G. Mussardo

• PhD Theoretical Physics
• International School for Advanced Studies

In this paper we study transitions of atoms between energy levels of several number-theory-inspired atom potentials, under the effect of time-dependent perturbations. First, we simulate in detail the case of a trap whose one-particle spectrum is given by prime numbers. We investigate one-body Rabi oscillations and the excitation lineshape for two r...

In this paper we show how to measure in the setting of digital quantum simulations the reflection and transmission amplitudes of the one-dimensional scattering of a particle with a short-ranged potential. The main feature of the protocol is the coupling between the particle and an ancillary spin-1/2 degree of freedom. This allows us to reconstruct...

A bstract We revisit and extend Fisher’s argument for a Ginzburg-Landau description of multicritical Yang-Lee models in terms of a single boson Lagrangian with potential φ ² ( iφ ) n . We explicitly study the cases of n = 1, 2 by a Truncated Hamiltonian Approach based on the free massive boson perturbed by PT symmetric deformations, providing clear...

In this paper we show how to measure in the setting of digital quantum simulations the reflection and transmission amplitudes of the one-dimensional scattering of a particle with a short-ranged potential. The main feature of the protocol is the coupling between the particle and an ancillary spin-1/2 degree of freedom. This allows us to reconstruct...

A bstract We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The S -matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann ζ ( s...

We compute the form factors of the order and disorder operators, together with those of the stress–energy tensor, of a two-dimensional three-state Potts model with vacancies along its thermal deformation at the critical point. At criticality, the model is described by the non-diagonal partition function of the unitary minimal model M 6 , 7 of confo...

A bstract We study a novel class of Renormalization Group flows which connect multicritical versions of the two-dimensional Yang-Lee edge singularity described by the conformal minimal models $$ \mathcal{M} $$ M (2 , 2 n + 3). The absence in these models of an order parameter implies that the flows towards and between Yang-Lee edge singularities ar...

Integrable Quantum Field Theories can be solved exactly using bootstrap techniques based on their elastic and factorisable S-matrix. While knowledge of the scattering amplitudes reveals the exact spectrum of particles and their on-shell dynamics, the expression of the matrix elements of the various operators allows the reconstruction of off-shell q...

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The $S$-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann $\zeta(s) $ fu...

We study a novel class of Renormalization Group flows which connect multicritical versions of the two-dimensional Yang-Lee edge singularity described by the conformal minimal models M(2,2n+3). The absence in these models of an order parameter implies that the flows towards and between Lee-Yang edge singularities are all related to the spontaneous b...

Integrable Quantum Field Theories can be solved exactly using bootstrap techniques based on their elastic and factorisable S-matrix. While knowledge of the scattering amplitudes reveals the exact spectrum of particles and their on-shell dynamics, the expression of the matrix elements of the various operators allows the reconstruction of off-shell q...

We report the experimental realization of the prime number quantum potential VN(x), defined as the potential entering the single-particle Schrödinger Hamiltonian with eigenvalues given by the first N prime numbers. Using computer-generated holography, we create light intensity profiles suitable to optically trap ultracold atoms in these potentials...

We study the decay of the false vacuum in the scaling Ising and tricritical Ising field theories using the truncated conformal space approach and compare the numerical results to theoretical predictions in the thin wall limit. In the Ising case, the results are consistent with previous studies on the quantum spin chain and the φ4 quantum field theo...

In this paper we study the non-unitary deformations of the two-dimensional Tricritical Ising Model obtained by coupling its two spin \mathbb {Z}_2 odd operators to imaginary magnetic fields. Varying the strengths of these imaginary magnetic fields and adjusting correspondingly the coupling constants of the two spin Z2 even fields, we establish the...

The quantum version of the free fall problem is a topic often skipped in undergraduate quantum mechanics courses, because its discussion usually requires wavepackets built on the Airy functions—a difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and for general many-body syste...

We study the decay of the false vacuum in the scaling Ising and tricritical Ising field theories using the Truncated Conformal Space Approach and compare the numerical results to theoretical predictions in the thin wall limit. In the Ising case, the results are consistent with previous studies on the quantum spin chain and the $\varphi^4$ quantum f...

