Interests and Co-authors

Currently, my interests are in phase transitions for spin systems, both classical and quantum. During my Ph.D., I studied multidimensional long-range Ising spin systems, where I introduced contours more suitable to give a direct proof of phase transition. This construction is highly based on Fröhlich-Spencer construction of multiscaled contours for one-dimensional long-range Ising spin systems. Also, during a year visiting University of Victoria, my coauthors and I developed a specification theory for a large class of quantum spin systems, in such a way that a general DLR approach to equilibrium states for quantum spin systems could be introduced. Below, you will find an up-to-date list of my coauthors.

Coauthors

• Rodrigo Bissacot
• Eric O. Endo
• Satoshi Handa
• Marcelo Laca
• João Maia

Publications

1. Phase Transitions in Long-Range Random Field Ising Models in Higher Dimensions-(2023) joint with Rodrigo Bissacot, and João Maia.
ArXiv Abstract
We extend the recent argument by Ding and Zhuang from nearest-neighbor to long-range interactions and prove the phase transition in the class of ferromagnetic random field Ising models. Our proof combines a generalization of Fröhlich-Spencer contours to the multidimensional setting, proposed by two of us, with the coarse-graining procedure introduced by Fisher, Fröhlich and Spencer. The result shows that the Ding-Zhuang strategy is also useful for interactions \(J_{xy}=|x-y|^{-\alpha}\) when \(\alpha > d\) in dimension \(d\geq 3\) if we have a suitable system of contours. We can consider i.i.d. random fields with Gaussian or Bernoulli distributions. Our main result is an alternative proof that does not use the Renormalization Group Method (RGM), since Bricmont and Kupiainen claimed that the RGM also works on this generality.

2. Quantum Statistical Mechanics via Boundary Conditions. A Groupoid Approach to Quantum Spin Systems-(2022) joint with Rodrigo Bissacot, and Marcelo Laca.

ArXiv Abstract
We use a groupoid model for the spin algebra to introduce boundary conditions on quantum spin systems via a Poisson point process representation. We can describe all the KMS states of quantum systems by means of a set of equations resembling the standard DLR equations, proving that we can characterize the set of KMS states as been the closed convex hull of thermodynamic limits, as in the classical statistical mechanics. We introduce a notion of quantum specification which recovers the classical DLR measures in the particular case of classical interactions. Our results are in the same direction as those obtained recently by Cha, Naaijkens, and Nachtergaele, differently somehow from the predicted by Fannes and Werner.
3. Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields-(2021) joint with Rodrigo Bissacot, Eric O. Endo, and Satoshi Handa.
ArXiv Abstract
Inspired by Fröhlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on \(\mathbb{Z}^d\), \(d\geq 2\). The argument, which is based on a multi-scale analysis, works for the sharp region \(\alpha>d\) and improves previous results obtained by Park for \(\alpha>3d+1\), and by Ginibre, Grossmann, and Ruelle for \(\alpha> d+1\), where \(\alpha\) is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial decaying magnetic field with power \(\delta>0\) as \(h^*|x|^{-\delta}\), where \(h^* >0\). For \(d < \alpha < d+1\), the phase transition occurs when \(\delta>\alpha-d\), and when \(h^*\) is small enough over the critical line \(\delta=\alpha-d\). For \(\alpha \geq d+1\), \(\delta>1\) it is enough to prove the phase transition, and for \(\delta=1\) we have to ask \(h^*\) small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.

Talks

1. Interfaces Between Quantum and Classical Statistical Mechanics:Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems. (2023). IME-USP, São Paulo, Brazil.
2. Probability Seminar:Contours and Peielrs Argument for long-range multidimensional Ising models. (2023). IME-USP, São Paulo, Brazil.
3. Spring Probability Seminar Series: DLR equations for quantum spin systems. (2023). New York University, Shanghai, China.
4. Dynamics and Probability Seminar: Phase transition in Ising systems with long-range interactions. (2022). University of Victoria, Victoria, Canada.
5. Operator Algebra Seminar: Equilibrium states in classical and quantum statistical mechanics (2021). University of Victoria, Victoria, Canada.
6. Probability Webinar: Long-range Ising model and decaying fields- a contour approach. (2021). UFRJ, Rio de Janeiro, Brazil (Online).
7. Applied Mathematics Seminar: Long-range multidimensional Ising models: A contour argument based on Fröhlich-Spencer (2021). University of Utrecht, Utrecht, Netherlands (Online).
8. Seminar on Probability and Stochastic Processes: A contour approach to multidimensional long-range Ising models (2021). IME-USP, São Paulo, Brazil (Online).
9. Statistical Mechanics: Recent Results: Fröhlich-Spencer revisited: a contour argument for long-range Ising models (2021). IME-USP, São Paulo, Brazil.
10. Probability, Statistical Mechanics and Large Deviations: Phase Transitions and Large Deviations Principle in Long Range Ising models (2020). IME-USP, São Paulo, Brazil.
11. Bernoulli IMS-One World Probability Symposium: Phase Transitions and Large Deviations Principle in Long Range Ising models (2020).
12. IMIA Operator Algebra and Noncommutative Geometry Seminar:A local characterization of Equilibrium for Quantum Spin Systems. (2019). UOW, Wollongong, Australia.
13. Dynamical Systems Workshop - Gibbs Condition: The relation between DLR equations and KMS states (2018). UFRGS, Porto Alegre, Brazil.