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There are a number of physical systems with phases described by topologically protected invariants (fractional quantum Hall, topological insulators) but what are the simplest mathematical models that exhibit topological phases? Is the toric code as simple as we can go?

Edit: Just to be clear, I'm talking about phases meaning states of matter, and not just the geometric phase that a wavefunction will pick up under parallel transport in a nontrivial configuration space. I'm looking for simple models where one can make a phase diagram, and as a function of the available couplings there is a change in some topological property of the system.

(For example, in the XY magnet, neglecting bound vortex-antivortex pair formation, there is an instability at finite temperature to creation a single vortex, which is topologically distinct from the vortex-free state.)

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  • $\begingroup$ I guess the Kosterlitz-Thouless transition would be one simple example where the quasi-particles are vortices - i.e. topological defects. But on further research it seems that is the physics of the XY phase which you've already mentioned. $\endgroup$
    – user346
    Jan 19, 2011 at 17:05
  • $\begingroup$ A closely related answer $\endgroup$
    – xiaohuamao
    Apr 21, 2015 at 2:21

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I think you need to define what you mean by a "topological state of matter", since the term is used in several inequivalent ways. For example the toric code that you mention, is a very different kind of topological phase than topological insulators. Actually one might argue that all topological insulators (maybe except the Integer Quantum Hall, class A in the general classification) are only topological effects rather than true topological phases, since they are protected by discreet symmetries (time reversal, particle-hole or chiral). If these symmetries are explicitly or spontaneously broken then the system might turn into a trivial insulator.

But one of the simplest lattice models (much simpler that the toric code, but also not as rich) I know of is the following two band model (written in k-space)

$H(\mathbf k) = \mathbf d(\mathbf k)\cdot\mathbf{\sigma},$

with $\mathbf d(\mathbf k) = (\sin k_x, \sin k_y, m + \cos k_x + \cos k_y)$ and $\mathbf{\sigma} = (\sigma_x,\sigma_y,\sigma_z)$ are the Pauli matrices. This model belongs to the same topological class as the IQHE, meaning that it has no time-reversal, particle-hole or chiral symmetry. The spectrum is given by $E(\mathbf k) = \sqrt{\mathbf d(\mathbf k)\cdot\mathbf d(\mathbf k)}$ and the model is classified by the first Chern number

$C_1 = \frac 1{4\pi}\int_{T^2}d\mathbf k\;\hat{\mathbf d}\cdot\frac{\partial \hat{\mathbf d}}{\partial k_x}\times\frac{\partial \hat{\mathbf d}}{\partial k_y},$

where $T^2$ is the torus (which is the topology of the Brillouin zone) and $\hat{\mathbf d} = \frac{\mathbf d}{|\mathbf d|}$. By changing the parameter $m$ the system can go through a quantum critical point, but this can only happen if the bulk gap closes. So solving the equation $E(\mathbf k) = 0$ for $m$, one can see where there is phase transitions. One can then calculate the Chern number in the intervals between these critical points and find

$C_1 = 1$ for $0 < m < 2$, $C_1 = -1$ for $-2 < m < 0$ and $C_1 = 0$ otherwise.

Thus there are three different phases, one trivial and two non-trivial. In the non-trivial phases the system has quantized Hall response and protected chiral edge states (which can easily be seen by putting edges along one axes and diagonalizing the Hamiltonian on a computer).

If one takes the continuum limit, the model reduces to a 2+1 dimensional massive Dirac Hamiltonian and I think the same conclusions can be reached in this continuum limit but the topology enters as a parity anomaly.

More information can be found here: http://arxiv.org/abs/0802.3537 (the model is introduced in section IIB).

Hope you find this useful.

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  • $\begingroup$ Neat. I'm not too worried about the continuum limit, but it's nice to see that on the lattice it's essentially just an anomalous Hall hamiltonion. $\endgroup$
    – wsc
    Jan 19, 2011 at 23:02
  • $\begingroup$ +1. Welcome to physics.SE @4tnemele. Looking forward to more awesome contributions such as this answer! $\endgroup$
    – user346
    Jan 20, 2011 at 6:10
  • $\begingroup$ That paper is darn hard to read. :P In any case, they only cite the more detailed study presented in their paper Electric-field modulation of the number of helical edge states in thin-film semiconductors. It appears to me that they don't solve this model analytically, though? $\endgroup$
    – Greg Graviton
    Feb 6, 2011 at 8:45
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This is a very good question. Let me give a little back ground first.

For a long time, physicists thought all different phases of matter are described by symmetry breaking. As a result, all continuous phase transitions between those symmetry breaking phases involve a change of symmetry.

Now we know that there are new kind of phases of matter beyond symmetry breaking -- topological order. So we should have new continuous phase transitions between those topologically ordered phases. Those new continuous phase transitions do not change any symmetry (ie the two phases connected by the transition have the same symmetry). To have an intuition about those new kind of the phase transition, one naturally ask, what are simple models that exhibit topological phase transitions?

Heidar has given a very good and simple model. Here I will list some research papers on this topic (please feel free to add if you know more papers)

  • X.-G. Wen and Y.-S. Wu, Phys. Rev. Lett. 70, 1501 (1993).
  • W. Chen, M. P. A. Fisher, and Y.-S. Wu, Phys. Rev. B 48, 13749 (1993).

The above two papers describe continuous phase transitions between FQH-FQH or FQH-Mott-insultor induced by periodic potentials.

