The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasi-particles. We here consider as an archetypical example for such a system the spin-$\frac{1}{2}$ XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization $m{=}0$. At a finite magnetization density $m$, the spin autocorrelation function $Π(x,t)$ (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasi-particle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a sub-diffusive $t^{1/3}$ scaling at large magnetization densities, $m {\rightarrow} \frac{1}{2}$, familiar from non-interacting models; it crosses over into a diffusive scaling $t^{1/2}$ upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient super-diffusion to ballistic relaxation at long times. In the limit $m{\to}0$, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent $z$ close to the value $\frac{3}{2}$ familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator $Π$ is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.
Equilibrium spatio-temporal correlation functions are central to understanding weak nonequilibrium physics. In certain local one-dimensional classical systems with three conservation laws they show universal features. Namely, fluctuations around ballistically propagating sound modes can be described by the celebrated Kardar-Parisi-Zhang (KPZ) universality class. Can such universality class be found also in quantum systems? By unambiguously demonstrating that the KPZ scaling function describes magnetization dynamics in the SU(2) symmetric Heisenberg spin chain we show, for the first time, that …
Using a functional renormalization group approach we show that at low temperatures and in a certain range of magnetic fields the longitudinal dynamic structure factor of quantum Heisenberg ferromagnets in dimensions $D\leq 2$ exhibits a well-defined quasi-particle peak with linear dispersion that we identify as zero-magnon sound. In $D>2$ the larger phase space available for the decay into transverse spin waves leads only to a broad hump centered at zero frequency whose width scales linearly in momentum.
We use the spin functional renormalization group recently developed by two of us [J. Krieg and P. Kopietz, Phys. Rev. B $\bf{99}$, 060403(R) (2019)] to calculate the magnetization $M ( H , T )$ and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H \rightarrow 0$ the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions our approach correctly predi…
We study the universal far from equilibrium dynamics of magnons in Heisenberg ferromagnets. We show that such systems exhibit universal scaling in momentum and time of the quasiparticle distribution function, with the universal exponents distinct from those recently observed in Bose-Einstein condensates. This new universality class originates from the SU(2) symmetry of the Hamiltonian, which leads to a strong momentum-dependent magnon-magnon scattering amplitude. We compute the universal exponents using the Boltzmann kinetic equation and incoherent initial conditions that can be realized with …
Spin-1/2 Heisenberg diamond-like decorated honeycomb lattice in a magnetic field from the perspective of localized magnons and exact diagonalization
Faculty of Science, P. J. Šafárik University（帕·约·沙法里克大学理学院）；Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy Paris Université（CY塞尔吉巴黎大学，CNRS UMR 8089理论物理与建模实验室）
Finite-temperature spin transport in the quantum Heisenberg spin chain is known to be superdiffusive, and has been conjectured to lie in the Kardar-Parisi-Zhang (KPZ) universality class. Using a kinetic theory of transport, we compute the KPZ coupling strength for the Heisenberg chain as a function of temperature, directly from microscopics; the results agree well with density-matrix renormalization group simulations. We establish a rigorous quantum-classical correspondence between the "giant quasiparticles" that govern superdiffusion and solitons in the classical continuous Landau-Lifshitz fe…
The bipartite entanglement across the magnetization process of a highly frustrated spin-1/2 Heisenberg octahedral chain is examined within the concept of localized magnons, which enables a simple calculation of the concurrence measuring a strength of the pairwise entanglement between nearest-neighbor and next-nearest-neighbor spins from square plaquettes. A full exact diagonalization of the finite-size Heisenberg octahedral chain with up to 4 unit cells (20 spins) evidences an extraordinary high precision of the localized-magnon theory in predicting measures of the bipartite entanglement at su…
Caloric Phenomena and Stirling-Cycle Performance in Heisenberg- Kitaev Magnon Systems
Pontificia Universidad Católica de Valparaíso（天主教瓦伊卡索大学）；Universidad Técnica Federico Santa María（费德里科·圣玛丽亚技术大学）；Universidad de La Frontera（拉弗龙特拉大学）；Universidad de Chile（智利大学）；Universidad de Tarapacá（塔拉帕卡大学）
