Quantum Critical Lattice Models

• Quantum critical lattice models are discrete many-body systems that undergo zero-temperature phase transitions driven by the competition between interactions and quantum fluctuations.
• They employ theoretical frameworks such as quantum field theories, large-N techniques, and dynamical mean-field methods to reveal universal scaling, critical exponents, and emergent excitations.
• These models have broad applications in correlated electron materials, ultracold atoms, and quantum information, offering insights into non-Fermi-liquid behavior and novel dynamical scaling phenomena.

Quantum critical lattice models are quantum many-body systems defined on discrete lattices that exhibit nonthermal continuous phase transitions at zero temperature, driven by parameters such as interaction strength, chemical potential, external fields, or lattice geometry. At these quantum critical points (QCPs), the competing interactions and quantum fluctuations give rise to critical behavior characterized by universal scaling, emergent symmetries, and unconventional excitations. This topic encompasses spin, bosonic, and electronic lattice systems, with broad implications for correlated electron materials, cold atom experiments, and quantum information.

1. Fundamental Mechanisms and Theoretical Frameworks

Quantum criticality in lattice models typically originates from the competition between distinct organizing tendencies, such as magnetic order favored by exchange interactions and singlet formation promoted by Kondo or pairing mechanisms. For example, the Kondo lattice model features local moments coupled to conduction electrons: the RKKY interaction favors antiferromagnetic or ferromagnetic order, while the Kondo exchange drives singlet formation and a paramagnetic heavy Fermi liquid with an enlarged Fermi surface. Varying a tuning parameter (often denoted $δ$, related to the ratio of characteristic energy scales) induces a quantum phase transition between these phases (Si, 2010).

The general framework relies on effective field theories, such as quantum Ginzburg–Landau or non-linear sigma models (QNLσM), large-$N$ techniques, and dynamical mean-field extensions (EDMFT). For fermionic systems, quantum criticality often aligns with Gross–Neveu–Yukawa universality, where Dirac fermions couple to fluctuating bosonic order parameters (e.g., charge-density, spin, or pairing fields) (Wang et al., 2014, Liu et al., 2021). For bosonic or spin models, quantum rotor or $\phi^4$ field theories describe the critical degrees of freedom (Sanders et al., 2019, Helmes et al., 2014).

Quantum criticality may be described within the Landau paradigm—where fluctuations of a local order parameter drive the transition (e.g., spin-density-wave (SDW) transitions (Si, 2010))—or in a beyond-Landau framework, as with deconfined quantum critical points (DQCP) where fractionalized excitations and emergent gauge fields are central (Zhao et al., 2020). Kondo breakdown transitions, Lifshitz points, and composite operator transitions (as in cubic* criticality) exemplify non-Landau universality classes (Si, 2010, Ran et al., 2023).

2. Prototypical Models and Universality Classes

Quantum critical lattice models span a variety of physical systems:

Model Type	Order Parameter/Phases	Universality Class
Kondo lattice (AF order vs. heavy FL)	Magnetization, Kondo singlet	SDW, local (Kondo-breakdown) QCP
Bose-Hubbard model (Mott–SF)	Superfluid order	$d$D XY ($d+1$D at tip)
Hubbard model on honeycomb lattice	AF order, Dirac mass	Gross–Neveu–Yukawa (chiral/Ising)
$J$–$Q$ spin models (AF–VBS DQCP)	Néel, VBS	Deconfined critical, multicritical
Bilayer/necklace XY models	Staggered magnetization	O(2) 3D $\phi^4$, area/log EE
Quantum loop/dimer models	Composite nematic/crystal order	Cubic* (rank‑2), vison condensation
1D $O(4)$ fermion chain	Dimer, spin order	1+1D Gross–Neveu/WZW [Wess-Zumino]
Ising/Kondo anisotropic systems	Longitudinal order, Kondo entanglement	Line of locally critical points

These models may exhibit continuous QCPs described by O($n$) or cubic universality classes, multicritical Lifshitz points, or transitions driven by the condensation of fractional excitations such as visons (Ran et al., 2023), spinons (Zhao et al., 2020), or holons (Komijani et al., 2017).

3. Scaling, Critical Exponents, and Correlation Functions

Quantum criticality leads to universal scaling behavior for observables such as susceptibilities, order parameters, correlation functions, and entanglement entropies. The scaling regime is characterized by diverging correlation length $\xi \sim |g-g_c|^{-\nu}$, vanishing or emergent energy scales (e.g., heavy Fermi temperature $T_F \rightarrow 0$), and power-law decay of correlation functions. Representative exponents extracted from large-scale simulations and analytic approaches include:

Model/Transition	$\nu$	$\eta$	Other Exponents
3D lattice Gross-Neveu (Chandrasekharan et al., 2013)	0.83–0.85	0.62–0.63	$\eta_\psi \approx 0.38$
Spinless Fermion Ising–Gross–Neveu (Wang et al., 2014)	0.80(3)	0.302(7)	$\tilde{\beta} \approx 0.52$
O(2) XY bilayer/necklace (Helmes et al., 2014)	0.6717	—	EE corner coeff. $-0.010(2)$ to $-0.009(2)$
Cubic* (rank-2) QLM (Ran et al., 2023)	—	$1.42$ (tensor)	$\eta \approx 0.04$ (vison); strong dichotomy
2D Bose-Hubbard (Sanders et al., 2019)	—	—	Exponents via $c_2, c_4$: $\varepsilon_2 = (2/7) \varepsilon_4$ at QCP
SU(2) Anderson/Kondo lattice (Hu et al., 2020, Si, 2010)	—	$\alpha \approx 0.7$ (log scaling)	$\chi_{\mathrm{loc}}(\omega) \sim \ln|T_K/\omega|$

Critical exponents may be highly non-classical at transitions involving composite operators or fractionalized excitations, as in cubic* universality or DQCPs.

