Giuseppe Mussardo

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization. In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased, and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.

A bstract: We develop a variational framework for addressing two-dimensional non-integrable quantum field theories through the exact structure of their integrable counterparts. Concentrating on the φ ⁴ Landau–Ginzburg model, we use the analytical Vacuum Expectation Values and Form Factors of local operators in the sinh-Gordon theory as the foundation of a variational ansatz. In this way, we obtain controlled estimates of central physical quantities of the φ ⁴ theory — such as the finite-volume ground-state energy and the physical mass as a function of the coupling constant. The strengths of the variational methods are leveraged in combination with the Hamiltonian truncation techniques and the LeClair-Mussardo formula, which also allow to probe the accuracy of the variational approximation varying the system size. Within the weak-coupling regime, a detailed numerical analysis reveals the behaviour of the finite-volume spectrum, the ground-state energy, and the elastic part of the scattering matrix, showing how the rigorous machinery of integrable models can serve as a guiding light into the complex landscape of non-integrable quantum field dynamics.

In this paper we study transitions of atoms between energy levels of several number-theory-inspired trapping potentials under the effect of time-dependent perturbations. First, we simulate in detail the case of a trap whose single-particle spectrum is given by the prime numbers. We investigate one-body Rabi oscillations and the excitation line shape for two resonantly coupled energy levels, and we show that quantum control is a faster method for state preparation than periodic perturbation. Next, we investigate cascades of such transitions, particularly whether one can construct a quantum system where the existence of a continuous resonant cascade from a given initial energy eigenstate is predicated by the validity of a given statement in number theory. We find that such resonance cascades, in a suitably designed one-body system, can be used to illustrate that the sequence of natural numbers is closed under multiplication. We further present ideas for two more resonance cascade experiments designed to illustrate certain statements within the Diophantus-Brahmagupta-Fibonacci identity and the Goldbach conjecture.

Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian—viewed simply as a matrix of its elements—does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of correlation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing between integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k -step gaps which can help in discriminating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group S N , thereby demonstrating both its effectiveness and its broad applicability.

The book provides an introduction to the analytic method of the S-matrix for dealing with strongly interacting models, with applications to quantum field theories and statistical physics.

The famous Goldbach conjecture states that any even natural number N N greater than 2 2 can be written as the sum of two prime numbers p^{\text{(I)}} p (I) and p^{\text{(II)}} p (II) . In this article we propose a quantum analogue device that solves the following problem: given a small prime p^{\text{(I)}} p (I) , identify a member N N of a \mathcal{N} ???? -strong set even numbers for which N-p^{\text{(I)}} N − p (I) is also a prime. A table of suitable large primes p^{\text{(II)}} p (II) is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a \sqrt{\mathcal{N}} ???? quantum advantage. Our numerical example involves a set of 51 even numbers just above the highest even classical-numerically explored so far [T. O. e Silva, S. Herzog, and S. Pardi, Mathematics of Computation 83, 2033 (2013)]. For a given small prime number p^{\text{(I)}}=223 p (I) = 223 , it took our quantum algorithm 5 steps to identify the number N=4× 10^{18}+14 N = 4 × 10 18 + 14 as featuring a Goldbach partition involving 223 223 and another prime, namely p^{\text{(II)}}=4× 10^{18}-239 p (II) = 4 × 10 18 − 239 . Currently, our algorithm limits the number of evens to be tested simultaneously to \mathcal{N} \sim \ln(N) ???? ∼ ln ( N ) : larger samples will typically contain more than one even that can be partitioned with the help of a given p^{\text{(I)}} p (I) , thus leading to a departure from the Grover paradigm.