Mermin-Wagner theorem

The Mermin-Wagner theorem or Mermin-Wagner-Hohenberg theorem is a theorem of theoretical , especially statistical physics , which states very generally that in one and two-dimensional systems at temperatures above absolute zero for systems with continuous symmetry and sufficiently short-range interactions cannot result in spontaneous symmetry breaking . It is named after N. David Mermin and Herbert Wagner , who derived the theorem based on the Bogoliubov inequality in the context of the gold stone theorem , for ferromagnetism and antiferromagnetism , and for low-dimensional crystals. At almost the same time, Pierre Hohenberg made the same considerations about quantum systems and showed that there should be no superfluidity and superconductivity in one and two dimensions. For the quantum field theory a corresponding theorem by Sidney Coleman was proven (non-existence of gold clay bosons in two dimensions). The lack of a symmetry break is often used synonymously that there must be no order in the system, e.g. B. no ferromagnetism , antiferromagnetism or no crystals. It has to be precisely that there can be no (perfect) long-range order, quasi-long-range order is not excluded. Application areas include a. the XY model ( n-vector model with -dimensional spin variable) and the Heisenberg model ( -dimensional spin variable ), which Mermin and Wagner originally considered in two dimensions. Even if the Mermin-Wagner-Hohenberg theorem prevents a classic phase transition in the XY model in two dimensions, phase transitions of a different kind can generally occur ( Kosterlitz-Thouless transition ). In contrast, there is no continuous symmetry in the Ising model ( -dimensional spin variable) (the spin variable takes the two discrete values ​​± 1), so that the theorem is not applicable. ${\displaystyle n=2}$ ${\displaystyle n=3}$ ${\displaystyle n=1}$

prehistory

Felix Bloch had already pointed out in 1930 when diagonalizing the Slater determinant for fermions that magnetism should not exist in 2D. Rudolf Peierls provided some clear arguments, which are listed below , and Lew Landau also worked on breaking symmetry in 2D.

Energetic argument

The sketch shows a chain of length L of magnetic moments, which are rotatable in a plane, in the lowest excited mode. The angle between adjacent moments is ${\displaystyle \gamma }$

One reason for the lack of long-range order is that long-range fluctuations (often called massless Goldstone modes in field theories ) are excited with very little expenditure of energy , which destroy the perfect periodicity. If we consider a magnetic model (such as the XY model in one dimension) to be a chain of the length of magnetic moments in harmonic approximation, i.e. H. the restoring forces when a moment is deflected are proportional to the deflection angle , then it follows that the energies are the square of the deflection angles . The total energy of the chain is thus . If you consider the lowest excited mode in one dimension (see sketch), then the moments just twist around along the length of the chain . If there are total magnetic moments on the chain, then the relative angle between all moments is the same and reads . The total energy for the lowest mode is . In the thermodynamic Limes , i.e. H. , , The energy of this mode disappears with system size . For systems of any size, the long-wave modes do not cost any energy and are consequently thermally excited. In this way the long-range order on the chain is destroyed. In two dimensions or on a surface is the number of mag. Moments that are energy for the longest wave of fashion . In the thermodynamic limit the energy for this is constant and the modes are excited at a sufficiently high temperature. The energy is in three dimensions or in volume . For systems of any size, the energy for the longest wave mode diverges and will consequently be suppressed. The long-range order is not disturbed in 3D. ${\displaystyle L}$ ${\displaystyle \gamma _{i}}$ ${\displaystyle E_{i}\propto \gamma _{i}^{2}}$ ${\displaystyle E_{ges}\propto \sum _{i}\gamma _{i}^{2}}$ ${\displaystyle L}$ ${\displaystyle 2\pi }$ ${\displaystyle N}$ ${\displaystyle \gamma _{i}=2\pi /N}$ ${\displaystyle E_{ges}\propto N\cdot \gamma _{i}^{2}=N{\frac {4\pi ^{2}}{N^{2}}}\propto L{\frac {4\pi ^{2}}{L^{2}}}}$ ${\displaystyle L\to \infty }$ ${\displaystyle N\to \infty }$ ${\displaystyle L/N=const.}$ ${\displaystyle \propto 1/L}$ ${\displaystyle N\propto L^{2}}$ ${\displaystyle E_{ges}\propto N^{2}\cdot \gamma _{i}^{2}=\propto L^{2}{\frac {4\pi ^{2}}{L^{2}}}}$ ${\displaystyle V=L^{3}}$ ${\displaystyle E_{ges}\propto N^{3}\cdot \gamma _{i}^{2}=\propto L^{3}{\frac {4\pi ^{2}}{L^{2}}}}$

