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Quantum critical point

A quantum critical point (QCP) is a continuous phase transition occurring at absolute zero temperature (T = 0 K), where quantum fluctuations—arising from the Heisenberg uncertainty principle—drive a qualitative change in the ground-state wavefunction of a quantum many-body system as a non-thermal tuning parameter, such as magnetic field, pressure, or doping, reaches a critical value.^[1]^[2] Unlike classical critical points at finite temperatures, which are dominated by thermal fluctuations, QCPs are governed exclusively by quantum effects, leading to divergent correlation lengths and long-range entanglement in the ground state.^[1]^[2] Although QCPs exist only at T = 0, their influence extends into a finite-temperature "quantum critical" regime, where quantum fluctuations persist and dominate the system's thermodynamics and dynamics over a broad range of tuning parameters and temperatures.^[1] This regime often manifests as non-Fermi liquid behavior, characterized by anomalous properties such as linear-in-temperature resistivity (ρ ∝ T) in strange metals, enhanced specific heat, and diverging susceptibilities like the magnetic Grüneisen parameter.^[1]^[2] For instance, in materials near a QCP, the phase boundary typically follows a power-law form T ∝ |g - g_c|^(νz), where g is the tuning parameter, g_c its critical value, and νz the dynamic critical exponent, reflecting the scale invariance of quantum fluctuations.^[2] QCPs have been experimentally realized and studied in diverse condensed matter systems, including antiferromagnets like TlCuCl₃ and CoNb₂O₆, where magnetic field tunes the transition from ordered to disordered phases, and heavy-fermion compounds exhibiting field-induced QCPs.^[1] They also play a pivotal role in unconventional superconductivity, as seen in iron-based superconductors like FeSe_{1-x}Te_x, where nematic QCPs—breaking rotational symmetry without magnetism—coexist with dome-shaped superconducting phases driven by fluctuating quasiparticles.^[3] Theoretical frameworks, advanced by researchers like Subir Sachdev, describe these phenomena using models such as the Hertz-Millis theory for itinerant systems and quantum rotor models for insulators, highlighting universal scaling behaviors across different materials.^[1]

Basic Concepts

Definition

A quantum critical point (QCP) is a special locus in the phase diagram of a quantum many-body system where a continuous, second-order phase transition occurs precisely at absolute zero temperature (T=0), separating two distinct quantum ground states. This transition is tuned by varying a non-thermal control parameter, such as pressure, magnetic field, chemical doping, or interatomic coupling strength, rather than temperature. Unlike classical phase transitions, the QCP arises solely from the competition between quantum ground states, with the critical behavior governed by the zero-point motion of the system's degrees of freedom.^[4] At the QCP, quantum fluctuations dominate completely, as there are no thermal excitations to mask or compete with them, leading to enhanced long-range correlations that extend over both spatial and temporal (imaginary-time) directions. These fluctuations often manifest through an order parameter φ that characterizes the broken symmetry (or other distinguishing feature) between the adjacent phases; φ vanishes continuously as the system approaches the QCP from the ordered side. Mathematically, the transition is described by a tuning parameter g, with the QCP located at g = g_c = 0 and T = 0 for simplicity, such that the order parameter behaves as φ ∝ |g|^\beta for g → 0^+ (where β is a critical exponent), while the free energy or susceptibility exhibits singular scaling. This setup highlights the purely quantum nature of the criticality, where the system's Hamiltonian H(g) = H_0 + g H_1 drives the instability without any finite-temperature broadening.^[4] In contrast to finite-temperature critical points, where thermal disorder smears the transition and drives symmetry breaking via entropy, a QCP involves no such thermal effects at T=0; instead, quantum zero-point fluctuations provide the effective "disorder" that tunes the system across phases, often resulting in universal scaling behaviors distinct from classical counterparts due to the extra imaginary-time dimension in the effective field theory. Quantum phase transitions, of which the QCP is the T=0 endpoint, thus probe the intrinsic quantum correlations in strongly interacting systems.^[4]

