Pomeranchuk and Density-Wave Instabilities

A Pomeranchuk instability and a density-wave instability are two different static eigenvalue problems. In the first, a forward-scattering eigenmode changes the energy and shape of an existing Fermi surface at $q=0$. In the second, a particle–hole bilinear mixes states separated by a nonzero crystal momentum $\mathbf Q$ and modulates charge or spin in space. When the lowest quadratic eigenvalue reaches zero, the assumed normal state reaches a local-stability boundary—a spinodal within that approximation. This does not, by itself, determine the ordered amplitude, the order of the transition, or whether another channel intervenes first.

Required background. Use Landau theory for the quasiparticle energy functional and the chapter’s total-density-of-states convention, and Fermi-liquid response for the static order of limits. Helpful background. Channel-resolved functional RG explains how several weak-coupling tendencies can be compared without mistaking a stopping scale for an ordered phase.

Uniform and modulated order answer different questions

Crystal momentum is defined modulo a reciprocal-lattice vector $\mathbf G$. Thus “finite wavevector” below means $\mathbf Q\not\equiv0\pmod{\mathbf G}$. Forward scattering and nested finite-transfer scattering are also distinct low-energy kinematic sectors in Fermi-surface renormalization Shankar 1994, §§ IX–X, pp. 126–138. The minimal distinction is:

Channel	Order parameter	Symmetry change	Direct diagnostic	What the quadratic test establishes
Pomeranchuk shape mode	$O_\Gamma(0)\propto\sum_{\mathbf k\sigma}f_\Gamma(\mathbf k)c^\dagger_{\mathbf k\sigma}c_{\mathbf k\sigma}$	Translations remain intact; a nontrivial rotational or point-group representation can break	Static shape susceptibility, spontaneous Fermi-surface or response anisotropy in the zero-field limit	Loss of positive stiffness in one forward-scattering eigenmode
Charge-density wave (CDW)	$\rho_r(\mathbf Q)$	Translations break; rotation can also break for a unidirectional state	Charge or structural satellite peak with form factor, phase, correlation length, and size scaling	A charge particle–hole eigenmode becomes soft in the stated approximation
Spin-density wave (SDW)	$\mathbf S_r(\mathbf Q)$	Translations break; spin rotation and time reversal can also break, subject to spin–orbit coupling	Magnetic structure factor, polarization, correlation length, and thermodynamic scaling	A spin particle–hole eigenmode becomes soft in the stated approximation

A $q=0$ mode is not automatically nematic. Uniform density and magnetization are the symmetric $\ell=0$ charge and spin modes, while a nematic is a translation-preserving mode in a nontrivial spatial representation. Conversely, a commensurate density wave enlarges the unit cell, whereas an incommensurate wave has no finite enlarged cell but still breaks translations and produces satellite wavevectors.

The chapter’s instability and validity map places the uniform and finite-$\mathbf Q$ branches beside their stopping tests. Its claim table gives the corresponding assumptions and evidence ceilings in a nonvisual form.

The Pomeranchuk test is a Fermi-surface eigenproblem

Work at $T=0$ in the static thermodynamic limit and assume that the normal state has a sharp, smooth interacting Fermi surface. On one spin-degenerate sheet, define the normalized density-of-states measure

$$\mathrm d\varpi_{\mathbf k} =\frac{2}{N(0)} \frac{\mathrm dS_{\mathbf k}}{(2\pi)^d v_F^*(\mathbf k)}, \qquad \int_{\mathrm{FS}}\mathrm d\varpi_{\mathbf k}=1,$$

where $N(0)$ is the total density of states including both spins. For several sheets, the measure acquires a sheet index and the interaction below becomes a matrix in sheet or orbital space.

