Final Report Summary - WEAKLY CONFINED RMT (Spontaneous Symmetry Breaking in Random Matrices: a model for the Anderson Transition from String Theory)
The main goal of this fellowship is to show that a certain class of Random Matrix Models, known as Weakly Confined Matrix Models (WCMM), can be used as the first analytically solvable toy models for the Anderson Metal-Insulator Transition.
This is the quantum phase transition which a standard conductor (such as copper) goes through, when disorder in the sample increases and renders it unable to sustain the kind of extended wave-functions required for conduction. This disorder is common in all materials and typically is due to imperfections of the crystalline structure or to the presence of impurities in the sample. More than half of the insulators on earth are Anderson insulators and this motivates any effort in understanding this phase transition, which was theoretically introduced more than 50 years ago. Despite this interest, the complexity of the problem is such that, so far, analytical methods reached only limited success and one mostly has to rely on numerical approaches, which are of limited help in explaining the true nature of these phases. The main obstacle lies in the lack of a general perturbative regime and hence on the need for methods amenable to non-perturbative approaches.
Random matrix models provide such non-perturbative structure, but traditionally are limited to the description of conductive systems. In these models, one mimics the disorder by sampling a large set of Hamiltonian interactions, taken from a given distribution. These Hamiltonians are taken as large NxN matrices and, in order to take full advantage of the analytical structure of the set-up, the distribution is usually assumed to be base invariant. Traditionally, this invariance has been taken to imply that the matrix eigenvectors are uniformly distributed over a N-dimensional sphere and hence, in average, have all non-zero entries. This property classifies them as extended and the system as conductive. A (partially) localized system would have a non-uniform distribution of eigenvectors over the N-dimensional sphere, corresponding to a breaking the full U(N) rotational invariance.
One fundamental fact random matrices have taught us is that the conductive regime (which is a property of the eigenvectors) is fundamentally linked to the distribution of the matrix eigenvalues, which universally follows the so called Wigner-Dyson statistics. Hence, to describe a non-extended phase, we need to look for matrix models whose eigenvalue distribution deviated from the Wigner-Dyson behavior.
One such example is provided by matrix models with a disconnected support for their eigenvalues, because the weight chosen for the invariant matrix distribution has several local minima and the eigenvalues remain trapped in these wells. The simplest case is that of a double well potential, where half of the eigenvalues lie close to one minimum and the other half close to the second minimum. A paper is being drafted at the moment, in which a Spontaneous Symmetry Breaking (SSB) mechanism is presented, for which the U(N) invariance is broken into a U(N/2) x U(N/2) by this non-trivial eigenvalue distribution.
An even more severe deviation from the Wigner-Dyson statistic is presented by the WCMM, which, despite being generated by a U(N) invariant measure, show an eigenvalue statistics similar to that observed for systems at the Anderson transition. Having introduced in the first paper the matrix model’s SSB mechanism caused by a non-trivial eigenvalue distribution, a second manuscript is in preparation to detail how the WCMM eigenvalue behavior can give rise to a pattern of different SSB of the U(N) symmetry into smaller groups. Since the symmetry results broken in all its possible ways, WCMM present what is called a critical phase, and hence is an excellent candidate to be the first analytically solvable toy model to describe the Anderson transition. Having such a toy model available will help us in understanding this phase transition better and devise new applications.
Originally, the close connection of WCMM to certain string theory models (most noticeably, the so called ABJM models) lead to the conjecture that the latter could have a formulation where the SSB is already manifest and possibly already discussed in the literature. SSB is a key ingredient in any fundamental (and effective) theory of nature, as it allows for new symmetries to emerge naturally and, through the Higgs mechanism, endows massless particles with a mass. The string-theory literature does not seem to have discussed a SSB mechanism in matrix models and hence it is possible that the SSB mechanism uncovered by this project has not been noticed before and could become a cornerstone in the applications of strongly interacting theories described by matrix models.
