Lecture 29: Quantum Critical Scaling

The material anchor is a CoNb${}_2$O${}_6$-like Ising-chain magnet in a transverse magnetic field. Inelastic neutron scattering can track the lowest magnetic excitation and its dispersion as the field tunes competition between Ising exchange and transverse polarization; thermodynamic probes test the finite-temperature crossover around the same field scale. A programmable cold-atom or Rydberg-array Ising simulator is the finite-size contrast: different system sizes and correlations can be measured under controlled preparation, but long-range couplings, open boundaries, readout error, and nonadiabatic ramps must still be audited. The nearest-neighbor periodic chain below supplies the scaling grammar, not a precision Hamiltonian for either platform.

Field tuning changes the ground-state competition even when the laboratory or simulator is finite and every measured gap remains nonzero. What makes the thermodynamic transition quantum rather than a thermal branch-selection problem, and how can finite data reveal it? Quantum-critical scaling supplies the inference connecting quantum magnets, cold-atom systems, and the thermodynamic limit.

We will diagonalize finite Ising rings and use their gap sequence and data collapse to test a thermodynamic critical claim while keeping finite- temperature crossover separate.

A numerical study of periodic Ising rings reports a nonzero even-sector gap for $L=4,6,8,10,12$, but the product $L{\mathrm{\Delta }}_{+}/J$ is nearly constant. Predict whether the thermodynamic gap closes and what should happen to the raw gap when $L$ doubles. Then predict whether the two sides of a finite-temperature fan are necessarily phase boundaries. These decisions are the practical use of scaling in quantum-magnet and cold-atom experiments: finite hardware never contains the thermodynamic singularity, so the inference must be earned from controlled size and temperature dependence.

Compute $[{\hat{\sigma }}^x,{\hat{\sigma }}^z]$, predict the ground states favored separately by Ising exchange and transverse field, and distinguish an eigenvalue gap from a probability weight. Free-energy selection from Lecture 17 is recalled only to compare thermal and quantum tuning. The relative thermal weight is introduced locally as a normalized probability ratio, not as a quantum amplitude; no partition function, entropy formula, or imaginary-time theory is presumed. The changed calculation replaces minimization at one size by a joint rescaling of data across sizes. The thermal time $\hslash /(k_{\mathrm{B}}T)$ is introduced only when it first cuts off the Ising correlation time.

29.1 Thermal minimization and quantum critical dynamics

The finite Ising rings are exact systems, but none contains a phase transition. The critical object is the coordinated thermodynamic and zero-temperature limit of a Hamiltonian family. Scaling collapse is therefore an asymptotic hypothesis tested by stable windows and residuals, not a theorem produced by plotting. Later in this lecture, linearizing a declared renormalization-group flow near a fixed point gives $\mathrm{d}{u}_a/\mathrm{d}\ell =y_a{u}_a$ and $\nu =1/y_r$; that local derivation is controlled for small deviations from the fixed point but is not a microscopic derivation of an interacting beta function. Irrelevant operators and the thermal cutoff provide the first corrections to finite-size inference.

Use finite spin-chain Hilbert spaces with a declared boundary condition, tuning parameter, critical value, reduced distance, energy and length units, system-size sequence, scaling functions and their domains, fit window, and uncertainty model. Scaling relations are asymptotic functions, not identities at all parameters.

A two-spin avoided crossing is easy to calculate but cannot be a critical point. The calculation must reveal what changes as the same Hamiltonian grows from two spins to a sequence of rings and finally to the thermodynamic limit. We will use the spectrum to introduce each new scale only when the preceding finite-size observation demands it.

The critical object is an asymptotic scaling structure of a Hamiltonian family near a zero-temperature singular limit. A finite-size gap, fitted exponent, or collapsed curve is an estimator of that structure, not the thermodynamic transition itself. We must state the order of the $L\to \mathrm{\infty }$, $T\to 0$, and tuning limits, the fit window, and the competing corrections before claiming universality. Likewise, the sides of a finite-temperature fan mark crossovers controlled by finite coherence time unless an independent thermal phase transition is established. The phase distinction is defined at $T=0$ after taking the thermodynamic limit; finite-size scaling approaches a joint limit with $L/\xi$ and the corresponding temporal ratio held fixed. Reversing this logic leaves a finite analytic system and therefore cannot by itself define the singular transition.