Historical Context

The concept of the quantum critical point emerged in the mid-1970s through the work of John A. Hertz, who introduced a theoretical framework for understanding critical phenomena in quantum-mechanical systems at low temperatures, particularly in itinerant fermionic systems like antiferromagnets where quantum fluctuations dominate. Hertz's approach treated the quantum phase transition as an effective classical problem in an extra dimension, focusing on the role of soft modes in driving the criticality at zero temperature. In 1993, A. J. Millis extended Hertz's theory by incorporating the effects of nonzero temperature using renormalization group techniques, distinguishing between clean and disordered (dirty) itinerant systems. This analysis revealed how thermal fluctuations modify the quantum critical behavior, leading to distinct scaling regimes and non-analytic corrections to observables like specific heat and resistivity in the vicinity of the critical point. Millis's contributions clarified the limitations of the Gaussian approximation in Hertz's original model and provided a more robust foundation for predicting experimental signatures in metallic quantum critical systems. Subir Sachdev generalized these ideas in the 1990s to insulating quantum magnets, developing effective field theories that capture the dynamics of order parameter fluctuations beyond itinerant electrons. His work emphasized the universal low-temperature properties of antiferromagnetic quantum critical points in two and three dimensions, highlighting damping mechanisms and the emergence of non-mean-field behaviors in the ordered phases. Sachdev's theories shifted focus toward strongly correlated insulators, where quantum fluctuations lead to novel scaling laws distinct from those in fermionic systems. By the 2000s, the quantum critical point framework gained prominence for explaining non-Fermi liquid behaviors observed in experiments on heavy-fermion compounds and other correlated materials, where anomalous transport and thermodynamic properties deviated from Landau Fermi liquid predictions near the critical tuning parameter. Concurrently, the proposal of deconfined quantum critical points by T. Senthil and collaborators in 2004 introduced a paradigm beyond the Landau-Ginzburg-Wilson framework, describing continuous transitions between topologically distinct ordered phases in two-dimensional antiferromagnets via emergent gauge fields and fractionalized excitations. More recently, in 2024, experimental confirmation of a broadened quantum critical ground state in two-dimensional superconductors was achieved through thermoelectric measurements on thin NbN films, revealing quantum fluctuations in the disordered superconducting state.^[5] The influence of quantum critical points extended significantly to high-temperature superconductivity and strange metal phases by the 2010s, with theoretical models linking critical fluctuations to the pseudogap and linear-in-temperature resistivity in cuprates.^[6] These connections positioned quantum criticality as a unifying concept for the "strange metal" regime, where Planckian dissipation and marginal Fermi liquid-like behaviors emerge from proximity to an underlying critical point.^[6]

Theoretical Framework

Quantum Phase Transitions

Quantum phase transitions (QPTs) occur at absolute zero temperature (T=0) and are induced by varying a non-thermal control parameter, such as a coupling constant g, rather than by thermal fluctuations as in classical phase transitions.^[7] In classical transitions, temperature drives the system across a critical point by exciting thermal disorder, but QPTs arise from quantum fluctuations that become dominant at T=0, leading to a change in the system's ground state without any thermal activation.^[8] This fundamental difference positions QPTs as a distinct class of critical phenomena, where the transition is tuned continuously through the parameter g, separating disordered and ordered phases.^[9] At the quantum critical point (QCP), the ground state undergoes a qualitative transformation, for instance, from a paramagnetic phase with no magnetic order to an antiferromagnetic phase exhibiting spontaneous staggered magnetization.^[4] These changes reflect a reorganization of the quantum ground state due to competing interactions, without reliance on finite-temperature entropy.^[7] Quantum fluctuations, particularly enhanced at the QCP, play a central role in destabilizing one phase and stabilizing another, though their detailed dynamics extend beyond the basic transition framework.^[9] In practice, non-thermal control parameters such as chemical doping x, applied pressure P, or magnetic field H are used to drive QPTs, with the QCP marking the value where the transition becomes continuous (second-order).^[8] For example, increasing pressure in certain materials can suppress magnetic order, tuning the system through a QCP where the ordered phase gives way to a disordered one.^[4] This tunability allows experimental access to the T=0 critical behavior, distinguishing QPTs from temperature-driven classical counterparts. A key feature of QPTs is adiabatic continuity: if the control parameter g is varied slowly enough—specifically, at a rate slower than the inverse of the correlation time near the QCP—the system remains in its instantaneous ground state without excitations.^[9] This principle ensures that the low-energy properties are preserved across the transition, enabling theoretical and experimental studies of the critical point. The singular part of the free energy, f_s, near a QPT exhibits scaling behavior adapted from classical critical phenomena but incorporating quantum effects through an effective dimensionality. It takes the form f_s \sim |g|^{2 - \alpha} where α is the specific heat exponent, characterizing the non-analyticity at the QCP.^[7] This form highlights the universal aspects of QPTs, analogous to thermal transitions yet rooted in zero-temperature quantum mechanics.^[8] Theoretical descriptions of QPTs vary depending on the nature of the system. For insulating magnets, models like the quantum rotor model, developed by Subir Sachdev, capture the physics of order parameter fluctuations in Mott insulators, leading to dynamic exponent z=1 and scaling behaviors distinct from itinerant cases.^[9]

