Krylov Complexity in $2d$ CFTs with SL $(2,\mathbb{R})$ deformed HamiltoniansKrylov Complexity in $2d$ CFTs with SL$(2,\mathbb{R})$ deformed Hamiltonians

Vinay Malvimat, Somnath Porey, Roy, Baishali

In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL $(2,\mathbb{R})$ 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 behavior 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 behavior from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL$(2,\mathbb{R})$ 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 behavior 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 behavior from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.

arXiv:2402.15835Feb 24, 20240 citations

Complexity trapped by simplicity

Francesca Ferlaino

NatureAug 19, 20142 citations

Krylov complexity in 2d CFTs with SL(2, ℝ) deformed Hamiltonians

Vinay Malvimat, Somnath Porey, Baishali Roy

A bstract In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL(2 , 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.A bstract In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL(2 , 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.

Journal of High Energy PhysicsFeb 6, 202516 citations

Complexity Theory

Jack D. Hidary

Springer eBooksJan 1, 20192 citations

Krylov complexity and the growth of the black hole interior in 3D gravity

Arpan Bhattacharyya, Sounak Pal, Juan F. Pedraza

We investigate the growth of the black hole interior in three-dimensional gravity from the boundary theory. For the two-sided BTZ black hole, we propose a boundary reconstruction of the time dependence of a codimension-one surface in terms of correlation functions of smeared operators in the thermofield double state, reproducing the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the Chern--Simons formulation of three-dimensional gravity, these nonlocal correlators are represented by bulk Wilson lines with smeared endpoints, extending the familiar connection between Wilson lines and codimension-two observables underlying holographic entanglement entropy to codimension-one observables. We then ask whether the same geometric growth is captured by Krylov complexity. For the smeared operators, we extract the Lanczos data from their correlation functions and find that operator Krylov complexity reproduces the late-time linear growth of the black hole interior, extending previous connections between operator growth and bulk geometry to AdS $_3$ . By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.We investigate the growth of the black hole interior in three-dimensional gravity from the boundary theory. For the two-sided BTZ black hole, we propose a boundary reconstruction of the time dependence of a codimension-one surface in terms of correlation functions of smeared operators in the thermofield double state, reproducing the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the Chern--Simons formulation of three-dimensional gravity, these nonlocal correlators are represented by bulk Wilson lines with smeared endpoints, extending the familiar connection between Wilson lines and codimension-two observables underlying holographic entanglement entropy to codimension-one observables. We then ask whether the same geometric growth is captured by Krylov complexity. For the smeared operators, we extract the Lanczos data from their correlation functions and find that operator Krylov complexity reproduces the late-time linear growth of the black hole interior, extending previous connections between operator growth and bulk geometry to AdS$_3$. By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.

hep-thCross-listedarXiv:2608.19373Aug 21, 2026

Relative Complexity

Leonard Susskind

SpringerBriefs in physicsJan 1, 20200 citations

Krylov Complexity in

2d

CFTs with SL

(2,\mathbb{R})

deformed Hamiltonians

In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL

(2,\mathbb{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 behavior 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 behavior from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.

_3

. By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.