physics2u
PhysicsMathsChemistryBiology
Tier
⌕ Search ⌘K
Derivation

Density of States and Van Hove Singularities

D-253 Home PU-303 Threads energy · waves · matter Depends on Tight-Binding Band Dispersion, Effective Mass and Semiclassical Electron Dynamics
Statement

For a single band with dispersion \(\varepsilon_n(\mathbf{k})\) on the Brillouin zone, the density of states per unit volume is the constant-energy surface integral \(g(\varepsilon)=\dfrac{1}{(2\pi)^d}\displaystyle\oint_{\varepsilon_n(\mathbf{k})=\varepsilon}\dfrac{dS_k}{\left|\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k})\right|}\). At critical points where \(\nabla_{\mathbf{k}}\varepsilon_n=\mathbf{0}\) the integrand diverges, and the topology of the Hessian there forces universal, dimensionality-dependent non-analyticities in \(g(\varepsilon)\) — the Van Hove singularities.

Why it matters

Almost every quantity that counts electronic states — specific heat, magnetic susceptibility, optical absorption edges, screening, the strength of instabilities toward magnetism or superconductivity — is an integral weighted by \(g(\varepsilon)\). Knowing where \(g\) spikes or kinks tells you which photon energies a crystal absorbs strongly and which fillings are prone to ordering.

The Van Hove theorem is a rare exact statement in a hard many-band problem: it fixes the analytic form of the singularities from nothing more than the fact that a smooth periodic function on a compact zone must have critical points. It is the bridge from a band structure \(\varepsilon_n(\mathbf{k})\) to a measurable spectrum.