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional E_7 E 7 Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation...

Entangled quantum states share properties that do not have classical analogs; in particular, they show correlations that can violate Bell inequalities. It is, therefore, an interesting question to see what happens to entanglement measures—such as the entanglement entropy for a pure state—taking the semiclassical limit, where the naive expectation i...

We study the leading and sub-leading magnetic perturbations of the thermal E7 integrable deformation of the tricritical Ising model. In the low-temperature phase, these magnetic perturbations lead to the confinement of the kinks of the model. The resulting meson spectrum can be obtained using the semi-classical quantisation, here extended to includ...

We report the first experimental realization of the prime number quantum potential $V_N(x)$, defined as the potential entering the single-particle Schr\"{o}dinger Hamiltonian with eigenvalues given by the first $N$ prime numbers. We use holographic optical traps and, in particular, a spatial light modulator to tailor the potential to the desired sh...

Entangled quantum states share properties that do not have classical analogs, in particular, they show correlations that can violate Bell inequalities. It is therefore an interesting question to see what happens to entanglement measures -- such as the entanglement entropy for a pure state -- taking the semi-classical limit, where the naive expectat...

We study the leading and sub-leading magnetic perturbations of the thermal $E_7$ integrable deformation of the tricritical Ising model. In the low-temperature phase, these magnetic perturbations lead to the confinement of the kinks of the model and the resulting meson spectrum can be obtained using the semi-classical quantization. An interesting fe...

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ -function is directly related to the growth of the Mertens function M ( x ) = ∑ k = 1 x μ ( k ) , where μ ( k ) is the Möbius coefficient of the integer k ; the RH is indeed true if the Mertens function goes asymptotically as M ( x ) ∼ x 1/2+ ϵ , w...

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional $E_7$ Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation op...

We report V51 NMR and inelastic neutron scattering (INS) measurements on a quasi-1D antiferromagnet BaCo2V2O8 under transverse field along the [010] direction. The scaling behavior of the spin-lattice relaxation rate above the Néel temperatures unveils a 1D quantum critical point (QCP) at Hc1D≈4.7 T, which is masked by the 3D magnetic order. With t...

The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann zeta-function is directly related to the growth of the Mertens function: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \simeq x^{1/2 + \epsilon}$. We show that this behavior can be established on the basis of a new probab...

A bstract One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/ b self-duality of its S -matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh( bϕ ). In this paper we develop truncated spe...

In this Letter we set up a suggestive number theory interpretation of a quantum ladder system made of N coupled chains of spin 1/2. Using the hard-core boson representation and a leg-Hamiltonian made of a magnetic field and a hopping term, we can associate to the spins σa the prime numbers pa so that the chains become quantum registers for square-f...

Characterizing the scaling with the total particle number (N) of the largest eigenvalue of the one-body density matrix (λ0) provides information on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting λ0∼NC0, then C0=1 corresponds in ODLRO. The intermediate case, 0<C0<1, corresponds in tran...

At sunset, the desert gave its best: the canyons with their laced peaks were slowly tinged with amaranth, the cactuses cast long shadows on the golden stones and the large fireball of the sun floated softly on the horizon, so close as to be able almost to touch it. On the desolate mesa of Los Alamos, the last violet and orange glow of dusk finally...

With the exception of a poet, a war hero or a rock star, dying young is a terrible mistake. James Clerk Maxwell made this mistake by dying in 1879 at the tender age of 48, and although, for physicists, he still remains one of the most significant figures in his field, his name means little or nothing to most people. It is unlikely, in fact, that an...

The history of science is full of episodes of discrimination against women: one of the most striking episodes remains that of Lise Meiter, the subject of one of our future chapters. Sometimes, it was the malice of words that took over, as in the case of the mathematician Maria Gaetana Agnesi, who lived in the second half of the eighteenth century a...

At the end of a conference by Albert Einstein, held in Lipsia in 1930 in front of the German Physical Society, the president, after warmly thanking the famous guest for his speech, looked around to welcome the questions of those present. From the last row of the room, a young man stood up. He appeared clumsy, with a bony face and a large rebel tuft...

