Showing papers on "Ising model published in 2007"

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

Abstract: In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.

Thermodynamic Formalism for Systems with Markov Dynamics

Abstract: The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism—a dynamical Gibbs ensemble construction—to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.

Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model

Abstract: We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.

Interacting anyons in topological quantum liquids: The golden chain

Abstract: We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial ("identity") channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional (2D) conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory

Abstract: The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter (q is not a root of unity and |q| = 1) is diagonalized using the representation theory of the q-Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2N. The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al. In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2N. Numerical checks performed for small values of N support the analysis.

Ising, Schelling and self-organising segregation

Abstract: The similarities between phase separation in physics and residential segregation by preference in the Schelling model of 1971 are reviewed. Also, new computer simulations of asymmetric interactions different from the usual Ising model are presented, showing spontaneous magnetisation (=self-organising segregation) and in one case a sharp phase transition.

Monte Carlo simulation results for critical Casimir forces

Abstract: The confinement of critical fluctuations in soft media induces critical Casimir forces acting on the confining surfaces. The temperature and geometry dependences of such forces are characterized by universal scaling functions. A novel approach is presented to determine them for films via Monte Carlo simulations of lattice models. The method is based on an integration scheme of free energy differences. Our results for the Ising and the XY universality class agree well with corresponding experimental results for wetting layers of classical binary liquid mixtures and of 4He, respectively.

Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices

Abstract: We report conjectures on the three-dimensional (3D) Ising model of simple orthorhombic lattices, together with details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK ' + KK '' + K ' K '' or sinh 2K - sinh 2(K ' + K '' + (K ' K ''/K)) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio x(c) = e(-2kc) = ((root 5 - 1)/2), as derived from K* = 3K or sinh 2K = sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization of the simple orthorhombic Ising ferromagnet is derived explicitly by the perturbation procedure, following the conjectures. The spin correlation functions are discussed on the terms of the Pfaffians, by defining the effective skew-symmetric matrix A(eff). The true range k(x) of the correlation and the susceptibility of the simple orthorhombic Ising system are determined by procedures similar to those used for the two-dimensional Ising system. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8 and nu = 2/3, showing the universality behaviour and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results based on the conjectures are compared with those of approximation methods and experimental findings. The putative solutions have been judged by several criteria. The deviations of the approximation results and the experimental data from the solutions are interpreted. Based on the solution, it is found that the 3D-to-2D crossover phenomenon differs with the 2D-to-1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D to the 2D values. Special attention is also paid to the extra energy caused by the introduction of the fourth curled-up dimension and the states at/near infinite temperature revealed by the weight factors of the eigenvectors. The physics beyond the conjectures and the existence of the extra dimension are discussed. The present work is not only significant for statistical and condensed matter physics, but also fill the gap between the quantum field theory, cosmology theory, high-energy particle physics, graph theory and computer science.

Fast mixing for independent sets, colorings, and other models on trees

Abstract: We study the mixing time of the Glauber dynamics for general spin systems on the regular tree, including the Ising model, the hard-core model (independent sets), and the antiferromagnetic Potts model at zero temperature (colorings). We generalize a framework, developed in our recent paper (Martinelli, Sinclair, and Weitz, Tech. Report UCB//CSD-03-1256, Dept. of EECS, UC Berkeley, July 2003) in the context of the Ising model, for establishing mixing time O(nlog n), which ties this property closely to phase transitions in the underlying model. We use this framework to obtain rapid mixing results for several models over a significantly wider range of parameter values than previously known, including situations in which the mixing time is strongly dependent on the boundary condition. We also discuss applications of our framework to reconstruction problems on trees. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 A preliminary version of this paper appeared in Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, January 2004. This work was done while the author was visiting the Departments of EECS and Statistics, University of California, Berkeley, supported in part by a Miller Visiting Professorship.

Spin-Flip Dynamics of the Curie-Weiss Model: Loss of Gibbsianness with Possibly Broken Symmetry

Abstract: We study the conditional probabilities of the Curie-Weiss Ising model in vanishing external field under a symmetric independent stochastic spin-flip dynamics and discuss their set of points of discontinuity (bad points). We exhibit a complete ana- lysis of the transition between Gibbsian and non-Gibbsian behavior as a function of time, extending the results for the corresponding lattice model, where only partial answers can be obtained. For initial temperature β −1 ≥ 1, we prove that the time-evolved measure is always Gibbsian. For 2 ≤ β −1 <1, the time-evolved measure loses its Gibbsian character at a sharp transition time. For β −1 < 2 , we observe the new phenomenon of symme- try-breaking in the set of points of discontinuity: Bad points corresponding to non-zero spin-average appear at a sharp transition time and give rise to biased non-Gibbsianness of the time-evolved measure. These bad points become neutral at a later transition time, while the measure stays non-Gibbs. In our proof we give a detailed description of the phase-diagram of a Curie-Weiss random field Ising model with possibly non-symmetric random field distribution based on bifurcation analysis.