Let $u_x(\mathbf k)$ be the energy by which the quasiparticle surface is displaced in the spin-symmetric or spin-antisymmetric channel $x=s,a$. With the spin normalization inherited from the Landau page, define the dimensionless integral operator

$$(\widehat F^x u_x)(\mathbf k) =\int_{\mathrm{FS}}\mathrm d\varpi_{\mathbf k'} F^x(\mathbf k,\mathbf k')u_x(\mathbf k').$$

The second-order change of $\mathcal K=E-\mu N$ per volume is then

$$\frac{\Delta\mathcal K_x^{(2)}}{V} =\frac{N(0)}{2} \left\langle u_x, (1+\widehat F^x)u_x\right\rangle_{\mathrm{FS}}.$$

The first term is the kinetic cost of moving the surface; the second is the residual quasiparticle interaction. If $\phi_\alpha$ is a normalized eigenfunction,

$$\widehat F^x\phi_\alpha =\lambda_\alpha^x\phi_\alpha, \qquad u_x=u_\alpha\phi_\alpha,$$

then

$$\frac{\Delta\mathcal K_{x,\alpha}^{(2)}}{V} =\frac{N(0)}{2} (1+\lambda_\alpha^x)\lvert u_\alpha\rvert^2.$$

Local stability therefore requires $1+\lambda_\alpha^x>0$ for every deformation allowed by the ensemble. At fixed particle number the symmetric deformation obeys

$$\langle 1,u_s\rangle_{\mathrm{FS}}=0,$$

so the constant density displacement is excluded. Its stiffness is instead tested from the curvature of $E(N)$, or from the grand-canonical static response with a declared neutralizing environment. Negative compressibility in a neutral model is loss of thermodynamic convexity and a tendency toward phase separation, not automatically a continuous $q=0$ transition. The eigenvalue form is the useful statement on an anisotropic or multiband surface, and it is the form used in lattice stability calculations such as Halboth and Metzner 2000, pp. 5164–5165.

The spherical criterion fixes every normalization factor

For an isotropic three-dimensional liquid, use the chapter convention

$$F^x(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}') =\sum_{\ell=0}^{\infty}F_\ell^x P_\ell(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}').$$

The addition theorem makes every spherical harmonic an eigenfunction:

$$\widehat F^xY_{\ell m} =\frac{F_\ell^x}{2\ell+1}Y_{\ell m}.$$

Consequently

$$1+\frac{F_\ell^{s,a}}{2\ell+1}>0, \qquad F_\ell^{s,a}>-(2\ell+1).$$

This is Pomeranchuk’s quadratic stability condition in the site’s total-DOS, unweighted-Legendre convention Pomeranchuk 1959, pp. 361–362, Official PDF. A source that expands the kernel with an extra factor $2\ell+1$, or uses a per-spin density of states, prints a different numerical $F_\ell$; the measured stiffness is unchanged after translation.

Two dimensions use Fourier harmonics

The factor $2\ell+1$ belongs to three-dimensional spherical harmonics. On a circular two-dimensional Fermi surface, take $\mathrm d\varpi=\mathrm d\theta/(2\pi)$ and define instead

$$F^x(\theta-\theta') =F_0^x+2\sum_{\ell=1}^{\infty} F_\ell^x\cos[\ell(\theta-\theta')].$$

Then both $\cos(\ell\theta)$ and $\sin(\ell\theta)$ have eigenvalue $F_\ell^x$, so the site-convention bound is

$$1+F_\ell^x>0.$$

Oganesyan, Kivelson, and Fradkin instead study spinless fermions and call a dimensional quadrupolar coupling $F_2^{\mathrm{OKF}}$. Their printed combination $2N_FF_2^{\mathrm{OKF}}$ is the dimensionless quadrupolar eigenvalue used here, so their condition $2N_FF_2^{\mathrm{OKF}}>-1$ is precisely $1+F_2>0$ after translation Oganesyan, Kivelson, and Fradkin 2001, § I, Eqs. (4)–(8), pp. 195109-2–3.

The low harmonics have different meanings:

• $\ell=0$, symmetric: density stiffness or compressibility of a neutral or irreducible electronic fluid.
• $\ell=0$, antisymmetric: uniform spin polarization, the ferromagnetic Pomeranchuk channel.
• $\ell=1$, symmetric in a Galilean continuum: a boost/current deformation tied to $m^*/m=1+F_1^s/3$. For order parameters equal to conserved charge or spin currents, a vertex zero can cancel the apparent susceptibility divergence; a generic $\ell=1$ form factor is not protected in this way Wu, Klein, and Chubukov 2018, §§ I–III.
• $\ell=2$, symmetric: a quadrupolar continuum deformation, the canonical Pomeranchuk route to a nematic Fermi fluid.

