QFT for quantum matter

Use this pathway when the central question is how collective quantum matter responds at low energy: which fields survive, which state or ensemble they inhabit, and which correlator a probe measures. The shared starting point is a nonrelativistic field description and a many-body response function; after that, choose one regime—Fermi liquid, superfluid or superconductor, quantum critical matter, or fractional quantum Hall matter—and carry a calculation far enough to test a scientific claim.

This is not a survey of condensed matter. Band-structure construction, magnetism, disorder, phonons, lattice numerics, and materials-specific inference enter only when the chosen question requires them. If your desired output is primarily a numerical reproduction rather than a field-theory response, compare the computational field theory pathway.

Decide whether this is the right field language

A nonrelativistic field is usually appropriate when particle number and chemical potential organize the state, antiparticles are absent at the energies of interest, and the leading dispersion is quadratic or tied to a lattice band rather than Lorentz symmetry. A representative continuum action contains

$$S_E=\int_0^\beta\mathrm d\tau\int\mathrm d^d x\, \bar\psi \left( \partial_\tau-\frac{\nabla^2}{2m}-\mu \right)\psi +S_{\mathrm{int}}.$$

Choose this language only after stating the spatial dimension, statistics, conserved charges, chemical potential, state, ultraviolet cutoff, and the degrees of freedom removed in reaching $S_E$. A lattice field, a patch field near a Fermi surface, a bosonic order parameter, and a topological gauge field are different low-energy choices; none follows merely from calling the system “many body.” Start with nonrelativistic fields and low-energy reduction when this choice is not yet controlled.

You need the many-body Green-function language when the requested output is a spectral function, susceptibility, transport coefficient, screening length, collective mode, or quasiparticle lifetime. These quantities differ by operator ordering and analytic boundary conditions. If the goal is only a static symmetry-allowed effective action, you may not need the full self-energy machinery, but you still need enough source and correlator language to identify the observable. The many-body correlator and response chapter is the common entry.

Preparation in capabilities, not course labels

You are ready to begin if you can do the following:

• vary an action, identify a continuous symmetry, and derive the associated density or current;
• Fourier transform a two-point kernel while keeping its time ordering or boundary condition attached;
• differentiate a generating functional and distinguish full, connected, Euclidean, time-ordered, and retarded correlators;
• use a chemical potential and thermal density operator without confusing a state parameter with a new interaction; and
• state an RG scaling assumption, an EFT expansion parameter, and the first effect omitted by an approximation.

Review only the capability that is missing. The relevant Core lessons are functional integrals and correlators, symmetry, currents, and Ward identities, renormalization and the RG, and effective field theory and matching. For thermal fluctuations, estimators, or covariance, use the statistical and probability check.

Your first three actions

1. Freeze one regime and one observable. Write the dimension, equilibrium state or initial density matrix, temperature, density or chemical potential, retained fields, cutoff, and quantity to be computed. “A two-dimensional Fermi gas at temperature $T$, probed by its retarded density response for specified $(\omega,\boldsymbol q)$” is sufficiently concrete.
2. Couple the observable to a source. Write the source term and derive the operator by differentiation. State whether the calculation produces a Matsubara, retarded, advanced, lesser, or statistical function; the symbol $G$ or $\chi$ alone is incomplete.
3. Calculate one limit before choosing a regime branch. For density response, obtain the free polarization, check compressibility and number conservation, then ask what new low-energy structure the project adds: stable quasiparticles, pairing, critical scaling, or topological order.

The first two actions prevent a formalism from being chosen before the question. The third produces a benchmark that survives when interactions or a new phase are introduced.

Running example: retarded density response

Work in units $\hbar=k_B=1$ with a translation-invariant equilibrium Fermi gas of mass $m$, chemical potential $\mu$, and spin or flavor degeneracy $g$. Let

$$\xi_{\boldsymbol p}=\frac{\boldsymbol p^2}{2m}-\mu, \qquad n(t,\boldsymbol x)=\psi^\dagger\psi, \qquad n(\tau,\boldsymbol x)=\bar\psi\psi.$$