Assumptions
A single band is treated in isolation, \(\varepsilon_n(\mathbf{k})\).If bands overlap in energy the total DOS is a sum \(\sum_n g_n(\varepsilon)\); each band still contributes its own singularities, but they can superpose and partially mask one another.
The dispersion \(\varepsilon_n(\mathbf{k})\) is smooth (analytic) on the Brillouin zone.Drop it and \(\nabla_{\mathbf{k}}\varepsilon\) can jump or fail to exist (e.g. at Dirac points or accidental band touchings), giving singularities not of the analytic Morse type derived here.
Bloch states are labelled by a real crystal momentum \(\mathbf{k}\) in a periodic potential (independent-electron picture).With strong disorder \(\mathbf{k}\) is no longer good; states localise and the sharp Van Hove features broaden into smooth humps.
Critical points are non-degenerate (Morse): the Hessian \(\partial^2\varepsilon/\partial k_i\partial k_j\) has no zero eigenvalue.At a degenerate critical point the leading expansion is higher than quadratic and the singularity exponent changes (e.g. monkey-saddle or higher-order Van Hove points with enhanced, sometimes power-law diverging, DOS in 2D).
Thermodynamic limit with periodic (Born–von Kármán) boundary conditions, so \(\mathbf{k}\) is quasi-continuous with density \(V/(2\pi)^d\).For a finite crystal the \(\mathbf{k}\)-sum is discrete and \(g(\varepsilon)\) is a comb of delta functions; the smooth surface integral is the \(N\to\infty\) envelope.
Derivation
1
\[ g(\varepsilon)\,d\varepsilon \;=\; \frac{1}{V}\sum_{\mathbf{k}\in\text{BZ}} \Theta\!\big(\varepsilon+d\varepsilon-\varepsilon_n(\mathbf{k})\big)-\Theta\!\big(\varepsilon-\varepsilon_n(\mathbf{k})\big) \]
Definition: \(g(\varepsilon)\) counts states per unit volume per unit energy; the number with energy in \([\varepsilon,\varepsilon+d\varepsilon]\) divided by crystal volume \(V\). Spin degeneracy is suppressed and restored at the end. A
2
\[ g(\varepsilon)=\frac{1}{V}\sum_{\mathbf{k}\in\text{BZ}}\delta\!\big(\varepsilon-\varepsilon_n(\mathbf{k})\big) \;\xrightarrow[\;V\to\infty\;]{}\; \frac{1}{(2\pi)^d}\int_{\text{BZ}} d^dk\;\delta\!\big(\varepsilon-\varepsilon_n(\mathbf{k})\big) \]
The difference of step functions is \(\delta(\varepsilon-\varepsilon_n)\,d\varepsilon\). In the thermodynamic limit the allowed \(\mathbf{k}\) fill the zone with density \(V/(2\pi)^d\), so \(\frac{1}{V}\sum_{\mathbf{k}}\to\frac{1}{(2\pi)^d}\int d^dk\). A
3
\[ \int_{\text{BZ}} d^dk\;\delta\!\big(\varepsilon-\varepsilon_n(\mathbf{k})\big) \;=\; \int d\varepsilon'\int_{\varepsilon_n(\mathbf{k})=\varepsilon'} dS_k \;\frac{\delta(\varepsilon-\varepsilon')}{\left|\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k})\right|} \]
Slice the zone into constant-energy surfaces. Foliate \(d^dk = dS_k\,dk_\perp\) where \(dk_\perp\) is the normal spacing between surfaces \(\varepsilon'\) and \(\varepsilon'+d\varepsilon'\). Since \(\nabla_{\mathbf{k}}\varepsilon_n\) is normal to the surface, \(d\varepsilon' = \left|\nabla_{\mathbf{k}}\varepsilon_n\right|\,dk_\perp\), i.e. \(dk_\perp = d\varepsilon'/\left|\nabla_{\mathbf{k}}\varepsilon_n\right|\). B
4
\[ g(\varepsilon)=\frac{1}{(2\pi)^d}\oint_{\varepsilon_n(\mathbf{k})=\varepsilon}\frac{dS_k}{\left|\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k})\right|} \]
Carry out the trivial \(\varepsilon'\) integral against \(\delta(\varepsilon-\varepsilon')\). This is the coarea (Federer) formula; it is valid wherever \(\left|\nabla_{\mathbf{k}}\varepsilon_n\right|\neq 0\), and its breakdown at critical points is exactly what generates the singularities. C
5
\[ \nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k}_c)=\mathbf{0}, \qquad \varepsilon_n(\mathbf{k})\approx \varepsilon_c+\frac{\hbar^2}{2}\sum_{i=1}^{d}\frac{(k_i-k_{c,i})^2}{m_i^*} \]
Group velocity \(\mathbf{v}=\hbar^{-1}\nabla_{\mathbf{k}}\varepsilon_n\) vanishes at a critical point \(\mathbf{k}_c\) (this is the semiclassical result invoked as a prior). Expand to quadratic order and diagonalise the Hessian; the eigenvalues define principal effective masses \(m_i^*=\hbar^2/(\partial^2\varepsilon/\partial k_i^2)\), which may be positive or negative. B
6
\[ \text{signature of }\{m_i^*\}:\quad \begin{cases} \text{all }>0 & \text{minimum } (M_0)\\ \text{all }<0 & \text{maximum } (M_d)\\ \text{mixed} & \text{saddle } (M_p,\;0<p<d) \end{cases} \]
Morse theory classifies each non-degenerate critical point by its index \(p\), the number of negative Hessian eigenvalues. A smooth periodic function on the compact torus \(T^d\) must possess critical points of every index (Morse inequalities), so saddles are unavoidable, not optional. C
7
\[ g(\varepsilon)\Big|_{\text{near }\varepsilon_c}=\frac{1}{(2\pi)^d}\oint\frac{dS_k}{\hbar^2\sqrt{\sum_i (k_i-k_{c,i})^2/(m_i^*)^2}} \]
Insert the quadratic form into \(\left|\nabla_{\mathbf{k}}\varepsilon_n\right|=\hbar^2\sqrt{\sum_i k_i^2/(m_i^*)^2}\) (measuring \(k_i\) from \(\mathbf{k}_c\)). The vanishing denominator at \(\mathbf{k}_c\) is integrable in \(d\ge 2\) but its interplay with the shape of the surface controls whether \(g\) or its derivative diverges. B
8
\[ d=3:\; g(\varepsilon)=g(\varepsilon_c)\pm C\sqrt{|\varepsilon-\varepsilon_c|}\;\;(\text{kink});\qquad d=2:\; g(\varepsilon)\sim -A\ln|\varepsilon-\varepsilon_c|\;(\text{saddle, log});\qquad d=1:\; g(\varepsilon)\sim \frac{B}{\sqrt{|\varepsilon-\varepsilon_c|}} \]
Evaluate the local surface integral for each index and dimension. In 3D the DOS stays finite but has a square-root cusp in slope; in 2D a saddle gives a symmetric logarithmic divergence; in 1D every band edge gives an inverse-square-root divergence. These are the Van Hove singularities. Derived explicitly for representative cases in the Worked examples. C
Result
\[ g(\varepsilon)=\frac{1}{(2\pi)^d}\oint_{\varepsilon_n(\mathbf{k})=\varepsilon}\frac{dS_k}{\left|\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k})\right|}, \qquad \nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k}_c)=\mathbf{0}\;\Rightarrow\; \text{Van Hove singularity} \]

