Manual Restricted-Orientation Convexity

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As the free energy is a linear function of energy and entropy, we expect similar convex sets as for the quantum case when making a scatter plot of the expectation values of energy, entropy and the order parameter with respect to all possible probability distributions see figure 2 for the classical two-dimensional Ising model.

Remarkably, we obtain a very similar picture as for the quantum case. The extreme points of the convex set now correspond to expectation values for Gibbs states which minimize the free energy. Figure 2. Convex set generated by the average nearest neighbor correlation , the entropy per site s , and the expectation value of the magnetization per site for all possible probability distributions of classical 2-state spin configurations on an infinite 2D square lattice. Beyond the critical point A an emerging ruled surface green again signals symmetry breaking. This set looks very similar to the quantum Ising case in 1D in figure 1 b , which is to be expected as both models lie in the same universality class.

In the following we work out the case for a paradigmatic example of a classical lattice spin model in full detail: the q -state Potts model [ 22 — 25 ] is a generalization of the ubiquitous -symmetric Ising model [ 26 , 27 ] to -symmetry.

It has been shown to be correspond to a lattice gauge theory of matter [ 28 , 29 ] and in certain parameter regimes to coloring problems [ 30 , 31 ] and hard-square lattice-gas models with nearest neighbor exclusion 1NN [ 32 ]. We consider the model in two spatial dimensions on a square lattice. At zero field, where the model possesses -symmetry, it undergoes a symmetry breaking phase transition at finite critical inverse temperature [ 23 , 33 ], above which the -symmetry is spontaneously broken. For general and zero field the model can be mapped onto a staggered six-vertex model, which can be solved exactly only at criticality [ 34 , 35 ].

The symmetry breaking phase transition in zero field is continuous for and of first order for [ 23 ]. The nature of the phase transition will be apparent from the geometrical features of the corresponding convex set phase diagrams which we construct below. Consider the space of all possible probability distributions of configurations of q -state spins with i the position on a two-dimensional square lattice with N sites, which form a convex set in some high-dimensional parameter space. In particular we consider three-dimensional projections of this set in the thermodynamic limit , parameterized by the three observables nearest neighbor interaction energy per site.

The convex set is then given by all possible points , such that , and s are compatible with each other, i. This is an instance of the classical marginal problem [ 36 — 39 ]. Notice that we are using a shifted magnetization with an offset , such that the convex set is reflection symmetric with respect to. The extreme points on the surface of this set are then naturally given by Gibbs states of 5. To see this, consider hyper planes in this three-dimensional parameter space, which are defined as families of points , related by a plane equation of the form.

Setting , and , this yields exactly the negative of the free energy per site of 5. For a given set of parameters i. Every point on the surface thus corresponds to a state of thermodynamic equilibrium, at parameters given by the orientation of the tangent plane and free energy proportional to the distance of the tangent plane to the origin.

Conversely, every point inside the convex set corresponds to a possible non-equilibrium state of the system.

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If the tangent plane touches the convex set at a unique point only, then the thermodynamic stable state is unique and exactly given by a Gibbs state which yields the observables given by the tangent point for the parameters defined by the orientation of the tangent plane, i. If however the tangent plane touches the set on an entire line or even a plane, then the state which minimizes the free energy for these parameters is not unique, which is a prerequisite of symmetry breaking.

The set of valid states can then be parameterized by one or more real parameters. Such ruled surfaces continuous sets of tangent lines or planes are thus the geometrical signatures that will enable us to detect symmetry breaking and the emergence of a connected order parameter.

These sets show interesting geometrical features from which a wealth of other information, such as the nature of phase transitions, locations of critical points, critical exponents, susceptibilities, etc can be extracted.

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The numerical data for plotting these surfaces has been obtained by means of tensor network techniques described in appendix A. For scatter plots of points obtained from random probability distributions, which approximate the convex set from the inside, see appendix B. Figure 3. Convex set generated by nearest-neighbor interaction energy , shifted magnetization and entropy per site s of all possible probability distributions of 3-state spins on a two-dimensional square lattice.

Due to reflection symmetry we only plot the upper half of the set. Blue lines denote points of constant and h and varying temperature T. At the critical point A the emergence of a green ruled surface signals a non-uniqueness of the thermal equilibrium state at zero field and thus symmetry breaking. As a guide to the eye we have plotted a few vertical lines on the ruled surface, along which the tangent plane touches the convex set.

There again the lowest energy state is exponentially degenerate, resulting in a finite residual entropy as described in section 3. This plane is only present for and does therefore not appear in the convex set drawn for the Ising model in figure 2. As a guide to the eye we have drawn two-dimensional grids onto the top and left plane, emphasizing the fact that there the tangent plane touches the set on the entire respective planes. Figure 4. For the phase transition is of first order and thus comes with a discontinuity of the three observables at the critical point.

This results in a coexistence region of the ordered and disordered phases and the critical point A gets stretched out into a gray flat triangular surface, where any mixture of the two phases is a valid state, i. This flat part then smoothly connects to the symmetry broken ordered phase represented by the green ruled surface. As a guide to the eye we have drawn a two-dimensional grid onto the flat triangular surface, emphasizing the fact that there the tangent plane touches the set on the entire triangular surface and we have also plotted a few vertical lines on the green ruled surface, along which the tangent plane touches the convex set.

