21 research outputs found
Rounding of First Order Transitions in Low-Dimensional Quantum Systems with Quenched Disorder
We prove that the addition of an arbitrarily small random perturbation of a
suitable type to a quantum spin system rounds a first order phase transition in
the conjugate order parameter in d <= 2 dimensions, or in systems with
continuous symmetry in d <= 4. This establishes rigorously for quantum systems
the existence of the Imry-Ma phenomenon, which for classical systems was proven
by Aizenman and Wehr.Comment: Four pages, RevTex. Minor correction
Continuum limit of random matrix products in statistical mechanics of disordered systems
We consider a particular weak disorder limit ("continuum limit") of matrix
products that arise in the analysis of disordered statistical mechanics
systems, with a particular focus on random transfer matrices. The limit system
is a diffusion model for which the leading Lyapunov exponent can be expressed
explicitly in terms of modified Bessel functions, a formula that appears in the
physical literature on these disordered systems. We provide an analysis of the
diffusion system as well as of the link with the matrix products. We then apply
the results to the framework considered by Derrida and Hilhorst [J. Phys. A
(1983)], which deals in particular with the strong interaction limit for
disordered Ising model in one dimension and that identifies a singular behavior
of the Lyapunov exponent (of the transfer matrix), and to the two dimensional
Ising model with columnar disorder (McCoy-Wu model). We show that the continuum
limit sharply captures the Derrida and Hilhorst singularity. Moreover we
revisit the analysis by McCoy and Wu [Phys. Rev. 1968] and remark that it can
be interpreted in terms of the continuum limit approximation. We provide a
mathematical analysis of the continuum approximation of the free energy of the
McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this
approximation, the free energy of the two dimensional Ising model with columnar
disorder is but not analytic at the critical temperature.Comment: 46 pages, one figure. Introduction reorganized, Proposition 1.5
corrects Proposition 1.6 of v2. Several other scattered modification
The Zeros of the Partition Function of the Pinning Model
We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments
Non-integrable Ising models in cylindrical geometry: Grassmann representation and infinite volume limit
In this paper, meant as a companion to arXiv:2006.04458, we consider a class
of non-integrable Ising models in cylindrical domains, and we discuss two
key aspects of the multiscale construction of their scaling limit. In
particular, we provide a detailed derivation of the Grassmann representation of
the model, including a self-contained presentation of the exact solution of the
nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic
estimates of the fermionic Green's function in the cylinder, required for the
multiscale analysis of the model. We also review the multiscale construction of
the effective potentials in the infinite volume limit, in a form suitable for
the generalization to finite cylinders. Compared to previous works, we
introduce a few important simplifications in the localization procedure and in
the iterative bounds on the kernels of the effective potentials, which are
crucial for the adaptation of the construction to domains with boundaries.Comment: 69 pages. Minor revisions. Version accepted by Annales Henri
Poincar\'e. arXiv admin note: substantial text overlap with arXiv:2006.0445
Proof of Rounding by Quenched Disorder of First Order Transitions in Low-Dimensional Quantum Systems
We prove that for quantum lattice systems in d<=2 dimensions the addition of
quenched disorder rounds any first order phase transition in the corresponding
conjugate order parameter, both at positive temperatures and at T=0. For
systems with continuous symmetry the statement extends up to d<=4 dimensions.
This establishes for quantum systems the existence of the Imry-Ma phenomenon
which for classical systems was proven by Aizenman and Wehr. The extension of
the proof to quantum systems is achieved by carrying out the analysis at the
level of thermodynamic quantities rather than equilibrium states.Comment: This article presents the detailed derivation of results which were
announced in Phys. Rev. Lett. 103 (2009) 197201 (arXiv:0907.2419). v3
incorporates many corrections and improvements resulting from referee
comment
Adiabatic Evolution of Low-Temperature Many-Body Systems
We consider finite-range, many-body fermionic lattice models and we study the
evolution of their thermal equilibrium state after introducing a weak and
slowly varying time-dependent perturbation. Under suitable assumptions on the
external driving, we derive a representation for the average of the evolution
of local observables via a convergent expansion in the perturbation, for small
enough temperatures. Convergence holds for a range of parameters that is
uniform in the size of the system. Under a spectral gap assumption on the
unperturbed Hamiltonian, convergence is also uniform in temperature. As an
application, our expansion allows us to prove closeness of the time-evolved
state to the instantaneous Gibbs state of the perturbed system, in the sense of
expectation of local observables, at zero and at small temperatures. As a
corollary, we also establish the validity of linear response. Our strategy is
based on a rigorous version of the Wick rotation, which allows us to represent
the Duhamel expansion for the real-time dynamics in terms of Euclidean
correlation functions, for which precise decay estimates are proved using
fermionic cluster expansion.Comment: The introduction and the discussion after the main result have been
improved, minor corrections. We added a new corollary, about the stronger
adiabatic convergence for switch functions with derivatives vanishing at
zero. 61 page
On Spin Systems with Quenched Randomness: Classical and Quantum
The rounding of first order phase transitions by quenched randomness is
stated in a form which is applicable to both classical and quantum systems: The
free energy, as well as the ground state energy, of a spin system on a
-dimensional lattice is continuously differentiable with respect to any
parameter in the Hamiltonian to which some randomness has been added when . This implies absence of jumps in the associated order parameter, e.g.,
the magnetization in case of a random magnetic field. A similar result applies
in cases of continuous symmetry breaking for . Some questions
concerning the behavior of related order parameters in such random systems are
discussed.Comment: 8 pages LaTeX, 2 PDF figures. Presented by JLL at the symposium
"Trajectories and Friends" in honor of Nihat Berker, MIT, October 200
The scaling limit of the energy correlations in non integrable Ising models
We obtain an explicit expression for the multipoint energy correlations of a
non solvable two-dimensional Ising models with nearest neighbor ferromagnetic
interactions plus a weak finite range interaction of strength , in a
scaling limit in which we send the lattice spacing to zero and the temperature
to the critical one. Our analysis is based on an exact mapping of the model
into an interacting lattice fermionic theory, which generalizes the one
originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising
model. The interacting model is then analyzed by a multiscale method first
proposed by Pinson and Spencer. If the lattice spacing is finite, then the
correlations cannot be computed in closed form: rather, they are expressed in
terms of infinite, convergent, power series in . In the scaling limit,
these infinite expansions radically simplify and reduce to the limiting energy
correlations of the integrable Ising model, up to a finite renormalization of
the parameters. Explicit bounds on the speed of convergence to the scaling
limit are derived.Comment: 75 pages, 11 figure
Effects of quenched randomness on classical and quantum phase transitions
This dissertation describes the e ffect of quenched randomness on fi rst order phase transitions in lattice systems, classical and quantum. It is proven that a large class of quantum lattice systems in low dimension (d <= 2 or, with suitable continuous symmetry, d <= 4) cannot exhibit first-order phase transitions in the presence of suitable ("direct") quenched disorder.Ph.D.Includes bibliographical referencesIncludes vitaby Rafael L. Greenblat