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Quantum phase transition
Quantum phase transition
Quantum phase transition
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In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter—such as magnetic field or pressure—at absolute zero temperature. The transition describes an abrupt change in the ground state of a many-body system due to its quantum fluctuations. Such a quantum phase transition can be a second-order phase transition.[1] Quantum phase transitions can also be represented by the topological fermion condensation quantum phase transition, see e.g. strongly correlated quantum spin liquid. In case of three dimensional Fermi liquid, this transition transforms the Fermi surface into a Fermi volume. Such a transition can be a first-order phase transition, for it transforms two dimensional structure (Fermi surface) into three dimensional. As a result, the topological charge of Fermi liquid changes abruptly, since it takes only one of a discrete set of values.

Classical description

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To understand quantum phase transitions, it is useful to contrast them to classical phase transitions (CPT) (also called thermal phase transitions).[2] A CPT describes a cusp in the thermodynamic properties of a system. It signals a reorganization of the particles; A typical example is the freezing transition of water describing the transition between liquid and solid. The classical phase transitions are driven by a competition between the energy of a system and the entropy of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. Their order is determined by the first discontinuous derivative of a thermodynamic potential.

A phase transition from water to ice, for example, involves latent heat (a discontinuity of the internal energy ) and is of first order. A phase transition from a ferromagnet to a paramagnet is continuous and is of second order. (See phase transition for Ehrenfest's classification of phase transitions by the derivative of free energy which is discontinuous at the transition). These continuous transitions from an ordered to a disordered phase are described by an order parameter, which is zero in the disordered and nonzero in the ordered phase. For the aforementioned ferromagnetic transition, the order parameter would represent the total magnetization of the system.

Although the thermodynamic average of the order parameter is zero in the disordered state, its fluctuations can be nonzero and become long-ranged in the vicinity of the critical point, where their typical length scale ξ (correlation length) and typical fluctuation decay time scale τc (correlation time) diverge:

where

is defined as the relative deviation from the critical temperature Tc. We call ν the (correlation length) critical exponent and z the dynamical critical exponent. Critical behavior of nonzero temperature phase transitions is fully described by classical thermodynamics; quantum mechanics does not play any role even if the actual phases require a quantum mechanical description (e.g. superconductivity).

Quantum description

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Diagram of temperature (T) and pressure (p) showing the quantum critical point (QCP) and quantum phase transitions.

Talking about quantum phase transitions means talking about transitions at T = 0: by tuning a non-temperature parameter like pressure, chemical composition or magnetic field, one could suppress e.g. some transition temperature like the Curie or Néel temperature to 0 K.

As a system in equilibrium at zero temperature is always in its lowest-energy state (or an equally weighted superposition if the lowest-energy is degenerate), a QPT cannot be explained by thermal fluctuations. Instead, quantum fluctuations, arising from Heisenberg's uncertainty principle, drive the loss of order characteristic of a QPT. The QPT occurs at the quantum critical point (QCP), where quantum fluctuations driving the transition diverge and become scale invariant in space and time.

Although absolute zero is not physically realizable, characteristics of the transition can be detected in the system's low-temperature behavior near the critical point. At nonzero temperatures, classical fluctuations with an energy scale of kBT compete with the quantum fluctuations of energy scale ħω. Here ω is the characteristic frequency of the quantum oscillation and is inversely proportional to the correlation time. Quantum fluctuations dominate the system's behavior in the region where ħω > kBT, known as the quantum critical region. This quantum critical behavior manifests itself in unconventional and unexpected physical behavior like novel non Fermi liquid phases. From a theoretical point of view, a phase diagram like the one shown on the right is expected: the QPT separates an ordered from a disordered phase (often, the low temperature disordered phase is referred to as 'quantum' disordered).

At high enough temperatures, the system is disordered and purely classical. Around the classical phase transition, the system is governed by classical thermal fluctuations (light blue area). This region becomes narrower with decreasing energies and converges towards the quantum critical point (QCP). Experimentally, the 'quantum critical' phase, which is still governed by quantum fluctuations, is the most interesting one.