The flat surfaces emerging from points B and C are of the same nature as described in figure 3. For zero field, and the thermodynamic state that minimizes the free energy is q -fold degenerate and the -symmetry can be spontaneously broken, such that. The maximum possible value can then be taken as the order parameter associated to this phase transition 6. For a given set of parameters any state within this q -fold degenerate space thus minimizes the free energy and is characterized by the same values for and s , but different 7.

This is nicely reflected in the convex sets through the emergence of a green ruled surface at the critical point. Zero field implies tangent planes with normal vectors lying in the plane, i. The tangent plane touches the convex set on a unique point in the plane everywhere except for and , where the tangent plane in fact touches the convex set along a whole line for each J and T , given by with and the maximum value of the order parameter.

An infinitesimal value of then immediately explicitly breaks the symmetry and causes the tangent plane to touch the set on a unique point of the set infinitesimally close to the edge of the ruled surface. Or equivalently, the curve of tangent points of a tangent plane given by as will end in a point with for.

This nicely reflects the fact that the order parameter can be obtained by first taking the thermodynamic limit at non-zero field before letting the field go to zero.

Symmetry breaking and the geometry of reduced density matrices

The nature of the phase transition changes from continuous to first order for , where a first order phase transition is characterized by a latent heat and a discontinuity of first derivatives of the free energy at the critical point. The internal energy and all other expectation values that can be written as a derivative of the free energy, such as the order parameter and also the entropy per site s therefore have a discontinuity at the critical point.

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In the convex set we can thus detect first order phase transitions through the appearance of flat hyperplanes at the boundary that arise even without additionally plotting the order parameter. At the critical point the thermal equilibrium state is not unique and any point on this hyperplane is a valid state of the system at the critical temperature. This corresponds to the coexistence of phases at the critical point which is characteristic for first order phase transitions. In the case of the Potts model, this flat hyperplane then smoothly connects to the ruled surface representing the symmetry broken phase see figure 4.

For continuous phase transitions the thermodynamic state at the critical point is still unique and there is no such additional hyperplane.

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We can thus already detect first order phase transitions in the lower dimensional convex set that does not include the order parameter. In the case of the Potts model, a two-dimensional convex set parameterized by and s thus already suffices to detect the phase transition for , it will however show no signature of the phase transition for see figure 5 , for which adding an additional axis corresponding to the order parameter is necessary.

Figure 5.

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  • As a guide to the eye we have extended this line to both sides to see that there is albeit very small curvature to both sides of the phase coexistence part. We want to emphasize here that these convex sets and thus also the ruled surfaces exist prior to making any references to any model Hamiltonian, we just consider finite dimensional projections of the convex set of all possible probability distributions of a system of physical degrees of freedom.

    This means that the reason for the occurrence of symmetry breaking phase transitions ultimately lies in the geometrical structure of the space of all possible probability distributions. It would therefore be interesting to investigate all possible projections of this set and classify all possible ruled surfaces that can arise on such projections. Exactly at this point, the two terms in the Hamiltonian become 'equally strong' in the following sense. If we start from the completely polarized state , the magnetic field term is minimized, whereas the interaction part has a positive energy contribution, resulting in a net energy of per site.

    If we now flip one spin at an arbitrary position from q to , we gain exactly the same amount of energy from the interaction term as we lose from the magnetic field term and the overall energy stays the same. In general, a cluster of N f flipped spins and a boundary of length N b results in a net energy change of , which is only zero for. Similarly, flipping from q to any always results in a net energy increase and the restricted space of lowest energy states is thus given by all configurations such that every is completely surrounded by see also figure 6.

    This restricted space is equivalent to the configuration space for the nearest-neighbor exclusion lattice-gas model 1NN [ 32 ] and grows exponentially with the system size. Figure 6. Starting from the fully polarized state with lowest possible energy, flipping single spins from q to leaves the overall energy invariant. Flipping two or more adjacent spins however results in a net energy increase, as does flipping from q to any. The resulting space consists of all configurations where such that every is completely surrounded by.

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    This symmetry of equal probability can however be spontaneously broken as any statistical mixture of such configurations is a valid state of the system with equal free energy. The entirety of all such mixtures is exactly given by the top blue plane in the convex sets, where point B marks the state of equal probability which has maximal entropy.

    To calculate the boundary of the top blue plane we consider tiny perturbations away from this point in parameter space, which immediately cause a jump onto the edge of the plane. Similar to degenerate perturbation theory we then simulate this perturbation Hamiltonian only within the restricted subspace of the top plane to lift the exponential degeneracy and determine its extreme points. The perturbation Hamiltonian is just the magnetic field term. We therefore wish to evaluate.

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    The entropy per site s is then given by. The other observables and are computed as usual but with respect to Note that entropy and are independent of q and for different q are related by just an offset. The top plane thus has the same shape for all q , but different vertical offset in. Our calculated value at this point reproduces the log of the value given in section 1.