Condensed Matter: Graduate-Level Formalism

Condensed Matter Physics » Graduate-Level Formalism

This page is the theory-first companion to the Condensed Matter hub. It collects the graduate-level machinery that the narrative pages use informally: second quantization and many-body Green’s functions; band theory beyond the nearly-free-electron picture, including the topological invariants of band insulators; the field theories of superconductors and quantum Hall fluids; the methods for strongly correlated electrons; topological order; the renormalization-group theory of quantum phase transitions; and the numerical methods that make all of this quantitative. Each section states definitions and key results compactly and in standard notation; derivations are left to the references at the end. The physical phenomena themselves are treated on Superconductivity, Quantum Hall & Topological Phases and Disorder & Localization, and the probes that test the theory on Experimental Techniques.

Conventions. $\hbar = k_B = 1$ in the Green’s-function, correlated-electron, and field-theory sections unless a formula is written out in full; $\beta = 1/T$; $\xi_{\mathbf{k}} = \epsilon_{\mathbf{k}} - \mu$ is the band energy measured from the chemical potential; $0^+$ is a positive infinitesimal.

How the pieces fit together

Most of the formalism below computes one of a small number of objects: a single-particle Green’s function (and hence the spectral function measured by photoemission and tunneling), a two-particle response function (measured by scattering and transport), or a topological invariant (measured by quantized responses). The diagram shows the dependencies.

flowchart TD
    SQ["Second quantization<br/>field operators, H in Fock space"] --> GF["Many-body Green's functions<br/>G, self-energy, Matsubara"]
    SQ --> BAND["Band theory<br/>k·p, Wannier functions"]
    BAND --> TOPO["Topological band theory<br/>Berry curvature, Chern, Z2, tenfold way"]
    GF --> SC["Superconductivity<br/>BdG, Ginzburg-Landau"]
    GF --> CORR["Strong correlation<br/>t-J, Anderson impurity, DMFT, slave particles"]
    TOPO --> QH["Quantum Hall field theory<br/>LLL projection, composite fermions, Chern-Simons"]
    QH --> TO["Topological order and SPT phases"]
    SC --> TOPO
    GF --> RG["Quantum criticality<br/>scaling, RG, effective field theory"]
    CORR --> RG
    GF --> NUM["Numerical methods<br/>DFT, QMC, tensor networks"]
    CORR --> NUM
    GF --> EXP["Observables<br/>A(k,ω), χ(q,ω), σ_xy"]
    TOPO --> EXP

Second quantization

Second quantization describes a many-fermion system in Fock space, with antisymmetry built into the operator algebra instead of into Slater determinants. Given a complete single-particle basis $\phi_\alpha(\mathbf{r})$ with fermion operators $c_\alpha$, the field operators are

obeying canonical anticommutation relations

A system of particles in an external potential $V(\mathbf{r})$ interacting through a two-body potential $U(\mathbf{r} - \mathbf{r}’)$ has the Hamiltonian

Note the operator ordering in the interaction term: annihilating in the reverse order of creation removes the self-interaction. In a discrete basis (plane waves, Wannier orbitals) the same Hamiltonian reads

and every lattice model on this page (Hubbard, t-J, Anderson, Kitaev) is a truncation of it to a few orbitals and short-range matrix elements.

Many-body Green’s functions

Definitions

The time-ordered, retarded, and imaginary-time single-particle Green’s functions are

$G^R$ is the causal response (analytic in the upper half $\omega$-plane) and is what experiments measure; the time-ordered $G$ is what zero-temperature perturbation theory produces; $\mathcal{G}$ is what finite-temperature perturbation theory and quantum Monte Carlo produce.

Spectral representation

Inserting a complete set of many-body eigenstates gives the Lehmann representation. At $T = 0$, with energies measured from $\mu$,

The spectral function

is the probability density for adding ($\omega > 0$) or removing ($\omega < 0$) an electron of momentum $\mathbf{k}$ with energy $\omega$. Its occupied part, $f(\omega)A(\mathbf{k},\omega)$, is what ARPES measures; its momentum integral is the density of states seen by STM.

Self-energy and quasiparticles

All interaction effects in $G$ are collected in the self-energy $\Sigma$ through the Dyson equation

If $\Sigma$ is small and smooth near the Fermi level, $G$ has a pole close to the real axis and can be written as a coherent quasiparticle part plus an incoherent background:

The quasiparticle residue $0 < Z \le 1$ is the weight of the coherent peak and equals the jump in the momentum distribution $n_{\mathbf{k}}$ at $k_F$; the mass is renormalized as $m^\ast/m = 1/Z$ for a momentum-independent self-energy; and the lifetime is $1/(2\Gamma)$, with $\Gamma \propto \omega^2 + \pi^2 T^2$ in a Fermi liquid. A non-Fermi liquid is a metal in which this structure fails: $Z \to 0$ at the Fermi surface, as in the marginal Fermi liquid $\text{Im}\,\Sigma \propto \max(|\omega|, T)$ used to describe the strange-metal phase of cuprates. The Fermi-liquid side is developed on Metals & Magnetism.