Let $K_0=H_0-\mu N$ be the grand-canonical generator used in the equilibrium correlators, and couple an external scalar potential with the sign

$$K_U(t)=K_0+\int\mathrm d^d x\,U(t,\boldsymbol x)n(t,\boldsymbol x).$$

Use the Fourier convention $F(\omega,\boldsymbol q)=\int\mathrm dt\,\mathrm d^d x\, e^{i\omega t-i\boldsymbol q\cdot\boldsymbol x}F(t,\boldsymbol x)$. A positive uniform $U$ therefore acts like a negative change in chemical potential. Linear response gives

$$\delta n(\omega,\boldsymbol q) =\chi^R_{nn}(\omega,\boldsymbol q) U(\omega,\boldsymbol q),$$

where

$$\chi^R_{nn}(t,\boldsymbol x) =-i\theta(t) \left\langle \left[ n(t,\boldsymbol x),n(0,\boldsymbol 0) \right] \right\rangle.$$

This is a causal real-time object: it vanishes before the perturbation. The source-to-response derivation and its assumptions are the content of the Kubo formula Kubo 1957, §§ 2–3, pp. 572–578.

The same equilibrium problem can first be calculated in imaginary time. Add $+U\bar\psi\psi$ to the Euclidean action and define $W_E[U]=\log Z_E[U]$. The source sign gives

$$\frac{\delta W_E}{\delta U(x)} =-\langle n(x)\rangle_U$$

and hence the Euclidean response kernel

$$\begin{aligned} \chi_E(x-y) &=\frac{\delta\langle n(x)\rangle_U}{\delta U(y)} \bigg|_{U=0}\\ &=-\left\langle \mathrm T_\tau\, \delta n(x)\delta n(y) \right\rangle. \end{aligned}$$

For bosonic Matsubara frequency $\Omega_m=2\pi mT$, Wick contraction and the fermionic frequency sum give

$$\chi_{E,0}(i\Omega_m,\boldsymbol q) =g\int\frac{\mathrm d^d p}{(2\pi)^d} \frac{ f(\xi_{\boldsymbol p})-f(\xi_{\boldsymbol p+\boldsymbol q}) }{ i\Omega_m+\xi_{\boldsymbol p} -\xi_{\boldsymbol p+\boldsymbol q} }.$$

For the exact equilibrium analytic function, continuation from the discrete imaginary frequencies to the upper edge of the real axis gives

$$\boxed{ \chi^R_0(\omega,\boldsymbol q) =g\int\frac{\mathrm d^d p}{(2\pi)^d} \frac{ f(\xi_{\boldsymbol p})-f(\xi_{\boldsymbol p+\boldsymbol q}) }{ \omega+\xi_{\boldsymbol p} -\xi_{\boldsymbol p+\boldsymbol q}+i0 } }.$$

This is the Lindhard density response Lindhard 1954, pp. 16–25, PDF. The $+i0$ is not decoration: it makes the function analytic for $\operatorname{Im}\omega>0$ and fixes the absorptive part. For $\omega>0$, $\operatorname{Im}\chi^R_0\leq0$ in this convention, so the associated dynamic structure factor is nonnegative.

Two limits check both the sign and conservation law. Taking the static limit before the uniform limit,

$$\begin{aligned} \lim_{\boldsymbol q\to0} \chi^R_0(0,\boldsymbol q) &=g\int\frac{\mathrm d^d p}{(2\pi)^d} f'(\xi_{\boldsymbol p})\\ &=-\frac{\partial n}{\partial\mu}. \end{aligned}$$

A positive potential lowers the density, as required. In contrast, $\chi^R_0(\omega,\boldsymbol 0)=0$ for nonzero $\omega$, because a spatially uniform perturbation couples to the conserved total number. Thus

$$\lim_{\omega\to0}\lim_{q\to0}\chi^R_0(q,\omega)=0, \qquad \lim_{q\to0}\lim_{\omega\to0}\chi^R_0(q,\omega) =-\frac{\partial n}{\partial\mu}.$$

The order of limits is part of the observable. The integral also has the correct dimension, $[\chi]=[\text{density}]/[\text{energy}]$, and in a Galilean continuum obeys the density $f$-sum rule

$$\int_0^\infty\frac{\mathrm d\omega}{\pi} \left[-\omega\operatorname{Im}\chi^R_{nn} (\omega,\boldsymbol q)\right] =\frac{nq^2}{2m}.$$

The detailed particle–hole continuum and limit checks are developed in polarization and the Lindhard function. On a lattice, the right side of the sum rule is replaced by the appropriate kinetic or stress expectation; $nq^2/(2m)$ should not be imported unchanged.