At equality only the quadratic coefficient has vanished. Cubic or quartic terms, coupling to the lattice, nonanalytic fermionic corrections, and competing finite-$\mathbf Q$ or Cooper channels determine whether the transition is continuous, first order, or preempted. The continuum two-dimensional quadrupolar theory and its overdamped collective fluctuations are worked out in Oganesyan, Kivelson, and Fradkin 2001, §§ I–IV, pp. 195109-1–5.

A crystal supplies representations, not angular momentum

On a square lattice with bond-aligned axes and lattice spacing one, two common even form factors are

$$f_{B_{1g}}(\mathbf k)=\cos k_x-\cos k_y, \qquad f_{B_{2g}}(\mathbf k)=\sin k_x\sin k_y.$$

A $90^\circ$ rotation changes the sign of each in the stated convention, so a nonzero expectation value selects an axis or diagonal while preserving translations. A fully symmetric $A_{1g}$ deformation can change the dispersion or Fermi-surface topology without breaking point-group symmetry; it is not nematic merely because its shape changes. A spin-antisymmetric shape mode is likewise not a charge nematic. With orbitals or spin–orbit coupling, the representation belongs to the product of momentum form factor and orbital–spin matrix, not to the scalar form factor alone. A multidimensional representation fixes a degenerate quadratic subspace; quartic terms choose its orientation and can change the transition order.

For a numerical patching of a regular Fermi surface, let

$$w_i=\int_{\text{patch }i}\mathrm d\varpi_{\mathbf k}, \qquad \sum_iw_i=1,$$

and let $F_{ij}^x$ sample the dimensionless Landau kernel. In the variables $\widetilde u_i=\sqrt{w_i}\,u_i$, the stability matrix is Hermitian,

$$M_{ij}^x =\delta_{ij}+\sqrt{w_i}\,F_{ij}^x\sqrt{w_j}.$$

Every allowed eigenvalue of $M^x$ must be positive. At fixed density, restrict the symmetric sector to vectors orthogonal to $(\sqrt{w_1},\sqrt{w_2},\ldots)$. This weighted matrix is the lattice replacement for the spherical $2\ell+1$ formula; simply diagonalizing an unweighted table of $F_{ij}$ gives the wrong stiffness. At a van Hove point, $N(0)$ and the normalized surface weights are singular, so one must retain finite temperature or finite energy–momentum cells rather than use this surface-only reduction.

External uniaxial strain is conjugate to a compatible nematic field. An anisotropic Fermi surface or resistivity measured at nonzero strain therefore shows a response, not necessarily spontaneous symmetry breaking; the thermodynamic limit must be taken before the conjugate field is removed. The weak-coupling square-lattice deformation found by Halboth and Metzner 2000, pp. 5164–5165 is a concrete lattice realization, not a universal phase diagram.

A finite-Q instability is a susceptibility eigenproblem

Take a lattice with $N_c$ unit cells and unit-cell volume one. Choose charge and spin generators $T^c=\mathbf 1$ and $T^i=\sigma^i/2$. For a form-factor label $r$, define the volume-normalized bilinear

$$O_r^A(\mathbf Q) =\frac{1}{\sqrt{N_c}} \sum_{\mathbf k,\alpha\beta} f_r(\mathbf k,\mathbf Q) c^\dagger_{\mathbf k+\mathbf Q,\alpha} (T^A)_{\alpha\beta} c_{\mathbf k,\beta}.$$

For $A=c$ this is a charge or bond-density operator; for $A=i$ it is one component of a spin-density operator. In a multiorbital system, $f_rT^A$ is replaced by a matrix $\Lambda_r^A$ in orbital, sublattice, and spin indices. Components at finite $\mathbf Q$ must be classified under the little group that leaves $\mathbf Q$ fixed and then combined over the full symmetry star of $\mathbf Q$. For a real charge or spin density, the order-parameter amplitudes obey $\Phi(-\mathbf Q)=\Phi(\mathbf Q)^*$.

The static Matsubara susceptibility is

$$\chi^A_{rs}(\mathbf Q) =\int_0^\beta\mathrm d\tau\, \left\langle \mathrm T_\tau O_r^A(\mathbf Q,\tau) O_s^A(\mathbf Q,0)^\dagger \right\rangle_{\!c}.$$

This is the thermodynamic, zero-bosonic-frequency object. It is not the uniform finite-frequency limit and not a collisionless collective-mode ray.