Hertz-Millis Theory

The Hertz-Millis theory provides a foundational framework for analyzing quantum critical points in itinerant electron systems, where magnetic or other ordering instabilities arise from weakly interacting fermions. Developed by John A. Hertz in 1976, the approach treats the order parameter fluctuations as bosonic modes coupled to a fermionic bath, deriving an effective field theory by integrating out the high-energy fermionic degrees of freedom. This results in a Landau-Ginzburg-like action with a non-analytic dynamical term that captures the dissipative effects from Landau damping.^[10] A. J. Millis extended this framework in 1993 by incorporating finite-temperature effects and performing a detailed renormalization group (RG) analysis, clarifying the conditions under which the fermionic integration is valid and revealing the scaling behavior near criticality. In the effective action, the order parameter ϕ couples to fermions via a term like g ϕ ψ† ψ, leading to a Gaussian propagator modified by the fermionic response. The dynamic exponent z emerges from this coupling: z=3 for ferromagnetic order (where momentum conservation conserves the total spin) and z=2 for antiferromagnetic order (where the ordering wavevector connects distinct Fermi surface points, enhancing damping).^[11] At the Gaussian fixed point of the RG flow, the upper critical dimension is d_c^+ = 4 - z, above which fluctuations are irrelevant and mean-field exponents apply; below d_c^+, interactions drive non-mean-field behavior, though the theory often predicts a runaway flow to strong coupling. For d + z > 4, the critical behavior is mean-field-like, with the correlation length exponent ν = 1/2 and the order parameter exponent β = 1/2. The theory assumes weak coupling and neglects vertex corrections or multi-orbital effects, focusing instead on the leading-order bosonic self-energy from the fermionic loop.^[12] A central prediction is the dynamical susceptibility of the order parameter, which takes the Ornstein-Zernike form with damping:

\chi(\mathbf{q}, \omega) \sim \frac{1}{r + q^2 + \frac{|\omega|}{q^{z-2}} + i \gamma \frac{\omega}{q^{z-1}}},

where r ∝ (g - g_c) tunes the distance to the quantum critical point g_c, q is the deviation from the ordering wavevector, and γ is a damping coefficient from the fermionic bath. For ferromagnetic cases (z=3), the |ω|/q term dominates low-frequency dynamics, while for antiferromagnetic (z=2), it simplifies to i γ ω, resembling overdamped modes. This form implies non-Fermi liquid signatures, such as linear-in-T resistivity in d=3 for antiferromagnets and T^{5/3} for ferromagnets, arising from scattering off critical fluctuations.^[12] Despite its successes in predicting scaling forms, the Hertz-Millis theory has notable limitations: it ignores higher-order interactions and strong-coupling regimes where the Gaussian fixed point is unstable, leading to first-order transitions or new fixed points in low dimensions. Additionally, while it captures non-Fermi liquid transport, it often fails to explain logarithmic divergences or marginal Fermi liquid phenomenology observed in experiments, necessitating extensions like the inclusion of disorder or Eliashberg-style resummations.^[12]

Quantum Critical Region

Phase Diagram Features

In the temperature-control parameter (T-g) plane, the phase diagram of a system near a quantum critical point (QCP) features an ordered phase for g < 0, where spontaneous symmetry breaking leads to long-range order at low temperatures, and a disordered phase for g > 0, characterized by the absence of such order. The QCP is located at the origin (g = 0, T = 0), marking the point of a continuous quantum phase transition at absolute zero. A line of finite-temperature phase transitions emanates from the QCP, separating the ordered and disordered phases at higher temperatures, with the critical temperature decreasing to zero as the QCP is approached.^[13] Above the QCP lies the quantum critical fan, a region where thermal energy exceeds the tuning scale set by the distance to criticality, specifically for temperatures T > |g|^{νz}, with ν the correlation length exponent and z the dynamical critical exponent. In this fan-shaped regime, quantum critical fluctuations dominate the physics, extending the influence of the T = 0 transition to finite temperatures and leading to non-Fermi-liquid behavior across a broad range of parameters. The boundaries of the fan are defined by crossover lines where the system transitions from Fermi-liquid-like behavior at high T and g > 0, through the quantum critical regime at intermediate T, to the ordered phase at low T and g < 0. These crossovers occur along lines scaling as T ~ |g|^{νz}, demarcating the onset of critical fluctuations.^[14] The phase diagram can be visualized as a fan opening upward from the QCP, with the width in the g direction expanding as T^{1/(νz)} due to the scaling of the correlation length ξ ~ |g|^{-ν} and the characteristic energy scale ~ ξ^{-z}. This geometry arises from the hyperscaling relation in the effective theory, where quantum fluctuations along the imaginary time direction contribute an additional dimension. In d spatial dimensions, the effective dimensionality is d_eff = d + z, which determines whether mean-field approximations hold (above the upper critical dimension d + z = 4) or if non-perturbative effects become important below it. For typical itinerant systems, such as antiferromagnets with z = 2 in d = 3, the fan reflects Gaussian fixed-point behavior with logarithmic corrections in some cases.^[13]^[14]

