Competing Orders

Competing orders are distinct collective organizations close enough in free energy that developing one changes the stability, amplitude, phase coherence, or spatial texture of another. Antiferromagnetism and superconductivity, charge order and superconductivity, nematicity and magnetism, or two flavor-polarized moiré states may share microscopic degrees of freedom while breaking different symmetries.

Competition does not mean that two orders can never be nonzero in the same region. A positive coupling can suppress both amplitudes yet still permit a homogeneous coexistence phase. Conversely, two anomalies in one bulk sample do not establish microscopic coexistence: different domains, surface and bulk regions, disorder distributions, or fluctuating order can contribute separate signals.

Required background. Order Parameters supplies the operators and symmetry transformations labeling each order. Landau Theory supplies symmetry-allowed free-energy expansions and equilibrium stability tests. Charge and Spin Density Waves supplies finite-wavevector order and its probe signatures, while Ginzburg–Landau Theory supplies the superconducting amplitude and gauge-covariant gradients entering the coupled examples.

Two coupled order parameters

Let $\phi$ and $\psi$ denote real scalar amplitudes for two orders whose symmetries forbid a bilinear coupling. The uniform free-energy density through quartic order is

$$f(\phi,\psi)=\frac{r_\phi}{2}\phi^2+\frac{u_\phi}{4}\phi^4 +\frac{r_\psi}{2}\psi^2+\frac{u_\psi}{4}\psi^4+\frac{g}{2}\phi^2\psi^2.$$

Here $u_\phi,u_\psi>0$. For vector or complex orders, replace squares by invariant norms and add the independent quartic invariants allowed by symmetry.

The stationary equations are

$$\begin{aligned} 0&=\phi\left(r_\phi+u_\phi\phi^2+g\psi^2\right),\\ 0&=\psi\left(r_\psi+g\phi^2+u_\psi\psi^2\right). \end{aligned}$$

The disordered solution has $\phi=\psi=0$ and is locally stable when both masses are positive. For $r_\phi<0$ or $r_\psi<0$, respectively, the pure stationary solutions are

$$\begin{aligned} \phi^2&=-\frac{r_\phi}{u_\phi}, &\psi&=0, &f_\phi&=-\frac{r_\phi^2}{4u_\phi},\\ \psi^2&=-\frac{r_\psi}{u_\psi}, &\phi&=0, &f_\psi&=-\frac{r_\psi^2}{4u_\psi}. \end{aligned}$$

Define

$$D=u_\phi u_\psi-g^2.$$

When $D\neq0$, the formal coexistence solution is

$$\begin{aligned} \phi^2 &= \frac{g r_\psi-u_\psi r_\phi}{D},\\ \psi^2 &= \frac{g r_\phi-u_\phi r_\psi}{D}. \end{aligned}$$

It is a stable homogeneous minimum only when both squared amplitudes are positive and $D>0$. The Hessian determinant at that solution is

$$\det H = 4\phi^2\psi^2D.$$

Competition can coexist

For $g>0$, either order raises the other’s effective quadratic coefficient:

$$\begin{aligned} r_{\phi,\mathrm{eff}}&=r_\phi+g\psi^2,\\ r_{\psi,\mathrm{eff}}&=r_\psi+g\phi^2. \end{aligned}$$

This is amplitude competition. If $0<g<\sqrt{u_\phi u_\psi}$, however, $D>0$ and a coexistence region is possible. Competition means mutual suppression relative to uncoupled amplitudes, not logical exclusion.

If $g>\sqrt{u_\phi u_\psi}$, the mixed stationary point is unstable. Two pure phases can then meet along a first-order boundary where their free energies are equal, on the branches with both $r_\phi,r_\psi<0$:

$$\frac{r_\phi^2}{u_\phi} = \frac{r_\psi^2}{u_\psi}.$$

For $g<0$, the orders reinforce one another. If $g<-\sqrt{u_\phi u_\psi}$, the quartic truncation is unbounded along a mixed direction and sixth-order terms are required. The resulting transition is often first order, but its topology depends on those higher terms.