The corresponding bare bubble is

$$\chi_{0,rs}^A(\mathbf Q) =-\frac{T}{N_c}\sum_{\mathbf k,\omega_n} \operatorname{Tr}\!\left[ \Lambda_r^A G(\mathbf k,i\omega_n) \Lambda_s^{A\dagger}G(\mathbf k+\mathbf Q,i\omega_n) \right].$$

For one spin-degenerate band this becomes

$$\chi_{0,rs}^A(\mathbf Q) =-C_A\int_{\mathrm{BZ}}\frac{\mathrm d^d k}{(2\pi)^d} f_r(\mathbf k,\mathbf Q)f_s(\mathbf k,\mathbf Q)^* \frac{n_F(\xi_{\mathbf k+\mathbf Q})-n_F(\xi_{\mathbf k})} {\xi_{\mathbf k+\mathbf Q}-\xi_{\mathbf k}},$$

where the coincident-energy ratio is defined by its derivative limit and

$$C_c=\operatorname{tr}[(T^c)^2]=2, \qquad C_i=\operatorname{tr}[(T^i)^2]=\frac12.$$

Thus, with these physical operator definitions, the raw charge bubble is four times the bubble for one spin component when the scalar form factors are identical. Rescaling an operator rescales its susceptibility and the corresponding interaction kernel inversely, leaving the instability eigenvalues unchanged. The positive thermodynamic response above equals minus the retarded polarization convention used on the Lindhard-function page at zero frequency.

As a concrete RPA starting point, define a bare instantaneous separable coupling $\mathbf g^A$ by

$$H_{\mathrm{int}}^A \supset-\frac12\sum_{\mathbf Q} O_r^A(\mathbf Q)^\dagger g^A_{rs}(\mathbf Q) O_s^A(\mathbf Q),$$

so that a positive eigenvalue of $\mathbf g^A$ enhances the declared channel in this sign convention. Static RPA sets the kernel $\mathbf I^A=\mathbf g^A$. A dressed Bethe–Salpeter calculation instead obtains a generally frequency-dependent particle–hole-irreducible vertex diagrammatically and projects it onto the stated static form-factor basis. Once that projection and its frequency prescription have been declared, both cases obey

$$\boldsymbol\chi^A =\boldsymbol\chi_0^A +\boldsymbol\chi_0^A\mathbf I^A \boldsymbol\chi^A.$$

When $\boldsymbol\chi_0^A$ is positive, write the solution in a manifestly Hermitian form,

$$\begin{aligned} \boldsymbol\chi^A &=(\boldsymbol\chi_0^A)^{1/2} [\mathbf 1-\mathbf K^A]^{-1} (\boldsymbol\chi_0^A)^{1/2},\\ \mathbf K^A &=(\boldsymbol\chi_0^A)^{1/2} \mathbf I^A (\boldsymbol\chi_0^A)^{1/2}. \end{aligned}$$

The normal-state response becomes singular when the largest eigenvalue of $\mathbf K^A(\mathbf Q)$ reaches one, equivalently when $\det(\mathbf 1-\boldsymbol\chi_0^A\mathbf I^A)=0$. The scalar expression $\chi=\chi_0/(1-g\chi_0)$ is only the commuting, one-form-factor RPA special case. In a nonorthogonal projected basis, first insert the overlap metric or pass to a dual orthonormal basis. Other sign conventions place minus signs in the kernel, so comparing denominators without comparing operator definitions is unsafe.

The symbol $\mathbf I^A$ here is irreducible in the particle–hole channel; the bare $\mathbf g^A$ is its RPA approximation, not a generic dressed vertex written into the microscopic Hamiltonian. If $\mathbf F^A$ is instead the full reducible vertex, the contraction is performed once,

$$\boldsymbol\chi^A =\boldsymbol\chi_0^A +\boldsymbol\chi_0^A\mathbf F^A\boldsymbol\chi_0^A;$$

putting $\mathbf F^A$ into another ladder would double-count reducible diagrams Tagliavini et al. 2019, § 2.1, Eqs. (1)–(10), pp. 5–6. The reusable Bethe–Salpeter treatment develops the channel, discretization, and convergence requirements.