  • N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).

The paper describes the continuous transition between strong and weak p-wave/d-wave BCS superconductors. Heidar's example is similar to this work.

  • Xiao-Gang Wen, Phys. Rev. Lett. 84, 3950 (2000). cond-mat/9908394.

The paper describes continuous transitions between double layer FQH states induced by interlayer tunneling and/or coupling

  • Maissam Barkeshli, Xiao-Gang Wen, Phys.Rev.Lett.105 216804 (2010).

The paper describes a continuous transition between between an Abelain FQH state and a non-Abelian FQH state induced by an anyon condensation.

I like to know more examples of topological phase transitions.

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  • $\begingroup$ In the context of topological phase transitions through condensation there is the work by Bais and Slingerland: arxiv.org/abs/0808.0627 . There is also some follow up work, such as arxiv.org/abs/0812.4596 and arxiv.org/abs/1108.0683 . $\endgroup$
    – Olaf
    May 31, 2012 at 18:27
  • $\begingroup$ @Olaf: Yes indeed, Bais and Slingerland give a general description of topological phase transition induced by condensation of quasi-particles in the topological phases, which is a very good work. Kitaev's honeycomb lattice model also has continuous topological phase transition which is similar what is described by Read and Green. $\endgroup$
    – Xiao-Gang Wen
    Jun 15, 2012 at 2:28
  • $\begingroup$ An important open question is that if the change of topological order described by condensible algebra (such as those described by Bais and Slingerland) correspond to continuous transition or not. The examples in my answer are all continuous transitions. The first order transitions are "trivial" in the sense that they can happen between any two phases. $\endgroup$
    – Xiao-Gang Wen
    Jun 8, 2017 at 12:07
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From what I understand, "quantum phases" or "Berry-Pancharatnam phases" are examples of "topological phases". See Y Ben-Aryeh 2004 J. Opt. B: Quantum Semiclass. Opt. 6 R1, "Berry and Pancharatnam Topological Phases of Atomic and Optical Systems", http://arxiv.org/abs/quant-ph/0402003 .

Under this definition of the terms, the simplest example of topological phase is the transition of states under unitary transformation in spin-1/2 for a single particle. This is the Berry or Pancharatnam phase.

Consider a spin-1/2 particle that begins in a state of spin-up (+z). Its spin is then measured in the x and y directions and then in the z direction. If the results of these measurements are that its spin is +x, +y, and then +z, its return to the +z case will be with a quantum phase of $\pi/4$ as I now show:

Let $\sigma_x$, $\sigma_y$, and $\sigma_z$ be the Pauli spin matrices. Then $(1+\sigma_x)/2$, $(1+\sigma_y)/2$, and $(1+\sigma_z)/2$ are the projection operators for spin in the +x, +y, and +z directions. The operator for a particle going from spin +z to +x to +y and back to +z in a series of measurements is the product of the projection operators which can be simplified as follows:

$(1+\sigma_z)/2\;(1+\sigma_y)/2\;(1+\sigma_x)/2\;(1+\sigma_z)/2 = (e^{i\pi/4}/\sqrt{8})\;(1+\sigma_z)/2$.

So if we let $|+>$ be spin up, then we have that the amplitude for a particle going through this sequence of states is: $<+|\; (1+\sigma_z)/2\;(1+\sigma_y)/2\;(1+\sigma_x)/2\;(1+\sigma_z)/2\; |+> = e^{i\pi/4}/\sqrt{8}$.

The $\pi/4$ is the topological phase. The $1/\sqrt{8}$ is the reduction in amplitude due to going through the measurements. I've typed this in from memory, it's not unlikely I got the sign wrong. :(

By the way, for spin-1/2 and spin-1, the topological phase is given by half the area (in steradians) carved out in the Bloch sphere by the sequence of states. For the above example, the area carved out is one octant. This has an area of $(4\pi) /8 = \pi/2$, so the topological phase is $\pi/4$.

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    $\begingroup$ While this is useful information, and geometric phases are an important tool to get at this question, I'm specifically asking about phases not in the sense of "that number of unit modulus that you multiply states by," but phase in the sense of collective state of matter. Particularly distinct phases that don't seem to admit any local order parameter description, but have robust "topologically protected" observable features. $\endgroup$
    – wsc
    Jan 19, 2011 at 2:12
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The so-called, lyotropic liquid crystals exhibit several topological transitions. The topology of a real space changes during such transitions. The most famous of them is the transition into the so-called, sponge phase. But there are also more simply ones. For example, lipid vesicles are known to transform into a string of beads (which still are topologically equivalent to a sphere) but then they split from one another (which is already a topological change). Living cells often split out vesicles formed by a part of the membrane. The topological transition is involved in this process.

It would be much better, if you specify what phenomena you have in mind, since various things may be thought of under this general name.

For example, the transitions of the order 2.5 have been considered once to explain Mott transitions in some materials. There the Fermi surface undergoes the topological phase transformation

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The topological phase transitions you ask about I presume is the breakdown of a Landau electron fluid where quantum fluctuations are comparable to thermal fluctuations. There is quite an extensive set of publications on this subject with heavy metals and quantum critical points. Cubrovic has found parallels with AdS~CFT in such systems.

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  • $\begingroup$ an actual reference would be helpful in the absence of any technical details. $\endgroup$
    – user346
    Jan 20, 2011 at 6:07