Reading. The density of states is the area of a constant-energy surface weighted by the inverse group speed: where electrons move slowly (flat bands, small \(\left|\nabla_{\mathbf{k}}\varepsilon\right|\)) they pile up in energy. Band extrema and saddle points, where the group velocity vanishes, are the sources of the non-analytic features. Their exact form — step, kink, log, or inverse-root — is fixed only by the spatial dimension \(d\) and the Morse index \(p\), independent of microscopic detail. Restore a factor of 2 for spin: \(g\to 2g\).

Units check. \(dS_k\) has units \(\text{m}^{-(d-1)}\) (reciprocal length to the \(d-1\)); \(\left|\nabla_{\mathbf{k}}\varepsilon\right|\) has units \(\text{J}\cdot\text{m}\) (energy \(\times\) length, since \(\mathbf{k}\) is inverse length). The prefactor \((2\pi)^{-d}\) has units \(\text{m}^{d}\). Product: \(\text{m}^{d}\cdot \text{m}^{-(d-1)}/(\text{J}\cdot\text{m})=\text{J}^{-1}\text{m}^{-d}\) — states per unit energy per unit \(d\)-volume, as required.

Limiting cases
  • Free electrons, 3D. \(\varepsilon=\hbar^2k^2/2m\) has a single minimum; the formula gives \(g(\varepsilon)=\frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{\varepsilon}\), the textbook \(\sqrt{\varepsilon}\) edge — a pure \(M_0\) Van Hove point.
  • Free electrons, 2D. \(g=m/(\pi\hbar^2)\), a constant step at the band bottom — the \(d=2\) minimum gives a jump discontinuity, not a divergence.
  • Free electrons, 1D. \(g(\varepsilon)=\frac{1}{\pi\hbar}\sqrt{m/(2\varepsilon)}\), the inverse-root divergence at the band edge.
  • Flat band, \(m^*\to\infty\). \(\left|\nabla_{\mathbf{k}}\varepsilon\right|\to 0\) over a whole region; \(g(\varepsilon)\to\) a delta-like peak — the extreme limit of state pile-up.
  • Isotropic parabolic band. All \(m_i^*\) equal; the surface integral reduces to the area of a sphere/circle and reproduces the free-electron results with \(m\to m^*\).
Breaks when
  • Degenerate (higher-order) critical points. If the Hessian has a zero eigenvalue the quadratic expansion fails; the singularity is no longer \(\sqrt{|\varepsilon-\varepsilon_c|}\) or \(\ln\). In 2D a monkey-saddle gives a power-law divergent DOS, driving stronger interaction instabilities than an ordinary log.
  • Massless / linear band touchings. Near a Dirac point \(\varepsilon\propto |\mathbf{k}|\), so \(\nabla_{\mathbf{k}}\varepsilon\neq 0\) but the surface area vanishes; graphene gives \(g(\varepsilon)\propto|\varepsilon|\), a linear zero, not a Morse singularity.
  • Strong disorder or short lifetime. A finite \(\tau\) replaces \(\delta(\varepsilon-\varepsilon_n)\) by a Lorentzian of width \(\hbar/\tau\); logarithmic and root singularities are smeared, and localisation can destroy the Bloch label entirely.
  • Strong correlations. When self-energy \(\Sigma(\mathbf{k},\varepsilon)\) is large the single-particle band picture breaks; the spectral function replaces \(g\) and features shift or split (Hubbard bands).
Failure modes
  • Confusing \(g(\varepsilon)\) with the constant-energy surface area alone. Students forget the \(1/\left|\nabla_{\mathbf{k}}\varepsilon\right|\) weight; the DOS is not just "how big the Fermi surface is."
  • Expecting a divergence in 3D. In 3D minima and maxima give only a \(\sqrt{|\varepsilon-\varepsilon_c|}\) edge and saddles give a kinked shoulder; \(g\) stays finite. Divergences require \(d\le 2\).
  • Assuming saddle points are optional. A periodic band on a torus is topologically forced to have saddles; a tight-binding band with only a min and max but no saddle is impossible.
  • Sign errors in the effective mass. Treating a maximum as if the curvature were positive puts the \(\sqrt{}\) tail on the wrong side of \(\varepsilon_c\).
  • Dropping spin and Brillouin-zone volume factors. Missing the factor 2 (spin) or misusing \((2\pi)^d\) vs \((2\pi)^d/V\) gives DOS wrong by orders of magnitude.
  • Using \(\varepsilon=\hbar^2k^2/2m\) globally in a lattice. The parabolic approximation only holds near a critical point; the true singularity count comes from the full periodic dispersion.
Discussion

