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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 tomographically the scattering amplitudes, which are in general complex numbers, from the readout of one qubit. Applications of our results are discussed.

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 φ²(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 evidence of the spontaneous breaking of PT symmetry. For n = 1, the symmetric and the broken phases are separated by the critical point corresponding to the minimal model M25, while for n = 2, they are separated by a critical manifold corresponding to the minimal model M25 with M27 on its boundary. Our numerical analysis strongly supports our Ginzburg-Landau descriptions for multicritical Yang-Lee models.

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) function along the axis of the complex s-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet L-functions again along the axis. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex s-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping s → 1 – s) and the Euler product representation of the Dirichlet L-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function, i.e. a function whose completion satisfies the duality equation χ(s) = χ(1 – s) but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.

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 conformal field theories and is accompanied by an internal S 3 symmetry. The off-critical thermal deformation is an integrable massive theory that is still invariant under S 3. The presence of infinitely many conserved quantities, whose spin spectrum is related to the exceptional Lie algebra E 6, allows us to determine the analytic S-matrix, the exact mass spectrum and the matrix elements of local operators of this model in an exact non-perturbative way. We use the spectral representation series of the correlators and the fast convergence of these series to compute several universal ratios of the renormalization group.

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 quantities such as two-point correlation functions with a high level of precision. In this review, we summarise results relevant to the contact point between theory and experiment providing a number of quantities that can be computed theoretically with great accuracy. We concentrate on universal amplitude ratios which can be determined from the measurement of generalised susceptibilities, and dynamical structure factors, which can be accessed experimentally e.g. via inelastic neutron scattering or nuclear magnetic resonance. Besides an overview of the subject and a summary of recent advances, we also present new results regarding generalised susceptibilities in the tricritical Ising universality class.

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 Yang-Lee edge singularities are all related to the spontaneous breaking of PT symmetry and comprise a pattern of flows in the space of PT symmetric theories consistent with the c-theorem and the counting of relevant directions. Additionally, we find that while in a part of the phase diagram the domains of unbroken and broken PT symmetry are separated by critical manifolds of class M (2, 2n + 3), other parts of the boundary between the two domains are not critical.

700 years after Dante Alighieri's death, this book intertwines the voice of the great poet with that of an exceptional contemporary, Marco Polo, who was equally curious about the geography of both earthly and celestial worlds. If Polo was the "ordinary genius" of the XIII century, the designation of "sorcerer genius" must go to Alighieri, the man with encyclopedic wisdom, at ease with his era's philosophy, theology, and science. The sorcerer genius-well versed in this world-must create their own, which he did with The Divine Comedy. On the other hand, The Travels of Marco Polo, the greatest classic in travel literature, offers wonder and provides delight. This book combines the unforgettable characters of both books, the darkness of the infernal landscapes with the immensity of the Asian deserts, the richness of the Mongol empire with the glamor of medieval philosophy, the aspirations and dreams of two great explorers with knowledge of the science of their time, as well as the ever-eternal cosmology. This is an accessible and entertaining book for high school students, scholars of scientific history and the history of ideas, and curious readers who want to know more about Dante and Marco Polo and their unquenchable thirst for knowledge. This book is a translation of an original Italian edition. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation.

In this paper we study the non-unitary deformations of the two-dimensional Tricritical Ising Model obtained by coupling its two spin ℤ2 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 presence of two universality classes of infrared fixed points on the critical surface. The first class corresponds to the familiar Yang-Lee edge singularity, while the second class to its tricritical version. We argue that these two universality classes are controlled by the conformal non-unitary minimal models M(2, 5) and M(2, 7) respectively, which is supported by considerations based on PT symmetry and the corresponding extension of Zamolodchikov’s c-theorem, and also verified numerically using the truncated conformal space approach. Our results are in agreement with a previous numerical study of the lattice version of the Tricritical Ising Model [1]. We also conjecture the classes of universality corresponding to higher non-unitary multicritical points obtained by perturbing the conformal unitary models with imaginary coupling magnetic fields.

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 for different N values. As a further application, we also implement a potential whose spectrum is given by the lucky numbers, a sequence of integers generated by a different sieve than the familiar Eratosthenes’s sieve used for the primes. Our results pave the way toward the realization of quantum potentials with arbitrary sequences of integers as energy levels and show, in perspective, the possibility to set up quantum systems for arithmetic manipulations or mathematical tests involving prime numbers.

