11.5 Quantum phase transitions

Fundamentals of quantum phase transitions

Quantum phase transitions (QPTs) happen at absolute zero temperature, where thermal fluctuations are completely frozen out and quantum fluctuations alone drive the system from one ground state to another. Unlike the phase transitions you encounter in classical thermodynamics, these are triggered not by heating or cooling but by tuning a non-thermal control parameter like pressure, magnetic field, or chemical doping. The physics near these transitions underpins some of the most puzzling and technologically promising phenomena in condensed matter, from unconventional superconductivity to non-Fermi liquid behavior.

Definition and characteristics

A QPT is an abrupt change in the ground state of a many-body quantum system at $T = 0$. You drive the transition by varying a coupling constant $g$ (pressure, field strength, composition) through a critical value $g_c$. At that critical value, the system hits a quantum critical point (QCP).

Key features of QPTs:

• The ground-state wavefunction restructures qualitatively at $g = g_c$
• Long-range quantum correlations and entanglement develop across the system
• The energy gap $\Delta$ between the ground state and first excited state vanishes at the QCP
• The correlation length $\xi$ diverges, making the system scale-invariant

Quantum vs. classical phase transitions

The last row is especially important. Because quantum mechanics couples spatial and temporal fluctuations, a QPT in $d$ spatial dimensions maps onto a classical transition in $d + z$ dimensions, where $z$ is the dynamic critical exponent. This mapping is a central tool in the theory.

Even though the QCP sits at $T = 0$, its influence fans out to finite temperatures in a quantum critical fan (more on this below), so QPTs have very real experimental consequences at accessible temperatures.

Role of quantum fluctuations

Quantum fluctuations originate from the Heisenberg uncertainty principle: conjugate variables like position and momentum, or different spin components, cannot simultaneously have sharp values. Near the QCP these fluctuations become the dominant source of correlations and can:

• Destroy an ordered phase (e.g., suppressing magnetic order by increasing quantum spin fluctuations)
• Induce long-range entanglement that has no classical analog
• Generate non-local correlations that extend over the entire system at $g = g_c$

Think of it this way: at a classical critical point, thermal energy shakes the system between competing configurations. At a QCP, the system tunnels between them quantum-mechanically, even at zero temperature.

Order parameters and critical exponents

An order parameter $\phi$ is a quantity that is zero in the disordered phase and nonzero in the ordered phase. For a ferromagnet, it's the magnetization $M$; for a superconductor, it's the pair condensate amplitude. Critical exponents describe how physical quantities diverge or vanish as $g \to g_c$.

For example, near a QCP you typically find power-law behavior:

• Correlation length: $\xi \sim |g - g_c|^{-\nu}$
• Energy gap: $\Delta \sim |g - g_c|^{z\nu}$
• Order parameter: $\phi \sim (g_c - g)^{\beta}$ (on the ordered side)

These exponents ($\nu$, $\beta$, $z$, and others) are not independent; they satisfy scaling relations analogous to those in classical critical phenomena.

Quantum critical points

At the QCP ($g = g_c$, $T = 0$):

• The correlation length $\xi \to \infty$
• The characteristic energy scale (gap) $\Delta \to 0$
• The system is scale-invariant: it looks statistically the same at all length scales
• No quasiparticle description survives, because excitations are critically damped

The QCP acts as an organizing center for the entire low-temperature phase diagram, even though you can never literally sit at $T = 0$ in the lab.

Universality classes

Systems with completely different microscopic physics can share the same set of critical exponents if they belong to the same universality class. What determines the class is not the detailed Hamiltonian but rather:

• The spatial dimensionality $d$
• The symmetry of the order parameter
• The range of interactions
• The dynamic critical exponent $z$

Common universality classes in QPTs include:

• Ising ($Z_2$ symmetry): e.g., transverse-field Ising model
• XY ($U(1)$ symmetry): e.g., superfluid-insulator transitions
• Heisenberg ($SU(2)$ symmetry): e.g., certain antiferromagnetic transitions

This is powerful because measuring the exponents in one system tells you the exponents for every system in that class.

Scaling behavior

Near the QCP, observables obey scaling forms. A generic thermodynamic quantity $\mathcal{O}$ at finite temperature satisfies:

$\mathcal{O}(g, T) = T^{p/z\nu} \, \Phi\!\left(\frac{g - g_c}{T^{1/z\nu}}\right)$

where $\Phi$ is a universal scaling function and $p$ depends on the observable. This means that data from different temperatures and tuning parameters should collapse onto a single curve when plotted in the right scaled variables. Experimentalists use this scaling collapse as a signature of quantum criticality.