The surface-integral formula reframes the DOS as a purely geometric object: it is the flux-weighted area of the isoenergy surface in reciprocal space. The weight \(1/\left|\nabla_{\mathbf{k}}\varepsilon\right| = 1/(\hbar|\mathbf{v}|)\) is the reciprocal of the semiclassical group speed, so the DOS literally measures how long, in \(\mathbf{k}\)-space, states linger near a given energy. Flat portions of a band — slow electrons — dominate \(g\), which is why nearly-dispersionless bands (heavy-fermion \(f\)-electrons, moiré flat bands) carry enormous densities of states and are hotbeds of correlated order.

Van Hove's 1953 insight was that the singularities are not accidents of a particular material but topological necessities. Morse theory guarantees that a smooth function on the \(d\)-torus has at least \(\binom{d}{p}\) critical points of index \(p\); each one stamps a universal singularity onto \(g(\varepsilon)\). The shape of that stamp depends only on \((d,p)\): a step in 2D at an extremum, a logarithm at a 2D saddle, a root cusp at a 3D extremum. This universality is why optical joint-DOS spectra of very different semiconductors show the same characteristic \(E_1,E_2\) critical-point lineshapes.

The 2D logarithmic singularity is physically the most consequential. When the Fermi level is tuned to a saddle-point energy, \(g(\varepsilon_F)\to\infty\) (logarithmically), and any weak interaction is amplified: the Stoner criterion \(I\,g(\varepsilon_F)>1\) for ferromagnetism, the Cooper log for superconductivity, and nesting-driven density waves all get a boost. This "Van Hove scenario" underlies proposals for high-\(T_c\) enhancement near optimal doping in cuprates and the correlated phases of magic-angle graphene, where a higher-order Van Hove point produces a power-law divergence stronger than \(\ln\). The renormalisation-group flow near such a point is genuinely non-trivial because the DOS enters the one-loop integrals nonperturbatively.

Common misconceptions. A Van Hove singularity is a feature of the density of states in energy, not of the wavefunction or of real space — nothing physical is singular; \(g\) is still integrable (its energy integral just counts states). "Singularity" means non-analytic, not infinite: in 3D the DOS is perfectly finite at a Van Hove energy, only its derivative misbehaves. And the singularity sits at the critical energy \(\varepsilon_c\), which need not be the Fermi energy — its physical impact is large only when doping brings \(\varepsilon_F\) to \(\varepsilon_c\).

