Research
The Kardar-Parisi-Zhang classical-to-quantum conjecture
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 as ξ~tᶻ with dynamical exponent z=3/2 and (ii) non-Gaussian temporal fluctuations δh~tᵝ with 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 Gaussian orthogonal ensemble (GOE) Tracy-Widom distribution, typical of random matrix ensembles.
Intermezzo: 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.
Surprisingly, numerical evidence suggests that the KPZ universal behavior emerges in isolated quantum spin chains, in transport phenomena at high (infinite) temperatures, i.e., far away from equilibrium. It has been recently conjectured that two-point correlation functions of Noether charges are described by scaling functions of the KPZ universality class. This is surprising because, in its quantum realization, it lacks canonical properties of classical systems: (i) the stochastic aspect (white noise) is absent, (ii) the scaling requires both integrability and non-abelian symmetries to manifest.
This phenomenology was missed by theory, numerics, and experiments, and it constitutes a recent breakthrough, which triggered a renewed interest in the dynamics of both classical and quantum integrable systems. The exponent z=3/2 can be derived within the framework of generalized hydrodynamics (see intermezzo above). 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. While it has been suggested that the underlying mechanism for anomalous diffusion could be ascribed to classical solitons, the relation between KPZ phenomenology and anomalous diffusion in many-body quantum transport is, to a large extent, not yet 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)
The conjecture has also found experimental verification, including:
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.
In cold-atom quantum simulators, the experimental setup allow detecting 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.
Related readings:
Scheie et al., Nat. Phys. 17, 726-730 (2021)
Wei et al., Science 376, 716-720 (2022)
In this respect, the one-dimensional Fermi-Hubbard model possesses a global U(1) ⊗ SU(2) symmetry in the charge and spin sectors, respectively. The global symmetry is raised to SU(2) ⊗ SU(2) at half-filling, making it a particularly interesting playground to explore the connection between non-abelian symmetries and KPZ scaling in fermionic systems.
Setting up a mixed domain-wall initial state, we investigated the Lindblad dynamics of the Hubbard chain in different regimes. Our numerical simulations show that: (i) in the absence of interactions, transport is ballistic, (ii) at half-filling both charge and spin diffusion profiles display a distinctive KPZ collapse upon a non-linear scaling of time, and transport is characterized by anomalous diffusion constants, (iii) lowering the symmetry in the charge sector from SU(2) to U(1), charge transport becomes ballistic, whereas spin transport remains superdiffusive and displays KPZ scaling [1].
[1] Moca et al., Phys. Rev. B 108, 235139 (2023)