Nesting is a kinematic enhancement, not an ordering theorem

Geometric nesting means that translating an extended portion of the Fermi surface by $\mathbf Q$ makes it coincide with another portion. At $T=0$, the associated nesting function samples only states on the surface,

$$\mathcal N(\mathbf Q) =\lim_{\omega\to0^+} \frac{-\operatorname{Im}\Pi_0^R(\mathbf Q,\omega)}{\omega} \propto \int\frac{\mathrm d^d k}{(2\pi)^d} \delta(\xi_{\mathbf k})\delta(\xi_{\mathbf k+\mathbf Q}).$$

The static real response $\chi_0(\mathbf Q)$ instead contains virtual transitions over an energy window and weights them by velocities, orbitals, and form factors. A peak in $\mathcal N$ can therefore occur at the wrong wavevector, while a peak in $\chi_0$ can receive substantial weight away from the Fermi surface. Explicit calculations for canonical charge-ordered metals demonstrate this distinction in Johannes and Mazin 2008, § I, pp. 165135-2–4.

At finite temperature, the double delta function is broadened to a factor proportional to $[-n_F'(\xi_{\mathbf k})]\delta(\xi_{\mathbf k+\mathbf Q}-\xi_{\mathbf k})$. The one-dimensional Peierls limit is special. Two nearly linear branches give, schematically,

$$\chi_0(2k_F+\delta q,0) =C\ln\!\left[ \frac{\Lambda} {\max(v_F\lvert\delta q\rvert,T,\gamma)} \right]+O(1), \qquad C>0,$$

where temperature, scattering rate $\gamma$, transverse warping, or imperfect nesting cuts off the logarithm. Intrinsic curvature of a strictly one-dimensional band does not remove its $2k_F$ logarithm, while a generic curved surface in two or three dimensions need not have this divergence. Even when $\chi_0$ is large, the interaction eigenfunction, self-energy, vertex corrections, phonons, and competing channels still decide whether order occurs.

A density wave reconstructs states separated by Q

For one form factor, a static mean field $\Phi_{\mathbf Q}$ produces the following two-state block when $2\mathbf Q=\mathbf G$. The same matrix is also the leading local avoided-crossing truncation when other translated states are far away:

$$H_{\mathbf k}= \begin{pmatrix} \xi_{\mathbf k} & \Phi_{\mathbf Q}f(\mathbf k)\\ \Phi_{\mathbf Q}^*f(\mathbf k)^* & \xi_{\mathbf k+\mathbf Q} \end{pmatrix}.$$

Its eigenvalues are

$$E_\pm(\mathbf k) =\frac{\xi_{\mathbf k}+\xi_{\mathbf k+\mathbf Q}}{2} \pm \sqrt{ \left(\frac{\xi_{\mathbf k}-\xi_{\mathbf k+\mathbf Q}}{2}\right)^2 +\lvert\Phi_{\mathbf Q}f(\mathbf k)\rvert^2}.$$

Thus a gap opens where the two translated dispersions cross and the form factor is nonzero; a density wave need not gap the entire Fermi surface. More generally, period-$p$ order with $p\mathbf Q=\mathbf G$ requires a $p\times p$ block spanning $\mathbf k+n\mathbf Q$ and permits a reduced Brillouin zone. A genuinely incommensurate wave couples an infinite momentum ladder, usually treated through controlled rational approximants or local truncations, and has no finite exact supercell.

In a CDW the mixing is spin independent; in a collinear SDW it changes sign between spin projections, or becomes a spin matrix for noncollinear order. Overhauser’s unrestricted Hartree–Fock result for the electron gas is a classic finite-$\mathbf Q$ SDW example Overhauser 1962, pp. 1437–1452. Its Hartree–Fock conclusion is not a theorem about the correlated electron gas, but the operator and reconstruction logic remain instructive.