Everything had happened so fast—in the last few days, in the last few hours—that she hadn't even had time to catch her breath. Of course, things had started to degenerate months earlier, when Hitler had ordered the German army to invade Austria and, immediately afterwards, proclaimed its annexation by Germany: in fact, from March 12, 1938, the Aust...

On the evening of Friday, June 20, 1862, a large crowd had crept into a street in the heart of the West End of London, a turbulent river of cylinder hats and pretty lace caps. Among the wide folds of the crinolines, there were the faces of several children, also excited by all of that turmoil and the strange agitation seemed to possess their parent...

Given the characters in question, this story could not have ended otherwise. A story that could be called The Great Misunderstanding, and its sequel, The Great Cold. The protagonists were two giants of twentieth-century mathematics, but it would have been difficult to find two men so profoundly different, so siderally distant. Their names were Paul...

1925 had been a year without any noteworthy events, and as it had passed, so did it smoothly proceed towards its end. The Swiss countryside was covered with snow, a thick and very white blanket lying on the tops and along the slopes of the mountains like a layer of marzipan: if you looked at the woods, with a little luck, you could spot a few bears...

The history of mathematics has lost track of Thomas Harriot. There is no well-known burial place or monument to remember him, as with Newton, buried in the Westminster Abbey with full honours. There is no formula or equation that bears his name: during his life, he published a single publication, dedicated to the study of the language and customs o...

In the summer of 1793, the small town of Scandiano, at the foot of the Apennines of Reggio Emilia, was the scene of a singular event that soon created turmoil not only in that small Emilian village, but also in the entire scientific world of the time, leading to a frenzy of heated controversies and animated discussions.

Johannes Kepler is one of the most atypical figures in the history of science. In fact, there is no other character who better embodies the passage from Renaissance magic to the dawn of modern science. Passionate about astrology and mysticism, prisoner of the religious and baroque culture of the time, he nevertheless managed to turn his gaze to the...

This is the story of a love, both tragic and crazy. A madness that lasted ten years, a passion that no war and no ocean was able to stop, a drama that took place between the South Seas and the coasts of India, between Mauritius and the rocks of Madagascar, between the beaches of the Philippines and the waves of the Indian ocean.

For Christmas of 1895, in addition to the usual greeting card, physicists from all over Europe received an envelope containing the X-ray of a long-fingered hand, adorned with a large ring: it was the hand of Mrs. Röntgen, portrayed in the world’s first X-ray images, taken by her husband, Wilhelm Conrad Röntgen, professor of physics at the Universit...

It is difficult to describe Bruno Touschek if you have not met him. In contrast to the stereotype of the Physics professor, absorbed in his thoughts and a little absent-minded, you have to imagine a restless type, inclined to make eccentric and extravagant jokes, to play hybrid word games (Austrian + Italian + English) and to respond the address of...

1938 was the year of racial laws in Italy, the last shameful act of a dictatorship that shortly thereafter dragged Italy, and the whole of humanity, into the immense tragedy of the Second World War. One afternoon, towards the end of that year, upon entering one of the rooms of the Physics Department in via Panisperna, Franco Rasetti found Enrico Fe...

Phase transitions are among the most fascinating phenomena in Nature. The world would be terribly flat and boring if the different phases of matter did not exist and if we did not witness the continuous flow of their transformations.

On April 28, 1954, in spite of an annoying breeze coming up from the Potomac River, in Washington DC, it was a beautiful spring morning. In a room on the second floor of the rather ugly building belonging to the Atomic Energy Commission, an exceptional witness was being heard. Besides the bench for the jury, consisting of three members, there was b...

According to a Chinese legend, reported by Jorge Luis Borges, there was a time when the world of men and the world of mirrors were two separate realities; between them, there was no coincidence of shapes or colors, and each had its own laws and customs. Peace reigned between the two worlds, and one could enter and exit the mirrors without any diffi...