Worked examples
1
Example 1 — 2D square-lattice tight binding: the logarithmic saddle. Locate the saddle point of \(\varepsilon(\mathbf{k})=-2t\left[\cos(k_x a)+\cos(k_y a)\right]\) and extract the DOS singularity at \(\varepsilon=0\), the half-filled band. Take \(t=1\ \text{eV}\), lattice constant \(a=3.0\ \text{Å}\). B
2
\[ \nabla_{\mathbf{k}}\varepsilon = 2ta\big(\sin k_x a,\;\sin k_y a\big)=\mathbf{0}\;\Rightarrow\; \mathbf{k}_c=\left(\frac{\pi}{a},0\right)\ \text{and}\ \left(0,\frac{\pi}{a}\right),\quad \varepsilon_c=0 \]
Group velocity vanishes at the zone-edge midpoints \(X\); these have \(\varepsilon_c=-2t[\cos\pi+\cos 0]=0\). B
3
\[ \text{Hessian at }(\tfrac{\pi}{a},0):\; \frac{\partial^2\varepsilon}{\partial k_x^2}=2ta^2\cos k_xa=-2ta^2,\quad \frac{\partial^2\varepsilon}{\partial k_y^2}=+2ta^2 \;\Rightarrow\; \varepsilon\approx ta^2\big(-q_x^2+q_y^2\big) \]
Opposite-sign curvatures (one negative mass, one positive) confirm a Morse index-1 saddle in \(d=2\). Here \(\mathbf{q}=\mathbf{k}-\mathbf{k}_c\). B
4
\[ g(\varepsilon)=\frac{1}{(2\pi)^2}\int d^2q\,\delta\!\big(\varepsilon-ta^2(-q_x^2+q_y^2)\big) \;=\;\frac{1}{2\pi^2 t a^2}\,\ln\!\frac{\Lambda}{|\varepsilon|} \]
The hyperbolic level sets of a saddle make the \(q\)-integral logarithmic; \(\Lambda\sim t\) is the bandwidth cutoff. Exact lattice result: \(g(\varepsilon)=\frac{1}{2\pi^2 t a^2}\ln\!\left|\frac{16t}{\varepsilon}\right|\). Include spin: multiply by 2. C
5
\[ ta^2=(1\ \text{eV})(3.0\times10^{-10}\ \text{m})^2=9.0\times10^{-20}\ \text{eV·m}^2,\quad \frac{2}{2\pi^2 ta^2}=\frac{1}{\pi^2\,ta^2}\approx 1.13\times10^{18}\ \text{eV}^{-1}\text{m}^{-2} \]
Prefactor with spin. Evaluate the log at, say, \(\varepsilon=0.1\ \text{eV}\): \(\ln(16/0.1)=\ln 160\approx 5.08\). A
\[ g(0.1\ \text{eV})\approx 1.13\times10^{18}\times 5.08 \approx 5.7\times10^{18}\ \text{eV}^{-1}\text{m}^{-2} \;\;(\text{per spin-summed 2D cell}) \]

Reading. The DOS diverges logarithmically as \(\varepsilon\to 0\); tuning the Fermi level to half filling places it on the singularity, the seat of magnetic and superconducting instabilities on the square lattice.

Units check. \(\text{eV}^{-1}\text{m}^{-2}\times(\text{dimensionless log})=\text{eV}^{-1}\text{m}^{-2}\), a 2D DOS per unit area. Correct.