Coulomb forces and the lattice reshape charge order

The $\ell=0$ symmetric Pomeranchuk coefficient tests the compressibility of a neutral fluid or the irreducible electronic subsystem. One must declare whether particle number, chemical potential, or electrochemical potential is controlled, together with the ionic background, gates, and other screening channels. The fully screened charge response of a charged material is different. In the positive-response convention used here,

$$[\mathcal R_c(\mathbf q,0)]^{-1} =[\mathcal R_{c,\mathrm{irr}}(\mathbf q,0)]^{-1} +V_C(\mathbf q),$$

with $V_C(q)=4\pi e^2/(\epsilon q^2)$ in a three-dimensional medium and $V_C(q)=2\pi e^2/(\epsilon q)$ for an ideal two-dimensional layer embedded in three dimensions. The divergence suppresses macroscopic charge separation. It can move the softest charge mode to a finite wavevector when a negative short-range stiffness competes with gradient energy, but it neither forbids every finite-$\mathbf Q$ CDW nor selects a universal wavelength. The result depends on dimension, surface tension, and screening; for example, the unscreened two-dimensional model of Jamei, Kivelson, and Spivak 2005, pp. 056805-1–4 produces intermediate textures, but that conclusion is not universal for gated or short-range-screened systems. Gates, other bands, and a neutralizing background change the long-distance kernel; the screening and RPA page fixes the corresponding sign convention.

Ionic motion supplies another coupled channel. If a displacement mode $u$ couples linearly through $g u O_c$, integrating out the electrons to quadratic order in $u$ gives, in a compatible static convention,

$$D_{\mathrm{eff}}^{-1}(\mathbf Q,0) =D_0^{-1}(\mathbf Q,0) -g(\mathbf Q)^\dagger \boldsymbol\chi_e(\mathbf Q,0) g(\mathbf Q).$$

Here $\boldsymbol\chi_e$ is the electronic response with phonon feedback removed—equivalently, irreducible with respect to the phonon line. Otherwise this expression would count the same hybridization twice. A soft phonon and an enhanced electronic response can therefore be parts of one hybrid instability. Observing a structural satellite or a Fermi-surface nesting vector does not establish that the electrons alone caused the transition. Momentum-dependent electron–phonon matrix elements are decisive in concrete materials Johannes and Mazin 2008, §§ I–III, while long-range Coulomb frustration of a short-range phase-separation tendency was developed by Emery and Kivelson 1993, pp. 597–621.

From enhanced response to established order

The strongest justified statement depends on what has actually been checked:

Observation or calculation	What it supports	What remains before claiming order or mechanism
Peak in a bare bubble or nesting function	Favorable particle–hole kinematics in the chosen band and form factor	Interacting kernel, matrix elements, self-energy, competing wavevectors, and convergence
Bethe–Salpeter or fRG eigenvalue approaching one	Strong channel-resolved normal-state tendency in a declared approximation	Basis, size, regulator, frequency, self-energy, and vertex checks; continuation beyond the stopping point
Structure factor growing proportionally to volume	Long-range order at the selected $\mathbf Q$ if the thermodynamic and field limits are controlled	Domains, form factor, polarization, commensurability, and alternative structural or magnetic origins
Soft phonon or elastic anomaly	Proximity to a coupled lattice instability	Whether the primary driver is electronic, ionic, or inseparable at the measured scales
Transport or spectral anisotropy	Nematic response or an anisotropic state	Zero-strain extrapolation, spontaneous domains, and exclusion of explicit crystal anisotropy

With the volume-normalized bilinear above, define

$$S_A(\mathbf Q) =\left\langle O^A(\mathbf Q)O^A(\mathbf Q)^\dagger\right\rangle, \qquad m_A^2 =\lim_{N_c\to\infty}\frac{S_A(\mathbf Q)}{N_c}.$$

The conjugate intensive source must scale consistently with the normalized operator. For $\mathbf Q\not\equiv-\mathbf Q$, use the Hermitian pair

$$H_h=H-\sqrt{N_c}\left[ h^*O^A(\mathbf Q)+hO^A(\mathbf Q)^\dagger \right].$$

At a self-conjugate wavevector, use one Hermitian component and a real source. Spontaneous order in that fixed direction then requires

$$\lim_{h\to0^+}\lim_{N_c\to\infty} \frac{\langle O^A(\mathbf Q)\rangle_h}{\sqrt{N_c}}\ne0.$$

A finite system has no exact spontaneous symmetry breaking. Correlation-length and structure-factor scaling are therefore essential: long-range order gives $S_A(\mathbf Q)\sim N_cm_A^2$, whereas a finite correlation length gives a nonextensive $S_A$ once the system is much larger than that length. A first-order transition can occur before a quadratic eigenvalue reaches zero.

