Natural Order Parameter Research Articles

Gradient-flowed order parameter for spontaneous gauge symmetry breaking

The gauge-invariant two-point function of the Higgs field at the same spacetime point can make a natural gauge-invariant order parameter for spontaneous gauge symmetry breaking. However, this composite operator is ultraviolet divergent and is not well defined. We propose using a gradient flow to cure the divergence from putting the fields at the same spacetime point. As a first step, we compute it for the Abelian Higgs model with a positive mass squared at the one-loop order in the continuum theory using the saddle-point method to estimate the finite part. The order parameter consistently goes to zero in the infrared limit of the infinite flow time.

Topological squashed entanglement: Nonlocal order parameter for one-dimensional topological superconductors

Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to log(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi 1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.

Entropic order parameters for the phases of QFT

We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT’s. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the ’t Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.

Replica symmetry breaking in neural networks: a few steps toward rigorous results

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are drawn according to Hebb’s learning rule): this is accomplished by grafting Guerra’s telescopic averages on the transport equation technique, recently developed by some of the authors. As an overture, we apply the technique to solve the Sherrington-Kirkpatrick model with i.i.d. Gaussian couplings centered at J0 and with finite variance J; the mean J0 provides a ferromagnetic contribution to be detected in a noisy environment tuned by J, hence making this model a natural test-case to be investigated before addressing the Hopfield model. For both the models, an explicit expression of their quenched free energy in terms of their natural order parameters is obtained at the Kth step (K arbitrary, but finite) of replica-symmetry-breaking. In particular, for the Hopfield model, by assuming that the overlaps respect Parisi’s decomposition (following the ziqqurat ansatz) and that the Mattis magnetization is self-averaging, we recover previous results obtained via replica-trick by Amit, Crisanti and Gutfreund (1RSB) and by Steffan and Kühn (2RSB).

Tensor-network approach to phase transitions in string-net models

We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops expectation values, and a natural order parameter detecting the breakdown of the topological phase. In the presence of a string tension, a quantum phase transition occurs between the topological phase and a trivial phase. We benchmark our approach for $\mathbb{Z}_2$ string nets and capture the second-order phase transition which is well known from the exact mapping onto the transverse-field Ising model. More interestingly, for Fibonacci string nets, we obtain first-order transitions in contrast with previous studies but in qualitative agreement with mean-field results.

On the Marchenko–Pastur law in analog bipartite spin-glasses**The authors are pleased to dedicate this paper to Giorgio Parisi on the occasion of his seventieth birthday.

In recent decades, statistical mechanics of disordered systems (mainly spin-glasses) has become one of the main tools to investigate complex systems, probably due to the celebrated Replica Symmetry Breaking scheme of Parisi Theory and its deep implications.In this work we consider the analog bipartite spin-glass (or real-valued restricted Boltzmann machine in a neural-network jargon), whose variables (those quenched as well as those dynamical) share standard Gaussian distributions. First, via Guerra’s interpolation technique, we express its quenched free-energy in terms of the natural order parameters of the theory (namely the self- and two-replica overlaps), then, we re-obtain the same result by using the replica-trick: a mandatory tribute, given the special occasion. Next, we show that the quenched free-energy of this model is the functional generator of the moments of the matrix whose entries are the correlation coefficients between the weights connecting the two layers of the spin-glass (i.e. the Wishart matrix in random matrix theory or the Hebbian coupling in neural networks): as weights are quenched stochastic variables, this acts as a powerful tool to inspect random matrices. In particular, we find that the Stieltjes transform of the spectral density of the correlation matrix is determined by the (replica-symmetric) quenched free-energy of the bipartite spin-glass model. In this setup, we re-obtain the Marchenko–Pastur law in a very simple way.

Interfaces in coexisting metals and Mott insulators

Motivated by the direct observation of electronic phase separation in first-order Mott transitions, we model the interface between the thermodynamically coexisting metal and Mott insulator. We show how to model the required slab geometry and extract the electronic spectra. We construct an effective Landau free energy and compute the variation of its parameters across the phase diagram. Finally, using a linear mixture of the density and double-occupancy, we identify a natural Ising order parameter which unifies the treatment of the bandwidth and filling controlled Mott transitions.

PT-symmetry breaking in waveguides with competing loss-gain pairs

We consider a periodic waveguide array whose unit cell consists of a $\mathcal{PT}$-symmetric quadrimer with two competing loss/gain parameter pairs which lead to qualitatively different symmetry-broken phases. It is shown that the transitions between the phases are described by a symmetry-adapted nonlocal current which maps the spectral properties to the spatially resolved field, for the lattice as well as for the isolated quadrimer. Its site-average acts like a natural order parameter for the general class of one-dimensional $\mathcal{PT}$-symmetric Hamiltonians, vanishing in the unbroken phase and being nonzero in the broken phase. We investigate how the beam dynamics in the array is affected by the presence of competing loss/gain rates in the unit cell, showing that the enriched band structure yields the possibility to control the propagation length before divergence when the system resides in the broken $\mathcal{PT}$ phase.