Matsubara formalism

At finite temperature $\mathcal{G}(\tau)$ is (anti)periodic on $\tau \in [0,\beta)$, so it has a Fourier series over discrete Matsubara frequencies

The free fermion propagator is $\mathcal{G}0(\mathbf{k}, i\omega_n) = 1/(i\omega_n - \xi{\mathbf{k}})$, and the Feynman rules are identical to $T = 0$ except that frequency integrals become sums $T\sum_n$. Real-frequency quantities follow by analytic continuation $i\omega_n \to \omega + i0^+$, which is exact for analytic expressions but ill-conditioned for numerical data (it is usually done with maximum-entropy or related regularized inversions).

Linear response

Two-particle correlation functions describe the response to weak external fields. The Kubo formula gives the retarded response of an observable $A$ to a perturbation $-B\,F(t)$:

Conductivity (current-current), magnetic susceptibility (spin-spin), and the dynamic structure factor measured by neutron and X-ray scattering are all of this form. The fluctuation-dissipation theorem relates the absorptive part $\chi’’(\mathbf{q},\omega)$ to the scattering cross-section, $S(\mathbf{q},\omega) = 2\,[1 + n_B(\omega)]\,\chi’’(\mathbf{q},\omega)$.

Band theory beyond free electrons

The k·p method

Writing a Bloch state as $\psi_{n\mathbf{k}} = e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}$ turns the crystal Hamiltonian into one for the periodic part,

and $\frac{\hbar}{m_0}\mathbf{k}\cdot\mathbf{p}$ can be treated as a perturbation about a band extremum (taken at $\mathbf{k} = 0$). Second-order perturbation theory gives the effective-mass tensor of band $n$:

Nearby bands with large momentum matrix elements therefore make masses light. The Kane model keeps the coupled conduction and valence bands exactly. In its simplest two-band form, with interband matrix element $P$ (units of energy times length) and gap $E_g = E_c - E_v$,

so $m^\ast \approx \hbar^2E_g/(2P^2)$ near the band edge: narrow-gap semiconductors (InSb, HgCdTe) have very light carriers and strongly non-parabolic bands. The full 8-band Kane model adds spin-orbit coupling and underlies the band-inversion mechanism of HgTe quantum wells.

Wannier functions

Wannier functions are the lattice Fourier transforms of Bloch states,

and give a localized orbital basis in which tight-binding and Hubbard models are defined. They are not unique: any $\mathbf{k}$-dependent unitary mixing $|u_{n\mathbf{k}}\rangle \to \sum_m U_{mn}(\mathbf{k})|u_{m\mathbf{k}}\rangle$ of a group of bands gives valid Wannier functions. Maximally localized Wannier functions (Marzari and Vanderbilt) fix this gauge freedom by minimizing the total spread

Wannier-function centres are the modern theory of electric polarization (their sum is the Berry phase of the occupied bands), and exponentially localized Wannier functions exist if and only if the bands are topologically trivial: a nonzero Chern number is an obstruction to Wannierization. This is the link between band theory and topology developed next. In practice Wannier functions are produced from DFT calculations by codes such as Wannier90 and used to build downfolded models for DMFT and transport.

Topological band theory

For a set of occupied bands, the Berry connection and Berry curvature are

The second form shows that curvature concentrates where bands nearly touch. Berry curvature enters semiclassical dynamics as an anomalous velocity, $\dot{\mathbf{r}} = \hbar^{-1}\nabla_{\mathbf{k}}E_n - \dot{\mathbf{k}}\times\boldsymbol{\Omega}_n$, which produces the intrinsic anomalous Hall effect.

Chern number and the TKNN formula. In two dimensions the integral of the curvature over the closed Brillouin zone is an integer, the Chern number, and the Kubo formula for the Hall conductance of a band insulator reduces to

This is the TKNN result (Thouless, Kohmoto, Nightingale, den Nijs, 1982), which explains the integer quantum Hall effect and, with Haldane’s 1988 lattice model, shows that quantized Hall conductance does not require a net magnetic field (Chern insulators).

The Z2 invariant. With time-reversal symmetry ($\Theta^2 = -1$ for spin-1/2) the total Chern number vanishes, but a $\mathbb{Z}2$ invariant survives. Fu and Kane expressed it through the sewing matrix $w{mn}(\mathbf{k}) = \langle u_{m,-\mathbf{k}}|\Theta|u_{n\mathbf{k}}\rangle$, which is antisymmetric at the time-reversal-invariant momenta $\Gamma_i$:

with the product over the 4 TRIM in 2D or the 8 TRIM in 3D (the latter gives the strong index $\nu_0$; products over TRIM in a plane give the three weak indices). If the crystal also has inversion symmetry the formula simplifies to a product of parity eigenvalues $\xi_{2m}(\Gamma_i) = \pm 1$ of the occupied Kramers pairs,

which is how Bi$_{1-x}$Sb$_x$ and Bi$_2$Se$_3$ were first identified as topological insulators.

