Gonihedric Ising Actions

TL;DR

This paper explores a generalized Ising model with complex interactions, exhibiting diverse phase transitions and a unique semi-global symmetry, potentially useful for discretizing string worldsheet theories.

Contribution

It introduces a new generalized Ising action with multiple interaction terms and analyzes its phase transition behavior and symmetry properties.

Findings

Displays both first and second order phase transitions.

Possesses a novel semi-global symmetry.

Potential application in string worldsheet discretization.

Abstract

We discuss a generalized Ising action containing nearest neighbour, next to nearest neighbour and plaquette terms that has been suggested as a potential string worldsheet discretization on cubic lattices by Savvidy and Wegner. This displays both first and second order transitions depending on the value of a ``self-intersection'' coupling as well as possessing a novel semi-global symmetry.

Equations

$$S=\frac{1}{2}\sum _{<ij>}|{\vec{X}}_i-{\vec{X}}_j|\theta ({\alpha }_{ij}),$$

$$H$$

$$M=⟨\frac{1}{L^3}\sum _i{\sigma }_i⟩.$$

$$h_c$$

$$\varphi (m_C)$$

$${\frac{\partial \varphi (m_C)}{\partial m_i}}_{(i=1\dots 8)}=0$$

$$H=\frac{1}{2}\sum _{[i,j,k,l]}{\sigma }_i{\sigma }_j{\sigma }_k{\sigma }_l.$$

Full text

1 INTRODUCTION

Savvidy et al. [1] recently suggested a novel discretized random surface theory, the so-called Gonihedric string,

[TABLE]

where the sum is over the edges of some triangulated surface, $\theta(\alpha_{ij})=|\pi-\alpha_{ij}|^{\zeta}$, $\zeta$ is some exponent, and $\alpha_{ij}$ is the dihedral angle between neighbouring triangles with common link $<ij>$. This definition of the action was inspired by the geometrical notion the linear size of a surface, originally defined by Steiner.

In equ.(1) it is the surface itself that is discretized, rather than the space in which it is embedded. An alternative approach to discretizing the linear size is to discretize the embedding space by restricting the allowed surfaces to the plaquettes of a (hyper)cubic lattice. This method was applied by Savvidy and Wegner [2, 3, 4, 5], who rewrote the resulting theory as a generalized Ising model by using the geometrical spin cluster boundaries to define the surfaces. The energy of a surface on a cubic lattice is given explicitly in the Savvidy-Wegner models by $E=n_{2}+4\kappa n_{4}$, where $n_{2}$ is the number of links where two plaquettes meet at a right angle, $n_{4}$ is the number of links where four plaquettes meet at right angles, and $\kappa$ is a free parameter which determines the relative weight of a self-intersection of the surface. In the limit $\kappa\rightarrow\infty$ the surfaces would be strongly self-avoiding, whereas the opposite limit $\kappa\rightarrow 0$ would be that of phantom surfaces that could pass through themselves without hindrance. This should be contrasted with the standard 3D Ising model with nearest neighbour interactions where the surfaces are weighted by their areas.

On a cubic lattice the generalized “Gonihedric” Ising Hamiltonian which reproduces the desired energy $E=n_{2}+4\kappa n_{4}$ contains nearest neighbour ($<i,j>$), next to nearest neighbour ($<<i,j>>$) and round a plaquette ($[i,j,k,l]$) terms

[TABLE]

Such generalized Ising actions, or their equivalent surface formulations, have quite complicated phase structures for generic choices of the couplings [6, 7, 8]. The particular ratio of couplings in equ.(2), however, is non-generic and introduces a novel symmetry into the model - it is possible to flip any plane of spins at zero energy cost.

Although not a local gauge symmetry, the flip symmetry of the model is intermediate between this and a global symmetry. This symmetry poses something of a problem when carrying out simulations, as it means that a simple ferromagnetic order parameter

[TABLE]

will be zero in general, because of the layered nature of the ground state. Even staggered magnetizations would not do as the interlayer spacing can be arbitrary. It is possible, however, to force the model into the ferromagnetic ground state, which is equivalent to any of the layered ground states, with a suitable choice of boundary conditions on a finite lattice. Fixed boundary conditions would do the job by penalizing any flipped spin planes by a boundary term, but they carry a higher price in finite size effects than the customary periodic boundary conditions. A more elegant solution is to fix any two internal planes of spins in the lattice, whilst retaining the periodic boundary conditions. This has the desired effect of picking out the ferromagnetic ground state, whilst minimizing any finite size effects. With fixed spin planes we can therefore still employ the simple order parameter of equ.(3).