Interactions dress both propagators and the density vertex. Dressing the lines while leaving an incompatible bare vertex can violate the Ward identity, compressibility, or sum rule. The self-energy and vertex must be treated consistently, as shown by Baym and Kadanoff 1961, pp. 291–295 and the site’s Ward-consistent response lesson.

Real time, Euclidean time, and nonequilibrium scope

These objects are related in equilibrium, but they are not interchangeable:

Object	Definition or data	What it answers
Euclidean response $\chi_E(i\Omega_m,\boldsymbol q)$	Imaginary-time-ordered correlator at discrete bosonic frequencies	Equilibrium functional-integral and Monte Carlo calculations
Retarded response $\chi^R(\omega,\boldsymbol q)$	Commutator with $-i\theta(t)$ and a real-frequency boundary value	Causal linear response, collective poles, and absorption
Dynamic structure factor $S(\boldsymbol q,\omega)$	Positive Lehmann weight	Scattering intensity after probe matrix elements and kinematics are included
Statistical or Keldysh correlator	Symmetrized fluctuations plus an initial density matrix	Occupations, noise, and evolution away from equilibrium

In equilibrium, the KMS condition and a spectral representation relate these objects. With the convention above,

$$-2\operatorname{Im}\chi^R_{nn} (\boldsymbol q,\omega) =\left(1-e^{-\beta\omega}\right) S(\boldsymbol q,\omega).$$

If the exact analytic function is known, the Matsubara boundary values determine the retarded one. Reconstructing it from finitely many noisy Euclidean data is an ill-posed inverse problem, not the substitution $i\Omega_m=\omega+i0$. The equilibrium dictionary is given in Mahan 2000, ch. 3.

A weak time-dependent probe of an equilibrium state is still equilibrium linear response. A quench, strong drive, aging state, or time-dependent distribution generally lacks imaginary-time periodicity and requires an initial density matrix on a closed real-time contour. In that setting, retarded and statistical correlators carry independent information. Continue to real-time contours and Keldysh response only when the question genuinely needs that additional structure Kamenev 2011, chs. 2–3.

Choose one regime branch

Use the feature that controls the low-energy calculation, not the material’s name:

Choose	When it is the right first branch	Result to carry forward
Fermi surfaces and Fermi liquids	A sharp Fermi surface and long-lived quasiparticles organize the state	Extract residue or lifetime criteria, Landau parameters, and a density, spin, or zero-sound response; identify an instability or non-Fermi-liquid failure signal Shankar 1994, §§ II–VI.
Pairing, superfluidity, and superconductivity	Pairing, phase stiffness, or a Meissner response is the defining low-energy structure	Derive the pairing or phase effective action, separate neutral global symmetry from electromagnetic gauge response, and test a gauge-invariant observable Nambu 1960, pp. 648–663.
Quantum phase transitions and critical metals	A zero-temperature tuning parameter and scale-invariant regime control the question	State the critical fields, $z$, relevant perturbations, crossover window, and a response scaling function; test alternatives to the proposed fixed point Sachdev 2011, chs. 1–2.
Fractional quantum Hall matter and anyons	Landau-level or Chern-band projection, a many-body gap, and topological data replace a local order parameter	Connect projected dynamics to Hall response, quasiparticle charge and statistics, boundary data, and at least one discriminator against competing order Nayak et al. 2008, §§ II–IV.

Do not force a project into one of these rows. A problem centered on disorder, ordinary band topology, magnetism, phonons, or a driven open system may need a different chapter or pathway. When two rows genuinely meet—for example, a paired state near a quantum critical point—complete one controlled baseline before adding the second set of low-energy modes.

Solved checks and templates

1. Fill the starting statement

Complete this sentence:

In ___ spatial dimensions, in the state ___, I retain the fields ___ below the scale ___ and will compute the ___ correlator of ___ in the limits ___, to compare with ___.

Worked template

For the running example:

In two spatial dimensions, in a translation-invariant grand-canonical equilibrium state at $(T,\mu)$, I retain nonrelativistic fermion fields $\psi_\sigma$ below a bandwidth cutoff $\Lambda$ and will compute the retarded density correlator $\chi^R_{nn}(\omega,\boldsymbol q)$, keeping the static and dynamic uniform limits distinct, to compare with a calibrated density-coupled scattering probe.

This statement fixes the field, state, correlator, cutoff, limit order, and probe class. It does not yet assume a Fermi liquid; that conclusion requires an interacting quasiparticle and response check.