Entropic argument

In one dimension there is only one path between neighboring particles, in two dimensions there are two paths and in three dimensions there are six paths

An entropic argument against perfect long-range order in crystals in goes as follows (see sketch): If one considers a chain of atoms / particles with the mean particle distance , then thermal fluctuations z. B. between particles and ponds , ensure that the distance by a length fluctuates: . Same size, the amplitude fluctuations, the distance between particles and include: . If the two distance fluctuations are statistically independent, as it is the case for thermal fluctuations, then the distance between the two particles fluctuations add and particles also statistically independent (i.e., at twice the average distance..) . For two particles in N times the average distance at statistically independent addition of fluctuations follows . Although the mean distance is well defined, the deviations from a perfectly periodic chain grow with the square root of the system size. In three dimensions you have to walk in at least three different spatial directions in order to sweep the entire volume; in a cubic crystal this would be the sum along the diagonal of a cube, from particle to particle . As can be seen in the sketch, there are a total of six different options. The fluctuations in the length of the six paths cannot now be statistically independent, since they prevail between the same particles and . They can only add up coherently and remain on the space diagonal of the cube of the order of magnitude . In two dimensions, Herbert Wagner and David Mermin have shown that the distance fluctuations logarithmically with the system size grow: . ${\displaystyle D<3}$ ${\displaystyle <a>}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle \xi _{0,1}}$ ${\displaystyle a=<a>\pm \xi _{0,1}}$ ${\displaystyle -1}$ ${\displaystyle 0}$ ${\displaystyle |\xi _{-1,0}|=|\xi _{0,1}|}$ ${\displaystyle -1}$ ${\displaystyle +1}$ ${\displaystyle \xi _{-1,1}={\sqrt {2}}\cdot \xi _{0,1}}$ ${\displaystyle \xi _{0,N}={\sqrt {N}}\cdot \xi _{0,1}}$ ${\displaystyle <a>}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle \xi }$ ${\displaystyle L}$ ${\displaystyle \xi \propto ln(L)}$

Example in a colloidal monolayer

Two-dimensional crystal with thermal fluctuations of the particles. The red lines symbolize crystal axes and the green arrows the deflection from the respective equilibrium position.

The photo shows a (quasi) two-dimensional crystal made of colloids (small particles in aqueous solution that are sedimented on an interface and can only make Brownian diffusion in the plane ). The hexagonal crystalline order is easy to see on medium scales because the deviations from the perfect crystal only grow logarithmically, i.e. very slowly. The fluctuations can also be seen clearly, as a deviation of the positions from the grid lines drawn in red here. These fluctuations are essentially the lattice vibrations of the crystal (acoustic phonons ). A direct experimental proof of the Mermin-Wagner-Hohenberg fluctuations would be if the deviations (green arrows in the enlargement) grow logarithmically with the distance from a locally adapted coordinate system (blue). This so-called logarithmic divergence is accompanied by an algebraic (slow) decay of (spatial) correlation functions. The order is then called quasi-long-range (see also hexatic phase ).

Interestingly, clear signs of Mermin-Wagner fluctuations have been found in amorphous, disordered systems. In this work, it was not the deviation from lattice positions but the size of the mean square displacement of the particles as a function of time that was investigated. The question was, as it were, shifted from the spatial area to the time domain. D. Cassi as well as F. Merkl and H. Wagner provided the theoretical background. In this work, a connection between the probability of return in the case of chance paths and the spontaneous breaking of symmetry in various dimensions has been shown. The non-vanishing return probability of a random path or random walk in one and two dimensions is dual to the lack of long-range order in one and two dimensions, while the vanishing return probability is dual to the existence of long-range order in 3D.