The tenfold way. For non-interacting gapped fermions the possible invariants are fixed by the dimension and by the presence or absence of three antiunitary or chiral symmetries: time reversal $T$, particle-hole $C$, and their product, chiral (sublattice) symmetry $S$. The Altland-Zirnbauer classes and their invariants (the “periodic table” of Kitaev and Schnyder-Ryu-Furusaki-Ludwig) are:

($\pm 1$ gives the square of the antiunitary operator; 0 means the symmetry is absent.) The table covers only internal symmetries. Crystalline symmetries (mirrors, rotations, glides) add many more topological crystalline phases, including higher-order phases with protected corner or hinge states. Topological quantum chemistry and the related symmetry-indicator method (2017 onward) diagnose band topology from the symmetry labels of Bloch states at high-symmetry momenta alone, which made high-throughput searches of materials databases possible; those surveys found that topological bands are common rather than exotic.

Superconductivity formalism

Bogoliubov-de Gennes formalism

Mean-field pairing theory is written in the doubled (Nambu) space of particles and holes. With $\Psi_{\mathbf{k}} = (c_{\mathbf{k}\uparrow}, c_{\mathbf{k}\downarrow}, c_{-\mathbf{k}\uparrow}^\dagger, c_{-\mathbf{k}\downarrow}^\dagger)^T$,

where $h$ is the normal-state Bloch Hamiltonian and the $2\times 2$ gap matrix satisfies $\Delta^T(\mathbf{k}) = -\Delta(-\mathbf{k})$ by Fermi statistics. The doubling builds in a particle-hole symmetry,

so eigenvalues come in $\pm E$ pairs. This is the symmetry $C$ of the tenfold way, and the reason a zero-energy BdG state can be its own antiparticle (a Majorana mode). For spin-singlet pairing in a single band the problem reduces to the reduced Nambu spinor $(c_{\mathbf{k}\uparrow}, c_{-\mathbf{k}\downarrow}^\dagger)$:

with $\Delta_{\mathbf{k}}$ determined self-consistently by the BCS gap equation (see Emergent Phases). In real space the same structure, the BdG equations, handles vortices, interfaces, and Andreev bound states.

Ginzburg-Landau theory

Near $T_c$ the free energy is expanded in a complex order parameter $\psi(\mathbf{r})$ coupled minimally to the vector potential, with pair charge $e^\ast = 2e$ and mass $m^\ast$ (conventionally $2m$):

Varying with respect to $\psi^\ast$ and $\mathbf{A}$ gives the two GL equations:

The two length scales are the coherence length and the penetration depth,

both diverging as $(T_c - T)^{-1/2}$, so their ratio $\kappa = \lambda/\xi$ is nearly temperature independent. $\kappa < 1/\sqrt{2}$ gives a type I superconductor (positive normal-superconductor surface energy); $\kappa > 1/\sqrt{2}$ gives type II, which admits Abrikosov vortices each carrying one flux quantum $\Phi_0 = h/2e$ between $H_{c1}$ and $H_{c2} = \Phi_0/(2\pi\mu_0\xi^2)$.

Josephson effects

Two superconductors coupled through a weak link carry a supercurrent set by the gauge-invariant phase difference $\varphi$:

A real junction is modelled by the resistively and capacitively shunted junction (RCSJ): a Josephson element in parallel with a resistance $R$ and capacitance $C$. Current conservation gives

the equation of a damped particle in the tilted-washboard potential $U(\varphi) = -E_J\cos\varphi - (\hbar I/2e)\varphi$ with Josephson energy $E_J = \hbar I_c/2e$. The Stewart-McCumber parameter $\beta_c = 2eI_cR^2C/\hbar$ decides whether the $I$-$V$ curve is hysteretic ($\beta_c \gg 1$). Under microwave irradiation at frequency $\omega$ the phase locks and the $I$-$V$ curve develops Shapiro steps at

which, because $K_J = 2e/h$ is exact in the 2019 SI, realize the volt. Quantizing $\varphi$ and its conjugate charge turns the same circuit into a superconducting qubit: the transmon is the $E_J \gg E_C$ limit of the Cooper-pair box.

Quantum Hall field theory

Landau levels and the lowest Landau level

In a field $B$ the kinetic energy of a 2D electron is quantized into Landau levels $E_n = \hbar\omega_c(n + \frac12)$, each with degeneracy $B/\Phi_0$ per unit area ($\Phi_0 = h/e$). The magnetic length is $\ell_B = \sqrt{\hbar/eB}$ and the filling factor is $\nu = 2\pi\ell_B^2 n$. In the symmetric gauge the lowest-Landau-level (LLL) states are

with $z = x - iy$ (or $x + iy$, depending on the sign of the field). State $m$ is a ring of radius $\sqrt{2m}\,\ell_B$. Any LLL wavefunction is therefore an analytic function of the $z_i$ times a Gaussian, which is why trial states such as Laughlin’s are written as polynomials.