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 theory; in particular, we confirm that while the theoretical predictions get the dependence of the bubble nucleation rate on the latent heat right, they are off by a model-dependent overall coefficient. The tricritical Ising model allows us on the other hand to examine more exotic vacuum degeneracy structures, such as three vacua or two asymmetric vacua, which leads us to study several novel scenarios of false vacuum decay by lifting the vacuum degeneracy using different perturbations.

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 systems by making use of a gauge transformation that corresponds to a change of reference frame from the laboratory frame to the one comoving with the falling system. Using this approach, the quantum mechanics problem of a particle in an external gravitational potential reduces to a much simpler one where there is no longer any gravitational potential in the Schrödinger equation. It is instructive to see that the same procedure can be used for many-body systems subjected to an external gravitational potential and a two-body interparticle potential that is a function of the distance between the particles. This topic provides a helpful and pedagogical example of a quantum many-body system whose dynamics can be analytically described in simple terms.

We present a one-dimensional multicomponent model, known to be partially integrable when restricted to the subspaces made of only two components. By constructing fully antisymmetrized bases, we find integrable excited eigenstates corresponding to the totally antisymmetric irreducible representation of the permutation operator in the otherwise nonintegrable subspaces. We establish rigorously the breakdown of integrability in those subspaces by showing explicitly the violation of the Yang-Baxter equation. We further solve the constraints from the Yang-Baxter equation to find exceptional momenta that allows Bethe ansatz solutions of solitonic bound states. These integrable eigenstates have distinct dynamical consequence from the embedded integrable subspaces previously known, as they do not span their separate Krylov subspaces, and a generic initial state can partly overlap with them and therefore have slow thermalization. However, this novel form of weak ergodicity breaking contrasts with that of quantum many-body scars in that the integrable eigenstates involved do not have necessarily low entanglement. Our approach provides a complementary route to arrive at exact excited states in nonintegrable models: instead of solving towers of single-mode excited states based on a solvable ground state in a nonintegrable model, we identify the integrable eigenstates that survive in a deformation of the Hamiltonian away from its integrable point.

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 include also mesonic excitations composed of two different kinks. An interesting feature of the integrable sub-leading magnetic perturbation of the thermal E7 deformation of the model is the possibility to swap the role of the two operators, i.e. the possibility to consider the model as a thermal perturbation of the integrable A3 model associated to the sub-leading magnetic deformation. Due to the occurrence of vacuum degeneracy unrelated to spontaneous symmetry breaking in A3, the confinement pattern shows novel features compared to previously studied models. Interestingly enough, the validity of the semi-classical description in terms of the A3 endpoint extends well beyond small fields, and therefore the full parameter space of the joint thermal and sub-leading magnetic deformation is well described by a combination of semi-classical approaches. All predictions are verified by comparison to finite volume spectrum resulting from truncated conformal space.

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 E7 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 operators, σ(x) and σ⁰(x), in either phase are accompanied by associated disorder operators, µ(x) and µ⁰(x). Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For σ(x) and σ⁰(x), the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest Z2 odd particles. Similarly, the equations for µ(x) and µ⁰(x) have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for σ(x)/σ⁰(x) are fixed in terms of those for µ(x)/µ⁰(x). We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the ∆-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

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 is that they may become singular or zero. This conclusion is, however, incorrect. In this paper, we determine the ℏ→0 limit of the bipartite entanglement entropy for a one-dimensional system of N quantum particles in an external potential and we explicitly show that this limit is finite. Moreover, if the particles are fermionic, we show that the ℏ→0 limit of the bipartite entanglement entropy coincides with the Shannon entropy of N bits.

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) = ςk=1xμ (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+ , where is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős-Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive 'experimental' confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.

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 the aid of accurate analytical analysis and numerical calculations, we show that the zone center INS spectrum at Hc1D is precisely described by the pattern of the 1D quantum Ising model in a magnetic field, a class of universality described in terms of the exceptional E8 Lie algebra. These excitations are nondiffusive over a certain field range when the system is away from the 1D QCP. Our results provide an unambiguous experimental realization of the massive E8 phase in the compound, and open a new experimental route for exploring the dynamics of quantum integrable systems as well as physics beyond integrability.

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 spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b ≪ 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.

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-free integers. The rung Hamiltonian involves permutation terms between next-neighbor chains and a coprime repulsive interaction. The system has various phases; in particular, there is one whose ground state is a coherent superposition of the first N prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics.

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

Physical Review B 102, 184510, (17 November 2020)