Condensed Matter: Superconductivity, Quantum Hall & Topological Phases

Condensed Matter Physics » Superconductivity, Quantum Hall & Topological Phases

This page surveys the phases of matter that exist only because very many particles act collectively: superconductors, quantum Hall fluids, topological insulators and semimetals, topological spin textures, strongly correlated metals and insulators, and the ordered phases of soft matter. For each it gives the defining phenomena, the standard theory at the level of its key equations, and the current experimental state of the field. Full derivations and the field-theoretic machinery (Bogoliubov-de Gennes, K-matrix Chern-Simons theory, DMFT, the tenfold way) are on Graduate-Level Formalism; the role of disorder in the quantum Hall effect is developed on Disorder & Localization.

Organizing principles

Landau’s paradigm classifies phases by spontaneous symmetry breaking: a magnet breaks spin-rotation symmetry, a crystal breaks translation symmetry, a superconductor breaks the U(1) phase symmetry of the electron wavefunction, and each is described by a local order parameter. Since the discovery of the quantum Hall effect in 1980 it has been clear that this is not the whole story. Some gapped phases differ not in any local order parameter but in a global, quantized property of their ground-state wavefunction, and that difference can only change if the energy gap closes.

flowchart TD
    G["Gapped phases of matter"] --> SSB["Symmetry breaking<br/>local order parameter<br/>(magnets, crystals, superfluids, superconductors)"]
    G --> SRE["Short-range entangled,<br/>no symmetry breaking"]
    G --> LRE["Long-range entangled:<br/>intrinsic topological order"]
    SRE --> TRIV["Trivial insulators<br/>(adiabatically connected to a product state)"]
    SRE --> INV["Invertible topological phases<br/>no symmetry needed<br/>(integer QH, Chern insulators)"]
    SRE --> SPT["Symmetry-protected topological<br/>(Z2 topological insulators, Haldane chain)"]
    LRE --> TO["Fractionalized phases with anyons<br/>(fractional QH, Z2 spin liquids)"]

Gapless phases (metals, Dirac and Weyl semimetals, critical points) sit outside this chart and are classified by the topology of their Fermi surfaces or band touchings. The recurring theme is the bulk-boundary correspondence: a nontrivial bulk invariant forces gapless states on the boundary, which is what most experiments detect.

Superconductivity

Superconductivity was discovered by Kamerlingh Onnes in 1911 in mercury below 4.2 K. It is a thermodynamic phase with a set of sharply defined properties:

London equations

The London brothers (1935) described the electrodynamics with a supercurrent proportional to the vector potential, $\mathbf{j}_s = -(n_se^2/m)\mathbf{A}$. Combined with Ampère’s law this gives

so a field decays exponentially over the London penetration depth $\lambda_L$, typically 50-500 nm. This is the Meissner effect.

Ginzburg-Landau theory and the two types of superconductor

Ginzburg-Landau (GL) theory (1950) writes the free energy near $T_c$ in terms of a complex order parameter $\psi(\mathbf{r})$, with $|\psi|^2$ the superfluid density:

Two lengths emerge: the coherence length $\xi = \hbar/\sqrt{2m^\ast|\alpha|}$, over which $\psi$ can vary, and the penetration depth $\lambda$, over which fields are screened. Their ratio, the GL parameter $\kappa = \lambda/\xi$, divides superconductors into two classes:

In the mixed state, flux enters as Abrikosov vortices, each carrying one flux quantum $\Phi_0$ around a normal core of radius $\sim\xi$, and they arrange into a triangular lattice. The upper critical field is $\mu_0H_{c2} = \Phi_0/(2\pi\xi^2)$, which exceeds 100 T in some cuprates. Pinning of vortices by defects is what lets type II wires (NbTi, Nb$_3$Sn, REBCO tapes) carry large currents in MRI, particle accelerator, and fusion magnets.