See also

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References

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Quantum phase transition

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from Grokipedia
A quantum phase transition is a continuous transformation in the of a many-body quantum system that occurs at temperature, as a nonthermal control parameter—such as strength, , chemical doping, or strain—is tuned through a critical value, leading to nonanalytic behavior in the ground-state energy and other thermodynamic properties due to quantum fluctuations. These transitions differ fundamentally from classical s, which are driven by at finite temperatures and involve the competition between energetic order and entropic disorder; in contrast, quantum phase transitions arise from zero-point quantum fluctuations and can exhibit avoided or actual level crossings in the system's energy spectrum. At the quantum critical point separating two distinct ground states, quantum fluctuations become long-ranged and dominant, resulting in quantum critical behavior characterized by universal scaling laws, power-law correlations, and emergent phenomena that can persist to finite temperatures in a quantum critical fan region of the . This criticality often maps onto effective field theories, such as the (2+1)-dimensional O(3) for antiferromagnetic transitions, where the extra dimension arises from , enabling the study of universality classes analogous to but distinct from classical ones. Notable features include the role of Berry phases in fermionic systems, which can induce secondary order like charge density waves, and the breakdown of Landau's near certain critical points in metals. Quantum phase transitions are observed across diverse condensed matter systems, including quantum magnets like LiHoF₄ where a magnetic field induces a ferromagnetic-to-paramagnetic shift, high-temperature such as La₂₋ₓSrₓCuO₄ where doping suppresses antiferromagnetic order at x ≈ 0.02, and heavy-fermion materials exhibiting transitions to unconventional or non-Fermi liquids. These transitions underpin the emergence of exotic states of matter, such as fractional quantum Hall phases and strange metals, and their study has advanced theoretical frameworks like the for quantum systems while enabling experimental probes via techniques including neutron scattering, specific heat measurements, and simulations.

Background on Phase Transitions

Classical Phase Transitions

Classical phase transitions refer to abrupt changes in the macroscopic properties of a material, such as , , or , that occur as is varied at finite temperatures. These transitions arise from the of many particles in a and are fundamental to understanding the thermodynamic states of matter. They are classified into and second-order types based on the nature of the changes in thermodynamic quantities. transitions are discontinuous, involving a and a jump in the first of the free energy, such as or , exemplified by the of into . In contrast, second-order or continuous transitions exhibit no but feature divergences in higher-order derivatives, like specific heat, with critical fluctuations becoming long-ranged, as seen in the ferromagnetic transition in iron where emerges continuously below the . The driving mechanism behind classical phase transitions is the competition between energetic ordering and thermal disorder, governed by the minimization of the G=H−TSG = H - T S, where HH favors ordered phases at low temperatures, and SS promotes disorder at high temperatures. play a central role, enabling the system to explore different configurations and leading to the emergence of distinct phases through the balance of these contributions. The theoretical framework for classifying these transitions was established by in 1933, who proposed ordering them by the highest of the thermodynamic potentials that shows a discontinuity. This was later advanced by in 1937, who developed a phenomenological theory using an order parameter to describe second-order transitions near criticality, emphasizing and expansion of the free energy in powers of the order parameter. A representative example is the -gas transition in , where below the critical of approximately 374°C, liquid and vapor coexist with a discontinuous change, but at the critical point (374°C, 218 atm), the distinction between phases vanishes, and fluctuations become isotropic and divergent. Quantum phase transitions extend this paradigm by incorporating quantum fluctuations at , replacing thermal effects as the dominant driver.

Transition to Quantum Regimes

As temperature decreases toward , thermal fluctuations in a system progressively diminish, allowing quantum zero-point motion to dominate the behavior of particles and fields. This shift arises fundamentally from the Heisenberg uncertainty principle, which enforces nonzero fluctuations in such as position and momentum even in the , preventing the system from settling into a classical equilibrium configuration. In contrast to classical phase transitions, where thermal agitation drives order-disorder changes at finite temperatures, the quantum regime at low temperatures reveals inherently dynamic phenomena governed by these quantum uncertainties. The crossover from classical to quantum regimes becomes evident as T→0T \to 0, where the system's dynamics transition from being controlled by thermally activated processes to quantum tunneling and coherent oscillations. In this limit, the effective dimensionality of the system increases from the spatial dimension dd to deff=d+zd_{\text{eff}} = d + z, with zz being the dynamic exponent that relates spatial and temporal correlations; for many models, z=1z = 1, effectively adding an extra dimension analogous to an imaginary-time direction in . This enhancement in effective dimensionality alters the critical behavior, making quantum phase transitions distinct from their classical counterparts, such as second-order transitions driven by . At exactly T=0T = 0, quantum phase transitions occur in the of the system, tuned by non-thermal parameters like , pressure, or chemical doping rather than . These parameters modify the Hamiltonian, driving the system across a where the ground-state exhibits singularities, such as a closing gap. The dominance of quantum fluctuations ensures that the transition reflects the zero-temperature , with properties like correlation lengths diverging as ξ∝∣g−gc∣−ν\xi \propto |g - g_c|^{-\nu}, where gg is the tuning parameter and gcg_c its critical value. Quantum effects thus provide a conceptual bridge to classical transitions by smearing sharp finite-temperature boundaries at low TT, extending influence into a quantum critical region above T=0T = 0 where quantum fluctuations still control long-time and long-length-scale properties. In this region, observables display scaling behaviors intermediate between classical and purely quantum limits, with thermal perturbations becoming relevant only at higher temperatures. This crossover highlights how unifies the understanding of phase transitions across temperature scales, revealing universal features emergent from the ground-state competition between order and disorder.