Ernst Mach’s imperious voice rang out in the main hall of the University of Vienna: “Professor Boltzmann, I don’t think his atoms exist!” He was standing among the wooden benches of the second row, rather agitated, his long grizzled beard shaken by the movement of his head, a strange light in his very black eyes, and his finger pointed towards that...

That Wolfgang Pauli was one of the greatest theoretical physicists of the twentieth century is a well-known fact: his discoveries—in particular, the famous exclusion principle (which earned him the Nobel Prize in 1945) and the hypothesis of the existence of the neutrino—are the basis of our current understanding of matter and among the most importa...

Königsberg, a city on the eastern border of Prussia, was one of the most flourishing ports in the Baltic Sea in the nineteenth century. Within the circles of philosophers, it was known as “the city of Immanuel Kant”: the great German thinker was born and had lived there, and he had always led an extremely regular and habitual life; the locals used...

Robert Oppenheimer once said that Wolfgang Pauli was the only person he knew who was equal to his caricature. Several years later, Jeremy Bernstein felt compelled to add that if Pauli was similar to his caricature, Paul Dirac was even more picturesque than his caricature! The impenetrable reserve, the austere lifestyle, an almost impenetrable secre...

The beginning of the twentieth century witnessed a great interest in the conquest of the earth’s poles, the most inhospitable places on our planet. In the early years of the century, there was, in fact, a rapid succession of adventurous enterprises, often with dramatic results, sometimes animated only by the taste of challenge and risk, by the desi...

There were only a few days before Christmas, and the streets of Paris were full of sleet: from the river, a freezing wind was blowing, and everyone seemed to hurry as if looking for shelter. Some carriages stood in front of Notre Dame, the coach drivers intent on calming the horses that were moving about restlessly. Along the Seine, a few barges fu...

From the narrow streets of the Latin Quarter of Paris to the chaotic and dusty arteries of Bombay, from the polar cold of the Finnish tundras to the tropical Brazilian climate, from the Alsatian boulevards of Strasbourg to the austere northeastern buildings of Princeton. A prison sentence for suspected espionage and an adventurous escape to the Uni...

On the morning of August 1, 1930, at the P & O Bombay pier, Peninsular and Oriental Lines, there was the usual confusion of all departures, with the porters loading baggage and large crates, port officials working to make the latest arrangements and sailors who went up and down from the piers between ropes and shrouds. Foreign travellers could be r...

We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particle systems, we derive the analogous results for the many-particle case in the presence of a general interaction two-body potential and the corresponding Floquet Hamiltonia...

The quantum version of the free fall problem is a topic usually skipped in undergraduate Quantum Mechanics courses because its discussion would require to deal with wavepackets built on the Airy functions -- a notoriously difficult computation. Here, on the contrary, we show that the problem can be nicely simplified both for a single particle and f...

We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrödinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of hi...

Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one--body density matrix ($\lambda_0$), provides informations on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting $\lambda_0\sim N^{\mathcal{C}_0}$, then $\mathcal{C}_0=1$ corresponds to ODL...

One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the $b \rightarrow 1/b$ self-duality of its $S$-matrix, of which there is no trace in its Lagrangian formulation. Here $b$ is the coupling appearing in the model's eponymous hyperbolic cosine present in its Lagrangian, $\cosh(b\phi)$. In this paper we develop trunc...

We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particles systems, we derive the analogous results for the many-particles case in presence of a general interaction two-body potential and the corresponding Floquet Hamiltonian....

Exotic excitations can emerge in the vicinity of a quantum phase transition. When the quantum critical point of the one-dimensional (1D) transverse field Ising model is perturbed by a longitudinal magnetic field, it was predicted that its massive excitations are precisely described by the exceptional E$_8$ Lie algebra. Here we show an unambiguous e...

In this paper we set up a suggestive number theory interpretation of a quantum ladder system made of ${\mathcal N}$ coupled chains of spin 1/2. Using the hard-core boson representation, we associate to the spins $\sigma_a$ along the chains the prime numbers $p_a$ so that the chains become quantum registers for square-free integers. The Hamiltonian...

Chapter 11 discusses the so-called minimal conformal models, all of which are characterized by a finite number of representations. It goes on to demonstrate how all correlation functions of these models satisfy linear differential equations. It shows how their explicit solutions are given by using the Coulomb gas method. It also explains how their...