A second objective of the fellowship was to continue the developing of the fellow’s existing interests on one-dimensional integrable models and their entanglement properties. Several published results have been achieved in this direction.
An intensive effort has been devoted in the past few years on studying the bi-partite entanglement of many-body systems, since it encodes properties of the ground state which are not approachable through standard local correlators and it could help us in understanding the foundation of quantum mechanics and how to harvest its power in quantum computers.
In a first paper, the character structure of the Renyi entropy of the XYZ chain was analyzed, finding that, surprisingly, it is the same as that of an Ising model. In a second article, the same was done for a family of models known as Restricted Solid-On-Solid (RSOS) and the role of the boundary conditions was identified in selecting different conformal characters. These results indicate that, in the massive regime, while the leading term in the entanglement entropy is universal, corrections are not.
In a third manuscript, the symmetries of the spin-½ XYZ chain were found to be encoded in the transformation properties of the coupling parameters of the model, as an extension of the modular group. The modular group, acting in real space, plays an important role in the classification of 1+1-dimensional critical models. Having found a similar structure, but in parameter space, and for a gapped, integrable system could help in understanding several properties of integrable models which have been observed during the years, but still lack a general explanation.
Finally, the entanglement properties of the quantum Ising model have been studied to highlight the role of edge states in breaking the so called local convertibility. It was proven that the entanglement of edge states can decrease approaching a quantum phase transition, while that of bulk states always increases as the correlation length grows. These findings have important consequences: for instance, they imply that, in the search for quantum many body systems that can be used as the kind of universal quantum simulator famously proposed by Feynmann, one should consider only systems that show lack of local convertibility. Also, non-local convertibility could emerge as an alternative, finite size estimator of long-range entanglement.
Finally, the fellow undertook a new and exciting project, in the very competitive field of out-of-equilibrium physics. Together with his collaborators, he asked what happens to a moving, localized excitation if the underlying Hamiltonian is suddenly changed in its interaction strength (this being a variation of the traditional quantum quench protocol). They study the dynamics of this set-up using a non-linear hydrodynamics approach and through numerical integration and they characterized the post-quench dynamics in terms of a reflected and a transmitted profile with universal shapes and velocities. They are also contacting experimental groups, which seem interested in studying this protocol.
This is the quantum phase transition which a standard conductor (such as copper) goes through, when disorder in the sample increases and renders it unable to sustain the kind of extended wave-functions required for conduction. This disorder is common in all materials and typically is due to imperfections of the crystalline structure or to the presence of impurities in the sample. More than half of the insulators on earth are Anderson insulators and this motivates any effort in understanding this phase transition, which was theoretically introduced more than 50 years ago. Despite this interest, the complexity of the problem is such that, so far, analytical methods reached only limited success and one mostly has to rely on numerical approaches, which are of limited help in explaining the true nature of these phases. The main obstacle lies in the lack of a general perturbative regime and hence on the need for methods amenable to non-perturbative approaches.
Random matrix models provide such non-perturbative structure, but traditionally are limited to the description of conductive systems. In these models, one mimics the disorder by sampling a large set of Hamiltonian interactions, taken from a given distribution. These Hamiltonians are taken as large NxN matrices and, in order to take full advantage of the analytical structure of the set-up, the distribution is usually assumed to be base invariant. Traditionally, this invariance has been taken to imply that the matrix eigenvectors are uniformly distributed over a N-dimensional sphere and hence, in average, have all non-zero entries. This property classifies them as extended and the system as conductive. A (partially) localized system would have a non-uniform distribution of eigenvectors over the N-dimensional sphere, corresponding to a breaking the full U(N) rotational invariance.
One fundamental fact random matrices have taught us is that the conductive regime (which is a property of the eigenvectors) is fundamentally linked to the distribution of the matrix eigenvalues, which universally follows the so called Wigner-Dyson statistics. Hence, to describe a non-extended phase, we need to look for matrix models whose eigenvalue distribution deviated from the Wigner-Dyson behavior.