Continuous atomic displacements and lattice distortion during fcc–bcc martensitic transformation

The continuous matrices of atomic displacements and lattice distortion from face-centered-cubic (fcc) to body-centered-cubic (bcc) phases compatible with the hard-sphere geometry of iron atoms are calculated for different possible final orientation relationships (ORs), such as Bain, Pitsch and Kurdjumov–Sachs (KS). The angular distortion introduced in the calculations appears as a natural order parameter of the fcc–bcc transitions. The average atomic displacement is lower with Pitsch and KS ORs than for Bain; one of these two distortions probably constitutes the natural mechanism for stress-free isolated small crystals. For matrix-embedded transformations that occur in bulk steels, the distortion associated with KS OR seems very appropriate: it has special mathematical properties and can explain the {225} habit planes with a simple O-lattice criterion. From these calculations, the fcc–bcc martensitic transformation appears to be “angular distortive”, and not of “shear” type as usually assumed.

On the relation between kinetically constrained models of glass dynamics and the randomfirst-order transition theory

In this paper we revisit and extend the mapping between two apparently different classes ofmodels. The first class contains the prototypical models described—at the mean-field level—bythe random first-order transition (RFOT) theory of the glass transition, called either the‘random XORSAT problem’ (in the information theory community) or the ‘dilutedp-spin model’ (in the spin glass community), undergoing a single spin-flip Glauber dynamics.The models in the second class are kinetically constrained models (KCM): theirHamiltonian is that of independent spins in a constant magnetic field, hence theirthermodynamics is completely trivial, but the dynamics is such that only groupsof spins can flip together, thus implementing a kinetic constraint that inducesa non-trivial dynamical behavior. A mapping between some representatives ofthese two classes has been known for a long time. Here we formally prove thismapping at the level of the master equation, and we apply it to the particular caseof Bethe lattice models. This allows us to show that an RFOT model can bemapped exactly into a KCM. However, the natural order parameter for the RFOTmodel, namely the spin overlap, is mapped into a very complicated non-localfunction in the KCM. Therefore, if one were to study the KCM without knowingthe mapping onto the RFOT model, one would guess that its physics is quitedifferent from the RFOT one. Our results instead suggest that these two apparentlydifferent descriptions of the glass transition are, at least in some cases, closelyrelated.

Contact-density analysis of lattice polymer adsorption transitions

By means of contact-density chain-growth simulations, we investigate a simple lattice model of a flexible polymer interacting with an attractive substrate. The contact density is a function of the numbers of monomer–substrate and monomer–monomer contacts. These contact numbers represent natural order parameters and allow for a comprising statistical study of the conformational space accessible to the polymer in dependence of external parameters such as the attraction strength of the substrate and the temperature. Since the contact density is independent of the energy scales associated to the interactions, its logarithm is an unbiased measure for the entropy of the conformational space. By setting explicit energy scales, the thus defined, highly general microcontact entropy can easily be related to the microcanonical entropy of the corresponding hybrid polymer–substrate system.

Physical replicas and the Bose glass in cold atomic gases

We study cold atomic gases in a disorder potential and analyse the correlations between different systems subjected to the same disorder landscape. Such independent copies with the same disorder landscape are known as replicas. While, in general, these are not accessible experimentally in condensed matter systems, they can be realized using standard tools for controlling cold atomic gases in an optical lattice. Of special interest is the overlap function which represents a natural order parameter for disordered systems and is a correlation function between the atoms of two independent replicas with the same disorder. We demonstrate an efficient measurement scheme for the determination of this disorder-induced correlation function. As an application, we focus on the disordered Bose–Hubbard model and determine the overlap function within the perturbation theory and a numerical analysis. We find that the measurement of the overlap function allows for the identification of the Bose-glass phase in certain parameter regimes.

Mean-field theory of Anderson localization: Asymptotic solution in high spatial dimensions

The Anderson model of noninteracting disordered electrons is studied in high spatial dimensions. We find that off-diagonal one- and two-particle propagators behave as Gaussian random variables with respect to momentum summations. With this simplification and with the electron-hole symmetry we reduce the parquet equations for two-particle irreducible vertices to a single algebraic equation for a local vertex. We find a disorder-driven bifurcation point in this equation signaling vanishing of diffusion and onset of Anderson localization. There is no bifurcation in $d=1,2$ where all states are localized. A natural order parameter for Anderson localization emerges from the construction.

