Surface Critical Exponents Research Articles

Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional Clifford circuits

Measurement-induced entanglement transitions in quantum dynamics represent a new class of nonequilibrium transitions, akin to thresholds in quantum error correction. In this work, the authors explore the universal properties of entanglement transitions in one- and two-dimensional monitored Clifford circuits, using a graph-state-based algorithm to unravel geometric properties of entanglement. A study of entanglement clusters in the steady state reveals that, despite similarities in the bulk, the surface critical exponents show strong deviations from classical percolation.

Scalable Probes of Measurement-Induced Criticality.

We uncover a local order parameter for measurement-induced phase transitions: the average entropy of a single reference qubit initially entangled with the system. Using this order parameter, we identify scalable probes of measurement-induced criticality that are immediately applicable to advanced quantum computing platforms. We test our proposal on a 1+1 dimensional stabilizer circuit model that can be classically simulated in polynomial time. We introduce the concept of a "decoding light cone" to establish the local and efficiently measurable nature of this probe. We also estimate bulk and surface critical exponents for the transition. Developing scalable probes of measurement-induced criticality in more general models may be a useful application of noisy intermediate scale quantum devices, as well as point to more efficient realizations of fault-tolerant quantum computation.

Engineering Surface Critical Behavior of (2+1)-Dimensional O(3) Quantum Critical Points.

Surface critical behavior (SCB) refers to the singularities of physical quantities on the surface at the bulk phase transition. It is closely related to and even richer than the bulk critical behavior. In this work, we show that three types of SCB universality are realized in the dimerized Heisenberg models at the (2+1)-dimensional O(3) quantum critical points by engineering the surface configurations. The ordinary transition happens if the surface is gapped in the bulk disordered phase, while the gapless surface state generally leads to the multicritical special transition, even though the latter is precluded in classical phase transitions because the surface is in the lower critical dimension. An extraordinary transition is induced by the ferrimagnetic order on the surface of the staggered Heisenberg model, in which the surface critical exponents violate the results of the scaling theory and thus seriously challenge our current understanding of extraordinary transitions.

Computer Simulation of Critical Behavior of Semi-Infinite Antiferromagnetic Material

Phase transitions have been computer-simulated using the Monte Carlo method in the three-dimensional antiferromagnetic Ising model for the various ratios of the exchange integrals calculated over the surface and bulk of the system. The phase diagram of the system has been constructed. It has been shown that the phase diagram contains a phase in which the ordering of spins located on the surface takes place at a temperature below the Neel temperature. Contrary to the ferromagnetic Ising model, lines of phase transitions intersect at the tetracritical rather than tricritical point. The dependence of the surface critical exponents on the ratio of the exchange integrals has been obtained. It has been discovered that, at a ratio of the exchange integrals less than a certain value, the surface phase transition exhibits attributes of the phase transition of the third order.

Determination of the solid surface critical exponent β1 from contact-angle variation on approach to a wetting transition: Cyclohexane/aniline/quartz.

The three-phase contact angle (θ) for the system cyclohexane/aniline/quartz has been measured from drop shapes as a function of temperature on approach to the cyclohexane/aniline upper consolute solution temperature Tc. The experiments employed exacting criteria previously established for thermodynamic-quality measurements at fluid interfaces. A first-order wetting transition from partial wetting to complete wetting was observed at a temperature Tw, 2.12 K below Tc. The contact angle vanishes at Tw, scaling as cos θ ∼ |T - Tc|(β1-μ) for T < Tw and cos θ = 1.0 for Tw < T < Tc. The experimental results give a value for β1 = 0.74 ± 0.03, in agreement with theoretical calculations. The data clearly rule out higher order contributions to the change in the contact angle near the critical point for this system. These results are in marked contrast to previous measurements on this system from measurements of capillary rise and meniscus curvature.

Statistical mechanics of polymer chains grafted to adsorbing boundaries of fractal lattices embedded in three-dimensional space

We study the adsorption problem of linear polymers, immersed in a good solvent, when the container of the polymer–solvent system is taken to be a member of the Sierpinski gasket (SG) family of fractals, embedded in the three-dimensional Euclidean space. Members of the SG family are enumerated by an integer b (2≤b<∞), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the surface critical exponents γ11,γ1, and γs which, within the self-avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with both, one, and no ends grafted to the adsorbing surface (adsorbing boundary), respectively. By applying the exact renormalization group method, for 2≤b≤4, we have obtained specific values for these exponents, for various types of polymer conformations. To extend the obtained sequences of exact values for surface critical exponents, we have applied the Monte Carlo renormalization group method for fractals with 2≤b≤40. The obtained results show that all studied exponents are monotonically increasing functions of the parameter b, for all possible polymer states. We discuss mutual relations between the studied critical exponents, and compare their values with those found for other types of lattices, in order to attain a unified picture of the attacked problem.