Scaling and Universality

Near a quantum critical point (QCP), the system's behavior is governed by scaling laws that emerge from the divergence of the correlation length \xi as the tuning parameter g approaches its critical value g_c, with \xi \sim |g - g_c|^{-\nu}, where \nu is the correlation length exponent.^[10] The imaginary time correlation length \xi_\tau scales as \xi_\tau \sim \xi^z, introducing the dynamic exponent z that relates spatial and temporal scales. These relations underpin the hyperscaling hypothesis, which connects thermodynamic exponents via $2 - \alpha = \nu (d + z), where \alpha is the specific heat exponent and d is the spatial dimension; this holds when fluctuations are relevant below the upper critical dimension. Universality in QCPs implies that systems sharing the same low-energy fixed point exhibit identical critical exponents, determined by the symmetry of the order parameter, dimensionality, and presence of disorder or itinerancy, rather than microscopic details. For instance, clean insulating antiferromagnets, described by models like the O(3) nonlinear sigma model, belong to a universality class with z = 1, reflecting Lorentz-invariant dynamics. In contrast, dirty itinerant ferromagnets fall into a class with z = 4.^[15] In the quantum critical fan region—where temperature dominates over the distance to the QCP—singular thermodynamic and transport properties deviate from Fermi liquid behavior. The electronic specific heat coefficient exhibits logarithmic divergence, C/T \sim -\ln T, due to the marginal relevance of bosonic fluctuations. Similarly, the resistivity follows \rho \sim T^{d/z} in d dimensions, capturing the scattering of quasiparticles by critical modes; for example, in three dimensions with z=3, this yields linear-in-T resistivity. The singular part of the free energy density obeys the scaling form

f \sim T^{(d+z)/z} \Phi\left( \frac{g - g_c}{T^{1/(\nu z)}} \right),

where \Phi is a universal scaling function that encodes the crossover between quantum critical and Fermi liquid regimes. This form derives from the renormalization group invariance and ensures that observables like entropy and susceptibility inherit the same scaling structure. Hyperscaling breaks down above the upper critical dimension, where mean-field exponents apply and dangerous irrelevant variables alter the scaling; for typical \phi^4 theories relevant to QCPs, this occurs when d + z > 4, leading to logarithmic corrections or Gaussian fixed points.^[10]

