What is the Haldane gap?

Question

The Haldane Phase is a topological phase of matter in which a Haldane gap opens due to the breaking of either time-reversal symmetry or inversion symmetry. Physically speaking, what is the "Haldane gap" a gap of and how can one detect such a gap?

Answer 1

Short answer: Haldane Gap is just a name for a gap in the band structure occurring at the K and K' points for the Haldane model. It tells us which sector of the phase diagram we are in and what properties of the system we can expect (edge modes, Chern number, etc.).

Alternatively, the Haldane gap can also refer to the gap in integer spin 1D Heisenberg models - where the spin wave excitations are gapped (as opposed to half-integer spin 1D heisenberg chains).

Long answer: The Haldane model is often introduced & studied in the context of Topological Phases, and that is the picture in which I will give the answer.

The starting point is a simple tight-binding model of Graphene (spinless for simplicity) with the hopping Hamiltonian on a hexagonal lattice: $$ \hat{H} = - t \sum_{\langle i,j \rangle} (c_i^\dagger c_j + h.c.) = \sum_{\textbf{k}}\begin{pmatrix}c^\dagger_{\textbf{k},A} & c^\dagger_{\textbf{k},B} \end{pmatrix} H_\textbf{k} \begin{pmatrix}c_{\textbf{k},A} \\ c_{\textbf{k},B} \end{pmatrix}$$

Diagonalizing it in momentum space leads to two energy bands forming Dirac Cones at the high symmetry K and K' points. It means at those points in the Brillouin zone, the bands exactly touch each other i.e. the energy expanded near those points can be written as: $$ E_\pm(\textbf{q}) = \pm \hbar v_F \vert{\textbf{q}}\vert,$$ with $\hbar v_F = 3ta/2$, and $\textbf{q}=\textbf{k}-\textbf{K}^{(} {'}^{)}$ is the relative momentum, and thus the gap in the band structure is equal to $\Delta = \min_\textbf{k}(E_+ - E_-)= 0$.

In this model, the zero points of the band structure are protected by a combination of lattice-inversion symmetry $\mathcal{I}$ and time-reversal symmetry, so to open the gap, we need to break one of the two symmetries (or both).

1. To break the lattice-inversion symmetry $\mathcal{I}$ we need to make A&B sublattices inequivalent, so we add a "mass" term which has an equal sign at both the K and K' points, namely: $$H_\text{k} = H^0_\text{k} + m_I \sigma^z $$ This changes the band structure, and the dispersion relations near K, K' points now take the following form: $$ E_\pm(\textbf{q}) = \pm \sqrt{ \vert\hbar v_F{\textbf{q}}\vert^2+m_I^2},$$ with the Haldane gap $\Delta = 2 \vert m_I\vert$

2. To break the time-reversal symmetry $\mathcal{T}$, we add a magnetic flux that is zero on average but has all of the spatial symmetries of the lattice (imaginary next-nearest-neighbor hopping term), which corresponds to a change: $$H_\text{k} = H^0_\text{k} + m_\tau \tau^z \sigma^z,$$ with a valley degree of freedom $\tau = \pm1$ at K, K' respectively. This again gives us $$ E_\pm(\textbf{q}) = \pm \sqrt{ \vert\hbar v_F{\textbf{q}}\vert^2+m_\tau^2},$$ with the Haldane gap $\Delta = 2 \vert m_\tau\vert$

Physically, one realizes the breaking through the Haldane model: $$ \hat{H} = - t \sum_{\langle i, j \rangle}c^\dagger_i + t_2 \sum_{\langle\langle i,j\rangle\rangle} e^{-\nu_{ij}\phi}c^\dagger_ic_j+M\sum_i \varepsilon_i c_i^\dagger c_i +h.c.,$$ (here $m_\tau = 3\sqrt{3}t_2$, $m_I=M$, and $\varepsilon=+1/-1$ for A/B sublattice) and we can construct the phase diagram as a function of $m_I/m_\tau$ versus $\phi$

With this background, we can now better answer your questions:

Physically speaking, what is the "Haldane gap" a gap of?

It is the separation of the two bands in the energy spectrum. It has a number of physical consequences: 1. the only way to go between different phases on the phase diagram is by closing and reopening the gap, i.e., a continuous change of parameters (deformation) will not change 'much' the properties of your phase unless you close the gap. 2. It ties back to the concept of topological invariants - by analyzing the relative gap sizes at K, K' point, we can determine the Berry Curvature/Chern numbers

How can one detect such a gap?