Conjectures on exact solution of three - dimensional (3D) simple orthorhombic Ising lattices

Abstract: We report the conjectures on the three-dimensional (3D) Ising model on simple orthorhombic lattices, together with the details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and the weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, by employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK' + KK'' + K'K'' or sinh 2K sinh 2(K' + K'' + K'K''/K) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio xc = exp(-2Kc) = (sq(5) - 1)/2, as derived from K* = 3K or sinh 2K sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization and the spin correlation functions of the simple orthorhombic Ising ferromagnet are derived explicitly. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are alpha = 0, beta = 3/8, gamma = 5/4, delta = 13/3, eta = 1/8 and nu = 2/3, showing the universality behavior and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results obtained based on the conjectures are compared with those of the approximation methods and the experimental findings. The 3D to 2D crossover phenomenon differs with the 2D to 1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D values to the 2D ones.

The Complexity of Ferromagnetic Ising with Local Fields

Abstract: We consider the complexity of approximating the partition function of the ferromagnetic Ising model with varying interaction energies and local external magnetic fields. Jerrum and Sinclair provided a fully polynomial randomized approximation scheme for the case in which the system is consistent in the sense that the local external fields all favour the same spin. We characterize the complexity of the general problem by showing that it is equivalent in complexity to the problem of approximately counting independent sets in bipartite graphs, thus it is complete in a logically defined subclass of nP previously studied by Dyer, Goldberg, Greenhill and Jerrum. By contrast, we show that the corresponding computational task for the $q$-state Potts model with local external magnetic fields and $q>2$ is complete for all of nP with respect to approximation-preserving reductions.

Finite-size scaling in complex networks.

Abstract: A finite-size-scaling (FSS) theory is proposed for various models in complex networks. In particular, we focus on the FSS exponent, which plays a crucial role in analyzing numerical data for finite-size systems. Based on the droplet-excitation (hyperscaling) argument, we conjecture the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process, all of which are confirmed reasonably well in numerical simulations.

Spontaneous symmetry breakings in graphene subjected to in-plane magnetic field

Abstract: Application of the magnetic field parallel to the plane of the graphene sheet leads to the formation of electron- and holelike Fermi surfaces. Such situation is shown to be unstable with respect to the formation of an excitonic condensate even for an arbitrary weak magnetic field and interaction strength. At temperatures lower than the mean-field temperature, the order parameter amplitude is formed. The order parameter itself is a U(2) matrix allowing for the combined rotations in the spin and valley spaces. These rotations smoothly interpolate between site and bond centered spin-density waves and spin-flux states. The trigonal warping, short-range interactions, and the three-particle umklapp processes freeze some degrees of freedom at temperatures much smaller than the mean-field transition temperature, and make either Berezinskii-Kosterlitz-Thouless [Sov. Phys. JETP 32, 493 (1971); J. Phys. C 5, L124 (1972); 6, 1181 (1973)] (driven either by vortices or half-vortices) or Ising type transitions possible. Strong logarithmic renormalization for the coupling constants of these terms by the Coulomb interaction is calculated within one-loop renormalization group. It is found that in the presence of the Coulomb interaction, some short-range interaction terms become much greater than one might expect from the naive dimensionality counting.

Renormalization of concurrence: The application of the quantum renormalization group to quantum-information systems

Abstract: We have combined the idea of renormalization group and quantum-information theory. We have shown how the entanglement or concurrence evolve as the size of the system becomes large, i.e., the finite size scaling is obtained. Moreover, we introduce how the renormalization-group approach can be implemented to obtain the quantum-information properties of a many-body system. We have obtained the concurrence as a measure of entanglement, its derivatives and their scaling behavior versus the size of system for the one-dimensional Ising model in transverse field. We have found that the derivative of concurrence between two blocks each containing half of the system size diverges at the critical point with the exponent, which is directly associated with the divergence of the correlation length.

Monte Carlo simulations of two-dimensional hard core lattice gases.

