Quantum Critical Wavefunctions

• Quantum critical wavefunctions are many-body states near zero-temperature phase transitions that exhibit universal scaling, multifractality, and emergent symmetries.
• They display algebraic correlations, divergent lengths, and multifractal spatial distributions, as demonstrated in models like impurity systems and quantum magnets.
• Analytical and numerical methods, including tensor networks and impurity model analyses, reveal robust universality and critical phenomena in these quantum states.

Quantum critical wavefunctions are many-body states at or near zero-temperature quantum phase transitions, where the ground state of the quantum system undergoes singular, non-analytic changes in response to a tuning parameter (e.g., interaction strength, disorder, or field). These wavefunctions display universal scaling properties, multifractality, emergent symmetries, and algebraic spatial or mode-correlation structure distinct from both ordered and disordered phases. Across diverse models—including quantum impurity systems (Blunden-Codd et al., 2016), disordered correlated lattices (Aguiar et al., 2012), quantum Hall trial states (Das et al., 2021), fractal tight-binding systems (Yao et al., 2024), tensor network critical points (Xu et al., 2020), and critical quantum magnets (Changlani et al., 2018)—quantum criticality imprints robust algebraic and statistical features onto the corresponding wavefunctions, enabling their identification and analysis both numerically and analytically.

1. Universal Properties and Signatures of Quantum Critical Wavefunctions

Quantum criticality manifests in ground-state wavefunctions through algebraic (power-law) correlations, divergent correlation lengths, and multifractal spatial structure. The paradigmatic indicators include:

• Scale invariance: Correlators and local observables display power-law decay in space, energy, or mode index, consistent with a diverging correlation length $\xi$ and absence of characteristic scale.
• Multifractality and amplitude fluctuations: Critical wavefunctions exhibit nontrivial distributions of amplitude intensity over space; their inverse participation ratios (IPRs) and generalized fractal dimensions $D_q$ interpolate between those of extended and localized phases, reflecting a complex, scale-invariant spatial structure (Aguiar et al., 2012, Das et al., 2021, Yao et al., 2024).
• Emergent symmetry and critical statistics: At critical points, symmetry properties not present in noncritical phases may emerge, as seen in the symmetry of displacement distributions in impurity models (Blunden-Codd et al., 2016) or the extensive ground-state degeneracy in frustrated magnets (Changlani et al., 2018).
• Universal distribution functions: The local amplitude (e.g., $|\psi|^2$) and two-point correlator distributions become universal, with power-law tails and critical exponent dependencies set by the universality class of the transition (Aguiar et al., 2012, Yao et al., 2024).
• Algebraic decay in mode basis: In quantum impurity problems, average coherent-state displacements $f_k$ vanish algebraically in frequency, and squeezing amplitudes develop universal plateaus at low energies (Blunden-Codd et al., 2016).

These features extend beyond specific microscopic realizations and are directly linked to the critical fixed point dictating the nontrivial scaling and statistics of the wavefunction.

2. Models and Methods: Realizations of Quantum Critical Wavefunctions

Quantum critical wavefunctions are studied across a variety of lattice and continuum models:

• Sub-Ohmic spin-boson impurity models: Ground states are expanded in multimode coherent states via the coherent-state expansion (CSE), revealing algebraic decay of displacement and universal squeezing at criticality (Blunden-Codd et al., 2016).
• Disordered interacting systems: At the Mott-Anderson transition, large-N statistical dynamical mean field theory (stat DMFT) on the Bethe lattice yields universal scaling forms for local quasiparticle amplitudes and power-law wavefunction intensity distributions (Aguiar et al., 2012).
• Quantum Hall states and non-unitary CFT: The Gaffnian trial state, representative of a quantum critical point at $\nu=2/5$, exhibits diverging correlation length, gaplessness, and multifractal entanglement signatures; its minimally modified variants interpolate between critical and gapped phases (Das et al., 2021).
• Fractal and quasicrystal lattices: Tight-binding wavefunctions on planar Sierpiński carpets, built via iterative self-similar generators, are universally in a critical phase for all energies, with multifractality and anomalous diffusive dynamics linked to the underlying geometry (Yao et al., 2024).
• Tensor network/PEPS states at RK points: Thermofield-double (TFD) states represented as PEPS lead to quantum critical wavefunctions whose norms map to partition functions of classical 3D models, precisely encoding the criticality (Xu et al., 2020).
• Quantum antiferromagnets with extensive critical manifolds: The kagome spin-$1/2$ XXZ model at $J_z/J=-1/2$ admits an exponentially large degeneracy of zero-energy "three-coloring" ground states, which map onto localized magnon eigenmodes and express quantum criticality in an unconventional, non-Landau sense (Changlani et al., 2018).

This broad representation underscores the structural and conceptual unity of quantum critical wavefunctions across different settings.

