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@@ -20,43 +20,272 @@ The model is very simple: two different hopping terms, corresponding to right- a
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```@example hatano-nelson
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tb_hamiltonian(cbr, [[1]], HERMITIAN)
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```
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The non-Hermitian model reduces to the Hermitian model when the left- and right-directed hopping amplitudes are equal. When the two are _not_ equal, the Hatano-Nelson model features exceptional points and spontaneous symmetry breaking of the real spectrum, as we can verify by example (using Brillouin.jl and GLMakie.jl for dispersion plotting):
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The non-Hermitian model reduces to the Hermitian model when the left- and right-directed hopping amplitudes are equal. When the two are _not_ equal, the Hatano--Nelson model features nontrivial spectral winding in the sense that its spectrum traces out a finite-area spectral loop in the complex plane as its momentum is varied across one loop.
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We can see this by visualizing the complex energy as we vary $k$ from -1/2 to 1/2:
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```@example hatano-nelson
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ptbm = tbm([0.9, 1.1]) # a model with 0.9 hopping amplitude to right, 1.1 to the left
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using Brillouin
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kp = irrfbz_path(1, directbasis(1, 1)) # a k-path in plane group 1
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kpi = interpolate(kp, 500) # interpolated over 500 points
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Es = spectrum(ptbm, kpi)
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Es_re = sort(real.(Es); dims=2)
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Es_im = sort(imag.(Es); dims=2)
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using GLMakie
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plot(kpi, Es_re, Es_im; color=[:blue, :red])
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ks = range(-1/2, 1/2, 500) # 500 sampling points
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Es = spectrum(ptbm, ks) # 500×1 matrix
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Es = vec(Es) # convert to vector
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plot(real(Es), imag(Es))
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```
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We can also explore the consequences of breaking time-reversal symmetry:
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```
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The loop has a winding $\nu = (2\pi \mathrm{i})^{-1}\oint \mathrm{d}k \partial_k \log E(k)$ quantized to $\pm 1$ when the two hopping amplitudes are unequal.
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### Breaking time-reversal symmetry
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We can also create models that do not assume time-reversal symmetry, which here introduces additional imaginary-prefactor counterparts to the standard Hatano--Nelson model:
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```@example hatano-nelson
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brs_notr = calc_bandreps(1, 1; timereversal=false) # EBRs in plane group 1, without time-reversal symmetry
## A more complicated example: exceptional surfaces in p4
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## Two-band Hatano--Nelson-like model with exceptional points
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We can generalize the single-band Hatano--Nelson model from above simply by constructing a two-band model, with two orbitals placed at the unit cell center.
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To do so, we simply include two copies of the same EBR in the composite band representation provided to `tb_hamiltonian`: each copy is treated as a separate physical orbital.
The resulting model has 10 free terms: the first 6 terms are self-couplings (self-energies and intercell hoppings between the same physical orbital): the last 4 are hoppings between the two distinct orbitals. We restrict the model to this subset in the interest of simplicity:
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```@example hatano-nelson
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tbm = tbm[7:10]
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```
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The four terms span a model of the kind:
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$$
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\mathbf{H}(k) =
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\begin{bmatrix}
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0 & t_1 + t_2 \mathrm{e}^{2\pi\mathrm{i}k} \\
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t_1' + t_2' \mathrm{e}^{-2\pi\mathrm{i}k} & 0
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\end{bmatrix},
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$$
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with intercell hoppings $t_2^{(\prime)}$ and intracell hoppings $t_1^{(\prime)}$.
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A specific instance of this model can be created from `tbm` via `tbm([t₁, t₂, t₁′, t₂′])`.
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An simple special case -- but interesting, as we will see -- is equal inter- and intracell hoppings from orbital 2 to orbital 1 ($2\rightarrow 1$ hopping), i.e., $t_1 = t_2 = 1$, fully suprressed intercell $1 \rightarrow 2$ hopping, i.e., $t_2' = 0$ and free intracell $1 \rightarrow 2$ hopping, i.e., $t_1' = t$.
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With this restriction, the model features an exceptional point (band degeneracy without a complete associated eigenfunction basis), occuring when $t_1 + t_2 \mathrm{e}^{2\pi\mathrm{i}k} = 1 + \mathrm{e}^{2\pi\mathrm{i}k} = 0$, i.e., when $\mathrm{e}^{2\pi\mathrm{i}k} = -1 \Leftrightarrow k = \pm 1/2$ (the BZ edge).