Dimension and symmetry also constrain the inference:

Strictly two-dimensional short-range case	Finite-temperature conclusion
Continuum nematic with continuous rotations	No true orientational long-range order; a Berezinskii–Kosterlitz–Thouless transition to quasi-long-range order is possible Oganesyan, Kivelson, and Fradkin 2001, p. 195109-1
Lattice nematic with $C_4\to C_2$	The broken symmetry is discrete, so finite-temperature Ising order is allowed; explicit strain is a conjugate field that rounds the transition
SDW with exact continuous spin symmetry	No finite-temperature magnetic long-range order in broad one- and two-dimensional Hubbard and $t$–$J$ classes Koma and Tasaki 1992, pp. 3248–3251. The classic short-range spin-model theorem is Mermin and Wagner 1966, pp. 1133–1136, and Loss, Pedrocchi, and Leggett 2011, pp. 107201-1–4 extend it to localized lattice spins coupled to interacting itinerant carriers. Spin–orbit anisotropy, long-range interactions, or interlayer coupling can change the conclusion
Commensurate CDW breaking only discrete translations	Finite-temperature long-range order is allowed
Ideal unpinned incommensurate CDW with short-range phase stiffness	Its continuous sliding phase is XY-like, so at most quasi-long-range order survives below a BKT transition Kosterlitz and Thouless 1973, pp. 1181–1203. Commensurability and impurity pinning McMillan 1975, pp. 1187–1196, Coulomb forces, or interlayer coupling alter this conclusion

Therefore a ladder or fRG denominator that vanishes marks a normal-state quadratic breakdown within that approximation. Where an exact theorem forbids conventional long-range order, it is not a conventional ordering temperature. In an XY-like channel, the exact susceptibility can diverge through quasi-long-range order at a BKT transition, but a ladder pole alone does not locate $T_{\mathrm{BKT}}$; that requires stiffness and vortex-defect analysis.

Finally, uniform nematicity need not be a primary Pomeranchuk bilinear. If two symmetry-related density-wave fields $\phi_x$ and $\phi_y$ fluctuate at $\mathbf Q_x$ and $\mathbf Q_y$, define the composite field

$$\mathcal N_{\mathrm{vest}} =\lvert\phi_x\rvert^2-\lvert\phi_y\rvert^2.$$

Vestigial order means $N_{\mathrm{vest}}=\langle\mathcal N_{\mathrm{vest}}\rangle\ne0$ while $\langle\phi_x\rangle=\langle\phi_y\rangle=0$. It preserves translations but breaks the same point-group symmetry as a primary nematic. Coincident onsets or identical symmetry therefore cannot identify which field is primary; the detailed free-energy and strain analysis belongs to competing orders and electronic nematicity. A microscopic preemptive example is given by Fernandes et al. 2012, §§ II–IV.

Common pitfalls

Calling every $q=0$ deformation nematic. Uniform density, magnetization, and fully symmetric band deformations are $q=0$ but do not have the defining nontrivial spatial transformation law. State the point-group representation and the conjugate field.

Identifying a nesting picture with a density wave. A translated Fermi-surface overlap is only one contribution to the bare response. The static interacting eigenvalue, form factor, and thermodynamic scaling decide whether a density wave forms.

Reading a ladder pole as the complete phase transition. The pole diagnoses loss of quadratic stability within a specified normal-state kernel. It does not determine nonlinear saturation, transition order, fluctuation effects, or whether another channel preempts it.

Equating negative electronic compressibility with macroscopic phase separation. Long-range Coulomb forces suppress a uniform charge mode and can favor an intermediate scale. State the neutralizing background, screening environment, and ensemble before interpreting $F_0^s$.

Using anisotropy without removing explicit fields. Strain, orthorhombicity, domains, probe matrix elements, and spin–orbit coupling can all create or select anisotropy. Spontaneous order requires the correct zero-field limit.

Exercises

Recover the spherical stability bound

Using

$$F(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}') =\sum_\ell F_\ell P_\ell(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}'),$$

show that $Y_{\ell m}$ has eigenvalue $F_\ell/(2\ell+1)$ under integration with $\mathrm d\Omega'/(4\pi)$. Find the symmetric $\ell=2$ stability bound.