Symmetry decides more than the sign of one coefficient

The biquadratic term is generic when $\phi$ and $\psi$ break unrelated symmetries. Other couplings carry stronger information:

• A bilinear $\lambda\phi\psi$ is allowed when the product is symmetry invariant. This can hold in the original symmetry group, or become possible after an external field lowers the symmetry. It hybridizes the modes, so the original fields need not label independent transitions.
• A trilinear $\lambda\phi\psi\eta$ is allowed when the product transforms as a scalar. Condensing two fields then acts as a source for the third.
• A term $\lambda\eta\phi^2$ can induce a noncritical scalar response, such as a fully symmetric strain. If $\eta$ breaks another symmetry, it must multiply a bilinear with the same symmetry, such as $|M_x|^2-|M_y|^2$ for an appropriate nematic channel.
• Gradient terms couple domain walls, vortices, strain textures, and finite-wavevector order even when the uniform amplitudes repel.
• Quenched disorder can randomize local masses $r_a(\mathbf r)$ and convert a sharp bulk topology into spatially distributed onsets.

Deriving this invariant list from symmetry is more reliable than choosing a coupling because it produces a desired phase diagram.

Worked example: suppression versus exclusion

Take dimensionless coefficients $r_\phi=r_\psi=-1$ and $u_\phi=u_\psi=1$. Without coupling, each squared amplitude is one. With $g=1/2$, the mixed solution has $\phi^2=\psi^2=2/3$. Its free energy is $-1/3$, lower than either pure-state value $-1/4$. Both orders are suppressed, yet coexistence is the stable equilibrium.

For $g=3/2$, the mixed stationary point has $\phi^2=\psi^2=2/5$ and free energy $-1/5$. It is a saddle, and either pure phase has lower free energy $-1/4$. Thus a positive coupling can yield either coexistence or exclusion depending on its magnitude.

The negative determinant in the second example does not make the quartic energy unbounded. The squared amplitudes cannot be negative. Indeed,

$$4f_4=(\sqrt{u_\phi}\,\phi^2-\sqrt{u_\psi}\,\psi^2)^2 +2\bigl(g+\sqrt{u_\phi u_\psi}\bigr)\phi^2\psi^2.$$

It is strictly stabilizing for $g>-\sqrt{u_\phi u_\psi}$. At equality a flat quartic direction appears, so the lower-order terms and higher powers must be examined. Below that value the quartic truncation fails along a mixed direction. This is distinct from the loss of a mixed minimum at large positive $g$.

Phase-Diagram Topologies

Write the leading masses near two bare transition lines as

$$\begin{aligned} r_\phi(T,x)&=a_\phi[T-T_\phi^{(0)}(x)],\\ r_\psi(T,x)&=a_\psi[T-T_\psi^{(0)}(x)]. \end{aligned}$$

The crossing of $T_\phi^{(0)}$ and $T_\psi^{(0)}$ does not by itself give the observed topology. The coupling and fluctuations decide what replaces that crossing.

Tetracritical behavior

For a stable mixed minimum, four phases can meet: disordered, pure $\phi$, pure $\psi$, and coexistence. Each boundary may be continuous at mean-field level. Entering the mixed phase from the pure-$\phi$ phase occurs when

$$r_\psi-\frac{g}{u_\phi}r_\phi=0,$$

because $\phi^2=-r_\phi/u_\phi$ renormalizes the $\psi$ mass. The analogous boundary follows by exchanging the fields.

Bicritical behavior

When mutual repulsion destabilizes the mixed minimum, the two continuous disorder-to-order lines meet a first-order line separating pure phases. Hysteresis, latent heat, discontinuous order-parameter jumps, and two-phase coexistence may occur along that line. Finite sweep rates and disorder can broaden these signatures, so absence of visible hysteresis is not proof of continuity.

Minimizing the stated quartic free energy with $u_\phi=u_\psi=1$ gives these regions in the $(r_\phi,r_\psi)$ plane. For $g=1/2$, the mixed boundaries are $r_\psi=r_\phi/2$ and $r_\phi=r_\psi/2$. For $g=3/2$, the dashed negative diagonal is the first-order boundary $r_\phi=r_\psi<0$. Solid boundaries are continuous in this mean-field model. Temperature and pressure trace material-specific paths through this coefficient plane.