Projected density algebra

Once the Landau-level spacing is the largest energy, the physics is that of the interaction projected onto a single level. The projected density operators $\bar\rho(\mathbf{q})$ no longer commute; they obey the Girvin-MacDonald-Platzman (GMP) algebra

(sign of the cross product depending on field orientation). The same algebra, approximately realized in flat Chern bands with near-uniform Berry curvature, is why fractional Chern and fractional quantum anomalous Hall states can form in moiré materials without a magnetic field.

Composite fermions

Jain’s construction attaches two flux quanta to each electron. The composite fermions see a reduced effective field

and fill an integer number $p$ of their own Landau levels (Lambda levels). The electron trial state is

which reproduces the principal FQH sequence $1/3, 2/5, 3/7, \ldots$ and $2/3, 3/5, \ldots$. At $\nu = 1/2$, $B^\ast = 0$ and the composite fermions form a Fermi sea (the Halperin-Lee-Read state), observed through commensurability oscillations and surface acoustic waves. Dirac composite-fermion theory (Son, 2015) reformulated this state in a particle-hole symmetric way.

Chern-Simons and K-matrix theory

The low-energy theory of an abelian quantum Hall fluid is a Chern-Simons theory of emergent gauge fields $a^I_\mu$ that encode the conserved current $j^\mu = \frac{1}{2\pi}\epsilon^{\mu\nu\lambda}\partial_\nu a_\lambda$:

The integer symmetric K matrix and charge vector $t$ determine all universal data:

A single level-$k$ Chern-Simons term also implements flux attachment (statistical transmutation): binding $k$ flux quanta to a particle changes its exchange phase by $k\pi$, so fermions plus an odd number of flux quanta behave as bosons and vice versa. Non-abelian states such as the Moore-Read Pfaffian proposed for $\nu = 5/2$ require non-abelian Chern-Simons theories (e.g. $SU(2)_2$).

Strongly correlated electrons

t-J and Anderson impurity models

At large on-site repulsion $U \gg t$ and near half filling, second-order perturbation theory in $t/U$ reduces the Hubbard model to the t-J model,

where $P$ projects out doubly occupied sites (a three-site hopping term of the same order is usually dropped). At half filling it is the Heisenberg antiferromagnet; doped, it is the standard minimal model for cuprates.

The single-impurity Anderson model describes one correlated orbital hybridized with a conduction band:

The bath enters only through the hybridization function $\Delta(\omega) = \sum_{\mathbf{k}}|V_{\mathbf{k}}|^2/(\omega - \epsilon_{\mathbf{k}})$. In the local-moment regime the Schrieffer-Wolff transformation maps it to the Kondo model, whose low-temperature screening of the moment below $T_K \sim D\,e^{-1/(\rho J_K)}$ is solved exactly by Wilson’s numerical renormalization group and the Bethe ansatz.

Dynamical mean-field theory

DMFT (Georges, Kotliar, and collaborators, 1989-1996) maps a lattice model onto an Anderson impurity embedded in a self-consistently determined bath. The approximation is that the self-energy is local, $\Sigma(\mathbf{k},\omega) \to \Sigma(\omega)$, which becomes exact in infinite dimensions. The self-consistency loop is:

flowchart LR
    S0["Initial guess<br/>Σ(ω)"] --> L["Lattice Dyson equation<br/>G_loc = (1/N) Σ_k [ω + μ − ε_k − Σ]⁻¹"]
    L --> W["Weiss field<br/>𝒢₀⁻¹ = G_loc⁻¹ + Σ"]
    W --> I["Solve impurity problem<br/>CT-QMC, ED, NRG → G_imp"]
    I --> N["Update self-energy<br/>Σ = 𝒢₀⁻¹ − G_imp⁻¹"]
    N --> C{"G_imp = G_loc?"}
    C -- no --> L
    C -- yes --> D["Converged local Σ(ω)<br/>→ A(k,ω), Mott transition"]

DMFT captures the Mott transition, including the coexistence of a quasiparticle peak with Hubbard bands, which static mean-field theory cannot. Combined with DFT (DFT+DMFT) it is the standard first-principles method for correlated d- and f-electron materials; cluster and diagrammatic extensions (cellular DMFT, DCA, dual fermions, $D\Gamma A$) restore momentum dependence needed for d-wave superconductivity and pseudogap physics. Continuous-time quantum Monte Carlo is the usual impurity solver.

Slave-particle methods

Slave-particle representations enforce the no-double-occupancy constraint by splitting the electron into auxiliary particles. In the $U \to \infty$ slave-boson representation

the holon $b$ carries charge and the spinon $f$ carries spin. The constraint is imposed with a Lagrange multiplier, and fluctuations around mean-field theory generate an emergent U(1) gauge field. Condensation $\langle b\rangle \neq 0$ gives a Fermi liquid with $Z \sim |\langle b\rangle|^2$; the uncondensed phase with a spinon Fermi surface is a U(1) spin liquid. Kotliar-Ruckenstein slave bosons (finite $U$), slave rotors, and the Schwinger-boson and Abrikosov-fermion representations of spins follow the same logic and are the standard language for parton constructions of fractionalized phases.