Solution

The addition theorem is

$$P_L(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}') =\frac{4\pi}{2L+1} \sum_{M=-L}^{L} Y_{LM}(\hat{\mathbf k})Y_{LM}(\hat{\mathbf k}')^*.$$

Orthogonality removes every term except $L=\ell$, $M=m$, leaving

$$\int\frac{\mathrm d\Omega'}{4\pi} F(\hat{\mathbf k}\mathbin{\cdot}\hat{\mathbf k}') Y_{\ell m}(\hat{\mathbf k}') =\frac{F_\ell}{2\ell+1}Y_{\ell m}(\hat{\mathbf k}).$$

Positive quadratic stiffness requires $1+F_\ell/(2\ell+1)>0$. For the symmetric quadrupole, $\ell=2$, so $F_2^s>-5$. Equality is a soft $q=0$ quadrupolar mode, not yet a proof of a continuous nematic transition.

Diagonalize a two-form-factor density-wave kernel

Suppose the Hermitian static kernel in two normalized form factors is

$$\mathbf K(g)=g \begin{pmatrix} 2&1\\ 1&2 \end{pmatrix}, \qquad g>0.$$

Find the leading eigenmode and the value at which the ladder response first becomes singular. What has and has not been established there?

Solution

The normalized eigenvectors are $(1,1)/\sqrt2$ and $(1,-1)/\sqrt2$, with eigenvalues $3g$ and $g$. The symmetric mixture is leading, and its resolvent factor $(1-3g)^{-1}$ becomes singular at $g_c=1/3$.

This establishes a quadratic instability of the specified two-form-factor normal-state ladder at the chosen $\mathbf Q$. It does not establish basis convergence, a thermodynamic transition, its order, the ordered amplitude, or dominance over wavevectors and channels omitted from the matrix.

Find the Coulomb-frustrated wavevector

A schematic three-dimensional inverse charge response is

$$\mathcal R_c^{-1}(q)=r+cq^2+\frac{A}{q^2}, \qquad c>0,\quad A>0.$$

Find its softest wavevector and the value of $r$ at which the quadratic mode first vanishes.

Solution

Differentiating the $q$-dependent part gives

$$2cq-\frac{2A}{q^3}=0,$$

so

$$q_*=(A/c)^{1/4}.$$

At this wavevector, $cq_*^2+A/q_*^2=2\sqrt{Ac}$. The quadratic instability therefore occurs at $r_c=-2\sqrt{Ac}$ rather than at $r=0$. Long-range Coulomb repulsion has removed the $q=0$ soft mode in this model and selected an intermediate scale. Higher gradients, screening, lattice commensurability, and nonlinear terms are still needed to identify the actual phase.

Distinguish primary and vestigial nematicity

On a square lattice, a $90^\circ$ rotation exchanges density-wave fields $\phi_x$ and $\phi_y$. Show how $\mathcal N_{\mathrm{vest}}=\lvert\phi_x\rvert^2-\lvert\phi_y\rvert^2$ transforms. Can $\langle\mathcal N_{\mathrm{vest}}\rangle$ be nonzero while translations remain unbroken, and does that make it a Pomeranchuk bilinear?

Solution

The rotation sends $\phi_x\leftrightarrow\phi_y$, hence $\mathcal N_{\mathrm{vest}}\to-\mathcal N_{\mathrm{vest}}$. A state with unequal fluctuation strengths,

$$\langle\lvert\phi_x\rvert^2\rangle \ne \langle\lvert\phi_y\rvert^2\rangle, \qquad \langle\phi_x\rangle=\langle\phi_y\rangle=0,$$

breaks $C_4$ to $C_2$ while preserving translations. It is nematic, but it is a composite of finite-wavevector fields, not the primary $q=0$ fermion bilinear that defines a Pomeranchuk route. Because both transform identically, symmetry permits them to mix; mechanism requires additional evidence.

Continue

Competing orders and electronic nematicity develops primary–vestigial mixing, strain, domains, and coexistence. Itinerant magnetism and spin fluctuations continues from the spin susceptibility to Stoner and paramagnon physics. Electron–phonon fields and retarded interactions treats the coupled lattice channel, while functional RG compares it with pairing and other weak-coupling tendencies.

References

• Victor J. Emery and Steven A. Kivelson, “Frustrated Electronic Phase Separation and High-Temperature Superconductors,” Physica C: Superconductivity 209 (1993) 597–621, doi:10.1016/0921-4534(93)90581-A.
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