Fluctuation corrections

Mean-field topology is a starting hypothesis. Near a multicritical point, order-parameter fluctuations renormalize $u_\phi$, $u_\psi$, and $g$. Depending on component numbers and dimension, the infrared behavior may approach an isotropic, biconical, or decoupled fixed point, or run toward a fluctuation-driven first-order transition. A proposed enlarged symmetry such as SO(5) requires matching susceptibilities, scaling dimensions, and symmetry-breaking perturbations; visual similarity between two domes is not enough.

Long-range strain, Coulomb interactions, metallic damping, quenched disorder, and coupling to gapless fermions can all invalidate the short-range classical functional. Quantum Criticality develops the zero-temperature and dynamical extensions.

First-order slopes are thermodynamic data

Let $f$ be free-energy density, $S=-\partial f/\partial T$ entropy density, and a tuning field $x$ have conjugate density $Y=-\partial f/\partial x$. Along a first-order line $x_c(T)$, equality of the two free-energy densities gives

$$\frac{dx_c}{dT} = -\frac{\Delta S}{\Delta Y}.$$

The sign and magnitude constrain proposed phase assignments. A kink in transport that violates known entropy and magnetization or volume jumps is unlikely to be the claimed equilibrium boundary.

What Coexistence Means

Homogeneous, microscopic coexistence

Both order parameters are nonzero within the same thermodynamic phase and the same microscopic volume. Examples include a magnetic superconductor in which the same electronic fluid supports a superconducting condensate and a static internal magnetic field. The strongest evidence combines a bulk superconducting volume fraction with a local magnetic probe whose entire spectral weight participates.

Phase separation

Different spatial regions realize different pure phases. Macroscopic separation obeys ordinary coexistence thermodynamics. Mesoscopic domains can arise from strain compatibility, disorder, interfacial energy, or long-range Coulomb frustration. A diffraction experiment whose beam covers many domains reports both Bragg signals even if no unit cell contains both orders.

Textured coexistence

Include gradients,

$$f_{\mathrm{grad}} = \frac{K_\phi}{2}|\nabla\phi|^2 + \frac{K_\psi}{2}|\nabla\psi|^2.$$

For $g>0$, suppressing $\phi$ in a vortex core, domain wall, defect halo, or surface lowers the local mass of $\psi$. The second order may therefore nucleate only in those textures. This is real spatial coexistence but not a uniform mixed phase.

Fluctuating coexistence

Two susceptibilities may be large while neither order has a static expectation value. A probe with integration time $\tau_{\mathrm p}$ can call fluctuations “static” when their correlation time exceeds $\tau_{\mathrm p}$. Every coexistence statement should therefore report spatial resolution, time window, and volume fraction.

Observation	What it establishes	What remains unresolved
Two bulk anomalies	Two thermodynamic or crossover scales may exist	Same microscopic volume or separate fractions
Two diffraction peaks	Both correlations occur in the illuminated volume	Co-location below the beam size
Full local-probe volume plus bulk order	Strong evidence for microscopic coexistence	Which microscopic electrons carry each order
Anticorrelated local maps	Spatial competition at the mapped resolution	Equilibrium coupling versus disorder landscape
One order rises when another is selectively suppressed	Evidence for positive effective coupling	Direct effect of the tuning field on both orders
Pump-induced transient enhancement	Nonequilibrium access to another basin	Equilibrium free-energy ordering

How tuning tests the coupling

Inside the stable mixed phase, holding every other bare coefficient fixed gives

$$\frac{\partial\phi^2}{\partial r_\psi}=\frac{g}{D}, \qquad \frac{\partial\psi^2}{\partial r_\psi}=-\frac{u_\phi}{D}.$$

For $g>0$, making $r_\psi$ less negative suppresses $\psi$ and increases $\phi^2$. A magnetic field or pressure rarely changes only one coefficient, however. Its direct effects on both orders must be distinguished from this indirect response. Peak intensity also depends on correlation length and ordered volume, so it need not equal a local squared amplitude.