3. Scaling Laws, Multifractality, and Universality

Quantum critical wavefunctions are characterized by specific scaling laws and universal exponents:

• Spatial scaling: At the Mott-Anderson transition, the critical wavefunction at distance $r$ from a reference site has the form $\psi_c(r) \simeq \xi^{-d/2} \Phi(r/\xi)$, with $\Phi(x)$ showing power-law and exponential decay in respective regions (Aguiar et al., 2012). The squared amplitude distribution $|\psi|^2$ obeys $P(x)\sim x^{-1+\alpha}$ with $\alpha\to 1$ at criticality.
• Multifractal spectrum: The $q$-th order IPR $P_q=\sum_i |\psi_i|^{2q}$ scales as $P_q\sim L^{-\tau(q)}$, giving generalized dimensions $D_q=\tau(q)/(q-1)$. In the Sierpiński carpet, $D_2$ lies between $0.6$ and $0.9$ depending on the generator pattern and Hausdorff dimension, with $D_q$ decreasing for larger $q$ indicating true multifractality (Yao et al., 2024). Similar moments characterize multifractal corrections near the Bethe-lattice transition (Aguiar et al., 2012).
• Mode-resolved scaling: In impurity problems, the critical displacement $f_k\propto \omega_k^{(1-s)/2}$, vanishing in the infrared, while the squeezing amplitude $\kappa_k$ attains a universal plateau (Blunden-Codd et al., 2016).
• Dynamics: Wavepacket spreading is subdiffusive, with mean-square displacement $\langle r^2(t)\rangle \sim t^\delta$ where $\delta=D_2/D_H<1$, and return probabilities $C(t)\sim t^{-\gamma}$ with $0<\gamma<1$ (Yao et al., 2024).

Critical exponents for correlation length ($\nu$), amplitude scaling ($\beta$), and multifractal corrections are dictated by the universality class (e.g., Ising, percolation), and can often be extracted numerically from scaling collapses.

4. Emergent Symmetry, Degeneracy, and Critical Manifolds

Quantum criticality frequently gives rise to enhanced symmetry or extensive degeneracy in the ground-state manifold:

• Emergent $\mathbb{Z}_2$ symmetry: In the sub-Ohmic spin-boson model at criticality, mode-displacement distributions are symmetric around zero, indicating a restoration of symmetry absent from ordered phases (Blunden-Codd et al., 2016).
• Extensive degeneracy and non-Landau criticality: At $J_z/J=-1/2$ on the kagome lattice, all magnetization sectors host exponentially many exact ground states realized via projected three-colorings, signaling an unconventional "first-order-like" transition characterized by a critical manifold rather than a single ordered or disordered state (Changlani et al., 2018).
• Non-unitary conformal symmetry: In quantum Hall trial wavefunctions (Gaffnian), the underlying non-unitary CFT is linked to gaplessness and critical correlations in the thermodynamic limit (Das et al., 2021).

Such symmetries and degeneracies are closely connected to the field-theoretic structure of the critical point, e.g., associated with higher symmetry groups, conformal invariance, or emergent gauge structure.

5. Quantum Critical Wavefunctions in Tensor Networks and Machine Learning

Recent frameworks employ tensor network formalisms and quantum machine learning circuits to model, classify, and extract quantum critical states:

• Tensor networks and TFD states: Purifying a density matrix via TFD, and expressing the resulting state as a PEPS, allows mapping of the wavefunction norm (or Rényi entropy) to a partition function of a 3D classical statistical model (e.g., $Z_2$ gauge-Higgs), thereby embedding the quantum criticality directly into the tensor network's structure (Xu et al., 2020).
• Quantum convolutional neural networks (QCNNs): QCNNs trained on variational quantum eigensolver wavefunctions can classify phases and identify the quantum critical point in 1D transverse-field Ising models, suggesting that network structure effectively encodes critical properties from labeled quantum data (Wrobel et al., 2021).

These methods provide powerful, scalable tools for representing critical many-body states and extracting universality class information from high-dimensional quantum data.

6. Physical Consequences and Experimental Relevance

Quantum critical wavefunctions carry implications for observable properties, including:

• Transport and dynamics: In fractals and disordered systems, the multifractal nature of critical wavefunctions leads to anomalous subdiffusive transport, nontrivial localization properties, and modified response to perturbations (Yao et al., 2024).
• Entanglement structure: Critical wavefunctions generally exhibit entanglement entropies scaling logarithmically with subsystem size, with universal subleading corrections determined by the underlying conformal structure or critical universality class (Xu et al., 2020, Das et al., 2021).
• Transition diagnostics: Energy-level statistics, wavefunction moments, and entanglement spectra all serve as precise markers to distinguish quantum critical phases and locate quantum critical points.
• Robustness and tunability: Critical features and exponents are robust under variations of microscopic model parameters, provided the system remains in the same universality class.

Understanding quantum critical wavefunctions thus underpins both theoretical classification and experimental exploration of strongly correlated quantum matter and quantum information systems.

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These features are corroborated across a broad landscape of models, methods, and observables, with growing interdisciplinary connections between condensed matter, quantum information, and statistical physics (Blunden-Codd et al., 2016, Aguiar et al., 2012, Changlani et al., 2018, Das et al., 2021, Xu et al., 2020, Yao et al., 2024).