Surface Roughness and Critical Exponent Analyses of Boron-Doped Diamond Films Using Atomic Force Microscopy Imaging: Application of Autocorrelation and Power Spectral Density Functions

The evolution of the surface roughness of growing metal or semiconductor thin films provides much needed information about their growth kinetics and corresponding mechanism. While some systems show stages of nucleation, coalescence, and growth, others exhibit varying microstructures for different process conditions. In view of these classifications, we report herein detailed analyses based on atomic force microscopy (AFM) characterization to extract the surface roughness and growth kinetics exponents of relatively low boron-doped diamond (BDD) films by utilizing the analytical power spectral density (PSD) and autocorrelation function (ACF) as mathematical tools. The machining industry has applied PSD for a number of years for tool design and analysis of wear and machined surface quality. Herein, we present similar analyses at the mesoscale to study the surface morphology as well as quality of BDD films grown using the microwave plasma-assisted chemical vapor deposition technique. PSD spectra as a function of boron concentration (in gaseous phase) are compared with those for samples grown without boron. We find that relatively higher boron concentration yields higher amplitudes of the longer-wavelength power spectral lines, with amplitudes decreasing in an exponential or power-law fashion towards shorter wavelengths, determining the roughness exponent (α ≈ 0.16 ± 0.03) and growth exponent (β ≈ 0.54), albeit indirectly. A unique application of the ACF, which is widely used in signal processing, was also applied to one-dimensional or line analyses (i.e., along the x- and y-axes) of AFM images, revealing surface topology datasets with varying boron concentration. Here, the ACF was used to cancel random surface “noise” and identify any spatial periodicity via repetitive ACF peaks or spatially correlated noise. Periodicity at shorter spatial wavelengths was observed for no doping and low doping levels, while smaller correlations were observed for relatively higher boron concentration. These semiquantitative spatial analyses may prove useful in comparing synthesis techniques and varying compositional makeups of diamond films and other technologically important electronic materials. These findings in terms of critical exponents are also correlated with traditional Raman spectroscopy and x-ray diffraction structural properties, thus helping to provide insight into the growth kinetics, albeit in reverse manner.

Surface critical exponents at a discontinuous bulk transition

Systems with a bulk first-order transition can display diverging correlation lengths close to a surface. This surface induced disordering yields a special type of surface criticality. Using extensive numerical simulations we study surface quantities in the two-dimensional Potts model with a large number of states $q$ which undergoes a discontinuous bulk transition. The surface critical exponents are thereby found to depend on the value of $q$, which is in contrast to prior claims that these exponents should be universal and independent of $q$. It follows that surface-induced disordering at first-order transitions is characterized by exponents that depend on the details of the model.

Off-critical parafermions and the winding angle distribution of the O(n) model

Using an off-critical deformation of the identity of Duminil-Copin and Smirnov, we prove a relationship between half-plane surface critical exponents γ1 and γ11 as well as wedge critical exponents γ2(α) and γ21(α) and the exponent characterizing the winding angle distribution of the O(n) model in the half-plane, or more generally in a wedge of wedge-angle α. We assume only the existence of these exponents and, for some values of n, the conjectured value of the critical point. If we assume their values as predicted by conformal field theory, one gets complete agreement with the conjectured winding angle distribution, as obtained by CFT and Coulomb gas arguments. We also prove the exponent inequality γ1 − γ11 ⩾ 1, and its extension γ2(α) − γ21(α) ⩾ 1 for the edge exponents. We provide conjectured values for all exponents for n ∈ [ − 2, 2).

Scaling behaviour of the time-fractional Kardar–Parisi–Zhang equation

The scaling behaviour of the time-fractional Kardar–Parisi–Zhang (TFKPZ) equation in (1 + 1) dimensions is investigated by scaling analysis and numerical simulations. The surface morphology and critical exponents with different fractional orders are obtained. The analytical results are consistent with the corresponding numerical solutions based on a Caputo-type fractional derivative. We find that, similar to the normal Kardar–Parisi–Zhang equation, anomalous behaviour does not appear in the TFKPZ model according to the scaling idea of local slope and numerical evidence. However, there exists significant finite-time effect of local scaling exponents in the TFKPZ system. Our results also imply that memory effects can affect the scaling behaviour of evolving fractional surface growth.

Scaling properties at the interface between different critical subsystems: The Ashkin-Teller model

We consider two critical semi-infinite subsystems with different critical exponents and couple them through their surfaces. The critical behavior at the interface, influenced by the critical fluctuations of the two subsystems, can be quite rich. In order to examine the various possibilities, we study a system composed of two coupled Ashkin-Teller models with different four-spin couplings $ϵ$ on the two sides of the junction. By varying $ϵ$, some bulk and surface critical exponents of the two subsystems are continuously modified, which in turn changes the interface critical behavior. In particular, we study the marginal situation, for which magnetic critical exponents at the interface vary continuously with the strength of the interaction parameter. The behavior expected from scaling arguments is checked by density matrix renormalization group calculations.