Variances as order parameter and complexity measure for random Boolean networks

Several order parameters have been considered to predict and characterize the transition between ordered and disordered phases in random Boolean networks, such as the Hamming distance between replicas or the stable core, which have been successfully used. In this work, we propose a natural and clear new order parameter: the temporal variance. We compute its value analytically and compare it with the results of numerical experiments. Finally, we propose a complexity measure based on the compromise between temporal and spatial variances. This new order parameter and its related complexity measure can be easily applied to other complex systems.

Investigating the phase diagram of finite extensive and nonextensive systems.

Microcanonical caloric curves at constant pressure and pressure versus volume curves at constant temperature are calculated using the probability distributions of a canonical ensemble at constant pressure. It is evidenced that the constant pressure path provides a natural order parameter even for small systems. While for large systems the Coulomb interaction is destroying the liquid-gas phase transition, for smaller systems the phase transition survives. Phase diagrams of a nuclear system without Coulomb and hard-core interactions are constructed and the critical point is identified.

Statistical mechanics of permanent random atomic and molecular networks: Structure and heterogeneity of the amorphous solid state

Under sufficient permanent random covalent bonding, a fluid of atoms or small molecules is transformed into an amorphous solid network. Being amorphous, local structural properties in such networks vary across the sample. A natural order parameter, resulting from a statistical-mechanical approach, captures information concerning this heterogeneity via a certain joint probability distribution. This joint probability distribution describes the variations in the positional and orientational localization of the particles, reflecting the random environments experienced by them, as well as further information characterizing the thermal motion of particles. A complete solution, valid in the vicinity of the amorphous solidification transition, is constructed essentially analytically for the amorphous solid order parameter, in the context of the random network model and approach introduced by Goldbart and Zippelius [Europhys. Lett. 27, 599 (1994)]. Knowledge of this order parameter allows us to draw certain conclusions about the stucture and heterogeneity of randomly covalently bonded atomic or molecular network solids in the vicinity of the amorphous solidification transition. Inter alia, the positional aspects of particle localization are established to have precisely the structure obtained previously in the context of vulcanized media, and results are found for the analog of the spin glass order parameter describing the orientational freezing of the bonds between particles.

Langmuir monolayers as disordered solids: Strain-tilt-backbone coupling and natural order parameters for the swiveling transitions

Multistability of Langmuir monolayers, in particular those composed of fatty acids, is reflected by a very rich and complicated phase diagram. We argue that strain-tilt-backbone coupling determines the behavior described by that diagram. Following the solid state approach, we show that a natural order parameter set is defined by thermal averages of spherical harmonics and strain tensor components. In addition, we show that the backbone order parameter can be conveniently represented by an elastic dipole tensor. Treating Langmuir monolayers as disordered solids, we have derived an orientational entropy contribution to the free energy. The swiveling transition between L2(L2h) and L2′(L2*) phases is discussed in detail and is conveniently described in terms of the proposed order parameters. We discuss why the change in the distortion direction of the 2D unit cell tracks the change in tilt direction of the molecules, and also why the cell dimensions involved in the swiveling transition are the same in both phases.

Quenched random graphs

Spin models on quenched random graphs are related to many important optimization problems. We give a new derivation of their mean-field equations that elucidates the role of the natural order parameter in these models.

The influence of external boundary conditions on the spherical model of a ferromagnet. I. Magnetization profiles

The spherical model of a ferromagnet is investigated for various (external) boundary conditions. It is shown that, besides the well-known critical point, a second one can be produced by the boundary conditions. Although the main asymptotic of the free energy is analytic at the new critical point, theO(N1−2/d) asymptotic possesses a singularity here. A natural order parameter of the model has singularities at both critical points. The magnetization profile is studied for the whole range of the model's parameters and at different scales. It is shown that (in an appropriate regime) below the second critical temperature the magnetization profile freezes, that is, becomes temperature independent. Distributions of the single spin variables and some macroscopic observables (including normalized total spin) are studied for the whole temperature range including the critical points.

Lattice analogues of N = 2 superconformal models via quantum-group truncation

We obtain lattice models whose continuum limits correspond to N = 2 superconformal coset models. This is done by taking the well-known vertex model whose continuum limit is the G×G/G conformal field theory, and twisting the transfer matrix and modifying the quantum-group truncation. We find that the natural order parameters of the new models are precisely the chiral primary fields. The integrable perturbations of the conformal field theory limit also have natural counterparts in the lattice formulation, and these can be incorporated into an affine quantum-group structure. The topological, twisted N = 2 superconformal models also have lattice analogues, and these emerge as an intermediate part of our analysis.