Three experiments illustrate different questions:

• In $\mathrm{YBa}_2\mathrm{Cu}_3\mathrm{O}_{6.67}$, Chang and collaborators found that a field suppressing superconductivity enhances charge order. This supports competitive coupling; it does not by itself reduce a spatially correlated density wave to one uniform scalar amplitude.
• In $\mathrm{CeRhIn}_5$ near $1.75\,\mathrm{GPa}$, Mito and collaborators combined magnetic splitting of the NQR spectrum with a bulk Meissner response. Local and bulk probes together support microscopic coexistence more strongly than two resistive anomalies would.
• In magic-angle graphene, Stepanov and collaborators found superconductivity could survive suppression of prominent insulating states by screening. A neighboring insulator is therefore not necessarily a required parent phase. Screening changes interactions and can affect several orders simultaneously.

For material details, see Charge and Spin Density Waves, Heavy Fermions, and Moiré Superconductivity. Phase locking and induced composite orders require more than the scalar repulsion in this model; Pair-Density Waves and Exotic Orders develops those examples.

Equilibrium and transient states answer different questions. A pump can change the two relaxation rates or trap a metastable state. Transient enhancement need not identify the lower equilibrium free energy; heating, recovery, and delay dependence must be included before assigning a new equilibrium boundary.

Exercises

1. Derive the coexistence solution

Starting from the two-order free energy, solve for a stationary point with $\phi\neq0$ and $\psi\neq0$, and determine its local stability condition.

Solution

Dividing the stationarity equations by the nonzero amplitudes gives

$$\begin{pmatrix} u_\phi & g\\ g & u_\psi \end{pmatrix} \begin{pmatrix} \phi^2\\ \psi^2 \end{pmatrix} = - \begin{pmatrix} r_\phi\\ r_\psi \end{pmatrix}.$$

Inverting the matrix yields

$$\begin{aligned} \phi^2 &= \frac{g r_\psi-u_\psi r_\phi}{D}, \\ \psi^2 &= \frac{g r_\phi-u_\phi r_\psi}{D}. \end{aligned}$$

where $D=u_\phi u_\psi-g^2$. Both expressions must be positive. At the stationary point,

$$H = 2 \begin{pmatrix} u_\phi\phi^2 & g\phi\psi\\ g\phi\psi & u_\psi\psi^2 \end{pmatrix}.$$

Its diagonal entries are positive and $\det H=4\phi^2\psi^2D$, so local stability requires $D>0$.

2. Classify two multicritical points

Take $u_\phi=u_\psi=1$. Compare $g=0.5$ and $g=1.5$ near a crossing of the bare transition lines.

Solution

For $g=0.5$,

$$D=1-0.25=0.75>0.$$

A stable mixed solution can occupy a wedge where both squared amplitudes are positive. The mean-field topology is tetracritical. For $g=1.5$,

$$D=1-2.25=-1.25<0.$$

The mixed stationary point is unstable. The pure phases exclude each other and meet along a first-order boundary, giving bicritical topology. Because $g>0$, the quartic energy remains bounded for nonnegative $\phi^2$ and $\psi^2$ despite the negative determinant.

3. Integrate out an induced secondary order

Let a fully symmetric, noncritical scalar distortion $\eta$ couple to a primary order through

$$f_\eta = \frac{r_\eta}{2}\eta^2 + \lambda\eta\phi^2, \qquad r_\eta>0.$$

Eliminate $\eta$ and state the effect on the quartic coefficient of $\phi$.

Solution

Minimization gives

$$\eta_* = -\frac{\lambda}{r_\eta}\phi^2.$$

Substitution yields

$$f_\eta(\eta_*) = -\frac{\lambda^2}{2r_\eta}\phi^4.$$

If the original quartic term is $u_\phi\phi^4/4$, then

$$u_{\phi,\mathrm{eff}} = u_\phi-\frac{2\lambda^2}{r_\eta}.$$

The distortion is induced and does not independently break a symmetry, and sufficiently strong coupling can make the effective quartic negative, requiring sixth-order stabilization and allowing a first-order transition.

4. Predict a selective-tuning response

Inside a stable coexistence phase with $g>0$, a perturbation raises $r_\psi$ while leaving the other bare coefficients fixed. What happens to both amplitudes?