Chapter 14 discusses how the identification of a class of universality is one of the central questions needing an answer for those in the field of statistical physics. This chapter discusses in detail the class of universality of several models, and provides examples that include the Ising model, the tricritical Ising model and its structure consta...

The Ising model in a magnetic field is one of the most beautiful examples of an integrable model. This chapter presents its exact S -matrix and the exact spectrum of its excitations, which consist of eight particles of different masses. Similarly, it discusses the exact scattering theory behind the thermal deformation of the tricritical Ising model...

The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, ZN transformations and current algebras. It also covers superconformal models, the Neveu–Schwarz and Ramond sectors, irreducible representations and minimal models, additional sy...

Chapter 4 begins by discussing the Peierls argument, which allows us to prove the existence of a phase transition in the two-dimensional Ising model. The remaining sections of the chapter deal with duality transformations (duality in square, hexagonal and triangular lattices) that link the low- and high-temperature phases of several statistical mod...

The Thermodynamic Bethe Ansatz (TBA) allows us to study finite size and finite temperature effects of an integrable model. This chapter investigates the integral equations that determine the free energy and gives their physical interpretation. It discusses Casimir energy, Bethe relativistic wave function, the derivation of thermodynamics, the meani...

Chapter 25 covers the Truncated Hilbert Space Approach provides a very efficient numerical algorithm to study many properties of a perturbed conformal field theory defined on a finite geometry, typically an infinite cylinder of radius R . These include the masses of the various excitations, their number below threshold, the presence of false vacua...

Free theories are usually regarded as trivial examples of quantum systems. This chapter proves that this is not the case of the conformal field theories associated to the free bosonic and fermionic fields. The subject is not only full of beautiful mathematical identities but is also the source of deep physical concepts with far reaching application...

Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects...

Chapter 16 covers the general properties of the integrable quantum field theories, including how an integrable quantum field theory is characterized by an infinite number of conserved charges. These theories are illustrated by means of significant examples, such as the Sine–Gordon model or the Toda field theories based on the simple roots of a Lie...

Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is l...

Chapter 10 introduces the notion of conformal transformations and the important topic of the massless quantum field theories associated to the critical points of the statistical models. The chapter establishes the important conceptual result that the classification of all possible critical phenomena in two dimensions consists of finding out all pos...

Two exact combinatorial solutions of the two-dimensional Ising model are the key topics of this chapter. Although no subsequent topic depends on them, both the mathematical and the physical aspects of these solutions are so elegant as to deserve special attention. Chapter 5 covers how the Ising model is a pathfinder in the field of critical phenome...

Chapter 2 discusses one-dimensional statistical models, for example, the Ising model and its generalizations (Potts model, systems with O ( n ) or Z n -symmetry, etc.). It discusses several methods of solution and covers the recursive method, the transfer matrix approach, and series expansion techniques. General properties of these methods, which a...

Chapter 3 discusses the approximation schemes used to approach lattice statistical models that are not exactly solvable. In addition to the mean field approximation, it also considers the Bethe–Peierls approach to the Ising model. Moreover, there is a thorough discussion of the Gaussian model and its spherical version, both of which are two importa...

This chapter deals with the exact solution of the two-dimensional Ising model as it is achieved through the transfer matrix formalism. It discusses the crucial role played by the commutative properties of the transfer matrices, which lead to a functional equation for their eigenvalues. The exact free energy of the Ising model and its critical point...

This book is an introduction to statistical field theory, which is an important subject within theoretical physics and a field that has seen substantial progress in recent years. The book covers fundamental topics in great detail and includes areas like conformal field theory, quantum integrability, S -matrices, braiding groups, Bethe ansatz, renor...

Chapter 22 introduces a perturbative technique based on the form factors to study non-integrable models. These models often include stumbling blocks like decays and production scattering processes, confinement phenomena and nucleation of false vacua, resonance peaks in the cross sections, etc. All these physical aspects are usually accompanied by a...