Core Concepts of Quantum Phase Transitions

Definition and Key Characteristics

A quantum phase transition is defined as a continuous change in the ground-state properties of a many-body quantum system occurring at temperature (T=0), induced by varying a non-thermal control parameter, such as , , or doping concentration, which crosses a critical value known as the . This transition separates distinct quantum phases characterized by different ground-state orders or properties, and it is fundamentally driven by quantum fluctuations arising from the , rather than thermal excitations. The concept was first systematically developed by J. A. Hertz in 1976, in the context of quantum critical phenomena in itinerant magnets, where he analyzed how quantum effects lead to phase instabilities at low temperatures. Key characteristics of quantum phase transitions include their potential to be either , involving a discontinuous change in the order parameter, or continuous (second-order), where the transition is smooth but accompanied by diverging correlation lengths. At exactly T=0, there is no change associated with the transition, as disorder is absent; however, the profoundly influences the and dynamics at finite but low temperatures, creating a quantum critical regime where properties exhibit non-Fermi-liquid-like behaviors. Unlike classical phase transitions, which are driven by and involve time-averaged static order, quantum phase transitions are "static" in the sense that they occur in the without thermal broadening, yet they are inherently dynamic due to the role of quantum correlations and zero-point motion in the fluctuations. In contrast to classical transitions, quantum phase transitions highlight the competition between quantum ground states, such as in metal-insulator transitions where varying disorder or interactions drives a system from a metallic state with delocalized electrons to an insulating state with localized charges, as exemplified by the Mott transition. Modern perspectives extend beyond Hertz's original framework to include topological quantum phase transitions, where phases are distinguished not by local order parameters but by global topological invariants, such as Chern numbers, leading to robust edge states and protected degeneracies without . These topological aspects, prominent since the 2000s, underscore the diversity of quantum criticality in strongly correlated and low-dimensional systems.

Order Parameters and Tuning Parameters

In quantum phase transitions, the order parameter is a that distinguishes the ordered and disordered phases, acquiring a nonzero expectation value in the ordered phase and vanishing continuously at the critical point. For instance, in antiferromagnets, the staggered magnetization serves as the order parameter, representing the alternating alignment of spins on sublattices, which breaks and signals long-range Néel order. This parameter, often denoted as ⟨ψ⟩\langle \psi \rangle, reflects the inherent to the transition, where the selects a particular direction in the order parameter space at zero . Tuning parameters are non-thermal control variables that drive the system across the gcg_c, where the order parameter vanishes, separating the disordered phase (where ⟨ψ⟩=0\langle \psi \rangle = 0) from the ordered phase (where ⟨ψ⟩≠0\langle \psi \rangle \neq 0). Common examples include , which couple to spin ; , which adjusts particle density; and lattice spacing, which modifies interactions in condensed matter systems. Near gcg_c, quantum fluctuations become dominant, suppressing the order parameter and leading to enhanced correlations that characterize the critical regime. A representative example occurs in high-TcT_c , where the superconducting gap acts as the order parameter for the paired electron state, tuned by doping concentration as the control parameter. In underdoped regimes, the gap emerges with doping away from the parent compound, while overdoping suppresses it, driving a transition to a nonsuperconducting phase. This zero-temperature symmetry breaking in the superconducting order is stabilized by quantum effects, analogous to classical transitions but governed by ground-state entanglement rather than thermal disorder.

