Quantum Phase Transitions and Competing Scales

A quantum phase transition changes the ground state as a nonthermal parameter is tuned. Its zero-temperature fixed point can organize finite-temperature and finite-size behavior, but only within a window bounded by microscopic scales, irrelevant couplings, and other instabilities. The first task is therefore to name every scale and the order of limits.

Required background. Universality Classes and Scaling Functions supplies RG eigenvalues and scaling functions; Thermodynamic Limits, Phases, and Ensemble Equivalence supplies the limits needed to define a phase transition.

Tuning, correlation length, and gap

Let $g$ be a Hamiltonian parameter and $r=(g-g_c)/g_0$ a dimensionless distance from criticality. A continuous transition has

$$\xi\sim \xi_0|r|^{-\nu}, \qquad \xi_\tau\sim \xi^z, \qquad \Delta\sim \xi_\tau^{-1}\sim |r|^{\nu z}.$$

$\nu$ is the correlation-length exponent and $z$ the dynamical exponent. The nonuniversal $\xi_0$ and the velocity or damping coefficient relating time to space must be retained when comparing different observables. If several modes have distinct $z$ or if a dangerously irrelevant coupling creates another time scale, a single $\xi_\tau$ is insufficient.

At $T=0$ the thermodynamic limit is taken before diagnosing spontaneous symmetry breaking. In a finite sample of length $L$, the critical gap scales as $L^{-z}$ only after aspect ratios, boundary conditions, and the leading irrelevant variables are fixed. A generic observable has

$$O(r,T,L,\{u_i\})= L^{-x_O}\mathcal F_O \left(rL^{1/\nu},\,TL^z,\,\{u_iL^{y_i}\}\right),$$

where $y_i<0$ for irrelevant couplings. This formula is a hypothesis to test, not permission to ignore drift: slowly irrelevant variables can dominate accessible sizes.

Temperature cuts off imaginary time

The equilibrium path integral has extent $\beta=1/T$, so temperature limits temporal correlations. Comparing $T$ with the zero-temperature gap defines the crossover scale

$$T^*(r)\sim |r|^{\nu z}.$$

When $T\ll T^*$, the system reflects one adjacent ground-state regime. When $T\gg T^*$ but $T$ remains below microscopic cutoffs, the quantum-critical fixed point can control scaling. This wedge is a crossover region, not generally a finite-temperature phase.

On an ordered side, a separate thermal transition $T_c(r)$ may occur in dimensions and symmetries that permit it. On a disordered side, activation can set in below $\Delta$. Disorder, interlayer coupling, superconductivity, or a first-order transition may truncate both crossovers. The boson-Hubbard transition provides a canonical example in which density, particle–hole symmetry, and the path through the phase diagram change $z$ Fisher et al. 1989, §§ III–V.

Competing infrared scales

An irrelevant coupling $u$ ordinarily gives corrections proportional to $\xi^{y_u}$. It is dangerously irrelevant when setting $u=0$ makes an observable singular or changes the ordered-state structure. Then an additional length $\xi'$ or thermal line can scale with an exponent not fixed by $\nu z$. Examples include the quartic coupling above an upper critical dimension and monopole fugacity near some fractionalized critical points.

Finite density, magnetic field, momentum relaxation, and probe frequency introduce further ratios:

$$\frac{\mu}{T},\qquad \frac{B}{T^{y_B/z}}, \qquad \frac{\Gamma_{\mathrm{mr}}}{T},\qquad \frac{\omega}{T}.$$

A proposed scaling collapse that varies only $r/T^{1/(\nu z)}$ is incomplete if any omitted ratio changes across the data set. Analytic backgrounds can also dominate thermodynamics even when the singular part scales correctly.

Continuous, first-order, and avoided transitions

Diverging $\xi$ and universal scale invariance characterize a continuous transition. A level crossing or discontinuous order parameter with finite $\xi$ is first order, although finite size rounds it. An avoided critical point can produce a large but finite scaling window. Establishing criticality therefore requires systematic growth of correlation scales, stable exponent estimates, and consistency between independent observables—not only a broad crossover.

Exercises

1. If $\nu=2/3$ and $z=1$, how does the crossover temperature scale with $r$?

Solution

$T^*\sim|r|^{\nu z}=|r|^{2/3}$. Doubling $|r|$ multiplies $T^*$ by $2^{2/3}$, up to a nonuniversal scale.

2. At criticality, how should the lowest gap scale in a finite box when $z=2$?

Solution

With fixed shape and boundary condition, $\Delta_L\sim L^{-z}=L^{-2}$. Corrections from irrelevant fields add subleading powers; a nonzero thermodynamic intercept would contradict the assumed continuous fixed point.

References

• Fisher, M. P. A., P. B. Weichman, G. Grinstein, and D. S. Fisher. “Boson Localization and the Superfluid–Insulator Transition.” Physical Review B 40 (1989): 546–570. DOI.