Experimental Examples

Heavy-Fermion Systems

Heavy-fermion systems, characterized by strongly correlated f-electron compounds, provide prototypical examples of antiferromagnetic quantum critical points (QCPs) tuned by non-thermal parameters such as chemical substitution or pressure. In the archetypal case of CeCu_{6-x}Au_x, the ground state transitions from non-magnetic to antiferromagnetically ordered for Au concentrations x > 0.1 at low temperatures below 1 K, with the QCP occurring at a critical concentration x_c \approx 0.1.^[16] This tuning suppresses the antiferromagnetic ordering temperature to zero, leading to a regime dominated by quantum fluctuations. Similar behavior is observed in other f-electron materials where doping or hydrostatic pressure drives the system through the QCP, altering the electronic structure profoundly. Near the QCP in these systems, non-Fermi-liquid (NFL) signatures emerge, deviating from conventional Fermi-liquid expectations. The electronic specific heat coefficient \gamma = C/T diverges logarithmically as \gamma \sim -\ln T, while the electrical resistivity exhibits linear temperature dependence \rho \sim T over wide ranges, reflecting scattering from critical spin fluctuations. These anomalies, observed in CeCu_{6-x}Au_x at x \approx x_c, indicate the breakdown of quasiparticle coherence and the influence of low-energy magnetic excitations. Key experimental evidence includes 1990s neutron scattering studies on CeCu_{6-x}Au_x, which revealed two-dimensional critical fluctuations at the QCP, with dynamic susceptibility showing \omega/T scaling consistent with the quantum critical fan.^[16] Pressure-tuned QCPs have also been identified, notably in CeRhIn_5, where hydrostatic pressure suppresses antiferromagnetism at a critical pressure P_c \approx 2.3 GPa, accompanied by NFL resistivity \rho \sim T^{1.1} and enhanced \gamma. The proximity to QCPs in heavy-fermion systems often promotes unconventional superconductivity, where pairing symmetries incompatible with conventional electron-phonon mechanisms arise due to critical fluctuations. In CeRhIn_5, pressure induces a superconducting dome peaking near P_c, with evidence for d-wave symmetry from phase-sensitive measurements and the absence of a Hebel-Slichter coherence peak in NMR relaxation. This suggests that antiferromagnetic fluctuations mediate the pairing, linking quantum criticality directly to emergent superconducting states. Recent studies in the 2020s, employing de Haas-van Alphen quantum oscillations, have further illuminated critical Fermi surface evolution near QCPs, revealing a crossover from localized to delocalized f-electrons and abrupt volume changes at the transition in materials like CeRhIn_5.^[17] These findings underscore the role of QCPs in driving topological changes in the electronic structure, with implications for understanding strange-metal behavior in correlated electron systems.

Cuprate Superconductors

Cuprate superconductors, exemplified by La_{2-x}Sr_xCuO_4, emerge from doping a parent Mott insulator with holes, transitioning from an antiferromagnetic insulator to a superconductor with critical temperatures up to approximately 40 K. At optimal hole doping p^* \approx 0.16 holes per Cu atom, a quantum critical point (QCP) is hypothesized, marking the boundary where the pseudogap phase terminates and a Fermi liquid-like state might emerge under suppressed superconductivity. In the normal state above the superconducting transition temperature T_c, the overdoped and optimally doped regimes exhibit a "strange metal" phase characterized by linear-in-temperature resistivity \rho \sim T, extending over a wide temperature range. This behavior is linked to Planckian scattering rates, where the electron scattering time reaches the fundamental limit \tau \sim \hbar / k_B T, indicative of strong quantum critical fluctuations at the proposed QCP.^[18] Experimental evidence from angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) in the 2010s demonstrates the pseudogap closing precisely at p^*, with the gap magnitude vanishing and the Fermi surface reconstructing around this doping level.^[19] Additionally, quantum oscillations observed in high magnetic fields in YBa_2Cu_3O_{6+x} reveal a sharp enhancement in quasiparticle effective mass diverging toward p^* \approx 0.16, providing thermodynamic support for the QCP. The superconducting phase diagram features a dome-shaped T_c(p) curve peaking near the QCP, where quantum fluctuations are believed to enhance electron pairing interactions, potentially mediating d-wave superconductivity. This non-Fermi liquid scaling manifests in transport as the observed linear resistivity. Recent theoretical developments from 2023 to 2025, incorporating Sachdev-Ye-Kitaev (SYK)-like models with spatially random interactions, have provided a framework for the universal strange metal properties in cuprates, capturing the Planckian dissipation without quasiparticles.^[20]^[21]

Advanced Topics

Deconfined Quantum Critical Points

Deconfined quantum critical points (DQCPs) represent a class of continuous quantum phase transitions between two symmetry-broken phases that break distinct symmetries not contained in each other as subgroups, challenging the conventional Landau-Ginzburg-Wilson paradigm of order parameter fractionalization.^[22] A prototypical example occurs in two-dimensional quantum antiferromagnets, where the transition separates a Néel antiferromagnetic phase, characterized by spin rotation symmetry breaking, from a valence bond solid (VBS) phase, which breaks lattice translation symmetry.^[22] Unlike standard transitions, the continuity arises from quantum interference effects that allow direct coupling without an intermediate phase.^[22] This concept was proposed by Senthil et al. in 2003 as a novel framework for quantum criticality in insulating antiferromagnets, resolving inconsistencies in applying traditional theories to experimentally observed transitions.^[22] The theory posits that at the critical point, the elementary degrees of freedom become fractionalized, leading to emergent phenomena beyond mean-field descriptions.^[22] Theoretically, DQCPs feature fractionalized excitations, such as spinons—neutral spin-1/2 quasiparticles representing fractionalized spins—and an emergent U(1) gauge field that mediates their interactions, resulting in a deconfined phase where these excitations propagate coherently over long distances.^[22] The dynamics are relativistic with dynamical exponent z=1, implying equal scaling of spatial and temporal fluctuations, and the criticality exhibits enlarged symmetries, such as an emergent SO(5) rotation symmetry unifying the Néel and VBS orders.^[22] This contrasts with standard universality classes by incorporating topological defects like monopoles, which play a crucial role in the VBS ordering.^[22] The effective field theory is formulated in terms of bosonic spinons z_\alpha (with \alpha = 1,2) coupled to a compact U(1) gauge field a_\mu, capturing both orders at the self-dual point where the theory is invariant under particle-vortex duality.^[22] The Néel order parameter emerges as the spinon bilinear \hat{n} = z^\dagger \vec{\sigma} z, where \vec{\sigma} are Pauli matrices, while the VBS order relates to the phase winding of the spinons, dual to vortex fields \psi.^[22] The core Lagrangian is