Quantum Critical Phenomena

Quantum Critical Point

The (QCP) represents the singular locus in the parameter space of a quantum many-body system where a continuous quantum phase transition occurs at temperature (T=0T = 0). It is defined by a critical value gcg_c of the tuning parameter gg, such as a or , at which the undergoes a qualitative change, becoming gapless and characterized by diverging quantum fluctuations. At this point, the energy gap Δ\Delta to the first closes, Δ∼∣g−gc∣νz\Delta \sim |g - g_c|^{\nu z}, where ν\nu is the correlation length exponent and zz is the dynamical exponent, leading to an instability driven by enhanced across the system. The order parameter, which distinguishes the phases on either side of the transition, vanishes continuously as g→gcg \to g_c. A key property of the QCP is the emergence of a quantum critical region in the spanned by the tuning parameter gg and TT, forming a fan-shaped wedge that extends from T=0T = 0 at g=gcg = g_c to finite temperatures. Within this region, quantum fluctuations dominate the low-energy physics, influencing observables over a broad range of T>0T > 0, even far from the exact QCP, due to the slow decay of critical modes. Hyperscaling relations hold in this regime for spatial dimensions d<d < upper critical dimension (typically d<3d < 3 for many models), connecting the singular part of the free energy density to the diverging scales near criticality. The spatial and temporal correlation functions exhibit characteristic divergences at the QCP, capturing the scale of quantum fluctuations. The spatial correlation length ξ\xi, which measures the extent of correlations in the ground state, diverges as the system approaches gcg_c: ξ∝∣g−gc∣−ν\xi \propto |g - g_c|^{-\nu} This form arises from scaling arguments applied to the effective quantum field theory describing the critical modes. Near the QCP, the tuning parameter deviation ∣g−gc∣|g - g_c| acts as a relevant perturbation that sets the only energy scale, and under renormalization group transformations, lengths rescale by a factor bb, transforming g−gcg - g_c to b1/ν(g−gc)b^{1/\nu}(g - g_c) to maintain invariance of the singular free energy. Choosing b∼∣g−gc∣−νb \sim |g - g_c|^{-\nu} yields the divergence of ξ\xi. Similarly, the characteristic temporal correlation time τc\tau_c, governing the dynamics of fluctuations, scales with the spatial length via the dynamical exponent zz, which relates energy and momentum scales (ω∼kz\omega \sim k^z): τc∝ξz∝∣g−gc∣−νz\tau_c \propto \xi^z \propto |g - g_c|^{-\nu z} The derivation follows from the same scaling framework: time rescales as bzb^z under RG transformations, so τc\tau_c transforms to b−zτcb^{-z} \tau_c. Setting b∼ξ∼∣g−gc∣−νb \sim \xi \sim |g - g_c|^{-\nu} ensures scale invariance, leading to the hyperscaling form above. At the QCP itself (g=gcg = g_c), both ξ\xi and τc\tau_c diverge, resulting in power-law correlations in space and imaginary time without intrinsic scales. These divergences at the QCP imply a non-analytic singular contribution to the ground-state free energy density, fs∼∣g−gc∣2−αf_s \sim |g - g_c|^{2 - \alpha}, where α\alpha is the specific heat exponent related by hyperscaling 2−α=(d+z)ν2 - \alpha = (d + z) \nu. At finite but low temperatures, this leads to singular thermodynamic behavior, such as power-law dependencies in specific heat C∼Td/zC \sim T^{d/z} and susceptibility within the quantum critical fan, reflecting the proliferation of gapless modes and long-lived quantum fluctuations.

