Timeline for Three versions of the classical $XY$ model and their low-temperature behaviors
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2022 at 15:24 | comment | added | Adam | @user196574 I find the $\phi/2\phi$ model less easy to understand, and prefer the coupled $XY$ model (the last one in the answer). There, one sees that the symmetry is broken from $O(2)\times O(2)\to O(2)\times \mathbb Z_2$ by the coupling between the two species of spins. However, and this is an important but subtle point, the $\mathbb Z_2$ symmetry is not associated with a local Ising spin variables that orders. (This would lead to contradictions, see the paper.) However, one can introduce non-local variables that do order. | |
| Dec 13, 2022 at 8:32 | comment | added | Yvan Velenik | That being said, I do not see what would be the mechanism behind the spontaneous symmetry breaking of a discrete symmetry (which one?) in the case of $H_4$ (and haven't had the time to look at the references @Adam has given in his answer). | |
| Dec 13, 2022 at 8:32 | comment | added | Yvan Velenik | @user196574 Note that, in the case of the variant I mention in my comment, the order parameter is not the magnetization. Indeed, the latter cannot be nonzero, by virtue of the Mermin-Wagner theorem. A picture of a typical low-temperature configuration can be found in Fig. A.1 of this book (p. 475 of the downloadable version). In the other state, typical configurations have the spins "rotating" in the other direction, when moving from a site to its neighbor. I am not sure a spin-wave argument can say anything about this type of ordering. | |
| Dec 13, 2022 at 7:19 | comment | added | user196574 | @YvanVelenik Thanks, the even-sublattice symmetry makes sense. I'm tempted that $H_4$ will struggle to have order by virtue of a Peierls argument for spin waves - low-momentum spin-waves will have a very low energy cost. I'm tempted that there will not even be an algebraic phase, because I expect single vortices have a finite energy cost (rather than logarithmically diverging in system size), allowing proliferation of vortices at any nonzero temperature. However, these are heuristic guesses. | |
| Dec 13, 2022 at 6:48 | comment | added | Yvan Velenik | Of course, the $O(2)$ symmetry cannot be spontaneously broken (Mermin-Wagner), but this discrete symmetry can (and is). I don't know what happens in the case of $H_4$. | |
| Dec 13, 2022 at 6:46 | comment | added | Yvan Velenik | @user196574 The variant I know of (the one discussed in the paper by Shlosman in this answer) has a different Hamiltonian: the interaction between nearest-neighbors is $J_2\cos(2\theta_i-2\theta_j)$, but the interaction between second-nearest-neighbors is $-J_1\cos(\theta_i-\theta_j)$. In this case, it is easy to see why there can be a discrete symmetry breaking at low temperatures: in addition to the $O(2)$ symmetry, the Hamiltonian is also invariant under the transformation that leaves the spins in the even sub-lattice invariant, but add $\pi$ to all spins in the odd sublattice. | |
| Dec 12, 2022 at 18:26 | comment | added | user196574 | +1 Thank you, these are very interesting results! I hadn't appreciated that the lack of long-range order followed from mere continuity of the potential (and even survives weak singularities). It appears then that a Peirls' style argument checking whether the energy cost of the lowest-momentum spin-wave diverges or not doesn't work? The $\phi/2\phi$ model is also very interesting, and I'll read those references in detail. In the meantime, could you comment on whether the low temperature symmetry broken phase is because of breaking a discrete symmetry in the model? It seems there's only $U(1)$. | |
| Dec 12, 2022 at 15:21 | comment | added | Adam | @YvanVelenik Thanks for the references, I didn't know those rigorous results. I've updated the answer to include them. | |
| Dec 12, 2022 at 15:20 | history | edited | Adam | CC BY-SA 4.0 |
added 890 characters in body
|
| Dec 12, 2022 at 10:50 | history | answered | Adam | CC BY-SA 4.0 |