Quantum field theory with boundaries is a very rich subject both from purely theoretical point of view or applicative aims. Chapter 21 presents the basis results relative to critical systems and massive integrable field theories. It covers stress-energy tensor in boundary conformal field theories and integrable deformations of these theories (inclu...

A crucial aspect of the Ising model is its fermionic nature and this chapter is devoted to this property of the model. In the continuum limit, a Dirac equation for neutral Majorana fermions emerges. The details of the derivation are much less important than understanding why it is possible. The chapter emphasizes the simplicity and the exactness of...

Chapter 23 discusses the setting of a semi-classical method—based on the Lagrangian density of the model, irrespective of whether or not it describes an integrable system—to address the computation of the particle spectrum of bound states in quantum field theory with a set of degenerate vacua connected by kink excitations. It begins by investigatin...

Chapter 24 discusses interacting fermions and supersymmetry (SUSY) models. The chapter addresses the semi-classical formalism relative to a fermion field in a bosonic background. It covers topics that include bosonic and fermionic bound states (both Dirac and Majorana), symmetric wells, supersymmetric theory, general results in SUSY theories, integ...

At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations t...

Chapter 15 introduces the notion of the scaling region near the critical points, identified by the deformations of the critical action by means of the relevant operators. The renormalization group flows that originate from these deformations are subjected to important constraints, which can be expressed in terms of sum-rules. This chapter also disc...

Chapter 17 discusses the S -matrix theory of two-dimensional integrable models. From a mathematical point of view, the two-dimensional nature of the systems and their integrability are the crucial features that lead to important simplifications of the formalism and its successful application. This chapter deals with the analytic theory of the S -ma...

We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrodinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of hi...

Science, with its inherent tension between the known and the unknown, is an inexhaustible mine of great stories. Collected here are twenty-six among the most enchanting tales, one for each letter of the alphabet: the main characters are scientists of the highest calibre mostly of whom, however, are unknown to the general public. This book goes fro...

An integrable model subjected to a periodic driving gives rise generally to a nonintegrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb-Liniger model in the presence of a linear potential with a periodic time-dependent strength is instead integrable and its quasienergies can be determined using the Bethe an...

An integrable model subjected to a periodic driving gives rise generally to a non-integrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb--Liniger model in presence of a linear potential with a periodic time--dependent strength is instead integrable and its quasi-energies can be determined using the Bethe an...

L functions based on Dirichlet characters are natural generalizations of the Riemann function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the generalized Riemann hypothesis relative to the non-trivial complex zeros of the Dirichlet L...

The scaling of the largest eigenvalue λ0 of the one-body density matrix of a system with respect to its particle number N defines an exponent C and a coefficient B via the asymptotic relation λ0∼BNC. The case C=1 corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well-known result also confirmed b...

L$ functions based on Dirichlet characters are natural generalizations of the Riemann $\zeta(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the...

The scaling of the largest eigenvalue $\lambda_0$ of the one-body density matrix of a system with respect to its particle number $N$ defines an exponent $\mathcal{C}$ and a coefficient $\mathcal{B}$ via the asymptotic relation $\lambda_0 \sim \mathcal{B}\,N^{\mathcal{C}}$. The case $\mathcal{C}=1$ corresponds to off-diagonal long-range order. For a...

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue $\lambda_0$ of the one-body-density matrix scales as $\lambda_0 \sim N$, where $N$ is the total number of particles. Putting $\lambda_0 \sim N^{{\cal C}}$ to define the scaling exponent ${\cal C}$, then ${\cal C}=1$ corresponds to ODLRO and ${\cal C}=0$ to t...

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue $\lambda_0$ of the one-body-density matrix scales as $\lambda_0 \sim N$, where $N$ is the total number of particles. Putting $\lambda_0 \sim N^{{\cal C}}$ to define the scaling exponent ${\cal C}$, then ${\cal C}=1$ corresponds to ODLRO and ${\cal C}=0$ to t...

We show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\mbox{Re}(s) > 1$ down to $\mbox{Re}(s) > 1/2$, without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior $C...

Using the Dirichlet theorem on the equidistribution of residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\Re(s) > 1$ down t...