I am not an expert on experimental methods, but I imagine that any method that allows you to map the band structure will give you information about a gap, e.g.:

1. Spectroscopic Measurements - band gap can be determined from the energy at which the absorption begins (the onset of absorption).
2. Electrical Conductivity measurements - through bulk edge correspondence, we know that the topological invariant in the insulating bulk will correspond to a conducting (metallic) mode at the edge of a sample. Conductivity measurements will then give you insight as to whether transport is occurring or not - in which phase in the diagram you are.
3. Microscope tools - some recent ideas, such as a quantum Twisting microscope, can also be able to map the band structure in momentum space.

This seems similar to the transition that happens in Mott insulators, correct?

There are some subtle differences between the two cases.

1. In the Graphene/Haldane model, the transition occurs between a conducting and an insulating state due to the opening up of a gap (breaking of the symmetries present in graphene). Here, the insulating behavior is understood as having no available states within the charge gap - requiring a large amount of energy to move an electron to the conduction band.

Note also that in the Haldane model, only an infinite-size system is entirely insulating - on the surface, one can expect metallic (conducting) edge modes provided that the Chern number is nonzero.

2. When it comes to Mott transitions, e.g., in Hubbard-type models, the transition happens when the Coulomb repulsion dominates over the kinetic energy, leading to the localization of electrons. Now, in contrast to a standard band insulator, there are plenty of low-energy degrees of freedom (in the case of a Hubbard model, you can think of it that electrons cannot move on the lattice but can still orient their spin degree of freedom $\leadsto$ AFM Heisenberg model)

Does this mean that Density-Functional Theory (DFT+U) could be used to generate Band-structure and Density of States plots to detect and quantify such Haldane gap?

You can already quantify the Haldane gap on the level of a tight-binding model (as shown above). We have not really taken electron interactions into account in our considerations. There are some related ideas where strong electron-electron correlations might be important, such as multilayer (twisted) graphene structures, fractional quantum Hall systems, etc.

As @Anyon correctly pointed out, the Haldane gap can also refer to the spin excitation gap in 1D integer-spin antiferromagnetic Heisenberg models. The general construction here can be quite involved, so I will showcase the most important steps.

We start from a 1D Heisenberg AFM: $$ \hat{H} = J \sum_i \hat{\vec{S}}_{i+1}+\hat{\vec{S}}_i$$

The simplest way to describe the fluctuations above the ground state is through the non-linear $\sigma$-model Lagrangian. $$ \mathcal{L}_{\text{eff}} = \frac{\rho_s}{2}\left[\left(\partial_t\vec{n}\right)^2 - v_s^2 \left(\vec{\nabla}\vec{n}\right)^2\right],$$ where $\vert\vec{n}\vert = 1$ and $\vec{n}$ represents the slow-varying fluctuations, $\rho_s$ is the spin stiffness and $v_s$ is the spin-wave velocity. If we do a more general derivation for a Heisenberg model with spin $S$ (without specifying whether it is integer or half-integer), we get an additional topological term $\gamma[\vec{n}]$, which in one dimension can be expressed as $$\gamma_{d=1}[\vec{n}] = 2\pi S \theta[\vec{n}],$$ where $\theta$ is the so-called Pontryagin index, and it is this term that leads to the spectrum being gapped - and here the gap is called a Haldane gap (for integer spins $e^{iY[\vec{n}]}=1$) or gapless (for half-integer spins).

How is the spin excitation gap related to the gap in the band-structure occurring at the K and K' points?

They were both discovered by Haldane. In the case of the spin chains, the Haldane gap arises from the topological properties of the nonlinear sigma model with a topological theta term. For the Haldane model in graphene, the gap results from a topological phase transition induced by breaking TRS. In both contexts, the presence of an energy gap is fundamentally tied to the topological properties of the system:

• In spin chains, it prevents low-energy excitations, leading to a unique gapped ground state
• In the graphene/Haldane model, the gap signifies the presence of a topologically non-trivial insulating state.

What is a spin excitation gap?

When we look at a kinetically frozen system like a Heisenberg Model or a Mott Insulator - the particles can no longer move but can still align their spins in various ways. Given a particular type of order of the spins (here, AFM order), we can think about low-energy excitations above the ordered state - e.g., spin waves. Then, the spin gap is nothing more than the minimal value of the energy of such an excitation. If we can only create excitations with energy above a certain non-zero value, then that means there is a gap in the spin-excitation spectrum. Meanwhile if we can create an excitation with an arbitrarily small energy above the gs - the spectrum is gapless.

Good resources for intro to Haldane physics might be:

1. Topocondmat website
2. Book by Kane - page 14 and onwards are relevant

and for the spin-1 Heisenberg chain physics:

1. Original paper by Haldane
2. A related answer on stackexchange