The Kardar-Parisi-Zhang universality class

In equilibrium, the concept of universality asserts that, near phase transitions, the physical behavior does not depend on the microscopic details of the system, e.g., the specific form of the interactions, but rather on its symmetries and dimensionality. Each universality class is characterized by a set of critical exponents, that govern the behavior of physical quantities.  

Classical physical phenomena belonging to the Kardar-Parisi-Zhang (KPZ) universality class are ubiquitous in nature, and they are essentially interface growth processes ranging from, e.g., ice, coffee stains, and tumor cells, to traffic. The underlying mechanism is described by a non-linear stochastic differential equation, whose solution — i.e., the height field h(x,t) — is characterized by (i) emerging transverse correlations over characteristic distances, thus linking space and time through the scaling parameter ξ=x/t¹ᐟᙆ with dynamical exponent z=3/2 and (ii) non-Gaussian fluctuations of the heigh field around its mean δh~tᵝ with exponent β=1/3. Therefore, within the KPZ universality class, time, space, and fluctuations scale with a 3:2:1 ratio.  

A typical representation of the emerging fractal pattern and the space-time correlations of the KPZ universality is given by the ballistic deposition model (or "sticky Tetris") a mathematical version of the game in which the blocks (plain 1x1 square blocks rather than the classical Tetrominos) do not fall to the bottom ("downstack") but stick to the first edge against which it becomes adjacent.

It turns out that sticky blocks radically change the growth process, and the height field remains correlated over long distances with a non-trivial algebraic transversal scale. Predictions for the limiting distributions indicate that the (properly scaled) heights should converge to the Tracy-Widom distribution of the Gaussian orthogonal ensemble (GOE) of random matrix theory. Note that, in general, the asymptotic distribution of a given process depends on the initial conditions. 

KPZ conjecture for integrable quantum spin chains

Numerical evidence suggests an emerging KPZ universal behavior in integrable quantum spin chains, in transport phenomena at high (infinite) temperatures, i.e., far away from equilibrium. Two-point correlation functions of Noether charges are described by scaling functions of the KPZ universality class with scaling parameter ξ=x/t¹ᐟᙆ and KPZ dynamic exponent z=3/2. 

This phenomenology was missed by previous theories, numerics, and experiments, and triggered a renewed interest in the dynamics of classical and quantum integrable systems. The dynamical exponent z=3/2 can be derived within the framework of generalized hydrodynamics (see intermezzo below). and eventually, the low- and high-temperature scenarios have been reconciled, showing that the onset of superdiffusion takes place above crossover length- and time-scales inversely proportional to temperature. It has been suggested that the underlying mechanism for anomalous diffusion could be ascribed to quasi-particles akin to classical solitons. However, the origin of KPZ scaling of two-point correlation functions and its link to SU(2) symmetry and integrability are, hitherto, not understood.

Related readings:

• Ljubotina et al., Nat. Comm. 8, 16117 (2017); Phys. Rev. Lett. 122, 210602 (2019)

• Gopalakrishnan et al., Phys. Rev. Lett. 122, 127202 (2019)

• Dupont et al., Phys. Rev. Lett. 127, 107201 (2021)

Experimental evidence

In the one-dimensional quantum antiferromagnet KCuF3, neutron scattering allows probing the structure factor — proportional to the Fourier image of the spin-spin correlation function, which was found to scale with the superdiffusive exponent z=3/2.

In cold-atom quantum simulators, the experimental setup allows measuring time- and spin-resolved snapshots of dynamical fluctuations, i.e., the full counting statistics of the spin transfer. By breaking the SU(2) symmetry (e.g., preparing an initial state with a net magnetization) or integrability (introducing a transverse coupling, realizing a dimensional crossover) it is also possible to verify that both are essential for observing superdiffusion.

The two-point spin-spin correlation function, which displays a KPZ-like scaling, is connected to the variance of the spin transfer full counting statistics. However, it has been a recent experiment performed by Google Quantum AI has shown that the full distribution — and specifically higher-order moments such as the skewness and the kurtosis, are incompatible with values from the Baik-Rains or the Tracy-Widom distribution, that describe classical phenomena in the KPZ universality class.

Related readings:

• Scheie et al., Nat. Phys. 17, 726-730 (2021)

• Wei et al., Science 376, 716-720 (2022)

• Rosenberg et al., 384, 48-53 (2024)

Generalized Hydrodynamics

In classical fluid dynamics does not describe individual particles with Newton's laws of motion but relies instead on phenomenological continuous differential equations — based on continuity equations of conserved quantities (i.e., mass, energy, momentum) that universally determine the long-time dynamics. Likewise, it can be expected that, in quantum systems with conservation laws, transport is described by coarse-grained hydrodynamics.

Typically, one would expect transport to be diffusive (z=2) and to follow Fick's law. However, in integrable systems, a macroscopic number of local conservation laws exists, which constrain the dynamics. For a quench from a typical inhomogeneous initial state, such generalized hydrodynamics predicts ballistic scaling (z=1) of densities and currents of conserved quantities. Noteworthy, due to symmetry reasons, the ballistic component to transport can vanish leaving behind modes that grow lower-than-linear with time — e.g., quasi-particle excitations that carry zero net magnetization contribute ballistically to energy transport but spin transport is dominated by slower magnetization fluctuations.

The equations of such a generalized hydrodynamics (GHD) can be solved with limited computational resources, in contrast to standard numerical approaches based, e.g., on tensor networks.

Related readings:

• Bertini et al., Phys. Rev. Lett. 117, 207201 (2016)

• Castro-Alvaredo et al., Phys. Rev. X 6, 041065 (2016)

• Doyon, SciPost Phys. Lect. Notes 18 (2020)