2. Recover the compressibility sign

Starting from the Lindhard integrand, take $\omega=0$ and then $\boldsymbol q\to0$. Why is the answer negative with the source convention used here?

Solution

Let $\Delta\xi=\xi_{\boldsymbol p+\boldsymbol q}-\xi_{\boldsymbol p}$. Then

$$f(\xi_{\boldsymbol p})-f(\xi_{\boldsymbol p+\boldsymbol q}) =-f'(\xi_{\boldsymbol p})\Delta\xi+O(\Delta\xi^2),$$

while the static denominator is $-\Delta\xi$. Their ratio tends to $f'(\xi_{\boldsymbol p})$, so

$$\chi^R_0(0,\boldsymbol q\to0) =g\int\frac{\mathrm d^d p}{(2\pi)^d}f'(\xi_{\boldsymbol p}) =-\frac{\partial n}{\partial\mu}.$$

The perturbation is $+U n$, equivalent to $\mu\mapsto\mu-U$. A positive $U$ must therefore reduce the density. With a source term $-U n$, both the response relation and this sign would change.

3. Select the first regime

Match each project to its first branch: (a) collisionless zero sound with a narrow quasiparticle peak; (b) a phase mode and superfluid stiffness; (c) an $\omega/T$ collapse near a tuned zero-temperature transition; (d) a fractional Hall plateau with candidate anyonic excitations.

Solution

(a) starts with Fermi-liquid theory, because the quasiparticle distribution and Landau interaction generate zero sound. (b) starts with the superfluid/superconductor branch, where the phase effective action and gauge-invariant stiffness are central. (c) starts with quantum criticality, but a scaling collapse alone does not identify a fixed point; field content, crossover boundaries, and competing fits must be tested. (d) starts with the fractional quantum Hall branch, where quantized response, projection, topological data, edge structure, and alternatives must agree. Fractional charge alone does not establish braiding.

Leave with a phase-to-response work product

Produce a short derivation or reproducible notebook for one selected regime. It should contain:

1. the dimension, state or initial density matrix, retained fields, symmetries, cutoff, and power counting;
2. the source coupling and exact definition of the Euclidean or real-time correlator;
3. one derived response or low-energy effective action, with continuation and limit order stated;
4. a map from that object to one probe, including matrix elements, resolution, and background assumptions; and
5. the approximation’s first failure boundary and the uncertainty or numerical convergence relevant to the claim.

For a density-response project, include at least the dimension check, upper-half-plane causality, spectral-sign check, static compressibility, dynamic uniform limit, and the appropriate sum rule. For an interacting calculation, add a Ward-identity or conserving-vertex check. For another regime, replace these with equally consequential tests: gauge-invariant stiffness and collective modes for a paired phase; scaling collapse plus corrections and crossover bounds for critical matter; or quantized response, ground-state and boundary consistency, and competing-order tests for fractional Hall matter.

Finish by connecting the calculated correlator to quantum-matter probes and inference. If no branch above matches the desired work product, return to learning pathways and choose by the result you actually need.

References

• Baym, Gordon, and Leo P. Kadanoff. “Conservation Laws and Correlation Functions.” Physical Review 124 (1961): 287–299. DOI.
• Kamenev, Alex. Field Theory of Non-Equilibrium Systems. Cambridge: Cambridge University Press, 2011. DOI.
• Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems.” Journal of the Physical Society of Japan 12 (1957): 570–586. DOI.
• Lindhard, Jens. “On the Properties of a Gas of Charged Particles.” Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 28, no. 8 (1954): 1–57. Open PDF.
• Mahan, Gerald D. Many-Particle Physics. 3rd ed. New York: Springer, 2000. DOI.
• Nambu, Yoichiro. “Quasi-Particles and Gauge Invariance in the Theory of Superconductivity.” Physical Review 117 (1960): 648–663. DOI.
• Nayak, Chetan, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma. “Non-Abelian Anyons and Topological Quantum Computation.” Reviews of Modern Physics 80 (2008): 1083–1159. DOI.
• Sachdev, Subir. Quantum Phase Transitions. 2nd ed. Cambridge: Cambridge University Press, 2011. DOI.
• Shankar, Ramamurti. “Renormalization-Group Approach to Interacting Fermions.” Reviews of Modern Physics 66 (1994): 129–192. DOI.