BCS theory

The microscopic theory of Bardeen, Cooper, and Schrieffer (1957) rests on Cooper’s observation that a filled Fermi sea is unstable to pairing under any net attraction, however weak. In conventional superconductors the attraction is phonon-mediated: an electron distorts the lattice, and the slowly relaxing distortion attracts a second electron. Pairs form between time-reversed states $(\mathbf{k}\uparrow, -\mathbf{k}\downarrow)$ within the Debye energy $\hbar\omega_D$ of the Fermi surface. The ground state is a coherent superposition of pair occupancies,

whose excitations are Bogoliubov quasiparticles with energy $E_{\mathbf{k}} = \sqrt{\xi_{\mathbf{k}}^2 + |\Delta_{\mathbf{k}}|^2}$, where $\xi_{\mathbf{k}}$ is the band energy measured from the Fermi level. The gap is determined self-consistently by the gap equation

For a constant attraction $V$ within $\hbar\omega_D$ of the Fermi surface and density of states $N(0)$ per spin, BCS theory gives universal weak-coupling results:

The non-analytic dependence on $V$ explains why pairing is invisible to perturbation theory. Coherence lengths range from about 1.6 μm in aluminium to 1-2 nm in cuprates. Strong electron-phonon coupling (Pb, Hg, hydrides) is treated by Eliashberg theory, which raises $2\Delta/k_BT_c$ above 3.53. The Anderson-Higgs mechanism, by which the photon acquires a mass inside a superconductor, is the condensed-matter prototype of the Higgs mechanism of particle physics.

Josephson effects and superconducting circuits

Two superconductors separated by a thin barrier (an insulator, normal metal, or constriction) exchange Cooper pairs coherently. Josephson (1962) predicted:

where $\varphi$ is the phase difference across the junction. A dc supercurrent flows with no voltage; a dc voltage $V$ produces an alternating current at $f = 2eV/h \approx 483.6\ \text{GHz per mV}$. Irradiating a junction with microwaves produces constant-voltage Shapiro steps at $V_n = nhf/2e$, which since the 2019 SI redefinition (exact $e$ and $h$) realize the volt.

Josephson junctions are the active element of superconducting electronics:

• SQUIDs. Two junctions in a loop form an interferometer whose critical current oscillates with period $\Phi_0$ in the enclosed flux, giving magnetometers sensitive to below $10^{-6}\,\Phi_0/\sqrt{\text{Hz}}$, used in materials characterization, geophysics, and magnetoencephalography.
• Superconducting qubits. The junction is a nonlinear, dissipationless inductor. Shunted by a capacitor it becomes an anharmonic oscillator whose lowest two levels form a qubit; the transmon, which operates with $E_J/E_C$ of order 50 to suppress charge noise, is the basis of most superconducting quantum processors. The 2025 Nobel Prize in Physics (Clarke, Devoret, Martinis) recognized the 1980s experiments demonstrating macroscopic quantum tunneling and energy quantization in such circuits.

Unconventional superconductors

In conventional superconductors the gap is nodeless and has the full symmetry of the crystal (s-wave). In unconventional superconductors the gap changes sign over the Fermi surface, which usually signals a pairing mechanism based on electronic (often spin-fluctuation) rather than phonon attraction.

No superconductor has been confirmed above ~250 K, and none near room temperature at ambient pressure. Widely publicized claims (the Lu-N-H hydride in 2023, LK-99 in 2023) did not survive replication.

Quantum Hall effects

Confine electrons to two dimensions, cool them to a few kelvin or below, and apply a strong perpendicular magnetic field: the Hall resistance locks onto $h/(\nu e^2)$ with $\nu$ an integer or a simple fraction, reproducible to parts in $10^{10}$ regardless of sample geometry, material, or disorder. That precision is the first laboratory signature of topology in a material: the Hall conductance counts a topological invariant that cannot change under smooth deformation. The von Klitzing constant $R_K = h/e^2 = 25\,812.807\,45\ldots\ \Omega$ is exact in the 2019 SI, and quantum Hall devices, increasingly made of graphene, realize the ohm.

