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...
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
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
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  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  the researcher extended the method of 
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  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 . 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 , 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 , 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
 V. Alba and P. Calabrese, PNAS 114 (30) 7947 (2017), Entanglement and thermodynamics after a quantum
quench in integrable systems.
 V. Alba and Pasquale Calabrese, SciPost Phys. 4, 017 (2018), Entanglement dynamics after quantum
quenches in generic integrable systems.
 V. Alba and P. Calabrese, Phys. Rev. B 96, 115421 (2017), Quench action and Renyi entropies in integrable
 V. Alba and P. Calabrese, J. Stat. Mech. (2017) 113105, Renyi entropies after releasing the Neel state in
the XXZ spin chain.
 V. Alba, Phys. Rev. B 97,245135 (2018), Entanglement and quantum transport in integrable systems.
 V. Alba and M. Fagotti, Phys. Rev. Lett. 119, 010601 (2017), Prethermalization at low temperature: Scent
of a spontaneously broken symmetry.
 V. Alba, Phys. Rev. E 95, 062132 (2017), Measuring the Renyi entropies: An out-of-equilibrium protocol
via the Jarzynski equality.
 G. Bigan Mbeng,V. Alba, and P. Calabrese, J. Phys. A: Math. Theor. 50, 194001 (2017), Negativity
spectrum in gapped phases of matter.
 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.
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