Renormalization group (RG) methods provide the theoretical machinery for deriving these scaling relations and computing critical exponents beyond mean-field approximations.

Types of quantum phase transitions

Continuous vs. discontinuous transitions

• Continuous (second-order): The order parameter grows smoothly from zero. The correlation length diverges, and critical fluctuations dominate. These are the transitions where universality and scaling apply most directly.
• Discontinuous (first-order): The order parameter jumps at the transition. There is no diverging correlation length, and the two phases can coexist. Hysteresis is common. First-order QPTs are less universal but still physically important (e.g., some metamagnetic transitions).

Most of the theoretical framework discussed here applies to continuous QPTs. First-order quantum transitions are conceptually simpler but harder to tune experimentally because of the abrupt jump.

Symmetry-breaking transitions

These are the "classic" QPTs, described within the Landau paradigm:

• Ferromagnetic: The ground state spontaneously breaks rotational spin symmetry. A nonzero magnetization appears below $g_c$.
• Antiferromagnetic: Translational symmetry (or time-reversal symmetry) is broken, producing a staggered magnetization with ordering wavevector $\mathbf{Q}$.
• Superfluid/superconducting: $U(1)$ gauge symmetry is broken, and a macroscopic phase-coherent condensate forms.

When a continuous symmetry is spontaneously broken, Goldstone modes (gapless excitations) appear. Spin waves in a ferromagnet and phonons in a superfluid are familiar examples.

Topological transitions

Not all QPTs involve symmetry breaking. Topological quantum phase transitions change a topological invariant of the ground state without any local order parameter:

• Integer quantum Hall transitions: The system jumps between plateaus with different Chern numbers $C$.
• Topological insulator transitions: A band inversion changes the $Z_2$ topological index.
• These transitions are often accompanied by the closing and reopening of a bulk energy gap.
• Protected gapless edge or surface states appear (or disappear) at the transition.

Topological QPTs lie outside the Landau symmetry-breaking framework and are an active area of research, especially in the context of symmetry-protected topological phases.

Quantum critical phenomena

Quantum critical region

The QCP at $T = 0$ generates a fan-shaped region in the $(g, T)$ phase diagram called the quantum critical fan. Inside this fan:

• Neither the ordered phase nor the disordered phase provides a good starting point for perturbation theory
• Well-defined quasiparticles typically do not exist
• Transport properties are anomalous (e.g., resistivity linear in $T$ instead of $T^2$)
• Universal scaling governs thermodynamic and dynamical responses

The fan is bounded by crossover lines (not sharp phase boundaries) where the system crosses over into conventional ordered or disordered behavior. The width of the fan grows with temperature, which is why quantum criticality is experimentally accessible even though the QCP itself is at $T = 0$.

Finite temperature effects

Temperature introduces a thermal length scale $L_T \sim T^{-1/z}$. The physics depends on how $L_T$ compares to the correlation length $\xi$:

• $L_T \ll \xi$ (inside the fan): Quantum critical scaling dominates. The system "doesn't know" which side of the QCP it's on.
• $L_T \gg \xi$ (outside the fan): The system behaves as a conventional ordered or disordered phase with thermal corrections.

As temperature increases, a quantum-to-classical crossover occurs: the imaginary-time direction in the path integral effectively shrinks, and the system's critical behavior reduces to that of a classical transition in $d$ dimensions.

Dynamical scaling

At a QCP, space and time scale differently. The dynamic critical exponent $z$ is defined by:

$\omega \sim k^z$

where $\omega$ is a characteristic frequency and $k$ a wavevector. This means:

• $z = 1$: Space and time scale equally (relativistic-like, as in many antiferromagnets)
• $z = 2$: Time scales quadratically with space (common for ferromagnetic transitions and the superfluid-insulator transition in the dirty limit)
• $z = 3$: Appears in itinerant ferromagnets (Hertz-Millis theory)

The effective dimensionality becomes $d_{\text{eff}} = d + z$, which determines whether mean-field theory is valid. If $d_{\text{eff}}$ exceeds the upper critical dimension $d_c^+$ (typically 4 for an Ising-like transition), mean-field exponents apply (possibly with logarithmic corrections).