4. Emergent Surface and Entanglement Criticality

Quantum critical points alter not only bulk behavior but also surface and entanglement properties. In coupled-ladder Heisenberg systems, surfaces terminating in gapless edge states (as in the SPT Haldane phase) display nonordinary critical behavior at the bulk QCP: the scaling dimension $y_{h_1}$ and anomalous surface exponents $\eta_\parallel, \eta_\perp$ are enhanced or even negative (Wang et al., 2022). These effects are direct consequences of pre-existing gapless surface modes, either from SPT states or from dangling spins in trivial phases, and do not require surface fine-tuning characteristic of classical multicritical points.

At criticality, the entanglement entropy exhibits leading area-law scaling, but with subleading universal logarithmic corrections associated with subsystem corners. In two-dimensional XY bilayer or necklace models, the coefficient of the $\ln(l)$ correction per $90^\circ$ boundary corner is approximately $-0.01$, nearly independent of microscopic details and matching results from other O(2)-critical systems and series expansion (Helmes et al., 2014). These invariants serve as sharp probes of the universality class.

5. Dynamical and Non-Fermi-Liquid Behavior

Quantum critical lattice models often exhibit non-Fermi-liquid behavior and anomalous dynamical scaling. In Kondo and Anderson lattice systems near a local QCP, the electronic self-energy acquires a power-law or logarithmic form (e.g., $$\mathrm{Im}\,\Sigma(\omega) \sim \omega^{d-1/2-\eta(2d+1)}$$), the effective mass $m^*(\omega) \sim \omega^{-\alpha}$ diverges at low frequency, and the quasiparticle residue vanishes ($Z(\omega) \to 0$). These features manifest as divergent specific heat coefficients ($C/T$), non-Fermi-liquid resistivity exponents, and $\omega/T$ scaling in dynamical susceptibilities (Vojta et al., 2015, Hu et al., 2020).

Spectroscopic signatures, such as power-law continua in dimer or vison dynamical structure factors (seen via stochastic analytic continuation of QMC data), emerge at QCPs with fractionalized excitations. In cubic* transitions, composite rank-2 order parameters display extremely large anomalous dimensions, with pronounced continua in both dimer and vison spectral functions (Ran et al., 2023).

6. Methodologies: Simulation and Analytical Techniques

A suite of unbiased numerical and analytical techniques underpins progress:

• Quantum Monte Carlo: Variants include determinant and continuous-time QMC, worm algorithms for bosons/fermions, the meron-cluster approach (for O(4) chains), and sweeping cluster QMC for constrained Hilbert spaces (loop/dimer models). These enable extraction of critical exponents, dynamic correlations, and entanglement entropy with high precision (Chandrasekharan et al., 2013, Helmes et al., 2014, Ganahl et al., 2019, Ran et al., 2023).
• Tensor Network Methods: Multi-scale entanglement renormalization ansatz (MERA) efficiently approximates ground states and extracts conformal data for 1D and 2D critical models, with dramatic GPU acceleration (Ganahl et al., 2019). iPEPS methods enable direct access to thermodynamic limit properties (e.g., CDW order in the honeycomb model (Wang et al., 2014)).
• Diagrammatic Extensions of DMFT: Dual Fermion approaches and EDMFT (with QMC/Bose-Fermi impurity solvers) incorporate both local and nonlocal quantum fluctuations beyond standard DMFT (Hirschmeier et al., 2017, Hu et al., 2020).
• Field-theoretical Analysis: RG (4–$\epsilon$ expansion), effective potentials, and mapping to sigma or Gross–Neveu/Heisenberg–Yukawa models characterize universality classes and multicritical structure (Zhao et al., 2020, Liu et al., 2021, Ran et al., 2023).
• Series Expansions and Hypergeometric Continuation: Exploit high-order perturbative data to determine phase boundaries and divergence exponents in Bose-Hubbard and related models (Sanders et al., 2019).

7. Experimental Relevance and Future Directions

Quantum critical lattice models describe or inspire numerous experimentally realized systems:

• Heavy Fermion Compounds: CeCu$_6$–Au, YbRh$_2$Si$_2$, and CeRhIn$_5$ exhibit signatures of local quantum criticality, abrupt Fermi surface reconstructions, and universal scaling exponents consistent with Kondo breakdown and EDMFT predictions (Si, 2010, Vojta et al., 2015, Yang et al., 2017).
• Ultracold Atoms in Optical Lattices: Direct simulations of Bose-Hubbard and Hubbard models with tunable interactions and geometry allow for the detection of Mott–superfluid transitions and critical exponents, with prospects for direct measurement of correlation function divergences and entanglement scalings (Sanders et al., 2019).
• Moiré Materials, Rydberg Arrays, and Engineered Quantum Systems: Quantum dimer/loop models and fractionalized QCPs can be simulated or emulated, providing dynamical fingerprints (e.g., spectral continua) for cubic* universality and vison condensation (Ran et al., 2023).
• Frustrated Magnets and Quantum Spin Liquids: Real materials with anisotropic or frustrated interactions manifest criticality driven by dimensional reduction, emergent 1D behavior, and spin liquid phases, as in certain organic salts and low-dimensional magnets (Yamaguchi et al., 2021).

This field remains at the forefront due to outstanding questions regarding the interplay of competing quantum fluctuations, the nature of non-Fermi liquid excitations, multicritical points (such as Lifshitz transitions in deconfined quantum critical models), the structure and measurement of entanglement invariants, and the synthesis or simulation of exotic quantum phases in experimental platforms.