## Periodic Reporting for period 2 - LoCoMacro (Local Control of Macroscopic Properties in Isolated Many-body Quantum Systems)

Reporting period: 2020-07-01 to 2021-12-31

A fundamental concept in statistical physics is that the equilibrium properties of systems with many elementary constituents can be described by few parameters, first and foremost the temperature. These can be tuned to modify the physical properties, and even the form in which matter manifests itself, so-called phases of matter. This generally requires a global control of the system, but there are also situations in which a local perturbation is sufficient to induce a phase transition. For example, pure water can be supercooled below its normal freezing point, remaining liquid; it is then sufficient to put the liquid in contact with a small piece of ice to induce global freezing.

When the system is not at equilibrium, its description becomes more complicated; nevertheless, a statistical description was shown to emerge when a quantum many-body system, isolated from the rest, is left to evolve for a long time. Being isolated, the system can not relax to an equilibrium state, but, when scrutinised locally, it appears as if it were prepared at an effective temperature (usually different from the actual temperature) or in some exotic state of matter. Arguably, the best understood situation is a quantum quench of a global parameter in a one dimensional quantum-many body system with translational invariance.

LoCoMacro aims at clarifying the effects of inhomogeneities in out-of-equilibrium systems, the main interest being in the global effects of inhomogeneities localised in a small part of the system.

The potential impact of the project can be summarised in the following points:

- It can teach new ways to realise exotic states of matter

- It can unveil strategies to control the macroscopic properties of an out-of-equilibrium system in a local way

- In the long run, it could have technological applications in devices for amplifying/switching signals.

Schematically, the various sub-objectives of LoCoMacro are listed below:

(a) Characterisation of the state at late times in the presence of different types of inhomogeneities.

(b) Exact calculations for the time evolution of correlation functions and entanglement measures.

(c) Analysis of the conservation laws in the presence of defects.

(d) Investigations into the issue of thermalisation in the presence of inhomogeneities.

(e) Development of efficient algorithms for quench dynamics.

(f) Role of emergent symmetries in intermediate-time dynamics.

(g) Effects of monitoring local observables in out-of-equilibrium states.

When the system is not at equilibrium, its description becomes more complicated; nevertheless, a statistical description was shown to emerge when a quantum many-body system, isolated from the rest, is left to evolve for a long time. Being isolated, the system can not relax to an equilibrium state, but, when scrutinised locally, it appears as if it were prepared at an effective temperature (usually different from the actual temperature) or in some exotic state of matter. Arguably, the best understood situation is a quantum quench of a global parameter in a one dimensional quantum-many body system with translational invariance.

LoCoMacro aims at clarifying the effects of inhomogeneities in out-of-equilibrium systems, the main interest being in the global effects of inhomogeneities localised in a small part of the system.

The potential impact of the project can be summarised in the following points:

- It can teach new ways to realise exotic states of matter

- It can unveil strategies to control the macroscopic properties of an out-of-equilibrium system in a local way

- In the long run, it could have technological applications in devices for amplifying/switching signals.

Schematically, the various sub-objectives of LoCoMacro are listed below:

(a) Characterisation of the state at late times in the presence of different types of inhomogeneities.

(b) Exact calculations for the time evolution of correlation functions and entanglement measures.

(c) Analysis of the conservation laws in the presence of defects.

(d) Investigations into the issue of thermalisation in the presence of inhomogeneities.

(e) Development of efficient algorithms for quench dynamics.

(f) Role of emergent symmetries in intermediate-time dynamics.

(g) Effects of monitoring local observables in out-of-equilibrium states.

The investigations carried out so far from the beginning of the project fall within the sub-objectives (a), (b), (e), (f), and (g).

The main achievement in (a) is the identification of subspaces that are invariant under time evolution in non-interacting spin chains with inhomogeneities [M. Fagotti, SciPost Phys. 8, 048 (2020)]. This has allowed for a non-perturbative definition of "locally quasi-stationary states", which are at the basis of generalised hydrodynamic descriptions. In particular, we have shown that the generalised hydrodynamic equation, which was obtained under a low-inhomogeneity assumption, can be lifted into an exact theory by reinterpreting it as the Schrödinger equation projected into the (invariant) subspace of the locally quasi-stationary states. In other words, we have worked out the theory of generalised hydrodynamics in non-interacting spin chains beyond perturbation theory, including all the classical, quantum, and lattice contributions.