Landau levels

A perpendicular field $B$ quantizes the cyclotron motion of 2D electrons into Landau levels

each with degeneracy $eB/h$ per unit area (one state per flux quantum $h/e$). The filling factor $\nu = nh/(eB)$ counts filled levels. In graphene the Dirac dispersion gives instead $E_n = \pm v_F\sqrt{2e\hbar B|n|}$ with a zero-energy level, producing plateaus at $\nu = \pm 2, \pm 6, \pm 10, \ldots$ and a quantum Hall effect visible at room temperature in strong fields.

Integer quantum Hall effect

Discovered by von Klitzing in 1980 (Nobel Prize 1985). When $\nu$ integer levels are filled the bulk is gapped and the Hall conductance is

Three complementary explanations exist:

• Topology. $\sigma_{xy}$ is $e^2/h$ times the sum of the Chern numbers of the filled bands (the TKNN formula), an integer by construction.
• Edge states. At the sample edge the confining potential bends Landau levels up through the Fermi energy, creating $\nu$ chiral edge channels that propagate in one direction only. Backscattering would require an electron to cross the sample, so transport along the edge is dissipationless and each channel contributes $e^2/h$ (Landauer-Büttiker).
• Disorder. Plateaus of finite width require localized states: as $B$ changes, the Fermi level moves through localized states that carry no current, so $\sigma_{xy}$ stays fixed. Only one energy per Landau level hosts extended states, where the plateau transition occurs (see Disorder & Localization).

Fractional quantum Hall effect

Tsui, Störmer, and Gossard found a plateau at $\nu = 1/3$ in 1982 (Nobel Prize 1998, shared with Laughlin). At fractional filling the non-interacting picture predicts a partially filled, massively degenerate level with no gap; the plateau exists only because Coulomb interactions select a unique, incompressible correlated state. Laughlin’s wavefunction for $\nu = 1/m$ ($m$ odd),

with $z = x + iy$, keeps electrons apart with an $m$-th order zero and is an excellent approximation to the exact ground state. Its excitations are fractionally charged quasiparticles of charge $e/m$ with anyonic exchange statistics $\theta = \pi/m$.

The composite-fermion picture (electrons bound to two flux quanta, moving in a reduced effective field) and the Chern-Simons effective theory are developed on Graduate-Level Formalism. Non-abelian anyons at $\nu = 5/2$ and related states are the original motivation for topological quantum computation, in which information is stored nonlocally and processed by braiding.

Anomalous quantum Hall effects

A quantized Hall conductance needs broken time-reversal symmetry and nonzero Chern number, not a magnetic field as such. Haldane showed in 1988 that a honeycomb lattice with complex hoppings realizes a Chern insulator at zero net field.

• Quantum anomalous Hall (QAH) effect. Observed in 2013 in magnetically doped (Cr- or V-doped) (Bi,Sb)$_2$Te$_3$ films, where ferromagnetism gaps the topological surface state, and later in the intrinsic magnetic topological insulator MnBi$_2$Te$_4$ (2020) and in moiré graphene.
• Fractional quantum anomalous Hall (FQAH) effect. Fractionally quantized Hall resistance at zero field was observed in 2023 in twisted bilayer MoTe$_2$ (at $\nu = -2/3$ and $-3/5$) and in 2024 in rhombohedral pentalayer graphene aligned with hBN. These are lattice analogues of FQH states (fractional Chern insulators), formed in flat Chern bands whose Berry curvature is nearly uniform, and they have become a leading platform for zero-field anyons.

Topological phases

Berry phase and Chern number

The mathematical engine is the Berry phase, the geometric phase a state acquires when its Hamiltonian is carried slowly around a closed loop in parameter space:

For Bloch electrons the parameter space is the Brillouin zone. The Berry curvature of band $n$ is $\Omega_n(\mathbf{k}) = \nabla_{\mathbf{k}}\times i\langle u_{n\mathbf{k}}|\nabla_{\mathbf{k}}u_{n\mathbf{k}}\rangle$, and in two dimensions its integral over the closed Brillouin zone is an integer, the Chern number:

A nonzero total Chern number of the filled bands implies $|C|$ chiral edge states and a quantized Hall conductance $Ce^2/h$. Berry curvature also has non-quantized consequences in metals: the intrinsic anomalous Hall effect, orbital magnetization, and the nonlinear Hall effect.