Abstract: Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two-dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, λ (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau–Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau–Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class—contradicting the predictions of the Landau–Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.

Spin transport in Heisenberg antiferromagnets in two and three dimensions

Abstract: We analyze spin transport in insulating antiferromagnets described by the $XXZ$ Heisenberg model in two and three dimensions. Spin currents can be generated by a magnetic-field gradient or, in systems with spin-orbit coupling, perpendicular to a time-dependent electric field. The Kubo formula for the longitudinal spin conductivity is derived analogously to the Kubo formula for the optical conductivity of electronic systems. The spin conductivity is calculated within interacting spin-wave theory. In the Ising regime, the $XXZ$ magnet is a spin insulator. For the isotropic Heisenberg model, the dimensionality of the system plays a crucial role: In $d=3$ the regular part of the spin conductivity vanishes linearly in the zero frequency limit, whereas in $d=2$ it approaches a finite zero frequency value.

Elastic interaction among transition metals in one-dimensional spin-crossover solids

Abstract: We present an exact examination of a one-dimensional (1D) spin-phonon model describing the thermodynamical properties of spin-crossover (SC) solids. This model has the advantage of giving a physical mechanism for the interaction between the SC units. The origin of the interaction comes from the fact that the elastic constant of the spring linking two atoms depends on their electronic states. This leads to local variation of the elastic constant. Up to now, all the statistical studies of this model have been performed in the frame of the mean-field (MF) approach, which is not adequate to describe 1D systems with short-range interactions. An alternative method, based on the variational approach and taking into account the short-range correlations between neighboring molecules, was also suggested, but it consists in an extension of the previous MF approximation. Here, we solve exactly this Hamiltonian in the frame of classical statistical mechanics using the transfer-matrix technique. The temperature dependence of the high spin fraction and that of the total energy are obtained analytically. Our results clearly show that there is a clear tendency to a sharp transition when we tune the elastic constants adequately, which indicates that first-order phase transition takes place at higher dimensions. In addition, we demonstrate the existence of an interesting isomorphism between the present model and Ising model under effective interaction and effective ligand field energy, in which both depend linearly on temperature and both come from the phonon contribution. We have also studied the effect of the pressure (the tension) on the thermodynamical properties of the high spin (HS) fraction and have found a nontrivial pressure effect that while for weak tension values, the low spin state is stabilized for the pressure above a threshold value, it enhances the interaction between the HS states. Finally, we have also introduced elastic interactions between the chains. Treating exactly (in mean field) the intrachain (interchain) contributions, we found that our model leads us to obtain first-order spin transitions when both short- and long-range interactions are ferroelastic. We show also that competing (antiferroelastic short-range and ferroelastic long-range) interactions between spin-state ions reproduce qualitatively the two-step-like spin-crossover transitions.

Planar Ising Correlations

Abstract: Preface.- I. Ising Model on a Finite Square Lattice.- II. Infinite Volume Limits.- III. Scaling Limits.- IV. Monodromy Preserving Deformations of the Euclidean Dirac Equation.- V. Analysis of Tau Functions.- VI. Holonomic Quantum Fields.- Appendix: Infinite Dimensional Spin Groups.- Bibliography.- Index.

Infinite-randomness fixed points for chains of non-Abelian quasiparticles.

Abstract: One-dimensional chains of non-Abelian quasiparticles described by SU(2)k Chern-Simons-Witten theory can enter random singlet phases analogous to that of a random chain of ordinary spin-1/2 particles (corresponding to k-->infinity). For k=2 this phase provides a random singlet description of the infinite-randomness fixed point of the critical transverse field Ising model. The entanglement entropy of a region of size L in these phases scales as S(L) approximately lnd/3 log(2)L for large L, where d is the quantum dimension of the particles.

Ising-like model for protein mechanical unfolding.

Abstract: The mechanical unfolding of proteins is studied by extending the Wako-Saito-Munoz-Eaton model This model is generalized by including an external force, and its thermodynamics turns out to be exactly solvable We consider two molecules, the 27th immunoglobulin domain of titin and protein PIN1 We determine equilibrium force-extension curves for the titin and study the mechanical unfolding of this molecule, finding good agreement with experiments By using an extended form of the Jarzynski equality, we compute the free energy landscape of the PIN1 as a function of the molecule length

Condensation in a capped capillary is a continuous critical phenomenon.