Scaling Laws and Universality

In quantum phase transitions, the scaling hypothesis describes the singular part of the ground-state free energy density as fs∼ξ−(d+z)f_s \sim \xi^{-(d+z)}, where ξ\xi is the correlation length diverging as ξ∼∣g−gc∣−ν\xi \sim |g - g_c|^{-\nu} near the critical point, with gg the tuning parameter, gcg_c its critical value, dd the spatial dimensionality, and zz the dynamic critical exponent. This implies fs∼∣g−gc∣ν(d+z)f_s \sim |g - g_c|^{\nu(d+z)}, capturing how quantum fluctuations dominate the critical behavior at zero temperature. Hyperscaling, which holds below the upper critical dimension, relates thermodynamic exponents via 2−α=ν(d+z)2 - \alpha = \nu (d + z), where α\alpha governs the singularity in the specific heat. Critical exponents characterize the universal aspects of these transitions, including ν\nu for the correlation length ξ∼∣g−gc∣−ν\xi \sim |g - g_c|^{-\nu}, zz relating spatial and temporal correlations via ξτ∼ξz\xi_\tau \sim \xi^z, and η\eta the anomalous dimension in the correlation function G(r)∼1/rd−2+ηG(r) \sim 1/r^{d-2+\eta} at criticality. These exponents satisfy scaling relations analogous to classical ones, such as γ=ν(2−η)\gamma = \nu (2 - \eta) for the susceptibility, but modified by the quantum dynamics. Universality asserts that systems sharing the same dd, symmetry group, range of interactions, and conservation laws belong to identical classes, exhibiting the same exponents regardless of microscopic details like lattice structure or short-range potentials. The renormalization group (RG) framework elucidates this universality by analyzing the flow of coupling constants under scale transformations, where relevant operators drive the system toward unstable fixed points that dictate long-distance behavior. A key insight is the quantum-to-classical mapping: a quantum phase transition in dd spatial dimensions corresponds to a classical transition in d+zd + z effective dimensions, with the imaginary-time direction (τ\tau) acting as an extra spatial dimension, allowing classical RG techniques to classify quantum universality classes. Beyond standard insulators or magnets, quantum critical points in itinerant electron systems, such as heavy-fermion metals, have revealed non-Fermi liquid universality classes since the early 2000s, where scaling leads to anomalous transport and thermodynamics without quasiparticles. For instance, experiments on YbRh₂Si₂ near its antiferromagnetic quantum critical point showed logarithmic divergences in resistivity and specific heat, consistent with marginal Fermi liquid behavior governed by critical bosonic modes. These findings, extended to other materials like CeCoIn₅, highlight how quantum criticality can destabilize Fermi liquids, yielding universal exponents like z≈3z \approx 3 in Hertz-Millis theory for antiferromagnetic fluctuations, though local criticality models better match observed anomalies.

Theoretical Models

Transverse-Field Ising Model

The transverse-field Ising model provides an exactly solvable paradigm for understanding quantum phase transitions, particularly in one dimension, where quantum fluctuations drive the system between ordered and disordered ground states at absolute zero temperature. The model describes a lattice of spin-1/2 particles with nearest-neighbor ferromagnetic interactions along one axis and a uniform magnetic field applied perpendicular to that axis. Its Hamiltonian is H=−J∑iσizσi+1z−h∑iσix,H = -J \sum_i \sigma_i^z \sigma_{i+1}^z - h \sum_i \sigma_i^x, where J>0J > 0 sets the scale of the Ising interaction, hh is the tunable transverse-field strength serving as the control parameter for the transition, and σix,z\sigma_i^{x,z} denote the Pauli matrices at site ii. This form captures the competition between the aligning tendency of the interactions and the disordering effect of the quantum transverse field. For h<Jh < J, the ground state exhibits spontaneous magnetization and long-range ferromagnetic order in the zz-direction, while for h>Jh > J, the system is in a gapped paramagnetic phase where spins preferentially align along the xx-direction, destroying the order. The second-order quantum phase transition occurs precisely at the critical value hc=Jh_c = J, marked by the closure of the single-particle excitation gap in the spectrum. This gapless point separates the two phases and leads to non-analytic behavior in thermodynamic quantities like the ground-state energy. The exact solvability of the one-dimensional model relies on the Jordan-Wigner transformation, which fermionized the spin operators into a set of non-interacting fermions via string operators, σiz=1−2ci†ci\sigma_i^z = 1 - 2 c_i^\dagger c_i and σix=(ci+ci†)∏j<i(1−2cj†cj)\sigma_i^x = (c_i + c_i^\dagger) \prod_{j<i} (1 - 2 c_j^\dagger c_j), where cic_i are fermionic annihilation operators. The resulting quadratic fermionic Hamiltonian is diagonalized in Fourier space, yielding a Bogoliubov quasiparticle spectrum ϵk=2J(1−λcos⁡k)2+(λsin⁡k)2\epsilon_k = 2J \sqrt{(1 - \lambda \cos k)^2 + (\lambda \sin k)^2}
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