1
Example 2 — 3D free-electron band minimum: the \(\sqrt{\varepsilon}\) edge. Compute the DOS of copper's conduction band modelled as free electrons (\(m^*=m_e\)) at \(\varepsilon=1.0\ \text{eV}\) above the band bottom, and confirm the square-root cusp characteristic of a 3D \(M_0\) Van Hove point. A
2
\[ \varepsilon=\frac{\hbar^2 k^2}{2m^*}\;\Rightarrow\; \nabla_{\mathbf{k}}\varepsilon=\frac{\hbar^2 k}{m^*}\hat{\mathbf{k}},\quad \left|\nabla_{\mathbf{k}}\varepsilon\right|=\frac{\hbar^2 k}{m^*};\qquad \oint dS_k = 4\pi k^2 \]
Isotropic parabolic band: constant-energy surface is a sphere of radius \(k=\sqrt{2m^*\varepsilon}/\hbar\); the group speed grows linearly with \(k\). Minimum at \(k=0\) is the \(M_0\) point. A
3
\[ g(\varepsilon)=\frac{2}{(2\pi)^3}\cdot\frac{4\pi k^2}{\hbar^2 k/m^*} =\frac{2}{(2\pi)^3}\cdot\frac{4\pi m^* k}{\hbar^2} =\frac{1}{2\pi^2}\left(\frac{2m^*}{\hbar^2}\right)^{3/2}\sqrt{\varepsilon} \]
Insert into the surface-integral formula, cancel one power of \(k\), substitute \(k=\sqrt{2m^*\varepsilon}/\hbar\). Factor 2 is spin. The explicit \(\sqrt{\varepsilon}\) is the 3D minimum Van Hove edge. B
4
\[ \left(\frac{2m_e}{\hbar^2}\right)^{3/2}=\left(\frac{2(9.11\times10^{-31})}{(1.055\times10^{-34})^2}\right)^{3/2} =\left(1.637\times10^{38}\ \text{J}^{-1}\text{m}^{-2}\right)^{3/2}=6.62\times10^{57}\ \text{J}^{-3/2}\text{m}^{-3} \]
Evaluate the mass prefactor in SI. \(\varepsilon=1.0\ \text{eV}=1.602\times10^{-19}\ \text{J}\), so \(\sqrt{\varepsilon}=4.00\times10^{-10}\ \text{J}^{1/2}\). A
\[ g=\frac{6.62\times10^{57}}{2\pi^2}\,(4.00\times10^{-10}) \approx 1.34\times10^{47}\ \text{J}^{-1}\text{m}^{-3} \approx 2.1\times10^{28}\ \text{eV}^{-1}\text{m}^{-3} \]

Reading. About \(2\times10^{28}\) states per eV per cubic metre — of order \(0.6\) states/eV per atom for a metal of atomic density \(\sim 8\times10^{28}\ \text{m}^{-3}\), the right ballpark for a simple metal. The DOS rises as \(\sqrt{\varepsilon}\): finite value, infinite slope at the band edge.

Units check. \(\text{J}^{-3/2}\text{m}^{-3}\times\text{J}^{1/2}=\text{J}^{-1}\text{m}^{-3}\); dividing \(1\ \text{eV}=1.602\times10^{-19}\ \text{J}\) converts to \(\text{eV}^{-1}\text{m}^{-3}\). Correct 3D DOS units.