Solution

Differentiating the mixed solution gives

$$\frac{\partial\phi^2}{\partial r_\psi} = \frac{g}{D}>0, \qquad \frac{\partial\psi^2}{\partial r_\psi} = -\frac{u_\phi}{D}<0.$$

The perturbation directly suppresses $\psi$ and releases $\phi$ from positive biquadratic competition. Observing this response supports $g>0$ only if the perturbation’s direct coupling to $\phi$ is independently bounded.

5. Distinguish two signals from one mixed phase

A sample shows a superconducting resistive transition and a magnetic diffraction peak. Is microscopic coexistence established? Design a stronger test.

Solution

No. A percolating superconducting path can produce zero resistance while a different volume fraction produces magnetic diffraction. Establish bulk superconductivity through heat capacity, magnetic shielding with demagnetization control, or superfluid stiffness. Then use a local magnetic probe such as NMR, NQR, or $\mu$SR to determine whether essentially the same sample volume develops static magnetism. Spatially resolved diffraction or microscopy should bound domain size, and pressure or field sweeps should test whether both signals follow a common equilibrium boundary.

6. Use a first-order slope

Two phases meet along $x_c(T)$. The entropy-density and conjugate-density jumps are $\Delta S=S_2-S_1$ and $\Delta Y=Y_2-Y_1$, using the same volume normalization as $f$. Derive the slope and interpret the case $\Delta S>0$, $\Delta Y>0$.

Solution

Along coexistence, $f_1=f_2$ and

$$df_i=-S_i\,dT-Y_i\,dx.$$

Equating differentials gives

$$\Delta S\,dT+\Delta Y\,dx=0,$$

so

$$\frac{dx_c}{dT} = -\frac{\Delta S}{\Delta Y}.$$

If both jumps are positive, the line has negative slope: increasing temperature favors phase 2 entropically, so coexistence requires a smaller value of the field $x$ that also favors phase 2.

References

• C. A. Balseiro and L. M. Falicov, “Superconductivity and charge-density waves”, Physical Review B 20, 4457–4464 (1979). Early microscopic treatment of competing superconducting and density-wave gaps.
• K. Binder, “Theory of first-order phase transitions”, Reports on Progress in Physics 50, 783–859 (1987). Metastability, nucleation, and finite-size signatures.
• P. Calabrese, A. Pelissetto, and E. Vicari, “Multicritical phenomena in O($n_1$)+O($n_2$)-symmetric theories”, Physical Review B 67, 054505 (2003). Renormalization-group fixed points beyond mean field.
• J. Chang et al., “Direct observation of competition between superconductivity and charge density wave order in YBa$_2$Cu$_3$O$_{6.67}$”, Nature Physics 8, 871–876 (2012). Magnetic-field release of charge order.
• J. Choi et al., “Spatially inhomogeneous competition between superconductivity and the charge density wave in YBa$_2$Cu$_3$O$_{6.67}$”, Nature Communications 11, 990 (2020). Distinct CDW regions and spatial competition.
• E. Demler, W. Hanke, and S.-C. Zhang, “SO(5) theory of antiferromagnetism and superconductivity”, Reviews of Modern Physics 76, 909–974 (2004). Enlarged-symmetry framework and experimental tests.
• E. Fradkin, S. A. Kivelson, and J. M. Tranquada, “Colloquium: Theory of intertwined orders in high temperature superconductors”, Reviews of Modern Physics 87, 457–482 (2015). Symmetry, stripes, pair-density waves, and defects.
• J. M. Kosterlitz, D. R. Nelson, and M. E. Fisher, “Bicritical and tetracritical points in anisotropic antiferromagnetic systems”, Physical Review B 13, 412–432 (1976). Multicritical topology and scaling.
• T. Mito et al., “Coexistence of antiferromagnetism and superconductivity near the quantum criticality of the heavy-fermion compound CeRhIn$_5$”, Physical Review Letters 90, 077004 (2003). NQR and bulk evidence for microscopic coexistence.
• P. Stepanov et al., “Untying the insulating and superconducting orders in magic-angle graphene”, Nature 583, 375–378 (2020). Screening test of the parent-insulator narrative.
• S. Sun and A. J. Millis, “Transient trapping into metastable states in systems with competing orders”, Physical Review X 10, 021028 (2020). Nonequilibrium basin selection and timescale effects.