaviour varies, with the K-Complexity once again displaying exponential growth in the heating phase and polynomial growth at the phase boundary. Notably, the precise exponent governing this growth depends on the heaviness of the state under examination. In the non-heating phase, we observe a transition in K-Complexity behaviour from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.

South Korea

Vinay Malvimat

India

Somnath Porey

Baishali Roy

Krylov complexity in 2$\textit{d}$ CFTs with SL(2$\textit{,}$ ℝ) deformed Hamiltonians

We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$ ,   <math> <mi>J</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{J}\overline{\textrm{T}} $$ , and   <math> <mi>J</mi> <mover> <mi>J</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{J}\overline{\textrm{J}} $$  deformations, focusing on first-order corrections in the deformation parameter. Under   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{T}\overline{\textrm{T}} $$  deformation, we demonstrate that the Lanczos coefficients b  n  exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both   <math> <mi>J</mi> <mover> <mi>J</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{J}\overline{\textrm{J}} $$  and   <math> <mi>J</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ \textrm{J}\overline{\textrm{T}} $$  deformations induce no first order correction to either the linear growth of Lanczos coefficients at large-n or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.

Belgium

Arghya Chattopadhyay

Arpita Mitra

Krylov complexity of deformed conformal field theories

We investigate the reflected entropy for various mixed state configurations in the two dimensional holographic conformal field theories sharing a common interface (ICFTs). In the AdS3/ICFT2 framework, we compute the holographic reflected entropy for the required configurations in the vacuum state of the ICFT2 which is given by twice the entanglement wedge cross section (EWCS) in a spacetime involving two AdS3 geometries glued along a thin interface brane. Subsequently, we evaluate the EWCS in the bulk geometry involving eternal BTZ black strings with an AdS2 interface brane, which is dual to an ICFT2 in the thermofield double (TFD) state. We explore the system from a doubly holographic perspective and determine the island contributions to the reflected entropy in the two dimensional semi-classical description involving two CFT2s coupled to an AdS2 brane. We demonstrate that the results from the island formula match precisely with the bulk AdS3 results in the large tension limit of the interface brane. We illustrate that the phase structure of the reflected entropy is quite rich involving many novel induced island phases and demonstrate that it obeys the expected Page curve for the reflected entropy in a radiation bath coupled to the AdS2 black hole.

Taiwan

Jaydeep Basak

Debarshi Basu

Himanshu Parihar

Gautam Sengupta

Holographic reflected entropy and islands in interface CFTs

We compute the Krylov Complexity of a light operator   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$   L  in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$   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   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$   H  is below or above the critical dimension h  H  = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS 3 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.

Arnab Kundu

United Kingdom

Ritam Sinha

State dependence of Krylov complexity in 2d CFTs

We compute the disruption of mutual information in a TFD state dual to a Kerr black hole with equal angular momenta in AdS 5 due to an equatorial shockwave. The shockwave respects the axi-symmetry of the Kerr geometry with specific angular momenta   <math> <msub> <mi>L</mi> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </msub> </math>  $$ {\mathcal{L}}_{\phi_1} $$  &amp;   <math> <msub> <mi>L</mi> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </msub> </math>  $$ {\mathcal{L}}_{\phi_2} $$ . The sub-systems considered are hemispheres in the left and the right dual CFTs with the equator of the S 3 as their boundary. We compute the change in the mutual information by determining the growth of the HRT surface at late times. We find that at late times leading up to the scrambling time the minimum value of the instantaneous Lyapunov index   <math> <msubsup> <mi>λ</mi> <mi>L</mi> <mfenced> <mo>min</mo> </mfenced> </msubsup> </math>  $$ {\lambda}_L^{\left(\min \right)} $$  is bounded by   <math> <mi>κ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>T</mi> <mi>H</mi> </msub> </mrow> <mfenced> <mrow> <mn>1</mn> <mo>−</mo> <mi>μ</mi> <msub> <mi>L</mi> <mo>+</mo> </msub> </mrow> </mfenced> </mfrac> </math>  $$ \kappa =\frac{2\pi {T}_H}{\left(1-\mu {\mathcal{L}}_{+}\right)} $$  and is found to be greater than 2πT  H  in certain regimes with T  H  and μ denoting the black hole’s temperature and the horizon angular velocity respectively while   <math> <msub> <mi>L</mi> <mo>+</mo> </msub> </math>  $$ {\mathcal{L}}_{+} $$  =   <math> <msub> <mi>L</mi> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </msub> </math>  $$ {\mathcal{L}}_{\phi_1} $$  +   <math> <msub> <mi>L</mi> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </msub> </math>  $$ {\mathcal{L}}_{\phi_2} $$ . We also find that for non-extremal geometries the null perturbation obeys   <math> <msub> <mi>L</mi> <mo>+</mo> </msub> </math>  $$ {\mathcal{L}}_{+} $$  &lt; μ  −1 for it to reach the outer horizon from the AdS boundary. The scrambling time at very late times is given by κτ * ≈ log   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  where   <math> <mi>S</mi> </math>  $$ \mathcal{S} $$  is the Kerr entropy. We also find that the onset of scrambling is delayed due to a term proportional to log(1 −   <math> <msub> <mrow> <mi>μ</mi> <mi>L</mi> </mrow> <mo>+</mo> </msub> </math>  $$ {\mu \mathcal{L}}_{+} $$ ) −1 which is not extensive and does not scale with the entropy of Kerr black hole.

