## Periodic Reporting for period 1 - OEMBS (Out-of-equilibrium entangled many-body systems)

Reporting period: 2016-09-01 to 2018-08-31

The aim of the project was to understand how complex behaviors arise in highly-entangled

isolated out-of-equilibrium quantum many-body systems. Entanglement is a key phenomenon

of quantum systems. Entangled systems cannot be described by knowing independently the

states of their elementary constituents. An important fact is that entanglement can be generated

by driving a system out of equilibrium, i.e. by applying to the system a sudden manipulation

(quantum quench).

A complete understanding of out-of-equilibrium behaviors in entangled many-body

systems is a fundamental challenge

in contemporary physics. The reason is that, in contrast with equilibrium, there is no

effective analytical framework to describe out-of-equilibrium phenomena yet. On the other

hand, numerical methods are effective only in describing the short-time dynamics.

In this context, the specific goal of the project was to clarify fundamental aspects of

out-of-equilibrium physics in strongly-entangled systems, focusing on the behavior

of quantities that can witness the presence of entanglement (entanglement measures).

In fact, in recent years, entanglement emerged as the interdisciplinary cornerstone

of several theoretical and numerical tools. Research on entanglement is intrinsically

interdisciplinary, lying at the interface between quantum information, quantum field theory,

statistical mechanics, and condensed matter physics. Progress in understanding

entanglement patterns in both equilibrium and out-of-equilibrium systems has the

potential to impact our theoretical understanding (both numerical and analytical) of

quantum systems. This could allow to discover exotic collective behaviors and

new phases of matter, which can find applications in quantum computing

technologies.

The researcher pursued a very interdisciplinary approach, by combining state-of-the-art

entanglement-based numerical tools as well as analytical methods . This allowed to

provide a full characterization of the out-of-equilibrium dynamics of entanglement measures in

quantum many body systems.

isolated out-of-equilibrium quantum many-body systems. Entanglement is a key phenomenon

of quantum systems. Entangled systems cannot be described by knowing independently the

states of their elementary constituents. An important fact is that entanglement can be generated

by driving a system out of equilibrium, i.e. by applying to the system a sudden manipulation

(quantum quench).

A complete understanding of out-of-equilibrium behaviors in entangled many-body

systems is a fundamental challenge

in contemporary physics. The reason is that, in contrast with equilibrium, there is no

effective analytical framework to describe out-of-equilibrium phenomena yet. On the other

hand, numerical methods are effective only in describing the short-time dynamics.

In this context, the specific goal of the project was to clarify fundamental aspects of

out-of-equilibrium physics in strongly-entangled systems, focusing on the behavior

of quantities that can witness the presence of entanglement (entanglement measures).

In fact, in recent years, entanglement emerged as the interdisciplinary cornerstone

of several theoretical and numerical tools. Research on entanglement is intrinsically

interdisciplinary, lying at the interface between quantum information, quantum field theory,

statistical mechanics, and condensed matter physics. Progress in understanding

entanglement patterns in both equilibrium and out-of-equilibrium systems has the

potential to impact our theoretical understanding (both numerical and analytical) of

quantum systems. This could allow to discover exotic collective behaviors and

new phases of matter, which can find applications in quantum computing

technologies.

The researcher pursued a very interdisciplinary approach, by combining state-of-the-art

entanglement-based numerical tools as well as analytical methods . This allowed to

provide a full characterization of the out-of-equilibrium dynamics of entanglement measures in

quantum many body systems.

During the duration of the fellowship the researcher focused on the study of both

equilibrium and out-of-equilibrium entangled many-body quantum systems.

Together with Professor Pasquale Calabrese, the researcher laid the basis

to describe the out-of-equilibrium behavior of entanglement measures

in integrable models, which are models that can be solved analytically.

For instance, in [1,2], the researcher obtained the first exact results

for the out-of-equilibrium dynamics of the entanglement entropy in

integrable interacting models. In [3,4] the researcher

extended the analysis of [1] to the post-quench dynamics of the

Renyi entropies, which are entanglement measures that

have the remarkable property of being experimentally

accessible with cold atomic gases. In [5] the researcher extended the method of [1]

to inhomogeneous systems. These are out-of-equilibrium experiments where two

one dimensional systems with macroscopically different properties are put

in contact and let to evolve.

