# Research Summary

My academic research is focused in many-body quantum systems out-of-equilibrium. These systems are characterized by a variety of novel phenomena, which are absent in equilibrium systems. This area has recently become the subject of intensive research, boosted in part by enormous progress in experimental techniques. However, in contrast to the equilibrium case, some of the most fundamental questions, such as whether the system attains a long-term stationary state, how this state is reached, and what is the nature of such a state, has been answered only for a limited number of cases. Furthermore, since there are several ‘’pathways” in which a system can be taken out of equilibrium, a plethora of different techniques are used to approach specific problems.

My long-term goal is to help establishing such unified framework, with the aim to provide a more systematic way to describe and classify those kinds of systems.

## Interacting Bosonic Quantum Field Theory

In Refs. [1, 2] we studied sudden quenches in 1D interacting bosonic field theories. In a quantum quench, the time-scale associated with an external variation of system parameters is much smaller than the typical relaxation time of the system (the change is effectively instantaneous). In Ref. [1], we considered the dynamics after the sudden switch-on of a lattice at the same time as the interaction strength was quenched.

We built a completely novel quantum Boltzmann equation (QBE) for the boson occupation number, with an interaction (sine-Gordon) term proportional to cos φ, substantially more complex than most previously studied QBE (e.g. using quartic theory and fermionic models), which accounts for multi-particle scattering.

Using a non-perturbative approximation scheme (2PI formalism) allowed us to naturally obtain expressions for the energy-momentum tensor (which describes both the density and flux of energy and momentum) and from that an expression for the conserved energy.

Since many different lattice models, such as spin-chains and the Bose-Hubbard model (which describes interacting bosons on a lattice) can be approximated by this same field theory, our results are very general.

In Ref. [2], we considered instead a sudden switching-on of a disordered potential. We found the dynamics to be quite rich with three different regimes.

A short-time regime, accessible via perturbation theory and dependent on microscopic parameters.

A pre-thermalization phase at intermediate times. In a pre-thermal regime integrated quantities (e.g. kinetic and potential energy) attain their correspondent thermal values much sooner than individual occupation numbers. We found, that within this stage of the dynamics, the density- density and single-particle correlation functions (related to the system’s superfluidity) displayed universal (i.e. independent of microscopic details) dynamical scaling behavior.

A long-time regime with thermalization caused by the combined effects of interactions and disorder.

We built a another novel quantum Boltzmann equation (QBE) accounting for both disorder and interactions. We found a “critical speeding up” of the thermalization. This is in marked contrast with the usual slowing down (where the relaxation time diverges) which happens upon approaching a critical point.

In Refs. [3, 4] we studied the quench of an isolated quantum system (in general dimension) described by a bosonic quartic theory with O(N) (group of rotations-reflections in N space dimensions) and showed the presence of universal physics in the dynamics for quenches tuned to the proper quantum critical point. We proved the existence of a novel universal exponent θ within a pre-thermal regime for the correlation functions and the magnetization. This result provided an interesting dynamical counterpart to the standard equilibrium renormalization group (RG) analysis. RG analysis allows one to investigate how parameters change, or “flow”, with the scale by gradually “integrating-out” short-scale physics in both space and time to obtain an effective theory at long space and time scales. At fixed-points the system is scale invariant (self-similar) and does not change as we “zoom out.” In our scenario, the time t after the quench is the new RG flow parameter. We can write schematically:

Thermal system ⇒ RG in thermal equilibrium ⇒ thermal, stable fixed point ⇒ thermal phase transition

Pre-thermalization phase ⇒ RG in out of equilibrium ⇒ non-thermal, unstable fixed point ⇒ dynamical phase transition (DPT).

In DPTs, qualitative different dynamical behaviors emerge upon changing parameters of the system’s Hamiltonian. We showed in particular that, in contrast to generic quenches to the gapped phase, the magnetization initially increased algebraically, with the universal exponent θ, and then started the usual exponentially decay. Our findings generalize to quantum closed systems the phenomenon of aging which was already well established in the classical regime and open quantum systems. Our results are very general and our Hamiltonian describes: for N = 0 the statistics of polymers; for N = 1 the Ising universality class, liquid-vapour and anisotropic magnet phase transitions among others; for N = 2 the superfluid Helium transition and for N = 3 isotropic ferromagnetic systems.'

## Development of New Criteria for Thermalization in Quantum Many-Body Systems

In Ref. [5] we studied quantum spin chains by analyzing the behavior of the so-called survival probability F(t) (the probability for finding the system in the initial state). We showed that:

In contrast to the previous consensus, for sufficiently strong interactions, the initial Gaussian decay of F(t), which is valid for any initial state and final Hamiltonian, does not switch to an exponential, but remains Gaussian at intermediate time-scales.

As the systems approaches many-body localization phase. We showed that, by increasing the level of disorder, the eigenstates and the initial state projected into the energy eigenbasis become multi-fractal. In both cases, the survival probability F(t) decays algebraically with exponent γ and this exponent agrees with the fractal dimension of the initial state.