Topological insulators

With time-reversal symmetry the Chern number vanishes, but spin-orbit coupling can produce a band inversion with a $\mathbb{Z}_2$ invariant $\nu = 0$ (trivial) or $1$ (topological).

Two dimensions: the quantum spin Hall effect. Kane and Mele (2005) and Bernevig, Hughes, and Zhang (2006) predicted insulators with a pair of counter-propagating, spin-polarized edge channels (helical edge states). The effect was observed in HgTe/CdTe quantum wells above a critical thickness of 6.3 nm (2007), and later in InAs/GaSb wells and monolayer WTe$_2$ (up to ~100 K). Time reversal forbids elastic backscattering between the two partners of a Kramers pair, so each edge contributes $e^2/h$ to the conductance.

Three dimensions. A 3D topological insulator (Bi$_{1-x}$Sb$_x$, then Bi$_2$Se$_3$, Bi$_2$Te$_3$, and related compounds, identified by ARPES from 2008) is a bulk insulator whose surface hosts an odd number of Dirac cones. Near the $\Gamma$ point the surface Hamiltonian is

with $\sigma$ the electron spin. The surface states have:

• a linear (Dirac) dispersion with a crossing at a time-reversal-invariant momentum, protected by Kramers’ theorem;
• spin-momentum locking: the spin lies in-plane and perpendicular to $\mathbf{k}$;
• a Berry phase $\pi$ around the Fermi surface, so direct backscattering $\mathbf{k}\to-\mathbf{k}$ is forbidden and the surface is immune to localization by non-magnetic disorder (it shows weak antilocalization).

The $\mathbb{Z}_2$ invariant is computed from the Fu-Kane formula or, with inversion symmetry, from the parities of occupied states at the eight time-reversal-invariant momenta (see Graduate-Level Formalism).

Topological semimetals

Band touchings in 3D can themselves be topological.

• Weyl semimetals have isolated, twofold band crossings that act as monopoles of Berry curvature with chirality $\pm 1$. They come in pairs of opposite chirality, require broken inversion or time-reversal symmetry, and are connected on the surface by open Fermi arcs. First observed in TaAs (2015).
• Dirac semimetals (Na$_3$Bi, Cd$_3$As$_2$, 2014) have fourfold crossings protected by crystal symmetry; breaking time reversal splits each into two Weyl points.
• Signatures include the chiral anomaly (negative longitudinal magnetoresistance for $\mathbf{E}\parallel\mathbf{B}$), large linear magnetoresistance, and quantized circular photogalvanic effects in chiral crystals.

Topological superconductors and Majorana modes

A superconductor’s BdG Hamiltonian has a built-in particle-hole symmetry, so superconductors fall into their own topological classes. The minimal example is Kitaev’s 1D chain of spinless fermions with p-wave pairing (2001), whose topological phase hosts an unpaired Majorana zero mode at each end, with $\gamma = \gamma^\dagger$. A pair of separated Majoranas stores one fermion parity bit nonlocally, and braiding Majoranas in 2D networks implements non-abelian exchange, the basis of proposed topologically protected qubits.

Practical proposals (2010) engineer effective p-wave pairing from s-wave superconductors proximity-coupled to spin-orbit-coupled nanowires (InAs, InSb) in a magnetic field, to topological-insulator surfaces, or to magnetic atom chains. Zero-bias conductance peaks consistent with Majoranas have been reported many times since 2012, but similar peaks arise from trivial Andreev bound states, and several high-profile claims were retracted. Microsoft reported interferometric parity measurements in InAs-Al “topological gap protocol” devices in 2025; the evidence that these devices host topological Majorana modes remains disputed, and no braiding of Majoranas has been demonstrated.