For an ordinary real order-parameter function $m:{\mathbb{R}}^d\to \mathbb{R}$, the Landau free-energy functional maps the whole function to a number:

The symbol $F$ is a functional, not an operator and not an ordinary function of position. At $h=0$, , and for uniform $m$, stationary points obey $m(r+u{m}^2)=0$. When , $m=0$ is stable. When , the stable minima are $m=\pm \sqrt{-r/u}$. In a thermal transition, temperature changes the effective coefficient $r(T)$, and thermal fluctuations explore configurations with equilibrium weights.

Only one statistical-mechanics idea is needed here. At temperature $T$, two coarse-grained configurations with free energies $F_1$ and $F_2$ have the relative equilibrium weight $p_1/p_2=\exp [-(F_1-F_2)/(k_{\mathrm{B}}T)]$. These weights define a probability distribution over configurations; they are not quantum amplitudes. The free energy includes the energetic and entropic preference of the degrees of freedom that have been coarse-grained away. Here entropy means the logarithmic measure of how many microscopic arrangements realize a given macroscopic configuration; no prior formula for it is required. Thus temperature can both change the effective coefficient $r(T)$ and populate fluctuations around the minima. No partition-function machinery is assumed below.

29.2 One Ising ring from competing terms to a closing gap

Consider

For and $\lambda =0$, the interaction favors $z$-ordered states. For $\lambda \gg J$, the field favors a product state polarized along $x$. Because $[{\hat{\sigma }}_i^x,{\hat{\sigma }}_i^z]\ne 0$, the two tendencies cannot be minimized simultaneously. The tuning parameter changes the ground-state quantum fluctuations even at $T=0$.

At $\lambda =0$, predict the two aligned ground states of a ferromagnetic ring. At $\lambda /J\to \mathrm{\infty }$, predict the unique $+x$-polarized product ground state. Because these states minimize incompatible operators, decide whether a finite matrix must show a singular crossing, a smooth avoided crossing, or either depending on symmetry sector.

For two spins with one Ising bond, $${\hat{H}}_2=-J{\hat{\sigma }}_1^z{\hat{\sigma }}_2^z-\lambda ({\hat{\sigma }}_1^x+{\hat{\sigma }}_2^x).$$ The odd combinations $(|\uparrow \uparrow ⟩-|\downarrow \downarrow ⟩)/\sqrt{2}$ and $(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$ have energies $-J$ and $+J$. In the even basis $$|{\varphi }_{+}⟩=\frac{|\uparrow \uparrow ⟩+|\downarrow \downarrow ⟩}{\sqrt{2}},|{\psi }_{+}⟩=\frac{|\uparrow \downarrow ⟩+|\downarrow \uparrow ⟩}{\sqrt{2}},$$ the numerical matrix is with eigenvalues $\pm \sqrt{J^2+4{\lambda }^2}$. Every finite eigenvalue is an analytic function away from exact symmetry crossings. This small system shows competition and avoided crossing, but not a thermodynamic singularity.

Nothing is hidden in the other symmetry sector. With $$|{\varphi }_{-}⟩=\frac{|\uparrow \uparrow ⟩-|\downarrow \downarrow ⟩}{\sqrt{2}},|{\psi }_{-}⟩=\frac{|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩}{\sqrt{2}},$$ the full four-dimensional problem separates as The dimensions add to four, and direct substitution gives ${\hat{H}}_2|{\varphi }_{-}⟩=-J|{\varphi }_{-}⟩$ and ${\hat{H}}_2|{\psi }_{-}⟩=+J|{\psi }_{-}⟩$. At $\lambda =0$, the even ground state and $|{\varphi }_{-}⟩$ are the two aligned ground states. For $\lambda /J\to \mathrm{\infty }$, the normalized lower even eigenvector tends to $$\frac{|{\varphi }_{+}⟩+|{\psi }_{+}⟩}{\sqrt{2}}=|+x,+x⟩,$$ as required by the field term. If instead $J=0$, its energy is $-2|\lambda |$, the sum of two independent one-spin ground-state energies. These sector, normalization, and limiting-state checks are more informative than quoting the two eigenvalues alone.

The block structure also exposes what the field actually does. The Ising term is diagonal in the $z$-basis, whereas each ${\hat{\sigma }}_i^x$ flips one spin. Consequently, the field couples the aligned even state $|{\varphi }_{+}⟩$ to the anti-aligned even state $|{\psi }_{+}⟩$, producing the off-diagonal matrix element $-2\lambda$. The lower even eigenvalue lies below $-J$ for every nonzero $\lambda$, so the finite-system ground state evolves smoothly from an aligned superposition toward an $x$-polarized state. The exact twofold degeneracy at $\lambda =0$ is tied to the symmetry and finite basis; it does not supply a diverging length. A quantum critical claim therefore requires scaling evidence across a size sequence and a declared thermodynamic limit.