The understanding of the early stage (prethermal) regimes of the out-of-equilibrium

dynamics after quantum quenches is an important priority of contemporary

physics. In [6] in collaboration with Maurizio Fagotti (from the Ecole Normale

Superieur in Paris) the researcher uncovered a new mechanism leading to interesting prethermal regimes.

These results have been published in Physics Review Letters (PRL). Another intriguing result at the

interface between equilibrium and out-of-equilibrium entangled systems

is in [7]. In the paper, the researcher presented a new method to calculate

the equilibrium values of entanglement measures by using out-of-equilibrium protocols.

A prominent theme of the research project was the investigation of novel

entanglement witnesses and their behavior in complex quantum many-body

phases of matter. In this respect an important quantity is the so-called

logarithmic negativity. In [8], together with the team at SISSA, the researcher

derived general results for the behavior of the entanglement negativity.

One of the novel and most useful entanglement-based theoretical tools is

the entanglement spectrum. In [9], in collaboration with Erik Tonni at SISSA and

Pasquale Calabrese the researcher investigated the structure of the entanglement

spectrum in systems having non-unique ground states.

All the results were communicated in conferences and workshops in prestigious institutions

worldwide.

References:

[1] V. Alba and P. Calabrese, PNAS 114 (30) 7947 (2017), Entanglement and thermodynamics after a quantum

quench in integrable systems.

[2] V. Alba and Pasquale Calabrese, SciPost Phys. 4, 017 (2018), Entanglement dynamics after quantum

quenches in generic integrable systems.

[3] V. Alba and P. Calabrese, Phys. Rev. B 96, 115421 (2017), Quench action and Renyi entropies in integrable

systems.

[4] V. Alba and P. Calabrese, J. Stat. Mech. (2017) 113105, Renyi entropies after releasing the Neel state in

the XXZ spin chain.

[5] V. Alba, Phys. Rev. B 97,245135 (2018), Entanglement and quantum transport in integrable systems.

[6] V. Alba and M. Fagotti, Phys. Rev. Lett. 119, 010601 (2017), Prethermalization at low temperature: Scent

of a spontaneously broken symmetry.

[7] V. Alba, Phys. Rev. E 95, 062132 (2017), Measuring the Renyi entropies: An out-of-equilibrium protocol

via the Jarzynski equality.

[8] G. Bigan Mbeng,V. Alba, and P. Calabrese, J. Phys. A: Math. Theor. 50, 194001 (2017), Negativity

spectrum in gapped phases of matter.

[9] V. Alba, P. Calabrese, and E. Tonni, J. Phys. A 51 024001 (2018), Entanglement spectrum degeneracy and

Cardy formula in 1+1 dimensional conformal field theories.

equilibrium and out-of-equilibrium entangled many-body quantum systems.

Together with Professor Pasquale Calabrese, the researcher laid the basis

to describe the out-of-equilibrium behavior of entanglement measures

in integrable models, which are models that can be solved analytically.

For instance, in [1,2], the researcher obtained the first exact results

for the out-of-equilibrium dynamics of the entanglement entropy in

integrable interacting models. In [3,4] the researcher

extended the analysis of [1] to the post-quench dynamics of the

Renyi entropies, which are entanglement measures that

have the remarkable property of being experimentally

accessible with cold atomic gases. In [5] the researcher extended the method of [1]

to inhomogeneous systems. These are out-of-equilibrium experiments where two

one dimensional systems with macroscopically different properties are put

in contact and let to evolve.

The understanding of the early stage (prethermal) regimes of the out-of-equilibrium

dynamics after quantum quenches is an important priority of contemporary

physics. In [6] in collaboration with Maurizio Fagotti (from the Ecole Normale

Superieur in Paris) the researcher uncovered a new mechanism leading to interesting prethermal regimes.

These results have been published in Physics Review Letters (PRL). Another intriguing result at the

interface between equilibrium and out-of-equilibrium entangled systems

is in [7]. In the paper, the researcher presented a new method to calculate

the equilibrium values of entanglement measures by using out-of-equilibrium protocols.