Bulk-boundary correspondence

Topological spin textures

The topological reasoning that protects quantum Hall plateaus also organizes the real-space arrangement of spins in magnets. In materials lacking inversion symmetry, or in thin films where an interface breaks it, competing interactions wind the magnetization into textures that cannot be unwound smoothly into a uniform state. These textures carry an integer charge, behave as stable particle-like objects, and can be moved by small currents.

Skyrmion number

Treat the magnetization direction as a unit vector field $\mathbf{m}(\mathbf{r})$. A 2D texture that is uniform far away maps the plane (compactified to a sphere) onto the unit sphere of directions, and such maps fall into classes labeled by an integer winding number, the skyrmion number:

The integrand is the solid angle swept by $\mathbf{m}$, so $N_{sk}$ counts how many times the texture wraps the sphere. No continuous deformation can change it, so a skyrmion ($N_{sk} = \pm 1$) is topologically protected against decay into the ferromagnet ($N_{sk} = 0$) in the continuum; on a lattice the protection becomes a finite energy barrier. $N_{sk}$ is the real-space analogue of the Chern number.

Skyrmions and the interactions that stabilize them

A magnetic skyrmion has its core spin antiparallel to the background, with the magnetization rotating through the plane in between. Writing the in-plane angle as $\phi = w\varphi + \gamma$ (winding $w$, helicity $\gamma$) and the core polarity as $p = \pm 1$, the charge is $N_{sk} = p\,w$ up to a sign convention. Two helicities are common:

The Dzyaloshinskii-Moriya interaction (DMI) is an antisymmetric exchange arising from spin-orbit coupling without inversion symmetry. It favours a fixed sense of spin rotation. A standard lattice model is

with exchange, DMI, Zeeman, and uniaxial-anisotropy terms. The competition between $J$ and $D$ sets a length $\ell \sim Ja/D$ that fixes the helix period and skyrmion size. At zero field the ground state is a helix; in a window of field and temperature skyrmions condense into a triangular skyrmion lattice, first seen by small-angle neutron scattering in MnSi (2009) and imaged in real space by Lorentz transmission electron microscopy in Fe${0.5}$Co${0.5}$Si (2010). Frustrated centrosymmetric magnets (for example Gd$_2$PdSi$_3$) host much smaller skyrmions, a few nanometres across, stabilized without DMI.

Merons and antiskyrmions

A meron covers only half the sphere: its core points out of plane while its boundary lies in the plane, giving $N_{sk} = p\,w/2 = \pm 1/2$. A meron has logarithmically divergent energy on its own and occurs in pairs or lattices; a meron and an antimeron can combine into a bimeron with $|N_{sk}| = 1$, the natural skyrmion analogue in in-plane magnets. Antiskyrmions ($w = -1$) are stabilized by anisotropic DMI in Heusler compounds with $D_{2d}$ symmetry.

Emergent electrodynamics: topological and skyrmion Hall effects

An electron moving through a smooth texture adiabatically aligns its spin with $\mathbf{m}$. The resulting Berry phase acts like a fictitious magnetic field, the emergent field $b_z = \frac{\hbar}{2e}\,\mathbf{m}\cdot(\partial_x\mathbf{m}\times\partial_y\mathbf{m})$, which carries one flux quantum $h/e$ per skyrmion (in magnitude). Two consequences follow.

• Topological Hall effect. The emergent field deflects carriers and adds a term to the Hall resistivity beyond the ordinary and anomalous contributions, $\rho_{xy} = \rho^{O}{xy} + \rho^{A}{xy} + \rho^{T}{xy}$, with $\rho^{T}{xy} \propto P\,R_0\,n_{sk}\,\Phi_0$ for spin polarization $P$, ordinary Hall coefficient $R_0$, and skyrmion density $n_{sk}$. A Hall anomaly confined to the skyrmion field window is a standard transport fingerprint, though it can be mimicked by two-component anomalous Hall signals and needs corroboration by imaging.
• Skyrmion Hall effect. Conversely, a current-driven skyrmion does not move parallel to the drive. Thiele’s equation for a rigid texture,

balances the gyrotropic (Magnus-like) force against Gilbert damping $\alpha$ and the driving force $\mathbf{F}$ (dissipation tensor $\mathcal{D}$). The skyrmion moves at a Hall angle set by $|\mathbf{G}|/(\alpha\mathcal{D})$, which can reach tens of degrees and was imaged directly in 2016-17.