Recently, studying an interacting integrable system (the dual folded XXZ model), we have also identified a different family of inhomogeneous states emerging at late times. They are locally equivalent to jammed states, in which the phase space available to the stable quasi-particles shrinks to zero.

The main achievement in (b) is the calculation of the time evolution of the entanglement entropy of subsystems in an interacting integrable model (XXZ spin chain) prepared by joining together two macroscopically different low-entangled states [V. Alba, B. Bertini, and M. Fagotti, SciPost Phys. 7, 005 (2019]. In particular, we have used generalised hydrodynamics to map the entanglement entropy at finite time back to the one at the initial time.

Sub-objective (e) has been attacked from a very general perspective. Specifically, we have estimated the size of the Hilbert space spanned by a generic state that time evolves with a local Hamiltonian under the unique assumption that the energy cumulants are extensive [M. Fagotti, SciPost Phys. 6, 059 (2019)]. The result suggests that the most efficient algorithms available at present to study quench dynamics have still a huge margin of improvement. We are currently working on an algorithm able to exploit this theoretical result, especially in the direction of storing information more efficiently.

Within (f), we have studied the strong coupling limit of the XXZ spin chain, in which the excited states become highly degenerate. We have identified an effective model that describes the limit (we called it "folded XXZ model", and it is equivalent to a special point of the Bariev model). We have worked out energies and eigenstates, [L. Zadnik and M. Fagotti, "The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties", SciPost Phys. Core 4, 010 (2021)], and investigated the dynamics when two macroscopic states are joined together [L. Zadnik, K. Bidzhiev, and M. Fagotti, "The Folded Spin-1/2 XXZ Model: II. Thermodynamics and Hydrodynamics with a Minimal Set of Charges", SciPost Phys. 10, 099 (2021)].

We are now working on Sub-objective (g). In particular, we are studying the dynamics after a quantum measurement of a localised observable in a jammed state of a quantum integrable spin chain.

The main achievement in (a) is the identification of subspaces that are invariant under time evolution in non-interacting spin chains with inhomogeneities [M. Fagotti, SciPost Phys. 8, 048 (2020)]. This has allowed for a non-perturbative definition of "locally quasi-stationary states", which are at the basis of generalised hydrodynamic descriptions. In particular, we have shown that the generalised hydrodynamic equation, which was obtained under a low-inhomogeneity assumption, can be lifted into an exact theory by reinterpreting it as the Schrödinger equation projected into the (invariant) subspace of the locally quasi-stationary states. In other words, we have worked out the theory of generalised hydrodynamics in non-interacting spin chains beyond perturbation theory, including all the classical, quantum, and lattice contributions.

Recently, studying an interacting integrable system (the dual folded XXZ model), we have also identified a different family of inhomogeneous states emerging at late times. They are locally equivalent to jammed states, in which the phase space available to the stable quasi-particles shrinks to zero.

The main achievement in (b) is the calculation of the time evolution of the entanglement entropy of subsystems in an interacting integrable model (XXZ spin chain) prepared by joining together two macroscopically different low-entangled states [V. Alba, B. Bertini, and M. Fagotti, SciPost Phys. 7, 005 (2019]. In particular, we have used generalised hydrodynamics to map the entanglement entropy at finite time back to the one at the initial time.

Sub-objective (e) has been attacked from a very general perspective. Specifically, we have estimated the size of the Hilbert space spanned by a generic state that time evolves with a local Hamiltonian under the unique assumption that the energy cumulants are extensive [M. Fagotti, SciPost Phys. 6, 059 (2019)]. The result suggests that the most efficient algorithms available at present to study quench dynamics have still a huge margin of improvement. We are currently working on an algorithm able to exploit this theoretical result, especially in the direction of storing information more efficiently.