Topological order and SPT phases

Topological quantum field theory

Gapped phases with long-range entanglement are described at low energy by topological field theories whose correlation functions do not depend on the metric. Two families cover the common cases:

Level-$k$ Chern-Simons theory describes chiral states such as the Laughlin fluid; level-$k$ BF theory describes the time-reversal-symmetric $\mathbb{Z}_k$ gauge theory, with $k = 2$ the toric code and the $\mathbb{Z}_2$ spin liquid. BF theory is the special case of the K-matrix theory with $K = \begin{pmatrix}0 & k\ k & 0\end{pmatrix}$.

Universal data of topological order

A topologically ordered phase is characterized by its anyons $a$, their quantum dimensions $d_a$, fusion rules $a \times b = \sum_c N^c_{ab}\,c$, and topological spins $\theta_a = e^{2\pi i h_a}$. From these:

Ground-state degeneracy. On a torus it equals the number of anyon types; on a genus-$g$ surface it is $\lvert\det K\rvert^g$ for abelian states.

Modular matrices. The $S$ matrix encodes mutual braiding and the $T$ matrix the self-statistics:

where $c_-$ is the chiral central charge. The fusion rules follow from $S$ by the Verlinde formula $N^c_{ab} = \sum_x S_{ax}S_{bx}S^\ast_{cx}/S_{0x}$.

Topological entanglement entropy. For a region with smooth boundary of length $L$,

so $\gamma = \ln 2$ for the toric code and $\gamma = \ln\sqrt{m}$ for the $\nu = 1/m$ Laughlin state. The constant $\gamma$ is isolated numerically by the Kitaev-Preskill or Levin-Wen constructions and is the standard diagnostic for spin liquids in DMRG studies.

Symmetry-protected topological phases

A symmetry-protected topological (SPT) phase has no intrinsic topological order (unique ground state, no anyons) but cannot be connected to a product state without breaking a symmetry $G$ or closing the gap. Bosonic SPT phases in $d$ spatial dimensions are classified, for most groups, by the group cohomology $H^{d+1}(G, U(1))$ (Chen, Gu, Liu, Wen), with beyond-cohomology phases in $d \ge 3$; fermionic and crystalline SPTs are classified by generalized cohomology or cobordism.

In one dimension the classification can be seen directly in matrix product states

An on-site symmetry acts on the physical index as a gauge transformation on the virtual (bond) index,

and the $V_g$ may form a projective representation of $G$. The classes of projective representations are $H^2(G, U(1))$. The spin-1 Haldane chain is the standard example: $V_g$ is a spin-1/2 representation of SO(3), so a chain with open ends carries two free spin-1/2 edge states.

Quantum phase transitions and scaling

A quantum phase transition (QPT) is a transition between distinct ground states of a many-body system, driven not by temperature but by a non-thermal coupling $g$ (pressure, magnetic field, doping, chemical potential) at $T = 0$. At the critical coupling $g_c$ the ground-state energy is non-analytic and a characteristic energy scale, the gap $\Delta$, vanishes. Although the transition sits at $T=0$, the quantum critical point (QCP) controls a wedge-shaped region of the finite-$T$ phase diagram, the quantum critical fan, where the only relevant scale is temperature itself. This is why a $T=0$ singularity leaves fingerprints ($T$-linear resistivity, anomalous specific heat, $\omega/T$ scaling) at experimentally accessible temperatures.

Diverging length and time

Approaching $g_c$ a correlation length and a correlation time both diverge:

The exponent $\nu$ is the correlation-length exponent, as in a classical transition, but space and time scale differently, as encoded in the dynamic critical exponent $z$. Equivalently the characteristic energy softens as

For a Lorentz-invariant critical theory $z = 1$; for a metallic QCP with overdamped order-parameter dynamics one typically finds $z = 2$ (antiferromagnet) or $z = 3$ (ferromagnet), reflecting Landau damping by the particle-hole continuum.

Quantum-to-classical mapping

The path integral of a $d$-dimensional quantum system is a statistical-mechanics problem in $d + z$ effective dimensions: imaginary time $\tau \in [0, \hbar\beta)$ acts as $z$ extra spatial directions. A $T=0$ quantum transition in $d$ dimensions therefore maps onto a classical transition in

dimensions, provided the resulting classical weights are positive (Berry phases and fermion signs can spoil the mapping). The transverse-field Ising chain maps onto the 2D classical Ising model with $z=1$. Finite temperature acts as a finite size $L_\tau = \hbar\beta$ in the imaginary-time direction, so

and the curves $T \sim |g - g_c|^{\nu z}$ trace the edges of the quantum critical fan.

Scaling hypothesis and exponents

Near the QCP the singular part of the free-energy density obeys a scaling form. With reduced coupling $t = (g - g_c)/g_c$ and a symmetry-breaking field $h$,

for an arbitrary rescaling factor $b$. Choosing $b = |t|^{-\nu}$ gives the order parameter, gap, and correlation functions; choosing $b = T^{-1/z}$ collapses finite-temperature data. The critical exponents are:

Given $z$, only two static exponents are independent; the rest follow from hyperscaling, which for a QPT uses the effective dimension $d + z$:

Finite-size and finite-temperature scaling. On a system of linear size $L$ the order parameter obeys

which is the practical route to extracting exponents from quantum Monte Carlo: data from different $L$ collapse onto one curve when the exponents are right.