Applications

• Racetrack memory. Parkin’s racetrack concept (2008) moves magnetic domain walls along a nanowire; Fert and co-workers proposed skyrmions as bits in 2013 because they can be driven by spin-orbit torques at low current densities and are less sensitive to pinning. The skyrmion Hall effect pushes bits toward the track edge, motivating textures with zero net gyrocoupling: antiferromagnetic and synthetic-antiferromagnet skyrmions and bimerons, whose sublattice contributions cancel.
• Unconventional computing. Thermally driven skyrmion diffusion and stochastic nucleation have been used for reservoir computing, probabilistic bits, and signal reshuffling.
• 2D magnets. Intrinsic magnetism in van der Waals monolayers (CrI$_3$, Cr$_2$Ge$_2$Te$_6$, Fe$_3$GeTe$_2$, 2017 onward) provides atomically thin hosts in which DMI, anisotropy, and texture stability can be tuned by gating, stacking, and twist. Néel skyrmions have been imaged in Fe$_3$GeTe$_2$-based heterostructures.

Strongly correlated systems

Band theory assumes electrons move independently in an average potential. That fails when the Coulomb repulsion between electrons is comparable to their kinetic energy (narrow d and f bands, flat moiré bands). Band theory can then be qualitatively wrong, predicting a metal where experiment finds an insulator, and the resulting phenomena include high-temperature superconductivity, heavy fermions, strange metals, and quantum spin liquids.

Hubbard model and the Mott transition

The minimal model keeps nearest-neighbour hopping $t$ and an on-site repulsion $U$:

For $U \ll t$ the system is a metal. For $U \gg t$ at half filling (one electron per site) the electrons localize one per site to avoid the penalty: a Mott insulator, insulating because of interactions and not band filling (NiO, V$_2$O$_3$, undoped cuprates). Virtual hopping then generates an antiferromagnetic exchange $J = 4t^2/U$ between the localized spins. The Mott metal-insulator transition at $U \sim W$ (the bandwidth) is described quantitatively by dynamical mean-field theory (DMFT). Despite its simplicity, the 2D Hubbard model has no exact solution, and large-scale numerical studies since the late 2010s have shown that its ground state near 1/8 hole doping is a delicate competition between d-wave superconductivity and stripe order that depends on details such as next-nearest hopping $t’$.

Heavy fermions and the Kondo effect

A single magnetic impurity in a metal is screened by conduction electrons below the Kondo temperature $T_K$, forming a many-body singlet (the Kondo effect, which explains the resistance minimum in dilute magnetic alloys). In rare-earth and actinide compounds (CeCu$_6$, CeCoIn$_5$, YbRh$_2$Si$_2$, UPt$_3$) a lattice of f-moments hybridizes with the conduction band, producing quasiparticles with effective masses up to several hundred times $m_e$. Doniach’s phase diagram describes the competition between Kondo screening, which favours a heavy Fermi liquid, and the RKKY interaction, which favours magnetic order; the quantum critical point between them shows non-Fermi-liquid behaviour and often unconventional superconductivity.

Cuprates and strange metals

The cuprates consist of CuO$2$ planes separated by charge-reservoir layers. The undoped parent is an antiferromagnetic Mott insulator; hole doping of a few percent destroys the antiferromagnetism and produces a superconducting dome peaking near 16% doping, with $d{x^2-y^2}$ pairing. Around it lie the pseudogap regime (a partial gap above $T_c$ in underdoped samples, with intertwined charge-density-wave and other orders) and, above optimal doping, a strange metal with resistivity linear in $T$ down to the lowest temperatures and a scattering rate close to the “Planckian” bound $\hbar/\tau \sim k_BT$. Similar $T$-linear resistivity appears in heavy fermions, pnictides, and twisted bilayer graphene, which suggests a common mechanism near quantum critical points. Explaining the pseudogap and the strange metal remains one of the central open problems of condensed matter physics.