Within (f), we have studied the strong coupling limit of the XXZ spin chain, in which the excited states become highly degenerate. We have identified an effective model that describes the limit (we called it "folded XXZ model", and it is equivalent to a special point of the Bariev model). We have worked out energies and eigenstates, [L. Zadnik and M. Fagotti, "The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties", SciPost Phys. Core 4, 010 (2021)], and investigated the dynamics when two macroscopic states are joined together [L. Zadnik, K. Bidzhiev, and M. Fagotti, "The Folded Spin-1/2 XXZ Model: II. Thermodynamics and Hydrodynamics with a Minimal Set of Charges", SciPost Phys. 10, 099 (2021)].

We are now working on Sub-objective (g). In particular, we are studying the dynamics after a quantum measurement of a localised observable in a jammed state of a quantum integrable spin chain.

- Characterisation of locally quasi-stationary states.

After having characterised locally quasi-stationary states in non-interacting spin chains, we aim at incorporating interactions in the structure. A remarkable by-product would be the development of a non-perturbative version of generalised hydrodynamics.

- Observables and entanglement measures.

The time evolution of the so-called Rényi entropies after quantum quenches is still an open problem, which we have planned to address. We indeed think that a solution could have a remarkable impact on our understanding of quantum correlations in the presence of interactions.

We also aim at clarifying the interplay between initial state and Hamiltonian in the formation of quantum correlations.

- Classification of the charges.

A localised defect in the Hamiltonian affects the local conservation laws. The new charges can be considered as the building blocks to describe the late-time dynamics after global quenches. Unfortunately their classification is hard even in the absence of interactions. We plan to make some progress in this respect.

- Algorithms for quench dynamics.

In the context of numerical simulation of quench dynamics, we aim at improving the state-of-the-art algorithms by feeding them with some theoretical knowledge.

- Thermalisation with inhomogeneities.

Non-equilibrium time evolution in generic models is expected to result in thermalisation. In translationally invariant integrable systems with weak integrability breaking this is usually captured by a quantum Boltzmann equation. We are working on assessing the generality of this picture, with the final goal of extending it to inhomogeneous systems.

- Emergent symmetries.

At particular values of the coupling constants, some models exhibit symmetries that strongly affect the intermediate-time dynamics. We are studying effective models that capture such limits and can also be used to understand the leading corrections to the asymptotic behaviour.

- Local monitoring.

By “local monitoring” we mean the repeated measurement of a local observable, as the spin at a given position. We are interested, in particular, in the global effects produced by the monitoring.

We are currently investigating the dynamics after a single quantum measurement, and we have already found a setting where the effect is macroscopic.

After having characterised locally quasi-stationary states in non-interacting spin chains, we aim at incorporating interactions in the structure. A remarkable by-product would be the development of a non-perturbative version of generalised hydrodynamics.

- Observables and entanglement measures.

The time evolution of the so-called Rényi entropies after quantum quenches is still an open problem, which we have planned to address. We indeed think that a solution could have a remarkable impact on our understanding of quantum correlations in the presence of interactions.

We also aim at clarifying the interplay between initial state and Hamiltonian in the formation of quantum correlations.

- Classification of the charges.

A localised defect in the Hamiltonian affects the local conservation laws. The new charges can be considered as the building blocks to describe the late-time dynamics after global quenches. Unfortunately their classification is hard even in the absence of interactions. We plan to make some progress in this respect.

- Algorithms for quench dynamics.

In the context of numerical simulation of quench dynamics, we aim at improving the state-of-the-art algorithms by feeding them with some theoretical knowledge.

- Thermalisation with inhomogeneities.

Non-equilibrium time evolution in generic models is expected to result in thermalisation. In translationally invariant integrable systems with weak integrability breaking this is usually captured by a quantum Boltzmann equation. We are working on assessing the generality of this picture, with the final goal of extending it to inhomogeneous systems.

- Emergent symmetries.

At particular values of the coupling constants, some models exhibit symmetries that strongly affect the intermediate-time dynamics. We are studying effective models that capture such limits and can also be used to understand the leading corrections to the asymptotic behaviour.

- Local monitoring.

By “local monitoring” we mean the repeated measurement of a local observable, as the spin at a given position. We are interested, in particular, in the global effects produced by the monitoring.

We are currently investigating the dynamics after a single quantum measurement, and we have already found a setting where the effect is macroscopic.