Renormalization group and universality

The exponents follow from the RG flow near the QCP. Linearizing the RG transformation about the fixed point, the eigenvalues $b^{y_i}$ of the relevant couplings fix $\nu = 1/y_t$ and $y_h$. The upper critical dimension for $\phi^4$-type theories is $d + z = 4$: above it ($d + z > 4$) the Gaussian fixed point and mean-field exponents apply, at it there are logarithmic corrections, and below it fluctuations produce non-trivial Wilson-Fisher exponents. For a 2D insulating antiferromagnet ($d=2$, $z=1$), $d + z = 3 < 4$, and the Néel-paramagnet transition is in the 3D classical O(3) universality class.

Universality is the payoff: $\nu$, $z$, and $\eta$ depend only on the dimension, the order-parameter symmetry, and the range of interactions, not on microscopic details.

Beyond Landau: deconfined quantum criticality

The Landau-Ginzburg-Wilson paradigm assumes a single order parameter, which makes a direct continuous transition between two phases with unrelated broken symmetries non-generic. The Néel-VBS transition of a 2D spin-1/2 antiferromagnet is the proposed exception (Senthil, Vishwanath, Balents, Sachdev, Fisher, 2004): both orders emerge from fractionalized spinons $z_\alpha$ (a $CP^{1}$ field) coupled to an emergent U(1) gauge field $a_\mu$,

Spinons are confined on both sides (into magnons in the Néel phase, into valence bonds in the VBS phase) and deconfined only at the critical point, where monopoles of the gauge field, which carry the VBS order, become irrelevant. Large-scale numerics on the J-Q model and related systems find drifting finite-size exponents and conflicting estimates, and recent work, including fuzzy-sphere studies since 2023, favours a weakly first-order or “pseudo-critical” transition controlled by nearby complex fixed points rather than a true conformal critical point. The question remains open.

Effective field theory in condensed matter

The renormalization group reframes condensed-matter problems as effective field theories (EFTs): rather than tracking every electron, one writes the most general local action consistent with the symmetries for the relevant low-energy degrees of freedom (an order parameter, a Goldstone mode, a gauge field, a Dirac cone), then organizes the terms by their importance under coarse-graining. Ginzburg-Landau theory is the prototype.

Symmetries fix the action

One identifies the slow fields and the exact and emergent symmetries (translations, rotations, time reversal, particle-hole, a global U(1) or O(N), gauge invariance) and writes every local term compatible with them. Symmetry decides which terms may appear; the RG decides which ones matter. Two consequences are immediate:

• Goldstone modes. A spontaneously broken continuous symmetry guarantees gapless modes whose action is fixed by the symmetry, for example the superfluid phase mode, $\mathcal{L} = \frac{\rho_s}{2}(\nabla\theta)^2 + \frac{\kappa}{2}(\partial_\tau\theta)^2$. Nonrelativistic systems can have fewer Goldstone modes than broken generators with quadratic dispersion (ferromagnetic magnons), as classified by Watanabe and Murayama.
• Forbidden couplings. Time reversal and parity forbid a Chern-Simons term; particle-hole symmetry constrains the BdG Hamiltonian; lattice symmetries protect band touchings (Dirac and Weyl points) against gapping.

Scaling dimensions and relevance

Under a coarse-graining step $x \to b\,x$, $\tau \to b^{z}\tau$ each field $\phi$ acquires a scaling dimension $\Delta_\phi$, fixed by demanding that the Gaussian part of the action be scale invariant. For a relativistic scalar in $D = d + z$ effective dimensions,

A coupling $g_n$ multiplying an operator $\mathcal{O}_n$ of dimension $\Delta_n$ has dimension $D - \Delta_n$ and flows as

Since $\Delta(\phi^{2k}) = k(D-2)$ at the Gaussian fixed point, this classifies every term:

The few relevant and marginal couplings constitute the universal data; the infinite tower of irrelevant operators only supplies corrections to scaling. This is why universality holds.

RG flow and fixed points

Adding loop corrections to the linear scaling gives beta functions $\beta_n = dg_n/d\ell$ with $\ell = \ln b$. Their zeros are fixed points, where the theory is scale invariant. For the O(N) model with interaction $\frac{u}{4}(\phi_a\phi_a)^2$, in $D = 4 - \epsilon$,

with the Gaussian fixed point $u = 0$ (stable for $\epsilon < 0$) and the infrared-stable Wilson-Fisher fixed point $u^{\ast} = 8\pi^2\epsilon/(N+8)$ for $\epsilon > 0$. Linearizing about $u^\ast$ gives, to first order in $\epsilon$,

connecting back to the scaling exponents above. Modern precision values for the 3D Ising and O(N) classes come from the conformal bootstrap and Monte Carlo rather than the $\epsilon$ expansion.