Motivated by the KPZ scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well-known to be non-integrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with $\log(1+ S_i\cdot S_j)$ interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons hav…
The mixed spin-1 and spin-1/2 Heisenberg octahedral chain with regularly alternating monomeric spin-1 sites and square-plaquette spin-1/2 sites is investigated using variational technique, localized-magnon approach, exact diagonalization (ED) and density-matrix renormalization group (DMRG) method. The investigated model has in a magnetic field an extraordinarily rich ground-state phase diagram, which includes the uniform and cluster-based Haldane phases, two ferrimagnetic phases of Lieb-Mattis type, two quantum spin liquids and two bound magnon crystals in addition to the fully polarized ferro…
Breakdown of the conventional spin-wave dynamics and its double-constraint modification in the spin-$\mathbf{\frac{1}{2}}$ triangular-prism Heisenberg antiferromagnet
Although the Bethe ansatz solution of the spin-1/2 Heisenberg model dates back nearly a century, the anomalous nature of its high-temperature transport dynamics has only recently been uncovered. Indeed, numerical and experimental observations have demonstrated that spin transport in this paradigmatic model falls into the Kardar-Parisi-Zhang (KPZ) universality class. This has inspired the significantly stronger conjecture that KPZ dynamics, in fact, occur in all integrable spin chains with non-Abelian symmetry. Here, we provide extensive numerical evidence affirming this conjecture. Moreover, w…
The rich ground-state phase diagram of the mixed spin-(1,1/2) Heisenberg octahedral chain was previously elaborated from effective mixed-spin Heisenberg chains, which were derived by employing a local conservation of a total spin on square plaquettes of an octahedral chain. Here we present a comprehensive analysis of the thermodynamic properties of this model. In the highly frustrated parameter region the lowest-energy eigenstates of the mixed-spin Heisenberg octahedral chain belong to flat bands, which allow a precise description of low-temperature magnetic properties within the localized-mag…
The Kardar-Parisi-Zhang (KPZ) universality class describes the coarse-grained behavior of a wealth of classical stochastic models. Surprisingly, it was recently conjectured to also describe spin transport in the one-dimensional quantum Heisenberg model. We test this conjecture by experimentally probing transport in a cold-atom quantum simulator via the relaxation of domain walls in spin chains of up to 50 spins. We find that domain-wall relaxation is indeed governed by the KPZ dynamical exponent $z = 3/2$, and that the occurrence of KPZ scaling requires both integrability and a non-abelian SU(…
In its original version the KPZ equation models the dynamics of an interface bordering a stable phase against a metastable one. Over past years the corresponding two-dimensional field theory has been applied to models with different physics. Out of a wide choice, the spin-spin time correlations for the Heisenberg chain will be discussed at some length, also the equilibrium time-correlations of the conserved fields for 1D fluids. An interesting recent theoretical advance is the construction of the scale-invariant asymptotic theory, the so-called KPZ fixed point.
Starting from the hamiltonian for the Heisenberg ferromagnet which comprise randomly distributed nonmagnetic ions as impurities in a Bravais lattice, we express the spin operators by means of the Dyson-Maleev transformation in terms of the Bose operators of the second quantization. Then by using methods of quantum statistical field theory, we derive the partition function and the free energy for the system. We adopt the Matsubara thermal perturbation method to a portion of the hamiltonian which describes the interaction between magnons and the stationary field of nonmagnetic ions. Upon averagi…
It is evidenced that effective lattice-gas models of hard-core monomers and dimers afford a proper description of low-temperature features of spin-1/2 Heisenberg diamond and octahedral chains. Besides monomeric particles assigned within the localized-magnon theory to bound one- and two-magnon eigenstates, the effective monomer-dimer lattice-gas model additionally includes dimeric particles assigned to a singlet-tetramer (singlet-hexamer) state as a cornerstone of dimer-tetramer (tetramer-hexamer) ground state of a spin-1/2 Heisenberg diamond (octahedral) chain. A feasibility of the effective d…
Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical int…
We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $S\geq 1/2$. We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.
