The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results

After a brief discussion of the Bogoliubov inequality and possible generalizations thereof, we present a complete review of results concerning the Mermin-Wagner theorem for various many-body systems, geometries and order parameters. We extend the method to cover magnetic phase transitions in the periodic Anderson Model as well as certain superconducting pairing mechanisms for Hubbard films. The relevance of the Mermin-Wagner theorem to approximations in many-body physics is discussed on a conceptual level.Comment: 33 pages; accepted for publication as a Topical Review in Journal of Physics: Condensed Matte

Holography and the Coleman-Mermin-Wagner theorem

In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of global symmetries is precluded by large thermal fluctuations of the order parameter. The holographic correspondence implies that analogous effects must also occur in 3+1 dimensional theories with gauged symmetries in certain curved spacetimes with horizon. By performing a one loop computation in the background of a holographic superconductor, we show that bulk quantum fluctuations wash out the classical order parameter at sufficiently large distance scales. The low temperature phase is seen to exhibit algebraic long range order. Beyond the specific example we study, holography suggests that IR singular quantum fluctuations of the fields and geometry will play an interesting role for many 3+1 dimensional asymptotically AdS spacetimes with planar horizon.Comment: 1+24 pages. 1 figur

Scalar Field Theory at Finite Temperature in D=2+1

We discuss the $\phi^6$ theory defined in $D=2+1$-dimensional space-time and assume that the system is in equilibrium with a thermal bath at temperature $\beta^{-1}$. We use the $1/N$ expansion and the method of the composite operator (CJT) for summing a large set of Feynman graphs.We demonstrate explicitly the Coleman-Mermin-Wagner theorem at finite temperature.Comment: 12 pages, 1 figure. To be published in Journal Mathematical Physics, typos adde

Dynkin isomorphism and mermin-wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin--Wagner theorem applies even though the symmetry groups of $\mathbb{H}^n$ and $\mathbb{H}^{2|2}$ are non-amenable

Holographic Symmetry-Breaking Phases in AdS3_3/CFT2_2

In this note we study the symmetry-breaking phases of 3D gravity coupled to matter. In particular, we consider black holes with scalar hair as a model of symmetry-breaking phases of a strongly coupled 1+1 dimensional CFT. In the case of a discrete symmetry, we show that these theories admit metastable phases of broken symmetry and study the thermodynamics of these phases. We also demonstrate that the 3D Einstein-Maxwell theory shows continuous symmetry breaking at low temperature. The apparent contradiction with the Coleman-Mermin-Wagner theorem is discussed.Comment: 15 pages, 7 figur