Boundary critical behaviour atm-axial Lifshitz points of semi-infinite systems with a surface plane perpendicular to a modulation axis

Semi-infinite $d$-dimensional systems with an $m$-axial bulk Lifshitz point are considered whose ($d-1$)-dimensional surface hyper-plane is oriented perpendicular to one of the $m$ modulation axes. An $n$-component $\phi^4$ field theory describing the bulk and boundary critical behaviour when (i) the Hamiltonian can be taken to have O(n) symmetry and (ii) spatial anisotropies breaking its Euclidean symmetry in the $m$-dimensional coordinate subspace of potential modulation directions may be ignored is investigated. The long-distance behaviour at the ordinary surface transition is mapped onto a field theory with the boundary conditions that both the order parameter $\bm{\phi}$ and its normal derivative $\partial_n\bm{\phi}$ vanish at the surface plane. The boundary-operator expansion is utilized to study the short-distance behaviour of $\bm{\phi}$ near the surface. Its leading contribution is found to be controlled by the boundary operator $\partial_n^2\bm{\phi}$. The field theory is renormalized for dimensions $d$ below the upper critical dimension $d^*(m)=4+m/2$, with a corresponding surface source term $\propto \partial_n^2\bm{\phi}$ added. The anomalous dimension of this boundary operator is computed to first order in $\epsilon=d^*-d$. The result is used in conjunction with scaling laws to estimate the value of the single independent surface critical exponent $\beta_{\mathrm{L}1}^{(\mathrm{ord},\perp)}$ for $d=3$. Our estimate for the case $m=n=1$ of a uniaxial Lifshitz point in Ising systems is in reasonable agreement with published Monte Carlo results.

Adsorption of long flexible polymer chains at planar surfaces: scaling analysis and criticalexponents

The process of adsorption on a planar repulsive, ‘marginal’ and attractivewall of long flexible polymer chains with excluded volume interactions in theframework of the renormalization group field-theoretical approach, directly ind = 3 dimensions up to two-loop order, for the semi-infinite|ϕ|4 n-vector model in the limit n → 0 is investigated. We perform Padé and Padé–Borel resummation of the series obtained forthe surface critical exponents, characterizing the process of adsorption of long flexiblepolymer chains at the surface. The polymer linear dimensions parallel and perpendicular tothe surface and the corresponding partition functions, as well as the behaviour of themonomer density profiles and the fraction of adsorbed monomers at the surface and in theinterior, are studied. The field-theoretical results obtained at fixed dimensionsd = 3 are in good agreement with recent Monte Carlo calculations.

Transfer‐matrix study of critical properties of the Ising model following from the three‐cluster MFRG approach

AbstractUsing a transfer matrix technique and the Ising model, the predictions of the MFRG concept for clusters with linear size up to 21 have been tested for the three‐cluster MFRG approach. Even for small sizes of the clusters, the three‐cluster estimates of critical couplings and bulk critical exponents give the accuracy level equal to that of two‐cluster renormalization for much greater sizes. Performing the asymptotic analysis for the new data, the convergence of the surface critical exponent yhs towards the exact value is illustrated for the first time. Our improved method has enabled us to obtain the results for clusters with substantially greater sizes, to accelerate the calculations and also to confirm the reliability of the MFRG approach for the Ising model. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Critical adsorption of polymers in a medium with long-range correlated quenched disorder

The process of adsorption on a planar wall of long flexible polymer chains in a medium with quenched long-range correlated disorder is investigated. We focus on the case of correlations between defects or impurities that decay according to the power law x(-a) for large distances x , where x= (r,z) . A field theoretical approach in d=4-epsilon and directly in d=3 dimensions up to one-loop order for the semi-infinite |phi|(4) m-vector model (in the limit m-->0 ) with a planar boundary is used. The whole set of surface critical exponents at the adsorption threshold T= T(a) , which separates the nonadsorbed region from the adsorbed one, is obtained. Moreover, we calculate the crossover critical exponent Phi and the set of exponents associated with it. We perform calculations in a double epsilon=4-d and delta=4-a expansion and also for a fixed dimension d=3 , up to one-loop order for different values of the correlation parameter 2< a < or =3 . The obtained results indicate that for systems with long-range correlated quenched disorder a different set of surface critical exponents arises. All the surface critical exponents depend on a . Hence, the presence of long-range correlated disorder influences the process of adsorption of long flexible polymer chains on a wall in a significant way.