Problems
  1. Derive the 1D DOS for a free-electron chain \(\varepsilon=\hbar^2k^2/2m\) from the surface-integral formula and identify the Van Hove index.
    Solution In 1D the "surface" is the pair of points \(\pm k\) with \(k=\sqrt{2m\varepsilon}/\hbar\); \(\left|\partial\varepsilon/\partial k\right|=\hbar^2 k/m\). With spin, \(g(\varepsilon)=\frac{2}{2\pi}\sum_{\pm}\frac{1}{\hbar^2 k/m}=\frac{2}{2\pi}\cdot\frac{2m}{\hbar^2 k}=\frac{1}{\pi\hbar}\sqrt{\frac{2m}{\varepsilon}}\cdot\frac{1}{2}\). Cleanly: \(g(\varepsilon)=\frac{1}{\pi\hbar}\sqrt{\frac{m}{2\varepsilon}}\) per spin, doubled for spin gives \(\frac{2}{\pi\hbar}\sqrt{\frac{m}{2\varepsilon}}\). It diverges as \(\varepsilon^{-1/2}\) at the band bottom — a \(d=1\), index-0 Van Hove singularity.
  2. For the 2D tight-binding band of Worked Example 1, show the band minimum at \(\Gamma=(0,0)\) gives a step (constant) DOS, and compute its height (per spin) for \(t=1\ \text{eV}\), \(a=3\ \text{Å}\).
    Solution Near \(\Gamma\), \(\varepsilon=-4t+ta^2(k_x^2+k_y^2)\), an isotropic 2D minimum with \(m^*=\hbar^2/(2ta^2)\). A 2D parabolic minimum gives \(g=\frac{m^*}{2\pi\hbar^2}=\frac{1}{4\pi t a^2}\) per spin (constant, a step). Numerically \(4\pi t a^2 = 4\pi(1\ \text{eV})(9\times10^{-20}\ \text{m}^2)=1.13\times10^{-18}\ \text{eV·m}^2\), so \(g=8.8\times10^{17}\ \text{eV}^{-1}\text{m}^{-2}\) per spin. The DOS jumps from 0 to this constant at \(\varepsilon=-4t\); no divergence at a 2D extremum.
  3. A 3D band has a saddle point (index 2, two negative masses) with \(\varepsilon\approx\varepsilon_c+\frac{\hbar^2}{2}\left(\frac{q_z^2}{m_3}-\frac{q_x^2}{m_1}-\frac{q_y^2}{m_2}\right)\). State the qualitative DOS behaviour on each side of \(\varepsilon_c\).
    Solution In 3D every non-degenerate critical point gives a finite DOS with a \(\sqrt{|\varepsilon-\varepsilon_c|}\) cusp in one branch. For an index-2 saddle (like an index-1 saddle by particle–hole symmetry of the argument) the DOS has a background plus a term \(\propto\sqrt{\varepsilon_c-\varepsilon}\Theta(\varepsilon_c-\varepsilon)\): the square-root tail appears on the low-energy side and the slope \(dg/d\varepsilon\) diverges as \(\varepsilon\to\varepsilon_c^-\), while \(g\) itself stays finite. No logarithm in 3D.
  4. Estimate the energy window over which a 2D logarithmic Van Hove singularity is "seen" at temperature \(T=100\ \text{K}\), and comment on whether the divergence is physically observable.
    Solution Thermal broadening smears \(g(\varepsilon)\) over \(\sim k_BT\). At \(T=100\ \text{K}\), \(k_BT=8.62\times10^{-5}\times100=8.6\ \text{meV}\). The log divergence \(\ln(16t/|\varepsilon|)\) is cut off at \(|\varepsilon|\sim k_BT\): with \(t=1\ \text{eV}\), the effective peak height is \(\propto\ln(16\ \text{eV}/8.6\ \text{meV})=\ln(1860)\approx 7.5\). So the divergence is never literally infinite; it saturates at a large but finite value set by \(\ln(t/k_BT)\). It is observable as a pronounced but rounded peak in ARPES/STS and in enhanced susceptibilities, sharpening as \(T\) falls.
  5. Use the Stoner criterion \(I\,g(\varepsilon_F)>1\) with the square-lattice result to estimate the interaction strength \(I\) needed for a ferromagnetic instability when \(\varepsilon_F\) is \(10\ \text{meV}\) from the saddle. Use \(t=1\ \text{eV}\), \(a=3\ \text{Å}\), per-area DOS from Example 1.
    Solution From Example 1 (spin-summed) \(g(\varepsilon)=\frac{1}{\pi^2 t a^2}\ln(16t/|\varepsilon|)\); for a Stoner estimate use per-spin \(g_\uparrow=\frac{1}{2\pi^2 ta^2}\ln(16t/|\varepsilon|)\). At \(|\varepsilon|=10\ \text{meV}\): \(\ln(16/0.01)=\ln 1600=7.38\). \(\frac{1}{2\pi^2 ta^2}=\frac{1}{2\pi^2(9\times10^{-20})}=5.63\times10^{17}\ \text{eV}^{-1}\text{m}^{-2}\), so \(g_\uparrow=4.16\times10^{18}\ \text{eV}^{-1}\text{m}^{-2}\). Stoner threshold \(I=1/g_\uparrow=2.4\times10^{-19}\ \text{eV·m}^2\). Converting to a per-site interaction with cell area \(a^2=9\times10^{-20}\ \text{m}^2\): \(I_{\text{site}}=I/a^2\approx 2.7\ \text{eV}\). Because the log makes \(g\) large, even a moderate on-site \(U\) of a few eV suffices — the essence of the Van Hove ferromagnetism scenario.