Austria

Rohan Poojary

Fast scrambling of mutual information in Kerr-AdS5

We compute the disruption of mutual information between the hemispherical subsystems on the left and right conformal field theories of a thermofield double state described by a Kerr geometry in <math><msub><mi>AdS</mi><mn>4</mn></msub></math> due to shock waves along the equatorial plane. The shock waves and the subsystems considered respect the axisymmetry of the geometry. At late times, the disruption of the mutual information is given by the lengthening of the Hubeny-Rangamani-Takayanagi surface connecting the two subsystems; we compute the minimum value of the Lyapunov index-<math><msubsup><mi>λ</mi><mi>L</mi><mrow><mo>(</mo><mrow><mi>min</mi><mo>)</mo></mrow></mrow></msubsup></math> at late times and find that it is bounded by <math><mi>κ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi><msub><mi>T</mi><mi>H</mi></msub></mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>μ</mi><mi>L</mi><mo>)</mo></mrow></mfrac></math>, where <math><mi>μ</mi></math> is the horizon velocity, and <math><mi>L</mi></math> is the angular momentum per unit energy of the shock wave. At very late times, we find the scrambling time for such a system is governed by <math><mi>κ</mi></math> with <math><mi>κ</mi><msub><mi>t</mi><mo>*</mo></msub><mo>=</mo><mi>log</mi><mi>S</mi></math> for large black holes with a large entropy <math><mi>S</mi></math>. We also find a term that increases the scrambling time by <math><mi>log</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>μ</mi><mi>L</mi><msup><mo>)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> but which does not scale with the entropy of the Kerr geometry.

Rohan R. Poojary

Fast scrambling of mutual information in Kerr-<math><msub><mi>AdS</mi><mn>4</mn></msub></math> spacetime

We investigate the time evolution of reflected entropy and entanglement negativity for mixed state configurations involving two adjacent and disjoint intervals in the radiation flux of moving mirrors by utilizing the AdS/BCFT duality. These measures are computed for the required mixed state configurations by using the respective replica techniques in the large central charge limit of the BCFT 1+1. We demonstrate that the results obtained exactly agree with the corresponding holographic computations in the dual bulk AdS 3 geometry with an end of the world brane. In this context, the analogues of the Page curves for these measures are obtained for the required configurations in the radiation flux of kink and escaping mirrors which mimic the Hawking radiation from evaporating and eternal black holes respectively.

Reflected entropy and entanglement negativity for holographic moving mirrors

We study the perturbation due to rotating shockwaves in Banados-Teitelboim-Zanelli geometries at late times and analyze the change in mutual information between the two subsystems belonging to the dual <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>L</mi></mrow></msub></mrow></math> and <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>R</mi></mrow></msub></mrow></math>. We find that the scrambling of mutual information is in general governed by the Lyapunov index <math><msub><mi>λ</mi><mi>L</mi></msub></math> which is bounded by <math><mi>κ</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>β</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>μ</mi><mi>L</mi><mo>)</mo></mrow></mfrac><mo>≥</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>β</mi></mfrac></math>, where <math><mi>μ</mi><mo>=</mo><msub><mi>r</mi><mo>−</mo></msub><mo>/</mo><msub><mi>r</mi><mo>+</mo></msub></math> and <math><mi>L</mi></math> is the angular momentum of the shockwave. For the special case of <math><mi>L</mi><mo>=</mo><mn>1</mn></math> we find the mutual information analytically and show that it is characterized by <math><msub><mi>λ</mi><mi>L</mi></msub><mo>=</mo><mi>κ</mi><mo>/</mo><mn>2</mn></math> with the scrambling time for large black holes given as <math><mrow><msub><mrow><mi>t</mi></mrow><mrow><mo>*</mo></mrow></msub><mo>=</mo><mfrac><mrow><mi>β</mi><mo>(</mo><mn>1</mn><mo>−</mo><mi>μ</mi><mo>)</mo></mrow><mrow><mi>π</mi></mrow></mfrac><mi>log</mi><mi>S</mi></mrow></math>.

Fast scrambling due to rotating shockwaves in BTZ

We establish the large central charge behavior of the entanglement negativity for a mixed state configuration of a single interval enclosed between two intervals in a holographic <math><msub><mi>CFT</mi><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></msub></math>. To this end we utilize the monodromy technique to elucidate the large central charge limit of a four point twist correlator which characterizes the negativity of the mixed state configuration. Subsequently we analyze the large central charge limit of the six point twist correlator which reduces to the above four point correlator in a specific limit. The results provide a strong consistency check for our recently proposed holographic entanglement negativity conjecture in the <math><msub><mi>AdS</mi><mn>3</mn></msub><mo>/</mo><msub><mi>CFT</mi><mn>2</mn></msub></math> scenario.

Entanglement negativity at large central charge

In this work, building up on [1] we present momentum space Ward identities related to broken higher spin symmetry as an alternate approach to computing correlators of spinning operators in interacting theories such as the quasi-fermionic and quasi-bosonic theories. The direct Feynman diagram approach to computing correlation functions is intricate and in general has been performed only in specific kinematic regimes. We use higher spin equations to obtain the parity even and parity odd contributions to two-, three- and four-point correlators involving spinning and scalar operators in a general kinematic regime, and match our results with existing results in the literature for cases where they are available.