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At the exceptional point, the Bloch Hamiltonian is defective in the sense that it is similar to a Jordan block $\big[\begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix}\big]$:
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```@example hatano-nelson
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t = .5 # intracell 1→2 hopping amplitude t₁′
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t₁, t₂, t₁′, t₂′ = 1, .8, t, 0
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ptbm = tbm([t₁, t₂, t₁′, t₂′])
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ptbm([1/2]) # evaluate the Bloch Hamiltonian at k = 1/2
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```
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We can visualize the resulting spectrum of the model over the Brillouin zone to learn more:
The spectrum is degenerate at $k = \pm 1/2$ as expected, generally complex, and exhibiting both time-reversal symmetry $E_1(k) = E_2(-k)^*$ and ``accidental'' particle-hole symmetry $E_1(k) = -E_1(k)$ (resulting from our restriction to a small set of hopping hoppings terms).
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### Exceptional points with PT symmetry
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We can also construct more complicated examples where symmetry plays a role. Consider for example the following simple extension of the Hatano-Nelson model to a 2D lattice with p4 symmetry:
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```@example hatano-nelson-p4
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Exceptional points are especially interesting in contexts where the Hamiltonian is not only non-Hermitian but also PT-symmetric (inversion and time).
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The previous Hatano--Nelson-like model, however, is T-symmetric (by default, `calc_bandreps` assumes time-reversal symmetry, and this assumption is propagated via `brs` and `cbr` to `tb_hamiltonian`) but inversion-broken, and so lacks PT-symmetry.
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We can build a variant, however, that breaks both P and T but retains PT symmetry.
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To do so, first construct the terms of a time-reversal model, starting now with a set of time-reversal broken EBRs:
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```@example PT-symmetry
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using Crystalline, SymmetricTightBinding, GLMakie # hide
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brs = calc_bandreps(1, 1; timereversal=false) # a single EBR, as before, but now without assumption of time-reversal
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pin_free!(brs, [1=>[0]]) # as before, pin free parameters of the EBR's Wyckoff position
The resulting model has no less than 20 possible terms: this is simply the result of being a fully unconstrained problem -- lack both hermitian, spatial, and time-reversal symmetry.
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We pick a small subset of these terms, with the aim of building a simple model. In particular, we retain imaginary onsite terms and the terms in our previous reduced model:
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```@example PT-symmetry
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onsite_terms = [2, 8]
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hopping_terms = [13, 15, 17, 19]
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tbm = tbm[vcat(onsite_terms, hopping_terms)]
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```
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The span of these terms result in a Bloch Hamiltonian:
The spectrum of the model is $E_\pm(k) = \sqrt{|t_1+t_2\mathrm{e}^{2\pi\mathrm{i}k}|^2 - \gamma^2}$, which is degenerate -- in fact, exceptional -- when $|t_1+t_2\mathrm{e}^{2\pi\mathrm{i}k}| = \gamma$ (assuming $\gamma>0$). The spectrum is qualitatively distinct before and after the exceptional point:
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- When $|t_1+t_2\mathrm{e}^{2\pi\mathrm{i}k}| > \gamma$: $E_\pm(k)$ is real (PT-unbroken phase).
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- When $|t_1+t_2\mathrm{e}^{2\pi\mathrm{i}k}| < \gamma$: $E_\pm(k)$ is imaginary (spontaneously broken PT-symmetry).
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We can see this readily by constructing the associated Hamiltonian from `tbm`, noting that `tbm([γ₁, γ₂, t₁, t₂, t₁′, t₂′])` corresponds to the general model and `tbm([γ, -γ, t₁, t₂, t₁, t₂])` to the PT-symmetric one:
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```@example PT-symmetry
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γ, t₁, t₂ = 0.5, 1, 1
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ptbm = tbm([γ, -γ, t₁, t₂, t₁, t₂])
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ks = range(-1/2, 1/2, 500)
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Es = spectrum(ptbm, ks)
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Es_re = real(Es) # real parts
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Es_im = imag(Es) # imaginary parts
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Es_re = sort(Es_re; dims=2) # necessary to explicitly sort for visualization, due to intrinsic
## A more complicated example: exceptional lines in p4
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We can also construct more complicated examples where symmetry plays a role.
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Consider for example a non-Hermitian model on a 2D lattice with p4 symmetry, obtained by placing *s*-like orbitals at the two symmetry-related edges of the unit cell (i.e., a (2c|A) orbital):
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```@example nonhermitian-p4
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using Crystalline, SymmetricTightBinding, GLMakie # hide
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