Limitation

With real magnets there is often no continuous symmetry, since the system becomes anisotropic even with an LS coupling . With atomic systems like graphs it can be shown that monolayers of cosmological size are necessary in order to be able to measure sufficiently large amplitudes of the long-wave fluctuations. Bertrand Halperin has summarized another discussion of the Mermin-Wagner-Hohenberg theorem and its limitations .

annotation

The contradiction between the Mermin-Wagner theorem (which forbids long-range order in crystals) and the first computer simulations (Alder & Wainwright) that indicated crystallization in 2D motivated Michael Kosterlitz and David Thouless to develop their work on topological phase transitions in 2D ( KTHNY Theory ), for which they received the Nobel Prize in Physics in 2016.

Footnotes

1. ↑ In the original work by Mermin and Wagner, interactions of finite range correspond to realistic short-range interactions. The condition is that the interaction between the magnetic moments or the particles can be integrated, i. H. fall faster than 1 / r. For the 1 / r potential in 2D, the next but one and the next but one neighboring correlations are sufficiently strong that e.g. B. the entropic argument does not work.
2. ↑ The article was submitted only two days later than the work on magnetism, but did not appear until half a year later

Individual evidence

1. ^ H. Wagner: Long-Wavelength Excitations and the Goldstone Theorem in Many-Particle Systems with "Broken Symmetries" . In: Journal of Physics . 195, April 26, 1966, pp. 273-299. doi : 10.1007 / bf01325630 .
2. ^ ND Mermin, H. Wagner: Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models . In: Physical Review Letters . 17, No. 22, November 28, 1966, p. 1133. doi : 10.1103 / PhysRevLett.17.1133 .
3. ^ ND Mermin: Crystalline Order in Two Dimensions . In: Physical Review . 176, June 6, 1968, pp. 250-254. doi : 10.1103 / PhysRev.176.250 .
4. ^ PC Hohenberg: Existence of Long-Range Order in One and Two Dimensions . In: Physical Review . 158, October 24, 1966, pp. 383-386. doi : 10.1103 / PhysRev.158.383 .
5. ^ Coleman, There are no Goldstone bosons in two dimensions, Commun. Math. Phys., Volume 31, 1973, p. 259
6. ^ F Bloch: On the theory of ferromagnetism . In: Journal of Physics . 61, Feb. 1, 1930, pp. 206-219. doi : 10.1007 / bf01339661 .
7. ^ RE Peierls: Remarks on transition temperatures . In: Helv. Phys. Acta . August 7, p. 81.
8. ↑ LD Landau: Theory of phase transformations II . In: Phys. Z. Soviet. . August 11, p. 545.
9. ↑ H. Shiba, Y. Yamada, T. Kawasaki, K. Kim: Unveiling Dimensionality Dependence of Glassy Dynamics: 2D Infinite Fluctuation Eclipses Inherent Structural Relaxation . In: Physical Review Letters . 117, No. 24, 2016, p. 245701. doi : 10.1103 / PhysRevLett.117.245701 .
10. ^ S. Vivek, CP Kelleher, PM Chaikin, ER Weeks: Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions . In: Proc. Nat. Acad. Sci . 114, No. 8, 2017, pp. 1850–1855. doi : 10.1073 / pnas.1607226113 .
11. ^ B. Illing, S. Fritschi, H. Kaiser, CL Klix, G. Maret, P. Keim: Mermin – Wagner fluctuations in 2D amorphous solids . In: Proc. Nat. Acad. Sci . 114, No. 8, 2017, pp. 1856–1861. doi : 10.1073 / pnas.1612964114 .
12. ↑ D. Cassi: Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals, and other discrete structures . In: Journal of Statistical Physics . 68, No. 24, 1992, pp. 3631-3634. doi : 10.1103 / PhysRevLett.68.3631 .
13. ↑ F. Merkl, H. Wagner: Recurrent random walks and the absence of continuous symmetry breaking on graphs . In: Journal of Statistical Physics . 75, No. 1, 1994, pp. 153-165. doi : 10.1007 / bf02186284 .
14. ^ RC Thompson-Flagg, MJB Moura, M. Marder: Rippling of graphene . In: Europhys. Lett . 85, No. 4, 2009, p. 46002. doi : 10.1209 / 0295-5075 / 85/46002 .
15. ^ BI Halperin: On the Hohenberg-Mermin-Wagner Theorem and Its Limitations . In: Journal of Statistical Physics . 175, No. 3-4, 2019, pp. 521-529. doi : 10.1007 / s10955-018-2202-y .