Emergent infrared symmetries are common: a lattice model with only discrete rotation symmetry can flow to a Lorentz- and conformally invariant fixed point, and gauge fields ($CP^1$, Chern-Simons, $\mathbb{Z}_2$) routinely emerge as the low-energy description of fractionalized phases even though the microscopic Hamiltonian has no gauge structure.

Computational methods

No single numerical method covers condensed matter; each trades system size, accuracy, dimensionality, and the fermion sign problem differently.

Density functional theory

DFT replaces the interacting problem by non-interacting Kohn-Sham electrons moving in an effective potential that reproduces the true ground-state density:

All the many-body physics is in the exchange-correlation functional, arranged on “Jacob’s ladder”: LDA uses $n$ only; GGA (PBE) adds $\nabla n$; meta-GGA (SCAN, r$^2$SCAN) adds the kinetic-energy density; hybrids (PBE0, HSE06) mix in a fraction of exact exchange and substantially improve band gaps. DFT+U adds a Hubbard correction on localized orbitals. Kohn-Sham eigenvalues are not formally quasiparticle energies, which is why band gaps from LDA/GGA are typically too small.

Quantum Monte Carlo

Variational Monte Carlo evaluates $E[\Psi_T] = \langle\Psi_T|H|\Psi_T\rangle/\langle\Psi_T|\Psi_T\rangle$ by sampling $|\Psi_T|^2$. Projector methods filter the ground state out of a trial state,

in real space (diffusion Monte Carlo, with a fixed-node approximation for fermions) or in a space of Slater determinants (auxiliary-field QMC). Determinant QMC and stochastic series expansion work at finite temperature. For fermions and frustrated magnets the sampled weights can be negative, and the average sign decays exponentially with system size and inverse temperature; Troyer and Wiese showed that a general solution of this sign problem would solve NP-hard problems. Sign-free models (half-filled Hubbard on bipartite lattices, many designer models) are therefore the testing ground for quantum criticality.

Tensor networks

Matrix product states (MPS) represent 1D states with area-law entanglement efficiently; DMRG optimizes them variationally, and time-evolving block decimation (TEBD) evolves them in real or imaginary time using Trotterized two-site gates. Projected entangled-pair states (PEPS, and iPEPS in the thermodynamic limit) generalize the idea to 2D, with observables computed by approximate contraction (corner transfer matrices or boundary MPS).

The code below is a complete, minimal TEBD for the transverse-field Ising chain. It finds the ground state by imaginary-time evolution from a product state and checks the energy against exact diagonalization. Tensors follow the convention A[left_bond, physical, right_bond].

import numpy as np

X = np.array([[0., 1.], [1., 0.]])
Z = np.array([[1., 0.], [0., -1.]])
I2 = np.eye(2)

def bond_hamiltonians(L, J=1.0, g=1.0):
    """H = -J sum Z_i Z_{i+1} - g sum X_i, split into two-site terms.
    Each interior site field is shared between its two bonds."""
    hs = []
    for i in range(L - 1):
        gl = g if i == 0 else g / 2
        gr = g if i == L - 2 else g / 2
        h = -J * np.kron(Z, Z) - gl * np.kron(X, I2) - gr * np.kron(I2, X)
        hs.append(h.reshape(2, 2, 2, 2))            # indices (s1', s2', s1, s2)
    return hs

def apply_gate(A, B, U, chi_max, tol=1e-12):
    """Contract A[a,s,b] B[b,t,c] with gate U[s',t',s,t], then SVD-truncate."""
    chi_l, d, _ = A.shape
    chi_r = B.shape[2]
    theta = np.tensordot(A, B, axes=(2, 0))                  # (a, s, t, c)
    theta = np.tensordot(U, theta, axes=([2, 3], [1, 2]))    # (s', t', a, c)
    theta = theta.transpose(2, 0, 1, 3).reshape(chi_l * d, d * chi_r)
    u, s, vh = np.linalg.svd(theta, full_matrices=False)
    keep = min(chi_max, int(np.sum(s > tol)))
    u, s, vh = u[:, :keep], s[:keep], vh[:keep, :]
    s /= np.linalg.norm(s)                                   # renormalize after non-unitary step
    return u.reshape(chi_l, d, keep), (np.diag(s) @ vh).reshape(keep, d, chi_r)

def tebd(mps, hs, dt, steps, chi_max, imaginary=True):
    """First-order Trotter TEBD: update even bonds, then odd bonds."""
    f = -dt if imaginary else -1j * dt
    gates = []
    for h in hs:                                  # exp(f h) via eigendecomposition
        w, v = np.linalg.eigh(h.reshape(4, 4))
        gates.append((v @ np.diag(np.exp(f * w)) @ v.conj().T).reshape(2, 2, 2, 2))
    for _ in range(steps):
        for parity in (0, 1):
            for i in range(parity, len(mps) - 1, 2):
                mps[i], mps[i + 1] = apply_gate(mps[i], mps[i + 1], gates[i], chi_max)
    return mps

def full_hamiltonian(hs, L):
    return sum(np.kron(np.kron(np.eye(2**i), h.reshape(4, 4)), np.eye(2**(L - i - 2)))
               for i, h in enumerate(hs))

def energy(mps, H):
    """<H> by contracting the MPS to a dense vector (fine for short chains)."""
    psi = mps[0]
    for A in mps[1:]:
        psi = np.tensordot(psi, A, axes=(-1, 0))
    psi = psi.reshape(-1)
    psi /= np.linalg.norm(psi)
    return np.real(psi.conj() @ H @ psi)

