Universality of Pattern Formation

We develop a theory of pattern formation in non-Hermitian scalar field theories. Patterned configurations show enhanced Fourier modes, reflecting a tachyonic instability. Multicomponent $\mathcal{PT}$-symmetric field theories with such instabilities represent a new universal class of pattern-forming models. The presence of slow modes and long-lived metastable behavior suggests a connection between the computational complexity of the sign problem and the physical characterization of equilibrium phases. Our results suggest that patterning may occur near the critical endpoint of finite density QCD.


INTRODUCTION
Pattern formation is a ubiquitous phenomenon throughout physics [1][2][3]. In patternforming systems, equilibrium phases exhibit complicated behaviors characterized by persistent inhomogeneous patterns such as stripes and dots. Conventional scalar field theories typically satisfy reflection positivity and thus have positive, self-adjoint transfer matrices [4]; such theories cannot display the modulated behavior associated with patterning. We show that multicomponent PT -symmetric scalar field theories with complex actions are a large, natural class of local field theories exhibiting patterning.
PT -symmetric field theories are invariant under the combined action of a discrete linear transformation P and complex conjugation T [5,6]. This symmetry implies that each eigenvalue of the transfer matrix is either real or part of a complex conjugate pair. It is this latter possibility that is responsible for modulated behavior and pattern formation. The prototypical example of a PT -symmetric field theory is the iφ 3 model, which is the field theory for the Lee-Yang transition [7]. In this case P takes φ → −φ. Many PT -symmetric models, including the iφ 3 model, have complex actions and therefore suffer from the sign problem [8]. QCD at nonzero chemical potential is of great interest in nuclear and particle physics but has a sign problem that has severely hampered its study [9,10].
In this paper we consider a PT -symmetric field theory that can be studied both analytically and with lattice simulations. Extensive simulations of this model indicate a smooth transition between pattern morphologies, in contrast to microphase behavior [2,11], which assumes clear distinctions between different morphologies. Instead we find a unifying principle: tachyonic instabilities drive pattern formation. We analytically derive a criterion describing when homogeneous phases are unstable to patterning. The pattern-forming region of parameter space exhibits increased computational complexity characterized by slow modes and long autocorrelation times in simulations. This suggests a connection between the computational difficulty associated with the sign problem and the physical characterization of equilibrium phases in scalar field theories.
A recent development in lattice simulations allows us to simulate the complex action where we set Eq. (1) represents a Hermitian scalar field φ(x) coupled to a PT -symmetric scalar field χ(x) by the imaginary strength ig. The dual action of Eq. (1) takes the form where the dual potentialṼ is given byṼ (φ, ∂ · π) = (∂ · π − gφ) 2 /2m 2 χ + λ(φ 2 − v 2 ) 2 + hφ. This model was recently studied for the case h = 0 in two and three dimensions [12]. Here we extend the study of this model to the full g −h plane. We present extensive simulations in d = 2, in which we vary the parameters g and h on a 64 2 lattice with parameters m 2 χ = 0.5, λ = 0.1 and v = 3. We have also observed similar phenomena in d = 3.
In figure 1 we show nine configuration snapshots of φ, each taken after 20, 000 lattice updates. These snapshots are taken from a large dataset, which extends from g = 0.0 to  Instead we see a smooth transition between long line segments to shorter ones as h increases, with a distribution of shapes in each configuration. The average action S varies smoothly as h and g are varied. In most of the two-dimensional simulations, we see a fairly complete ring in momentum space, consistent with pattern formation without preferred directions. In some cases, however, a smaller number of modes on the ring are excited, and the absence of isotropy is evident in the configurations. This may be related to finite size effects or to locking into an atypical but long-lived pattern. For many systems, it is known that the energy is minimized by regular patterns, typically stripes [13,14].
Our model is also amenable to analytical treatment. Because χ enters quadratically in the action S, it can easily be integrated out, yielding a nonlocal effective action of the form This model has been extensively studied in the case m χ = 0; see, e.g. We determine the value of the order parameter φ 0 at tree level by minimizing the potential or equivalently by minimizing the effective potential associated with S eff : The effect of χ on φ 0 is to restore the symmetric phase φ 0 = 0 for h = 0 at sufficiently large values of g. Given our simulation results, the details of the phase structure follow from the inverse φ propagator obtained at tree level: Three different behaviors are possible, depending on the zeros of G −1 . If both zeros occur at q 2 < 0, then the propagator decays exponentially. If the zeros are complex, they must form a complex conjugate pair and the propagator decays exponentially with sinusoidal modulation. The boundary between these two behaviors is by definition a disorder line [16].
If one or both zeros are real and positive, then the linearized theory predicts that these modes will grow exponentially, indicating that the homogeneous phase is unstable. This is the region where pattern formation occurs. This region is determined by noting that G −1 (q) has a minimum at q 2 > 0 provided g > m 2 χ . The propagator has tachyonic modes if the minimum lies below zero, corresponding to 2g − m 2 χ − 4λv 2 + 12λφ 2 0 < 0. The region predicted to have tachyonic modes is in reasonable agreement with the boundaries of the pattern-forming region observed in simulation, subject to the limitations imposed by lattice size and spacing.