Experimental observations

Materials exhibiting quantum criticality

Several material families serve as key experimental platforms:

• Heavy fermion compounds: $\text{CeCu}_{6-x}\text{Au}_x$ is a textbook example. Tuning the Au concentration $x$ suppresses antiferromagnetic order to $T = 0$, revealing non-Fermi liquid behavior (e.g., $C/T \sim \ln(T_0/T)$) in the quantum critical fan.
• Cuprate superconductors: Near optimal doping, the strange-metal phase shows $T$-linear resistivity over a wide temperature range, widely attributed to proximity to a QCP (though the nature of that QCP remains debated).
• Iron pnictides: Compounds like $\text{BaFe}_2(\text{As}_{1-x}\text{P}_x)_2$ display magnetic and structural QPTs, with superconductivity peaking near the QCP.
• Quantum magnets: $\text{TlCuCl}_3$ under pressure shows a QPT from a gapped spin-singlet phase to an ordered antiferromagnet, interpretable as Bose-Einstein condensation of magnon excitations.

Measurement techniques

• Neutron scattering: Directly probes the spin excitation spectrum $S(\mathbf{q}, \omega)$, revealing softening of modes and critical fluctuations near the QCP.
• ARPES (angle-resolved photoemission spectroscopy): Maps the single-particle spectral function $A(\mathbf{k}, \omega)$, showing how the electronic structure evolves through the transition.
• NMR (nuclear magnetic resonance): The spin-lattice relaxation rate $1/T_1$ is sensitive to low-energy spin fluctuations and can detect the onset of magnetic order.
• Quantum oscillation measurements (de Haas-van Alphen, Shubnikov-de Haas): Track changes in the Fermi surface topology as the system is tuned through a QCP.

Challenges in detection

Reaching the QCP experimentally is difficult for several reasons:

• Temperatures in the millikelvin range (dilution refrigerators) are often needed to enter the quantum critical regime
• Disorder and impurities can smear the transition, round the QCP, or nucleate a glassy phase (Griffiths effects)
• Fine-tuning the control parameter to $g_c$ requires high-precision sample preparation or continuous in-situ tuning (e.g., pressure cells)
• Competing instabilities (such as superconductivity) can mask the QCP by opening a gap before the critical point is reached

Theoretical approaches

Landau theory for quantum transitions

The starting point is often a quantum generalization of Landau-Ginzburg theory. You write a free-energy functional in terms of the order parameter field $\phi(\mathbf{x}, \tau)$, where $\tau$ is imaginary time (ranging from $0$ to $\hbar / k_B T$):

$S[\phi] = \int d^d x \int_0^{\hbar/k_BT} d\tau \left[ |\partial_\tau \phi|^2 + |\nabla \phi|^2 + r\,|\phi|^2 + u\,|\phi|^4 \right]$

Here $r \propto (g - g_c)$ tunes through the transition and $u > 0$ stabilizes the theory. This action looks like a classical Landau-Ginzburg theory in $(d + 1)$ dimensions (with the extra dimension being imaginary time), which is the formal basis for the $d_{\text{eff}} = d + z$ mapping.

Mean-field theory (saddle-point approximation of this action) predicts standard exponents ($\beta = 1/2$, $\nu = 1/2$, etc.) but neglects fluctuations. Near the QCP, fluctuations are strong, and you need RG methods to get the correct exponents.

Renormalization group methods

The RG approach systematically integrates out short-wavelength fluctuations to see how the effective theory flows as you zoom out:

1. Start with the action $S[\phi]$ at a microscopic cutoff $\Lambda$.
2. Integrate out modes with momenta in a shell $\Lambda/b < k < \Lambda$.
3. Rescale momenta ($k \to bk$), frequencies ($\omega \to b^z \omega$), and the field to restore the original cutoff.
4. Read off how the couplings $r$, $u$, etc. change under this transformation (the beta functions).
5. Fixed points of the flow correspond to universality classes; the eigenvalues at the fixed point give the critical exponents.

The RG reveals which perturbations are relevant (grow under coarse-graining and drive the system away from criticality) and which are irrelevant (shrink and don't affect universal properties). For quantum transitions, the imaginary-time direction means the RG must handle anisotropic scaling when $z \neq 1$.

Numerical simulations

When analytical methods hit their limits, numerical techniques step in:

• Quantum Monte Carlo (QMC): Samples the imaginary-time path integral stochastically. Very powerful for bosonic systems and unfrustrated magnets, but suffers from the sign problem for fermionic and frustrated systems.
• Density matrix renormalization group (DMRG): Extremely accurate for 1D systems. Variationally optimizes a matrix product state representation of the ground state.
• Tensor network methods: Generalize DMRG to 2D and higher dimensions using projected entangled pair states (PEPS) and related ansätze. Capturing long-range entanglement at a QCP remains computationally demanding.
• Exact diagonalization: Solves the Hamiltonian exactly for small system sizes (typically up to ~40 spins). Useful for benchmarking and for studying spectral properties, but finite-size effects are significant.