In Refs. [6, 7] we proposed a new, very general criterion (which includes integrable, chaotic, interacting, noninteracting, clean and disordered systems) for thermalization in isolated lattice many-body quantum systems. We showed that, at long times, power-law decays of the survival probability occur also in the ergodic (chaotic) phase, but in this case it is caused by the unavoidable bounds in the spectrum of quantum systems, in which case thermalization necessarily occurs, and the correlation between the eigenstates of the Hamiltonian.

While previous works focused either on the link between quantum chaos and short-time decays or on the onset of the power-law decay at long times, our works unifies both perspectives into a single framework. Furthermore, the decay exponent, which contains a large amount of information about the system (structure of the initial state, the spectrum, number of interacting particles and other properties) is an extremely useful quantity allowing us to group many different systems into only a few classes from which the long-term behavior can be predicted.

In Ref. [8] we studied the dynamics of excited state quantum phase transitions (ESQPTs), which generalize standard quantum phase transitions (abrupt changes of the ground state of a system when a control parameter reaches a critical point) to excited states, using the Lipkin-Meshkov-Glick (LMG) model (the latter describes a spin-chain with a XY-type Hamiltonian in an external transverse magnetic). We provided many alternative ways (besides the known divergences in the density of states) to investigate the presence of ESQPTs by examining the structure of the Hamiltonian, the localization of the eigenstates, the behavior of the magnetization at the critical point and the quench dynamics. We note that before our work, there were very few results concerning the dynamical effects of ESQPTs. As an unexpected bonus we identified interesting similarities between the LMG and the XX spin model.

In Ref. [9] we used full random matrices to obtain exact analytical expressions for the survival probability and the growth of the Shannon entropy and von Neumann entanglement entropies. Full random matrices are not realistic, but they serve as a reference for the analysis of realistic many-body quantum systems. In this work, we find that the von Neumann entanglement entropy and the Shannon entropy provide very similar information, which is convenient given that the former is more expensive to compute numerically.

In Ref. [10] we are studying the static and dynamical properties at the crossover between few- and many- body physics. For very small and very large number of particles, analytical results are known. There are much fewer results for systems in the borderline region, i.e. systems where the number of particles is somewhere in between those two limits. In our study we show that, for the case of spin-chains, very few particles are already sufficient for behavior of the system to display many-body-like behavior. Our results corroborate recent experimental results.

## Analytical and Numerical Methods

My research has involved the several analytical and numerical techniques:

Analytical techniques include bosonization, Schwinger-Keldysh field-theory formalism, Wilson and Callan- Symanzik renormalization-group methods (including the dynamical generalization associated with non- thermal fixed points), large N expansions, diagrammatic methods, two particle irreducible (2PI) formalism, ε-expansion.

Numerical methods include exact diagonalization, which allows for reliable computations of the quantum dynamics for very long time-scales and software packages such as Expokit, which can be used to study the dynamics of larger system sizes than ED.

## References

[1] **Marco Tavora** and A. Mitra, Quench dynamics of one-dimensional bosons in a commensurate periodic potential: A quantum kinetic equation approach, **Phys. Rev. B 88, 115144 (2013)**

[2] **Marco Tavora**, A. Rosch and A. Mitra, Quench dynamics of one-dimensional interacting bosons in a disordered potential: Elastic dephasing and critical speeding-up of thermalization, **Phys. Rev. Lett. 113, 010601 (2014).**

[3] A.Chiocchetta, **Marco Tavora**, A.Gambassi, A.Mitra, Short-time universal scaling in an isolated quantum system after a quench, **Phys. Rev. B 91, 220302(R) (2015)**

[4] A. Chiocchetta, **Marco Tavora**, A. Gambassi, A. Mitra, Short-time universal scaling and light-cone dynamics after a quench in an isolated quantum system in d spatial dimensions, **Phys. Rev. B 94, 134311 (2016)**

[5] E.J. Torres-Herrera, **Marco Tavora**, L.F. Santos, Survival Probability of the Neel State in Clean and Disordered Systems: An Overview (invitation for special issue), **Brazilian Journal of Physics 46, Issue 3 (2016).**

[6] **Marco Tavora**, E. J. Torres-Herrera and Lea F. Santos, Inevitable power-law behavior of isolated many-body quantum systems and how it anticipates thermalization, **Phys. Rev. A 94, 041603(R)**

[7] **Marco Tavora**, E. J. Torres-Herrera and Lea F. Santos, Powerlaw Decay Exponents: a Dynamical Criterion for Predicting Thermalization, **Phys. Rev. A 95, 013604 (2016)**

[8] Lea F. Santos, **Marco Tavora**, F. Perez-Bernal, Excited state quantum phase transitions in many-body systems with infinite-range interaction: localization, dynamics, and bifurcation, **Phys. Rev. A 94 (1), 012113**

[9] E.J. Torres-Herrera, J. Karp, **Marco Tavora**, Lea F. Santos, Realistic many-body quantum systems vs full random matrices: static and dynamical properties, **Entropy Special Edition Quantum Information 18 (10), 359**

[10] Mauro Schiulaz, **Marco Tavora** and Lea F. Santos, From few- to many-body quantum systems, available as preprint **here**.