# ProjectQuantum reaction-diffusion dynamics in discrete time

## Basic data

Title:

Quantum reaction-diffusion dynamics in discrete time

Duration:

01/11/2024 to 31/10/2026

Abstract / short description:

Many-body systems can display collective emergent behaviour that is not apparent from the physics of their individual constituents. This motivates the ongoing interest in the quantum community that focuses on understanding the role of quantum effects on such emergent behaviour. This is a challenging problem due to the large number of degrees of freedom that are involved: unlike for classical systems, there are no efficient numerical methods for solving quantum many-body dynamics. Therefore, there is a need to develop analytical approaches and theoretical models that may be tractable, and from which intuition and insight can be gained.

Specifically, my planned research will address the dynamics of quantum generalised reaction-diffusion systems. In their simplest manifestations they are composed of fermions or bosons that coherently hop through a lattice and that react once they meet. Such a reaction could be for example annihilation. In this case the system will approach an empty state at long times. However, the way in which this state is approached can display emergent collective behaviour. This manifests, for example, through a power-law decay of the particle density in time with a power that depends on dimensionality and the relative strength between the hopping and reaction rate.

In Tübingen these systems have been investigated over the past couple of years through a hydrodynamic approach constructed by exploiting the integrability (i.e., the presence of multiple local conserved charges) of the hopping dynamics in a limit in which the reaction rate is much slower than the hopping. Already in this regime novel features have been identified which are of genuine quantum origin. This concerns for instance the power-law of the density decay, which in some parameter regimes and for certain initial states differs to that of the classical system and displays altered universality properties. However, investigating the opposite regime in which the reaction rate exceeds the hopping strength remains a challenge.

The path that I will chose in my research builds on discrete time representation of quantum reaction diffusion dynamics. Here the coherent hopping dynamics is implemented by quantum unitary gates and (irreversible) reactions are represented by probabilistic gates or Kraus maps in general. Simple examples of unitary gates can be derived from local Hamiltonians through the so-called Trotter-Suzuki decomposition.

The discrete-time approach has several advantages: the dynamics is local, which gives rise to a well-defined “light”-cone for propagation of local observables, and there are analytical techniques for calculating correlation functions when the unitary gates fulfil particular relations. Examples are quantum (generalised) dual unitary circuits or kinetically contained models such the so-called Quantum Floquet East model for particular initial states. These analytical techniques allow to go beyond integrability, and the presence of conservation laws in the model.

Building on this, we seek to construct analytically solvable models which allow to control the relative strength between quantum coherent processes and dissipation. We anticipate to uncover critical phenomena and to be able to study their dependence on quantum effects. In particular, the questions we are going to address are the following: do universality properties of the diffusion-reaction dynamics change in a discrete-time setting? What is the role of conserved charges? Can we construct analytical models that allow to access reaction-diffusion dynamics beyond perturbed integrable systems?

Answering these questions will contribute towards our understanding of complex dynamical processes in quantum matter in regimes which are notoriously challenging to treat.

Specifically, my planned research will address the dynamics of quantum generalised reaction-diffusion systems. In their simplest manifestations they are composed of fermions or bosons that coherently hop through a lattice and that react once they meet. Such a reaction could be for example annihilation. In this case the system will approach an empty state at long times. However, the way in which this state is approached can display emergent collective behaviour. This manifests, for example, through a power-law decay of the particle density in time with a power that depends on dimensionality and the relative strength between the hopping and reaction rate.

In Tübingen these systems have been investigated over the past couple of years through a hydrodynamic approach constructed by exploiting the integrability (i.e., the presence of multiple local conserved charges) of the hopping dynamics in a limit in which the reaction rate is much slower than the hopping. Already in this regime novel features have been identified which are of genuine quantum origin. This concerns for instance the power-law of the density decay, which in some parameter regimes and for certain initial states differs to that of the classical system and displays altered universality properties. However, investigating the opposite regime in which the reaction rate exceeds the hopping strength remains a challenge.

The path that I will chose in my research builds on discrete time representation of quantum reaction diffusion dynamics. Here the coherent hopping dynamics is implemented by quantum unitary gates and (irreversible) reactions are represented by probabilistic gates or Kraus maps in general. Simple examples of unitary gates can be derived from local Hamiltonians through the so-called Trotter-Suzuki decomposition.

The discrete-time approach has several advantages: the dynamics is local, which gives rise to a well-defined “light”-cone for propagation of local observables, and there are analytical techniques for calculating correlation functions when the unitary gates fulfil particular relations. Examples are quantum (generalised) dual unitary circuits or kinetically contained models such the so-called Quantum Floquet East model for particular initial states. These analytical techniques allow to go beyond integrability, and the presence of conservation laws in the model.

Building on this, we seek to construct analytically solvable models which allow to control the relative strength between quantum coherent processes and dissipation. We anticipate to uncover critical phenomena and to be able to study their dependence on quantum effects. In particular, the questions we are going to address are the following: do universality properties of the diffusion-reaction dynamics change in a discrete-time setting? What is the role of conserved charges? Can we construct analytical models that allow to access reaction-diffusion dynamics beyond perturbed integrable systems?

Answering these questions will contribute towards our understanding of complex dynamical processes in quantum matter in regimes which are notoriously challenging to treat.

## Involved staff

### Managers

Institute for Theoretical Physics (ITP)

Department of Physics, Faculty of Science

Department of Physics, Faculty of Science

### Other staff

Institute for Theoretical Physics (ITP)

Department of Physics, Faculty of Science

Department of Physics, Faculty of Science

## Local organizational units

Institute for Theoretical Physics (ITP)

Department of Physics

Faculty of Science

Faculty of Science

## Funders

Bonn, Nordrhein-Westfalen, Germany