L, chi = 10, 16
hs = bond_hamiltonians(L, J=1.0, g=1.0)                      # critical point g = J
H = full_hamiltonian(hs, L)
mps = [np.array([1., 0.]).reshape(1, 2, 1) for _ in range(L)]   # product state |up...up>
for dt in (0.1, 0.02, 0.005):                                # shrink the Trotter step
    mps = tebd(mps, hs, dt, steps=int(5 / dt), chi_max=chi)
print("TEBD  E0 =", energy(mps, H))                          # about -12.38139
print("exact E0 =", np.linalg.eigvalsh(H)[0])                # -12.38149

The residual error (about $10^{-5}$ relative) is Trotter error; a second-order (Strang) splitting, a smaller final step, or DMRG would remove it. For production work use a maintained library (ITensor, TeNPy, quimb), which provide DMRG, TEBD, TDVP, conserved quantum numbers, and infinite-system algorithms.

Research frontiers

The narrative list of “hot topics” that used to open this page is collected here with the developments that have shaped the field through 2026.

Experimental probes of the formalism

Each standard probe measures a quantity computed directly by the formalism above. The full methodology is on Experimental Techniques.

Three relations are used constantly in data analysis.

ARPES self-energy. Fitting momentum distribution curves at fixed $\omega$ gives the peak position $k_m(\omega)$ and half width $\Delta k(\omega)$. With a bare dispersion $\epsilon^0_{\mathbf{k}}$ and bare velocity $v_0$,

Lifshitz-Kosevich. For each extremal Fermi-surface orbit the oscillatory magnetization is

so the temperature dependence of the amplitude gives $m^\ast$, its field dependence gives the quantum lifetime $\tau_q$, and the phase $\phi$ contains the Berry phase of the orbit.

Tunneling. For a tip with a flat density of states $\rho_t$ and energy-independent transmission,

and Fourier-transformed $dI/dV$ maps reveal scattering vectors $\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i$ between regions of high spectral weight.

References and further reading

Textbooks

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics (1976).
2. G. D. Mahan, Many-Particle Physics, 3rd ed. (2000).
3. A. A. Abrikosov, L. P. Gorkov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (1963).
4. A. Altland and B. Simons, Condensed Matter Field Theory, 3rd ed. (2023).
5. P. Coleman, Introduction to Many-Body Physics (2015).
6. X.-G. Wen, Quantum Field Theory of Many-Body Systems (2004).
7. S. M. Girvin and K. Yang, Modern Condensed Matter Physics (2019).
8. S. Sachdev, Quantum Phase Transitions, 2nd ed. (2011); Quantum Phases of Matter (2023).
9. B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors (2013).
10. D. Vanderbilt, Berry Phases in Electronic Structure Theory (2018).
11. M. Tinkham, Introduction to Superconductivity, 2nd ed. (1996).
12. T. Giamarchi, Quantum Physics in One Dimension (2004).

Reviews

1. A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, “Dynamical mean-field theory of strongly correlated fermion systems,” Rev. Mod. Phys. 68, 13 (1996).
2. M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
3. C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, S. Ryu, “Classification of topological quantum matter with symmetries,” Rev. Mod. Phys. 88, 035005 (2016).
4. N. Marzari et al., “Maximally localized Wannier functions: Theory and applications,” Rev. Mod. Phys. 84, 1419 (2012).
5. B. Keimer et al., “From quantum matter to high-temperature superconductivity in copper oxides,” Nature 518, 179 (2015).
6. N. P. Armitage, E. J. Mele, A. Vishwanath, “Weyl and Dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).
7. L. Balents, C. R. Dean, D. K. Efetov, A. F. Young, “Superconductivity and strong correlations in moiré flat bands,” Nat. Phys. 16, 725 (2020).
8. U. Schollwöck, “The density-matrix renormalization group in the age of matrix product states,” Ann. Phys. 326, 96 (2011).

See Also

• Superconductivity, Quantum Hall & Topological Phases — the phenomena this formalism describes.
• Disorder & Localization — Anderson localization, scaling theory, and many-body localization.
• Metals & Magnetism — Fermi-liquid theory and magnetic order.
• Experimental Techniques — full methodology for the probes that test this formalism.
• Condensed Matter Physics (Hub) — crystal structure, band theory, and semiconductors.
• Quantum Field Theory — field-theoretic methods for collective excitations.
• Renormalization — the renormalization group in field theory and statistical mechanics.
• Computational Physics — DFT, Monte Carlo, and tensor-network simulations in more depth.