Abstract
The Hohenberg–Mermin–Wagner (HMW) theorem states that infrared (IR) fluctuations prevent long-range order which breaks continuous symmetries in two dimensions (2D), at finite temperatures. We note that the theorem becomes physically effective for superconductivity (SC) only for astronomical sample sizes, so it does not prevent 2D SC in practice. We systematically explore the sensitivity of the magnetic and SC versions of the theorem to finite-size and disorder effects. For magnetism, finite-size effects, disorder, and perpendicular coupling can all restore the order parameter at a non-negligible value of T c equally well, making the physical reason for finite T c sample-dependent. For SC, an alternative version of the HMW theorem is presented, in which the temperature cutoff is set by Cooper pairing, in place of the Fermi energy in the standard version. It still allows 2D SC at 2–3 times the room temperature when the interaction scale is large and Cooper pairs are small, the case with high-T c SC in the cuprates. Thus IR fluctuations do not prevent 2D SC at room temperatures in samples of any reasonable size, by any known version of the HMW argument. A possible approach to derive mechanism-dependent upper bounds for SC T c is pointed out.
Highlights
The Hohenberg-Mermin-Wagner (HMW) theorem [1, 2] is probably the best-known mathematically exact result in the theory of phase transitions. It forbids ordered phases which break continuous symmetries in less than three dimensions, by showing that they are destabilized by infrared (IR) fluctuations
Despite its wide dissemination in textbooks and the research literature [4,5,6,7,8], we have been able to find only one comment, in unpublished lecture notes by Leggett [9], on the actual numbers appearing when the theoretical bound is evaluated: a 2D sample would have to be “the size of the Moon’s orbit” for its superconductivity (SC) to be suppressed below the temperatures at which it is observed in three dimensions
Because the HMW suppression is only logarithmic in the size of the sample in 2D, it cannot preclude 2D SC in reasonably-sized samples even at twice the room temperature
Summary
The Hohenberg-Mermin-Wagner (HMW) theorem [1, 2] is probably the best-known mathematically exact result in the theory of phase transitions. It forbids ordered phases which break continuous symmetries in less than three dimensions, by showing that they are destabilized by infrared (IR) fluctuations. In the formalism, the energy cost of the IR fluctuations suppressing Tc is not set by the SC mechanism, as one might imagine, but by the much larger Fermi energy. Because the HMW suppression is only logarithmic in the size of the sample in 2D, it cannot preclude 2D SC in reasonably-sized samples even at twice the room temperature
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