AB-INITIO DYNAMICS IN STRONGLY INTERACTING MANY-BODY SYSTEMS

Understanding strongly interacting quantum many-body systems remains a key problem in modern physics due to the quantum correlations they develop, which result in highly nontrivial macroscopic phenomena which still defy our efforts of a thorough theoretical understanding, like quantum phase transitions, collective excitations, topological quantum order, and high-temperature superconductivity. Among these challenges, a special place is reserved to the description of the dynamics of strongly correlated systems, which is key to understanding out-of-equilibrium phenomena and the evolution of entanglement. Gaining insight into the latter is crucial for the advancement of quantum technologies, including computation and communication protocols, where a major concern is the control and preservation of coherence and entanglement. This Thesis explores the numerical simulation of the dynamical properties of strongly interacting quantum many-body systems. A recent technique that tackle this problem is the time-dependent variational Monte Carlo (t-VMC), developed by G. Carleo [1, 2]. This method builds on the basic idea of the standard variational Monte Carlo (VMC): the optimization of a variational Ansatz wave function depending on a set of parameters. While the VMC looks for the set of parameters that provide the best approximation of the ground state, the t-VMC, given a starting set of parameters, prescribes how they should be updated at each time step in order to produce an approximation of the dynamics of the true quantum state. The t-VMC technique has been successfully applied especially to the class of neural-network quantum states (NNQS), which employs machine learning to represent the variational wave function. In the first part of this work, encompassing the first four chapters, we extend the t-VMC to the class of shadow wave functions (SWF) [3, 4], that incorporate many-body correlations by introducing a set of auxiliary “shadow” variables. SWFs can be interpreted as particular neural networks in which the shadow variables represent a layer of neurons having a specific connectivity. SWFs are especially attractive because they are related to the path integration formalism (a SWF can be seen as a single step of path integration [5]), so they are solidly grounded in the physics of a system. In Chapter 2 we mainly address the main novelty of our work, that is the extension of t-VMC to SWFs, which presents nontrivial problems, like the need of a reweighting procedure. We also introduce, in Chapter 3, a specific example of SWF, which we call the Baeriswyl-Shadow wave function (BSWF). Its structure takes a cue from the Physics of the transverse-field Ising model (TFI) and it is very concise, having only three parameters in the translation-invariant case. We develop from scratch a C++ code that allows us to perform t-VMC simulations of spin systems using various Ansatzë, including SWFs, and in particular the BSWF. In Chapter 4 we present some of the results of the simulations we performed with our code, and discuss some nontrivial details of the implementation. The results of our computations allow us to assess the strengths and weaknesses of the BSWF Ansatz, by checking its results against the ones yielded by other popular Ansatzë, like the Jastrow wave function or the restricted Boltzmann machine (RBM), the latter being a particular case of NNQS. We find that the BSWF achieves a description of the one-dimensional (1D) TFI ground state which is competitive with the translation-invariant RBM. This is remarkable since the BSWF only uses a very small number of parameters, that moreover do not scale with the size of the system. This comes at the cost of performing the Monte Carlo sampling of the “shadow” configurations of spins along with the “real” ones. On the other hand, the real-time simulation of a quantum quench in 1D shows that for short times the BSWF is very accurate, but it loses accuracy as time progresses. We attribute this result to the lack of explicit long-range correlations in the BSWF functional form. However, we find that the BSWF can significantly improve other Ansatzë, if used in conjunction with them as a multiplicative factor. We also generalize the BSWF architecture by introducing additional explicit long-range terms. The Ansatzë thus