Quantum spin liquids and flat bands

Frustrated magnets can avoid ordering down to $T = 0$ and form quantum spin liquids with fractionalized spinon excitations and emergent gauge fields. Kitaev’s exactly solvable honeycomb model (2006) motivated work on $\alpha$-RuCl$_3$; a half-quantized thermal Hall effect reported in 2018 has not been consistently reproduced, and no material is yet accepted as a spin liquid beyond reasonable doubt. Moiré materials (twisted graphene, twisted transition-metal dichalcogenides) create flat bands whose kinetic energy can be tuned by twist angle, displacement field, and gating, providing Hubbard-model physics, correlated insulators, superconductivity, and fractional Chern insulators in a single tunable device.

Soft condensed matter

Soft matter (liquid crystals, polymers, colloids, membranes, foams) has characteristic energies comparable to $k_BT$ at room temperature, so thermal fluctuations, entropy, and mesoscale structure dominate.

Liquid crystals

Liquid crystals are fluids of anisotropic molecules with partial order:

The elastic energy of a slowly varying director is the Frank free energy,

with splay, twist, and bend constants $K_i$ of order $10^{-11}$ N. Liquid-crystal displays switch pixels by using an electric field to reorient a twisted nematic against these elastic forces (the Fréedericksz transition). Topological defects (disclinations) in the director field are classified by homotopy theory, like the spin textures above.

Polymers

A flexible polymer of $N$ segments of length $l$ is modelled as a random walk. An ideal chain has mean-square end-to-end distance

In a good solvent, excluded volume swells the coil. Flory’s argument gives $R_F \sim N^{\nu}$ with $\nu = 3/(d+2) = 3/5$ in three dimensions, remarkably close to the renormalization-group value $\nu \approx 0.588$ (de Gennes showed the problem maps onto the $N \to 0$ limit of the O(N) model). In melts and at the theta temperature chains are again ideal. Entanglement between long chains produces viscoelasticity, described by the reptation model.

Colloids

Colloids are particles of roughly 1 nm to 1 μm suspended in a fluid. The DLVO theory (Derjaguin, Landau, Verwey, Overbeek) treats their stability as a competition between van der Waals attraction and screened electrostatic repulsion between charged double layers. The repulsion is screened over the Debye length

for a 1:1 electrolyte of number density $n_0$; in water at room temperature $\lambda_D \approx 0.304\ \text{nm}/\sqrt{I}$ with ionic strength $I$ in mol/L, so about 1 nm at 0.1 M. Adding salt collapses the barrier and causes aggregation. Hard-sphere colloids crystallize by entropy alone above a volume fraction of about 0.494, and colloids serve as model systems for studying crystallization, glass formation, and phase separation directly under a microscope.

References

1. M. Tinkham, Introduction to Superconductivity, 2nd ed. (1996).
2. J. F. Annett, Superconductivity, Superfluids and Condensates (2004).
3. S. M. Girvin and K. Yang, Modern Condensed Matter Physics (2019).
4. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
5. N. P. Armitage, E. J. Mele, A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).
6. N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nat. Nanotechnol. 8, 899 (2013).
7. B. Keimer et al., “From quantum matter to high-temperature superconductivity in copper oxides,” Nature 518, 179 (2015).
8. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (1995).

See Also

• Graduate-Level Formalism — Bogoliubov-de Gennes, K-matrix Chern-Simons theory, DMFT, topological order, and the tenfold way.
• Disorder & Localization — why quantum Hall plateaus need localized states.
• Metals & Magnetism — Fermi liquids, exchange, and magnetic order.
• Experimental Techniques — ARPES, STM, neutron scattering, and transport.
• Condensed Matter Physics (Hub) — crystal structure, band theory, and semiconductors.
• Quantum Field Theory — field-theoretic methods for collective excitations.