Aging phenomena in critical semi-infinite systems

Nonequilibrium surface autocorrelation and autoresponse functions are studied numerically in semi-infinite critical systems in the dynamical scaling regime. Dynamical critical behavior is examined for a nonconserved order parameter in semi-infinite two- and three-dimensional Ising models as well as in the Hilhorst--van Leeuwen model. The latter model permits a systematic study of surface aging phenomena, as the surface critical exponents change continuously as function of a model parameter. The scaling behavior of surface two-time quantities is investigated and scaling functions are confronted with predictions coming from the theory of local scale invariance. Furthermore, surface fluctuation-dissipation ratios are computed and their asymptotic values are shown to depend on the values of surface critical exponents.

Boundary critical behaviour atm-axial Lifshitz points: the special transition for the case of a surface plane parallel to the modulation axes

The critical behaviour of $d$-dimensional semi-infinite systems with $n$-component order parameter $\bm{\phi}$ is studied at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. Field-theoretic renormalization group methods are utilised to examine the special surface transition in the case where the $m$ potential modulation axes, with $0\leq m\leq d-1$, are parallel to the surface. The resulting scaling laws for the surface critical indices are given. The surface critical exponent $\eta_\|^{\rm sp}$, the surface crossover exponent $\Phi$ and related ones are determined to first order in $\epsilon=4+\case{m}{2}-d$. Unlike the bulk critical exponents and the surface critical exponents of the ordinary transition, $\Phi$ is $m$-dependent already at first order in $\epsilon$. The $\Or(\epsilon)$ term of $\eta_\|^{\rm sp}$ is found to vanish, which implies that the difference of $\beta_1^{\rm sp}$ and the bulk exponent $\beta$ is of order $\epsilon^2$.

Surface critical behaviour of semi-infinite systems with cubic anisotropy at the ordinary transition

The critical behavior at the ordinary transition in semi-infinite n-component anisotropic cubic models is investigated by applying the field theoretic approach in d=3 dimensions up to the two-loop approximation. Numerical estimates of the resulting two-loop series expansions for the critical exponents of the ordinary transition are computed by means of Pade resummation techniques. For $n<n_{c}$ the system belongs to the universality class of the isotropic n-component model, while for $n>n_{c}$ the cubic fixed point becomes stable, where $n_{c}<3$ is the marginal spin dimensionality of the cubic model. The obtained results indicate that the surface critical behavior of the semi-infinite systems with cubic anisotropy is characterized by a new set of surface critical exponents for $n>n_{c}$.

Boundary critical behavior atm-axial Lifshitz points for a boundary plane parallel to the modulation axes

The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bm{\phi}$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an $m$-dimensional subspace of $\mathbb{R}^d$. The associated $m$ modulation axes are presumed to be parallel to the surface, where $0\le m\le d-1$. An appropriate semi-infinite $|\bm{\phi}|^4$ model representing the corresponding universality classes of surface critical behavior is introduced. It is shown that the usual O(n) symmetric boundary term $\propto \bm{\phi}^2$ of the Hamiltonian must be supplemented by one of the form $\mathring{\lambda} \sum_{\alpha=1}^m(\partial\bm{\phi}/\partial x_\alpha)^2$ involving a dimensionless (renormalized) coupling constant $\lambda$. The implied boundary conditions are given, and the general form of the field-theoretic renormalization of the model below the upper critical dimension $d^*(m)=4+{m}/{2}$ is clarified. Fixed points describing the ordinary, special, and extraordinary transitions are identified and shown to be located at a nontrivial value $\lambda^*$ if $\epsilon\equiv d^*(m)-d>0$. The surface critical exponents of the ordinary transition are determined to second order in $\epsilon$. Extrapolations of these $\epsilon$ expansions yield values of these exponents for $d=3$ in good agreement with recent Monte Carlo results for the case of a uniaxial ($m=1$) Lifshitz point. The scaling dimension of the surface energy density is shown to be given exactly by $d+m (\theta-1)$, where $\theta=\nu_{l4}/\nu_{l2}$ is the anisotropy exponent.

Crossover between special and ordinary transitions in random semi-infinite Ising-like systems.

We consider the crossover behavior between special and ordinary surface transitions in three-dimensional semi-infinite Ising-like systems with random quenched bulk disorder. We calculate the surface crossover critical exponent Phi, the critical exponents of the layer alpha(1), and local specific heats alpha(11) by applying the field theoretic approach directly in three spatial dimensions (d=3) up to the two-loop approximation. The numerical estimates of the resulting two-loop series expansions for the surface critical exponents are computed by means of Padé and Padé-Borel resummation techniques. We find that Phi, alpha(1), alpha(11) obtained in the present paper are different from their counterparts of pure Ising systems. The obtained results support the idea that in a system with random quenched bulk disorder the plane boundary is characterized by a new set of critical exponents.