Our simulations and complementary analytical studies point to a common origin for patterning. The observed ring in Fourier space appears independently of the particular morphology of the configuration in real space. Combined with the gradual transition between different morphologies, this suggests that all pattern-forming behavior is associated with tachyonic modes. We see no indication of a first-or second-order thermodynamic phase transition between supposed microphases. It is of course possible that some currently unknown operator might serve as an order parameter for what are referred to as geometric transitions associated with percolative behavior. It is known in the case of the d=1 Ising model that there is an infinite class of nonlocal string operators, each with its own disorder line. This is associated with the behavior of the model in an imaginary magnetic field, which is the prototypical PT -symmetric problem [17], so it is plausible that such behavior may exist in other PT -symmetric models.
As a first step towards approximating the behavior of the equilibrium patterning state, we consider a simple model that provides additional insight into the transition between where the momenta k j are constant in magnitude but uniformly distributed in direction; the phases δ j are also uniformly distributed. The patterning we observe in simulations is strikingly similar to that in phase transition dynamics, although it is associated with equilibrium behavior. The dynamics of φ can be modeled by a Langevin equation where as usual Γ is a decay constant and η(x) is a white noise term. The difference between this model and a standard φ 4 field theory is the nonlocal term in S eff induced by χ, which stabilizes φ in what would otherwise be an unstable region of the phase diagram. It is easy to show that the dynamics of pattern formation have the same enhanced modes in momentum space as the equilibrated configurations do. The dichotomy between a tachyonic origin of patterned phases and the microphase model is reminiscent of the distinction between spinodal decomposition and nucleation and growth in phase transition dynamics. Spinodal decomposition is the mechanism by which unstable states equilibrate while nucleation and growth is associated with the decay of metastable states. We now know [18][19][20] that there is typically no sharp boundary between these two mechanisms in phase transition dynamics.
Because of the close connection between dynamics and statics in this model, we propose that the relation of our tachyonic picture to the microphase picture is essentially the same as that of spinodal decomposition to nucleation and growth.
We now demonstrate that multicomponent PT -symmetric scalar field theories with complex actions form a natural class of models associated with pattern formation. Consider a general field theory of this class in d dimensions where both φ and χ may have more than one component. The action has the form given by Eq. (1), but with V (φ, χ) an arbitrary potential satisfying the PT symmetry condition. As before, we find homogeneous equilibrium phases by minimizing V , with (φ 0 , χ 0 ) the global minimum. We assume that PT symmetry is maintained, which implies that φ 0 is real, χ 0 is imaginary and V (φ 0 , χ 0 ) is real.
The one-loop effective potential V eff (φ, χ) is given by where the mass matrix M in block form is given by This mass matrix evaluated at (φ 0 , χ 0 ) is not necessarily Hermitian but is PT -symmetric.
In the two-component case, we have This generalizes to the multicomponent case as where Σ is a diagonal matrix with entries ±1. The characteristic equations for M and M * are the same, so they have the same eigenvalues. As a consequence, the eigenvalues of M are either both real or form a complex conjugate pair.
We calculate the one-loop contribution to V eff at the tree-level minimum (φ 0 , χ 0 ). We see easily that det (q 2 + M) can be negative if and only if one or more of the eigenvalues of M are real and negative. If one or more eigenvalues are negative, then V eff will have an imaginary part, indicating instability of the homogeneous phase, and the equilibrium phase is inhomogeneous. The decay rate of the homogeneous phase is [21] where R is the region of q space where det M < 0. Note that this decay rate is perturbative, representing a fast decay, in contrast to the slow modes associated with changing pattern morphology. This indicates that relaxation from a given initial condition takes little simulation time relative to autocorrelation time, Adequately sampling the equilibrium state may require a great deal of time depending on the target observables. Note that in the general case, there may be more than one homogeneous solution which is unstable to pattern formation. We cannot necessarily predict which inhomogeneous phase has the lowest free energy. In the case of a theory with three or more components, pattern formation may be quite complicated [13,14].
The observation of pattern formation in field theories with complex actions raises interesting issues about computational complexity in bosonic models with sign problems. Some of the characteristics observed in our simulations, such as large numbers of metastable configurations and very slow quasizero modes, are reminiscent of glassy behavior. It is known that the problem of finding the ground state of an Ising model with general couplings is NP-hard [22]. Certain fermionic models with sign problems have been mapped to the Ising spin glass, a known NP-hard problem [23]. In the pattern-forming region of the model studied here, the computational complexity, as measured by our ability to adequately sample equilibrium behavior, increases dramatically; this is similar to the behavior of a spin glass, but without the random character of spin glass interactions. In PT-symmetric scalar field theories, imaginary couplings can change the fundamental behavior of interactions, making attractive couplings repulsive. This in turn can set up a conflict between attractive and repulsive forces, a well-known cause of pattern formation. For example, nuclear pasta, believed to occur in neutron star crusts, arises from the attractive nuclear force and the repulsive Coulomb force [11]. Thus the connection between the sign problem and pattern formation is in hindsight natural.
Our original interest in field theories with sign problems was motivated by QCD at finite density, a multi-component field theory with a generalized PT symmetry. The widelyconjectured phase structure of finite density QCD is characterized by a first-order line with a critical end point in the Z(2) universality class, similar to the model studied here. This raises the interesting possibility that finite-density QCD might exhibit pattern formation around its critical end point, composed of regions of confined and deconfined phase. As discussed above, patterns may also form out of equilibrium, an interesting feature from an experimental point of view.