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Topological order are sometimes defined in opposition with the order parameter originating from a symmetry breaking. The latter one being possibly described by a Landau theory, with an order parameter.

Then, one of the distinctions would be to say that topological order can not be described by a local order parameter, as e.g. in this answer by Prof. Wen. I thus suppose that a Landau theory describes a local order parameter.

I have deep difficulties to understand what does local mean ? Does it mean that the order parameter can be inhomogeneous (explicit position dependency), as $\Delta\left(x\right)$ ? Does it mean that one can measure $\Delta$ at any point of the system (say using a STM tip for instance) ? (The two previous proposals are intimately related to each other.) Something else ?

A subsidiary question (going along the previous one of course) -> What is the opposite of local: is it non-local or global ? (Something else ?) What would non-local mean ?

Any suggestion to improve the question is warm welcome.

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I'm no expert on topological order. But I found this article very enlightening: dx.doi.org/10.1063/PT.3.1641 This article, at least for me, represents the big moment of epiphany when topological order first made sense to me; it would be a bold (probably arrogant) claim to say it made me "understand" it. I actually find it very convenient to think of topological order from a disorder perspective: a system with topological order can look disordered from a Landau perspective, i.e. no measurable local order parameter but a finite topological order parameter. –  NanoPhys Jul 15 '13 at 13:04
    
@NanoPhys Thanks so much for this nice pedagogical review by Read. It definitely helps, even if I need more time to (start to) appreciate some details about the topological order. Particularly the point [Quote: There remains the possibility of degenerate ground-state that cannot be mapped onto one another by application of any local operator and that cannot be distinguished by the expectation value of any local operator. ] I think I'll need several lives to understand –  FraSchelle Jul 15 '13 at 19:54
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1 Answer

up vote 8 down vote accepted

In theories with spontaneous symmetry breaking, the phase transition can usually be characterized by a local order parameter $\Delta(x)$, which is not invariant under the relevant symmetry group $G$ of the Hamiltonian. The expectation value of this field has to be zero outside the ordered phase $\langle\Delta(x)\rangle = 0$, but non-zero in the phase $\langle\Delta(x)\rangle \neq 0$. This shows that there has been a spontaneous breaking of $G$ to a subgroup $H\subset G$ (where $H$ is the subgroup that leaves $\Delta(x)$ invariant).

What local means in this context, is usually that $\Delta(x)$ at point $x$, can be constructed by looking at a small neighborhood around the point $x$. Here $\Delta(x)$ can be dependent on $x$ and need not be homogeneous. This happens for example when you have topological defects, such as vortices or hedgehogs. One powerful feature of these Landau-type phases, is that there will generically be gapless excitations in the system corresponding to fluctuations of $\Delta(x)$ around its expectation value $\langle\Delta(x)\rangle$ in the direction where the symmetry is not broken (unless there is a Higgs mechanism). These are called Goldstone modes and their dynamics are described by a non-linear $\sigma$-model with target manifold $G/H$.

An example is the order parameter for s-wave superconductors $\langle\Delta(x)\rangle = \langle c_{\uparrow}(x)c_{\downarrow}(x)\rangle$, which breaks a $U(1)$ symmetry down to $\mathbb Z_2$. But there are no Goldstone modes due to the Higgs mechanism, the massive amplitude fluctuations are however there (the "Higgs boson"). [Edit: see EDIT2 for correction.]

A non-local order parameter does not depend on $x$ (which is local), but on something non-local. For example, a non-local (gauge-invariant) object in gauge theories are the Wilson loops $W_R[\mathcal C] = \text{Tr}_R{\left(\mathcal Pe^{i\oint_{\mathcal C}A_\mu\text dx^\mu}\right)},$ where $\mathcal C$ is some closed curve. The Wilson loop thus depends on the whole loop $\mathcal C$ (and a representation $R$ of the gauge group) and cannot be constructed locally. It can also contain global information if $\mathcal C$ is a non-trivial cycle (non-contractible).

It is true that topological order cannot be described by a local order parameter, as in superconductors or magnets, but conversely a system described by a non-local order parameter does not mean it has topological order (I think). The above mentioned Wilson loops (and similar order parameters, such a the Polyakov and 't Hooft loop), is actually a order parameter in gauge theories which probe the spontaneous breaking of a certain center-symmetry. This characterizes the deconfinement/confinement transition of quarks in QCD: in the deconfined phase $W_R[\mathcal C]$ satisfies a perimeter law and quarks interact with a massive/Yukawa type potential $V(R)\sim \frac{e^{-mR}}R$, while in the confined phase it satisfy an area law and the potential is linear $V(R)\sim \sigma R$ ($\sigma$ is some string tension). There might be other examples of spontaneous symmetry breaking phases with non-local order parameter. [Edit: see EDIT2.]

Let me just make a few comments about topological order. In theories with with spontanous symmetry breaking, long-range correlations are very important. In topological order the systems are gapped by definition, and there is only short-range correlation. The main point is that in topological order, entanglement plays the important role not correlations. One can define the notion of long-range entanglement (LRE) and short-range entanglement (SRE). Given a state $\psi$ in the Hilbert space, loosely speaking $\psi$ is SRE if it can de deformed to a product state (zero entanglement entropy) by LOCALLY removing entanglement, if this is not possible then $\psi$ is LRE. A system which has a ground state with LRE is called topological order, otherwise its called the trivial phase. These phases have many characteristic features which are generally non-local/global in nature such as, anyonic excitations/non-zero entanglement entropy, low-energy TQFT's, and are characterized by so-called modular $S$ and $T$ matrices (projective representations of the modular group $SL(2,\mathbb Z)$).