obtained are even more accurate than the BSWF in the description of the 1D TFI ground state, and in real-time 1D simulations are more consistent with the exact evolution, but, after a certain amount of simulation time, they suffer from an explosion of noise due to the reweighting technique needed to simulate the evolution of SWFs. We also perform some computations of the 2D TFI ground state, finding that the BSWF is accurate also in this dimensionality, considering its very low number of parameters. The second part of this work, represented by Chapter 5, is devoted to the use of different techniques to study the dynamical correlations of a two-dimensional dilute system of attractive fermions. We are motivated by a recent paper [6] suggesting the possibility of detecting the celebrated Higgs (or amplitude) mode of Fermi superfluids in the spin density dynamical structure factor. Specifically, we leverage generalized random phase approximation (GRPA) [7, 8, 9, 10, 6] and the unbiased auxiliary-field quantum Monte Carlo method (AFQMC) [11, 12, 13] to compute the dynamical structure factors in the density and spin density channels. Our AFQMC calculations, which are exact within the error bars, provide a benchmark for the approximate results of GRPA, although the computational cost does not allow us to achieve the very large sizes needed to test the suggestion of Ref. [6] because of the computational cost. The AFQMC results for smaller systems show good qualitative accordance with GRPA, the main effect being related to a renormalization of the superfluid gap due to many-body correlations. The results are also compatible with the presence of the peak in the spin mode that was first observed by the Authors of Ref. [6], and identified by them with the Higgs mode. However, a more detailed analysis, based on the GRPA dynamical correlations of the order parameter for large system sizes, shows that this peak cannot be identified with the Higgs mode. We suggest a different interpretation of its origin, rooted in the kinematics of the system. References [1] Giuseppe Carleo. Spectral and dynamical properties of strongly correlated systems. PhD thesis, SISSA, October 2011. [2] Federico Becca and Sandro Sorella. Time-Dependent Variational Monte Carlo, page 156–163. Cambridge University Press, 2017. [3] Silvio Vitiello, Karl Runge, and M. H. Kalos. Variational Calculations for Solid and Liquid 4 He with a ”Shadow” Wave Function. Phys. Rev. Lett., 60:1970–1972, May 1988. [4] S. A. Vitiello, K. J. Runge, G. V. Chester, and M. H. Kalos. Shadow wave-function variational calculations of crystalline and liquid phases of 4 He. Phys. Rev. B, 42:228–239, Jul 1990. [5] L. Reatto and G. L. Masserini. Shadow wave function for many-boson systems. Phys. Rev. B, 38:4516–4522, Sep 1988. [6] Huaisong Zhao, Xiaoxu Gao, Wen Liang, Peng Zou, and Feng Yuan. Dynamical structure factors of a two-dimensional Fermi superfluid within random phase approximation. New J. Phys., 22(9):093012, sep 2020. [7] R. Ganesh, A. Paramekanti, and A. A. Burkov. Collective modes and superflow instabilities of strongly correlated Fermi superfluids. Phys. Rev. A, 80:043612, Oct 2009. [8] Patrick Kelly and Ettore Vitali. On the Accuracy of Random Phase Approximation for Dynamical Structure Factors in Cold Atomic Gases. Atoms, 9(4), 2021. [9] Yoshihiro Yunomae, Daisuke Yamamoto, Ippei Danshita, Nobuhiko Yokoshi, and Shunji Tsuchiya. Instability of superfluid Fermi gases induced by a rotonlike density mode in optical lattices. Phys. Rev. A, 80:063627, Dec 2009. [10] R. Combescot, M. Yu. Kagan, and S. Stringari. Collective mode of homogeneous superfluid Fermi gases in the BEC-BCS crossover. Phys. Rev. A, 74:042717, Oct 2006. [11] S. Zhang. Auxiliary-Field Quantum Monte Carlo for Correlated Electron Systems. In E. Pavarini, E. Koch, and U. Schollwöck, editors, Emergent Phenomena in Correlated Matter: Modeling and Simulation Vol. 3. Verlag des Forschungszentrum Jülich, Jülich, Germany, 2013. [12] Ettore Vitali, Hao Shi, Mingpu Qin, and Shiwei Zhang. Computation of dynamical correlation functions for many-fermion systems with auxiliary-field quantum Monte Carlo. Phys. Rev. B, 94:085140, Aug 2016. [13] M. Motta, D. E. Galli, S. Moroni, and E. Vitali. Imaginary time correlations and the phaseless auxiliary field quantum Monte Carlo. J. Chem. Phys., 140(2):024107, 01 2014.