One such example is provided by matrix models with a disconnected support for their eigenvalues, because the weight chosen for the invariant matrix distribution has several local minima and the eigenvalues remain trapped in these wells. The simplest case is that of a double well potential, where half of the eigenvalues lie close to one minimum and the other half close to the second minimum. A paper is being drafted at the moment, in which a Spontaneous Symmetry Breaking (SSB) mechanism is presented, for which the U(N) invariance is broken into a U(N/2) x U(N/2) by this non-trivial eigenvalue distribution.
An even more severe deviation from the Wigner-Dyson statistic is presented by the WCMM, which, despite being generated by a U(N) invariant measure, show an eigenvalue statistics similar to that observed for systems at the Anderson transition. Having introduced in the first paper the matrix model’s SSB mechanism caused by a non-trivial eigenvalue distribution, a second manuscript is in preparation to detail how the WCMM eigenvalue behavior can give rise to a pattern of different SSB of the U(N) symmetry into smaller groups. Since the symmetry results broken in all its possible ways, WCMM present what is called a critical phase, and hence is an excellent candidate to be the first analytically solvable toy model to describe the Anderson transition. Having such a toy model available will help us in understanding this phase transition better and devise new applications.
Originally, the close connection of WCMM to certain string theory models (most noticeably, the so called ABJM models) lead to the conjecture that the latter could have a formulation where the SSB is already manifest and possibly already discussed in the literature. SSB is a key ingredient in any fundamental (and effective) theory of nature, as it allows for new symmetries to emerge naturally and, through the Higgs mechanism, endows massless particles with a mass. The string-theory literature does not seem to have discussed a SSB mechanism in matrix models and hence it is possible that the SSB mechanism uncovered by this project has not been noticed before and could become a cornerstone in the applications of strongly interacting theories described by matrix models.
A second objective of the fellowship was to continue the developing of the fellow’s existing interests on one-dimensional integrable models and their entanglement properties. Several published results have been achieved in this direction.
An intensive effort has been devoted in the past few years on studying the bi-partite entanglement of many-body systems, since it encodes properties of the ground state which are not approachable through standard local correlators and it could help us in understanding the foundation of quantum mechanics and how to harvest its power in quantum computers.
In a first paper, the character structure of the Renyi entropy of the XYZ chain was analyzed, finding that, surprisingly, it is the same as that of an Ising model. In a second article, the same was done for a family of models known as Restricted Solid-On-Solid (RSOS) and the role of the boundary conditions was identified in selecting different conformal characters. These results indicate that, in the massive regime, while the leading term in the entanglement entropy is universal, corrections are not.
In a third manuscript, the symmetries of the spin-½ XYZ chain were found to be encoded in the transformation properties of the coupling parameters of the model, as an extension of the modular group. The modular group, acting in real space, plays an important role in the classification of 1+1-dimensional critical models. Having found a similar structure, but in parameter space, and for a gapped, integrable system could help in understanding several properties of integrable models which have been observed during the years, but still lack a general explanation.
Finally, the entanglement properties of the quantum Ising model have been studied to highlight the role of edge states in breaking the so called local convertibility. It was proven that the entanglement of edge states can decrease approaching a quantum phase transition, while that of bulk states always increases as the correlation length grows. These findings have important consequences: for instance, they imply that, in the search for quantum many body systems that can be used as the kind of universal quantum simulator famously proposed by Feynmann, one should consider only systems that show lack of local convertibility. Also, non-local convertibility could emerge as an alternative, finite size estimator of long-range entanglement.
Finally, the fellow undertook a new and exciting project, in the very competitive field of out-of-equilibrium physics. Together with his collaborators, he asked what happens to a moving, localized excitation if the underlying Hamiltonian is suddenly changed in its interaction strength (this being a variation of the traditional quantum quench protocol). They study the dynamics of this set-up using a non-linear hydrodynamics approach and through numerical integration and they characterized the post-quench dynamics in terms of a reflected and a transmitted profile with universal shapes and velocities. They are also contacting experimental groups, which seem interested in studying this protocol.