Now place the Hamiltonian of Equation (29.2) on an even-length one-dimensional ring of $L$ sites with periodic boundary conditions, ${\hat{\sigma }}_{L+1}\equiv {\hat{\sigma }}_1$. The global Ising symmetry operator $\hat{P}=\prod _{i=1}^L{\hat{\sigma }}_i^x$ commutes with $\hat{H}$ and has eigenvalues $P=\pm 1$. We will follow the even-sector gap $${\mathrm{\Delta }}_{+}(L,\lambda )=E_{1,+}(L,\lambda )-E_{0,+}(L,\lambda ),$$ where both energies belong to $P=+1$. Declaring the sector matters. In the ordered regime, the lowest states in opposite parity sectors form a nearly degenerate symmetry pair, so the unrestricted first gap can become exponentially small even away from the bulk critical point. That splitting is not the bulk excitation gap whose critical scaling we seek.

The two-spin construction already gives an exact-diagonalization algorithm for the ring. In the $z$-basis the Ising term is diagonal, the transverse field connects bit strings that differ by one spin flip, and parity divides the matrix into two blocks. Direct diagonalization of the even block at $\lambda /J=1$ gives the following reproducible finite-size sequence:

The raw gap is nonzero for every listed ring. Yet the nearly constant last column predicts ${\mathrm{\Delta }}_{+}\propto L^{-1}$ and hence a zero thermodynamic gap. Removing the smallest size should change the fitted exponent less than retaining only two small sizes. This is the first concrete difference between a finite-size minimum and a critical closing.

Which column supplies evidence for a thermodynamic closing, and which other finite-system gap can become very small throughout the ordered regime without measuring the bulk excitation? The expected answer is the nearly constant $L{\mathrm{\Delta }}_{+}/J$ column and the opposite-parity symmetry-doublet splitting. The plausible wrong answer “the smallest raw gap proves the critical point” ignores both the size law and the symmetry sector. If it was tempting, recover the definition of ${\mathrm{\Delta }}_{+}$, fit $\ln {\mathrm{\Delta }}_{+}$ against $\ln L$ once with and once without $L=4$, and state the inferred exponent before introducing $\xi$ or $z$.

For orientation, the exact Jordan–Wigner and Bogoliubov solution—stated here, not derived—has one-particle excitation energy $$\epsilon (k)=2J\sqrt{1+{\gamma }^2-2\gamma \cos k},\gamma =\frac{\lambda }{J}.$$ Here $\epsilon$ is an ordinary nonnegative dispersion function of momentum, not an operator or a probability distribution. The lattice spacing is set to one, so $k$ is dimensionless. Boundary condition and parity determine the allowed finite-size momenta. In the thermodynamic limit, the minimum at $k=0$ is ${\mathrm{\Delta }}_{\mathrm{bulk}}=2J|1-\gamma |$, so it closes only at $\gamma =1$. There, $\epsilon (k)\simeq 2J|k|$, and momentum spacing of order $L^{-1}$ explains $z=1$. The even-sector gap contains the lowest allowed pair and therefore approaches $4\pi J/L$ for this periodic convention, consistent with the table. A different boundary condition or sector changes the prefactor and allowed momenta, not the asymptotic exponent.

29.3 The closing spectrum demands length, time, and energy scales

For the Ising chain, take ${\hat{A}}_{\mathbf{0}}={\hat{\sigma }}_i^z$ and ${\hat{B}}_{\mathbf{r}}={\hat{\sigma }}_{i+r}^z$, so $C_{zz}(r)=⟨{\hat{\sigma }}_i^z{\hat{\sigma }}_{i+r}^z⟩-⟨{\hat{\sigma }}_i^z⟩⟨{\hat{\sigma }}_{i+r}^z⟩$. Away from criticality its envelope decays on a length $\xi$. The exact chain solution gives $\xi \propto {|1-\gamma |}^{-1}$ near $\gamma =1$; this is the same model whose spectral gap was followed above, not a separately postulated critical system.

The exponent $\nu$ controls spatial divergence; $z$ converts spatial scale to correlation time. The gap $\mathrm{\Delta }$ is the characteristic low-energy scale associated with ${\xi }_{\tau }$. The proportionalities omit nonuniversal microscopic units, so a fit must define the dimensionless reduced parameter and energy normalization.