Note that, unlike popular belief, topological insulators and superconductors are SRE and are NOT examples of topological order!

If one requires that the system must preserve some symmetry $G$, then not all SRE states can be deformed to the product state while respecting $G$. This means that SRE states can have non-trivial topological phases which are protected by the symmetry $G$. These are called symmetry protected topological states (SPT). Topological insulators/superconductors are a very small subset of SPT states, corresponding to restricting to free fermionic systems. Unlike systems with LRE and thus intrinsic topological order, SPT states are only protected as long as the symmetry is not broken. These systems typically have interesting boundary physics, such as gapless modes or gapped topological order on the boundary. Characterizing them usually requires global quantities too and cannot be done by local order parameters.


EDIT: This is a response to the question in the comment section.

I am not sure whether there are any reference which discuss this point explicitly. But the point is that you can continuously deform/perturb the Hamiltonian of a topological insulator (while preserving the gap) into the trivial insulator by breaking the symmetry along the way (they are only protected if the symmetry is respected). This is equivalent to locally deforming the ground state into the product state, which is the definition of short range entanglement. You can find the statement in many papers and talks. See for example the first few slides here. Or even better, see this (slide with title "Compare topological order and topological insulator" + the final slide).

Let me make another comment regarding the distinction between intrinsic topological order and topological superconductors, which at first seems puzzling and contrary to what I just said. As was shown by Levin-Wen and Kitaev-Preskill, the entanglement entropy of ground state for a gapped system in 2+1D has the form $S = \alpha A - \gamma + \mathcal O(\tfrac 1A)$, where $A$ is the boundary area (this is called the area law, not the same area law I mentioned in the case of confinement), $\alpha$ is a non-universal number and $\gamma$ is universal and called the topological entanglement entropy (TEE). What was shown in the above papers is that the TEE is equal to $\gamma = \log\mathcal D$, where $\mathcal D\geq 1$ is the total quantum dimension and is only strictly $\mathcal D>1$ ($\gamma\neq 0$) if the system supports anyonic excitations.

Modulo some subtleties, LRE states always have $\gamma\neq 0$, which in turn means that they have anyonic excitations. Conversely for SRE states $\gamma = 0$ and there are no anyons present.

This seems to be at odds with the existence of 'Majorana fermions' (non-abelian anyons) in topological superconductors. The difference is that, in the case of topological order you have intrinsic finite-energy excitations which are anyonic and the anyons correspond to linear representations of the Braid group. While in the case of topological superconductors, you only have non-abelian anyons if there is an extrinsic defect (vortex, domain wall etc.) which the zero-modes can bind to, and they correspond to projective representation of the Braid group. The latter type anyons from extrinsic defects can also exist in topological order, but intrinsic finite-energy ones only exist in topological order. For more details, see the recent set of papers from Barkeshli, Jian and Qi.


EDIT2: Please see my comments below for some corrections and subtleties. Such as, it is in a sense not correct that superconductors are described by a local order parameter. It only appears local in a particular gauge. Superconductors are actually examples of topological order, which is rather surprising.

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Thanks a lot for this interesting answer, especially the discussion about long and short-range entanglement. Would you please give us some reference(s) about your statement that topological insulator and superconductor are short-ranged entangled ? Would be nice. Thanks again. –  FraSchelle Jul 15 '13 at 20:12
    
@Oaoa No problem, I'm glad if somebody finds it interesting and useful. See the edit for my answer to your question and some comment about a related subtlety. –  Heidar Jul 15 '13 at 21:43
    
Thanks again. The last slide of the Wen's presentation physics.utoronto.ca/~colloq/2012_2013/120913_XiaoGangWen.pdf says that superconductors are long-ranged-entangled... I'll try to verify, reading the papers cited there. Note that superconductors can be everywhere in the Wen's classification, depending on the gap symmetry. Thanks again for your beautiful answer. –  FraSchelle Jul 16 '13 at 3:58
    
@Oaoa: I'm happy you point out that Wen puts superconductors under LRE since I had forgotten those details completely, since this invalidates a few of my comments in the answer due to some deep subtleties (which I don't understand as clearly as I would like). However, note that this has nothing to do with the gap symmetry and the essential points of my comments about topological superconductors and Majorana zero modes still stand. Let me try to elaborate. –  Heidar Jul 16 '13 at 21:24
    
@Oaoa: Superconductivity is much more subtle than usually presented. The slightly wrong comment of mine above is that I claimed superconductors can be described by a local order parameter $\langle \Delta(x)\rangle=\langle c_{\uparrow}(x)c_{\downarrow}(x)\rangle$, this is however not correct in a sense. According to Elitzurs theorem [prd.aps.org/abstract/PRD/v12/i12/p3978_1 ], it is impossible to find a gauge invariant and local order parameter in the presence of the electromagnetic gauge field. (continued) –  Heidar Jul 16 '13 at 21:34
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