I am a little confused about the Mermin-Wagner theorem, and the assumptions it makes, and I would like to better understand the 'exceptions' to the rule that occur in practical (non-ideal) scenarios.
As is my understanding,
The Mermin-Wagner theorem states that there cannot be spontaneous symmetry breaking in 2d (or less) systems at any finite temperature T > 0. Essentially what this means is that you cannot have, for example, ferromagetic order in 2D systems since the continuous rotational (and translational?) symmetry of the spins are broken by the formation of ordered domains where one direction is preferred over another. The reason for this is that, in such low-dimensionality systems, the energy cost to excite a defect/disordered in the lattice is infinitesimal small and thus infinitely many excited modes are created that destroy the order.
But there are exceptions to the rule, so to speak. Since this theorem only applies to continuous symmetries, and only says that the ordered states are forbidden in 2 or fewer dimensions. In real life, though, it is is very difficult to achieve such ideal circumstances. 'Accidentally' symmetry breaking might occur to do a small biasing fields, the presence of a substrate that supports the 2d material, or the fact that a thin film still has a small, but finite, thickness and is therefor not really two dimensional.
My question is about what constitutes enough of a symmetry breaking for the Mermin-Wagner theorem to not apply. Does the non-continuous symmetry from the lattice break the symmetry? Spins located at discretely spaced lattice points are not continuously translationally symmetric. And the lattice type should also break the symmetry as well, whether it is a square, triangular, honeycomb, … lattice, would break rotational symmetry. Is this enough? I think the answer is “no,” (explained below) but I am not exactly sure why.
The Ising model is often used as an example of an 'exception' to the Mermin-Wagner theorem in that it has a discrete symmetry, and therefore has a 2d ferromagnetic state, compared to the 2d Heisenberg/XY model, where does not exhibit as ferromagnetic state in 2 dimensions. Both of these models involve spins being located at discrete lattice points, but while the Ising model constrains the spins to point only “up” or “down”, the Heisenberg model allows the spins to point any direction.
As is my understanding,
The Mermin-Wagner theorem states that there cannot be spontaneous symmetry breaking in 2d (or less) systems at any finite temperature T > 0. Essentially what this means is that you cannot have, for example, ferromagetic order in 2D systems since the continuous rotational (and translational?) symmetry of the spins are broken by the formation of ordered domains where one direction is preferred over another. The reason for this is that, in such low-dimensionality systems, the energy cost to excite a defect/disordered in the lattice is infinitesimal small and thus infinitely many excited modes are created that destroy the order.
But there are exceptions to the rule, so to speak. Since this theorem only applies to continuous symmetries, and only says that the ordered states are forbidden in 2 or fewer dimensions. In real life, though, it is is very difficult to achieve such ideal circumstances. 'Accidentally' symmetry breaking might occur to do a small biasing fields, the presence of a substrate that supports the 2d material, or the fact that a thin film still has a small, but finite, thickness and is therefor not really two dimensional.
My question is about what constitutes enough of a symmetry breaking for the Mermin-Wagner theorem to not apply. Does the non-continuous symmetry from the lattice break the symmetry? Spins located at discretely spaced lattice points are not continuously translationally symmetric. And the lattice type should also break the symmetry as well, whether it is a square, triangular, honeycomb, … lattice, would break rotational symmetry. Is this enough? I think the answer is “no,” (explained below) but I am not exactly sure why.
The Ising model is often used as an example of an 'exception' to the Mermin-Wagner theorem in that it has a discrete symmetry, and therefore has a 2d ferromagnetic state, compared to the 2d Heisenberg/XY model, where does not exhibit as ferromagnetic state in 2 dimensions. Both of these models involve spins being located at discrete lattice points, but while the Ising model constrains the spins to point only “up” or “down”, the Heisenberg model allows the spins to point any direction.
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