Abstract: We show that condensation in a capped capillary slit is a continuous interfacial critical phenomenon, related intimately to several other surface phase transitions. In three dimensions, the adsorption and desorption branches correspond to the unbinding of the meniscus from the cap and opening, respectively, and are equivalent to 2D-like complete-wetting transitions. For dispersion forces, the singularities on the two branches are distinct, owing to the different interplay of geometry and intermolecular forces. In two dimensions we establish precise connection, or covariance, with 2D critical-wetting and wedge-filling transitions: i.e., we establish that certain interfacial properties in very different geometries are identical. Our predictions of universal scaling and covariance in finite capillaries are supported by extensive Ising model simulation studies in two and three dimensions.

A statistical mechanics view on Kitaev's proposal for quantum memories

Abstract: We compute rigorously the ground and equilibrium states for Kitaev's model in 2D, both the finite and infinite versions, using an analogy with the 1D Ising ferromagnet. Next, we investigate the structure of the reduced dynamics in the presence of thermal baths in the Markovian regime. Special attention is paid to the dynamics of the topological freedoms which have been proposed for storing quantum information.

Lace Expansion for the Ising Model

Abstract: The lace expansion has been a powerful tool for investigating mean-field behavior for various stochastic-geometrical models, such as self-avoiding walk and percolation, above their respective upper-critical dimension. In this paper, we prove the lace expansion for the Ising model that is valid for any spin-spin coupling. For the ferromagnetic case, we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace-expansion coefficients for self-avoiding walk. As a result, we obtain Gaussian asymptotics of the critical two-point function for the nearest-neighbor model with \({d\,\gg\,4}\) and for the spread-out model with d > 4 and \({L\,\gg\,1}\) , without assuming reflection positivity.

Numerical renormalization group for continuum one-dimensional systems.

Abstract: We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.

Quantum noise analysis of spin systems realized with cold atoms

Abstract: We consider the use of quantum noise to characterize many-body states of spin systems realized with ultracold atomic systems These systems offer a wealth of experimental techniques for realizing strongly interacting many-body states in a regime with a large but not macroscopic number of atoms In this regime, fluctuations of an observable such as the magnetization are discernible compared to the mean value The full distribution function is experimentally relevant and encodes high order correlation functions that may distinguish various many-body states We apply quantum noise analysis to the Ising model in a transverse field and find a distinctive even versus odd splitting in the distribution function for the transverse magnetization that distinguishes between the ordered, critical, and disordered phases We also discuss experimental issues relevant for applying quantum noise analysis for general spin systems and the specific results obtained for the Ising model

Holonomy of the Ising model form factors

Abstract: We study the Ising model two-point diagonal correlation function C(N, N) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f(j)N,N, in the form factor expansion. The corresponding λ-extension of the two-point diagonal correlation function, C(N, N; λ), is shown, for arbitrary λ, to be a solution of the sigma form of the Painleve VI equation introduced by Jimbo and Miwa in their isomonodromic approach to the Ising model. Fuchsian linear differential equations for the form factors f(j)N,N are obtained for j ≤ 9 and shown to have both a 'Russian-doll' nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second-order linear differential operator associated with the elliptic integral E. From this, we show that each f(j)N,N is unexpectedly simple, being expressed polynomially in terms of the elliptic integrals E and K. In contrast, we exhibit some mathematical objects, built from these form factors f(j)N,N, which break the direct sum of symmetric powers decomposition, with its associated polynomial expressions. First we show that the scaling limit of these differential operators, and form factors, breaks the direct sum structure but not the 'Russian-doll' structure. Secondly, we show that the previous λ-extension of two-point diagonal correlation functions, C(N, N; λ), is, for singled-out values λ = cos(πm/n), (m, n integers), also solutions of Fuchsian linear differential equations. These solutions of Painleve VI are not polynomial in E and K but are actually algebraic functions, being associated with modular curves.

Size dependence of ferromagnetism in gold nanoparticles: Mean field results

Abstract: In this paper, a simple spin-spin Ising interaction model for the surface ferromagnetism is combined with the bulk Au diamagnetic response to model the size dependence of the magnetization of a Au nanoparticle. Using the maximum entropy formalism, we obtain the average temperature dependent magnetization within a mean field model. Our results qualitatively reproduce recent experimental observations of size-dependent magnetization of Au nanoparticles in which the ferromagnetic moment of thiol-capped nanoparticles is seen to increase for diameters larger than $0.7\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, peaking at approximately $3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, and subsequently decreasing as the particle diameter increases further.