2 ZERO-TEMPERATURE AND MEAN FIELD

As the Gonihedric model is a special case of the general action considered in [6] we can apply the methods used there for both the zero temperature phase diagram and mean field theory. For the zero temperature case this involves writing the full lattice Hamiltonian as a sum over individual cube Hamiltonians

[TABLE]

and observing that if the lattice can be tiled by a cube configuration minimizing the individual $h_{c}$ then the ground state energy density is $\epsilon_{0}=min\;h_{c}$.

This approach reveals that a layered ground state with parallel layers of flipped spins perpendicular to one of the lattice axes and arbitrary interlayer spacing is degenerate with the ferromagnetic ground state for all $\kappa$. This might have been expected from the flip symmetry of the hamiltonian itself. In addition, at $\kappa=0$ an extra ground state corresponding to diagonal flipped planes appears.

In the mean field approximation the spins are in effect replaced by the average site magnetizations. The calculation of the mean field free energy is an elaboration of the zero temperature approach in which the energy is decomposed into a sum of individual cube terms. The next to nearest neighbour and plaquette interactions in the Gonihedric model give the total mean field free energy as the sum of elementary cube free energies $\phi(m_{c})$, given by

[TABLE]

where $m_{C}$ is the set of the eight magnetizations of the elementary cube. This gives a set of eight mean-field equations

[TABLE]

one for each corner of the cube. Numerical iteration of these equations shows a single transition from a paramagnetic high temperature state to a layered, or the equivalent ferromagnetic, low temperature state. The $\beta_{c}$ determined in this fashion decreases quite sharply with $\kappa$.

3 MONTE-CARLO SIMULATIONS

In order to see how the full theory tallies with the zero-temperature and mean-field results we carried out Monte-carlo simulations [9] for various $\kappa$ values and lattices of size $10^{3},12^{3},15^{3},18^{3},20^{3}$ and $25^{3}$. Periodic boundary conditions were imposed in the three directions and three internal perpendicular planes of spins fixed to be $+1$. A simple Metropolis update was used because of the difficulty in concocting a cluster algorithm for a Hamiltonian with such complicated interaction terms. The program was tested on the standard nearest neighbour Ising model and the some of the parameters used in the generalized Ising models of [6] to ensure it was working. We measured the usual thermodynamic quantities for the model: the energy $E$, specific heat $C$, (standard) magnetization $M$, susceptibility $\chi$ and various cumulants.

We can clearly see the second order nature of the peak in the specific heat in Fig.1

Although extracting the exponent $\alpha$ from the scaling of the peak is not particularly reliable because of the constant term that appears in both the direct and finite-size scaling forms an approach using the energy measurements gives $(\alpha-1)/\nu=-1.3(2)$. Binder’s magnetization cumulant and the scaling of the susceptibility $\chi$ which is shown in Fig.2

give an estimate of $\nu=1.2(1)$. One also finds $\gamma/\nu=1.79(4)$ from the FSS of $\chi$. All these exponents, rather remarkably given that the model is defined in three dimensions, are close to the Onsager values of the two-dimensional Ising model with nearest neighbour interactions. The question of whether this is simply a numerical coincidence or a true equivalence awaits an answer from further numerical work. The exponents and the transition temperature vary little, if at all, as $\kappa$ is increased.

Simulations of the $\kappa=0$ model [10]

[TABLE]

show that this is a special case, displaying a first order transition. The transition still appears first order at $\kappa=0.1$ but softens rapidly as $\kappa$ increases, so the crossover to the second order behaviour seen at $\kappa=1$ is quite sharp.

4 CONCLUSIONS

We have investigated a class of Ising-like models suggested by Savvidy and Wegner as a lattice discretization of a particular random surface action that contains no bare surface tension term. The models have a novel flip symmetry that allows entire spin pnaes to be flipped at zero energy cost. We have found that the phase diagram displays only a single transition to a low temperature layered state that, as a consequence of the flip symmetry, is equivalent to a ferromagnetic ground state. For non-zero $\kappa$ the transition is second order, whereas at, and probably close to, $\kappa=0$ a first order transition is manifest. The presence of some degree of self-avoidance, in the form of a non-zero value for $\kappa$, would thus appear to have an important influence on the universality properties of the Savvidy-Wegner/gonihedric models.

It is also possible to define open surface variants of the models described here, which now incorporate link spins as well as vertex spins, and to define models with similar properties in higher dimensions. All of these merit investigation both from the string theory and statistical mechanical point of view.