Applications and implications

Quantum materials design

Understanding QPTs gives materials scientists a design principle: tune a system toward a QCP to enhance desired properties. Near a QCP, fluctuations are strong and can mediate new types of interactions. Specific strategies include:

• Engineering antiferromagnetic QCPs in heavy fermion systems to boost superconducting $T_c$
• Designing topological materials where a band inversion (a topological QPT) produces robust surface states for spintronics
• Searching for quantum spin liquid ground states, which host fractionalized excitations (spinons, visons) with potential applications in fault-tolerant quantum memory

Quantum computing

QPTs connect to quantum computation in several ways:

• Adiabatic quantum computing: The algorithm works by slowly tuning a Hamiltonian from an easy-to-prepare ground state to one encoding the solution. If the path crosses a QPT, the gap closes and the algorithm slows down exponentially. Avoiding or understanding these transitions is essential for algorithm design.
• Quantum annealing: Exploits quantum fluctuations (transverse fields) to explore energy landscapes. The anneal passes through or near a QCP, and the gap structure at that point controls performance.
• Topological quantum computation: Uses anyonic excitations from topological phases (which can be reached via topological QPTs) to perform inherently fault-tolerant gate operations.

High-temperature superconductivity

A major motivation for studying QPTs is the connection to unconventional superconductivity:

• In cuprates, the superconducting dome in the $(x, T)$ phase diagram is centered near a putative antiferromagnetic QCP. Quantum critical spin fluctuations may provide the pairing glue.
• In iron pnictides, superconductivity peaks where the antiferromagnetic/structural transition is suppressed to $T = 0$.
• The non-Fermi liquid normal state ($\rho \propto T$) above optimal doping is naturally explained by quantum critical scattering.
• Identifying and characterizing the relevant QCP could guide the search for superconductors with even higher $T_c$.

Current research directions

Non-equilibrium quantum criticality

Equilibrium QPTs are now well-established, so the frontier has moved to non-equilibrium settings:

• Dynamical quantum phase transitions (DQPTs): After a sudden quench of a parameter across a QCP, the Loschmidt amplitude $\mathcal{G}(t) = \langle \psi_0 | e^{-iHt} | \psi_0 \rangle$ can show non-analytic behavior at critical times, analogous to thermal free-energy singularities.
• Many-body localization (MBL): Strongly disordered interacting systems can fail to thermalize entirely. The MBL-to-thermal transition is itself a type of QPT, but one that defies conventional scaling theories.
• Floquet engineering: Periodic driving can stabilize phases with no equilibrium analog (e.g., discrete time crystals). The transitions between these Floquet phases are a new class of QPTs.

Quantum phase transitions in driven systems

Driven and open quantum systems add dissipation and periodic forcing to the picture:

• Dissipative QPTs: In open quantum systems coupled to a bath, competition between coherent dynamics and dissipation can produce steady-state phase transitions described by Lindblad master equations.
• Periodically driven (Floquet) systems: Time-periodic Hamiltonians can induce topological phase transitions (e.g., Floquet topological insulators) that have no static counterpart.
• Light-induced phases: Ultrafast laser pulses can transiently drive materials through QPTs, enabling the study of quantum criticality on femtosecond timescales.

Exotic quantum phases

Some of the most exciting open questions involve QPTs that go beyond the Landau paradigm:

• Quantum spin liquids: These are phases with long-range entanglement but no symmetry breaking. Candidate materials include herbertsmithite ($\text{ZnCu}_3(\text{OH})_6\text{Cl}_2$) and $\alpha\text{-RuCl}_3$.
• Deconfined quantum critical points: A proposed scenario where a direct continuous transition occurs between two phases that break different symmetries, something forbidden in Landau theory. Fractionalized excitations (spinons, monopoles) emerge right at the critical point.
• Strange metals: The anomalous $T$-linear resistivity observed in cuprates and other systems may signal a fundamentally new type of quantum critical state, possibly described by emergent gauge theories or holographic (AdS/CFT) duality.
• Emergent gauge theories: Near certain QCPs, the low-energy effective theory can contain gauge fields that were not present in the microscopic Hamiltonian, leading to deconfined phases and fractionalization.