\mathcal{L}_z = \sum_{\alpha=1}^2 |( \partial_\mu - i a_\mu ) z_\alpha |^2 + s |z|^2 + u (|z|^2)^2 + \kappa (\epsilon_{\mu\nu\lambda} \partial_\nu a_\lambda)^2,

with the Maxwell term for the gauge field ensuring z=1 dynamics; criticality occurs at s=0, where spinons and vortices interchange roles under duality, unifying the description of both phases.^[22] Experimental hints for DQCPs have emerged in quantum dimer models, particularly in the Shastry-Sutherland lattice realized in the material SrCu₂(BO₃)₂, where field-tuned transitions from a VBS to a Néel phase exhibit signatures of proximity to a deconfined critical point, including anomalous scaling in magnetization and specific heat.^[23] Simulations and proposals in Rydberg atom arrays on triangular lattices further support DQCP realizations, with 2025 studies demonstrating tunable interactions that simulate the fractionalized excitations and emergent gauge fields via van der Waals blockade mechanisms.^[23] These platforms offer controllable access to the self-dual regime, providing indirect evidence through entanglement entropy and correlation functions consistent with deconfined criticality.^[23]

Quantum Critical Endpoints

A quantum critical endpoint (QCEP) represents the zero-temperature termination of a finite-temperature line of critical transitions, where a second-order classical critical line meets the T=0 axis under quantum tuning parameters such as pressure or magnetic field.^[24] Unlike a standard quantum critical point (QCP) that directly separates ordered and disordered phases at T=0, the QCEP acts as the closure of a thermal critical line, often involving metamagnetic or multicritical behavior where quantum fluctuations dominate beyond the endpoint, suppressing magnetic order.^[24] This configuration arises when tuning suppresses the critical temperature to absolute zero, leading to non-Fermi-liquid properties in the surrounding quantum critical regime.^[25] Key characteristics of QCEPs include their role in closing thermal critical lines, where quantum effects enhance fluctuations that destabilize order parameter symmetry breaking above the endpoint, analogous to but distinct from direct QCP symmetry changes.^[26] In metamagnets, for instance, the QCEP often manifests as the end of a first-order metamagnetic transition line, with quantum criticality emerging from the suppression of this endpoint, resulting in divergent thermodynamic responses like specific heat and susceptibility.^[24] These endpoints differ from conventional QCPs by involving additional control parameters that fan out the phase diagram, incorporating tricritical wings where second- and first-order lines meet.^[26] Prominent examples include tricritical endpoints in metamagnets like Sr₃Ru₂O₇, where hydrostatic pressure tunes the metamagnetic quantum critical endpoint, revealing non-Fermi-liquid behavior and enhanced quantum fluctuations in the critical wing.^[25] In the heavy-fermion compound CeNi₂Ge₂, pressure tuning in the 1990s suppressed antiferromagnetic order to a magnetic endpoint at low temperatures, exhibiting quantum critical signatures in specific heat and resistivity near 8 kbar.^[27] Theoretically, QCEPs frequently occur at multicritical points, such as quantum tricritical points, where scaling involves modified exponents for the order parameter and correlation length due to the interplay of thermal and quantum fluctuations, distinct from mean-field values.^[26] A recent advancement is the 2024 observation of QCEP signatures in the Kitaev candidate material Na₂Co₂TeO₆, where high-resolution torque magnetometry revealed metamagnetic quantum criticality in a honeycomb antiferromagnet, expanding QCEP physics to systems with potential Kitaev interactions under magnetic fields.^[28] This phase diagram extension includes endpoint geometry within the broader quantum critical fan, highlighting universal scaling behaviors.^[28]

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