A prominent theme of the research project was the investigation of novel

entanglement witnesses and their behavior in complex quantum many-body

phases of matter. In this respect an important quantity is the so-called

logarithmic negativity. In [8], together with the team at SISSA, the researcher

derived general results for the behavior of the entanglement negativity.

One of the novel and most useful entanglement-based theoretical tools is

the entanglement spectrum. In [9], in collaboration with Erik Tonni at SISSA and

Pasquale Calabrese the researcher investigated the structure of the entanglement

spectrum in systems having non-unique ground states.

All the results were communicated in conferences and workshops in prestigious institutions

worldwide.

References:

[1] V. Alba and P. Calabrese, PNAS 114 (30) 7947 (2017), Entanglement and thermodynamics after a quantum

quench in integrable systems.

[2] V. Alba and Pasquale Calabrese, SciPost Phys. 4, 017 (2018), Entanglement dynamics after quantum

quenches in generic integrable systems.

[3] V. Alba and P. Calabrese, Phys. Rev. B 96, 115421 (2017), Quench action and Renyi entropies in integrable

systems.

[4] V. Alba and P. Calabrese, J. Stat. Mech. (2017) 113105, Renyi entropies after releasing the Neel state in

the XXZ spin chain.

[5] V. Alba, Phys. Rev. B 97,245135 (2018), Entanglement and quantum transport in integrable systems.

[6] V. Alba and M. Fagotti, Phys. Rev. Lett. 119, 010601 (2017), Prethermalization at low temperature: Scent

of a spontaneously broken symmetry.

[7] V. Alba, Phys. Rev. E 95, 062132 (2017), Measuring the Renyi entropies: An out-of-equilibrium protocol

via the Jarzynski equality.

[8] G. Bigan Mbeng,V. Alba, and P. Calabrese, J. Phys. A: Math. Theor. 50, 194001 (2017), Negativity

spectrum in gapped phases of matter.

[9] V. Alba, P. Calabrese, and E. Tonni, J. Phys. A 51 024001 (2018), Entanglement spectrum degeneracy and

Cardy formula in 1+1 dimensional conformal field theories.

The main source of innovation of the project resided in the challenging topics and the fundamental

physics questions addressed.

The main outcome of the research was a powerful framework based on the combination of a

simple picture for the entanglement spreading and exact solutions available for integrable systems.

This allowed to understand the dynamics of entanglement-related quantities in

integrable models. This result represents a remarkable progress compared to the state-of-the-art, as

entanglement is the most surprising fingerprint of quantum matter. For instance, at a conceptual

level, this result allows to understand the transformation between the entanglement measures and

``classical'' quantities, such as the thermodynamic entropy. This is an important

contribution to unveil how simple descriptions of matter, such as statistical mechanics or

thermodynamics, emerge in out-of-equilibrium quantum many-body systems. Moreover,

exact calculations of entanglement dynamics are challenging. At a technical level, the results of the

project provided the first analytical calculation of entanglement dynamics in interacting

integrable systems. Finally, this new framework of ideas proved to be quite fruitful, and

it constitutes a robust starting point for further interdisciplinary research.

physics questions addressed.

The main outcome of the research was a powerful framework based on the combination of a

simple picture for the entanglement spreading and exact solutions available for integrable systems.

This allowed to understand the dynamics of entanglement-related quantities in

integrable models. This result represents a remarkable progress compared to the state-of-the-art, as

entanglement is the most surprising fingerprint of quantum matter. For instance, at a conceptual

level, this result allows to understand the transformation between the entanglement measures and

``classical'' quantities, such as the thermodynamic entropy. This is an important

contribution to unveil how simple descriptions of matter, such as statistical mechanics or

thermodynamics, emerge in out-of-equilibrium quantum many-body systems. Moreover,

exact calculations of entanglement dynamics are challenging. At a technical level, the results of the

project provided the first analytical calculation of entanglement dynamics in interacting

integrable systems. Finally, this new framework of ideas proved to be quite fruitful, and

it constitutes a robust starting point for further interdisciplinary research.