Integrable matrix models in discrete space-time

-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau-Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

Related papers

Žiga Krajnik, Enej Ilievski, Tomaž Prosen

We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigma σ -models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigma σ -models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

arXiv:2003.05957SciPost PhysicsSep 14, 202040 citations

Anisotropic Landau-Lifshitz model in discrete space-time

Žiga Krajnik, Enej Ilievski, Tomaž Prosen +1 more

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

Sep 10, 202112 citations

Anisotropic Landau-Lifshitz model in discrete space-time

Žiga Krajnik, Enej Ilievski, Tomaž Prosen +1 more

arXiv:2104.13863SciPost PhysicsSep 9, 202113 citations

Modeli mnogodelčne dinamike v obliki klasičnih in kvantnih vezij

Kasim, Yusuf

Treating the time evolution of many-body systems in discrete time steps is very useful in both classical and quantum physics, simplifying exact and numerical calculations of different observables. In this thesis, we study different classical and quantum discrete-time models. In the first part, we focus on classical systems through the lens of cellular automata which, despite being simple in construction, exhibit a multitude of differing integrable and chaotic behaviors. We study a specific subset of momentum-conserving parity check automata (MCPCA), which are cellular automata defined on bipartite graphs that conserve momentum and parity along their interaction vertices. We show that these automata describe systems with non-ergodic behavior in the spectra of their autocorrelation functions for different types of lattices. Studying their conserved charges, we demonstrate that the information spreading in such systems undergoes a phase transition between a localized and a delocalized phase. Additionally, we investigate the transport properties of the conserved charges in a specific sector of charge configuration of the MCPCA, we observe that the transport is anomalous in this sector. Moreover, we provide a classification of all possible two-states 3-to-3 cellular automata rules defined on the honeycomb lattice based on their dynamics and spectral properties. In the final part of the thesis, we focus on the quantum dual-unitary gates, which are a special kind of quantum gates that preserve their unitarity condition under space-time exchange. We give exact results on the dynamical correlation functions of dual-unitary circuits defined on random geometries that we dubbed Mikado circuits. We show different random geometries realizations and the exact results for the first two moments of the averaged correlation functions.Treating the time evolution of many-body systems in discrete time steps is very useful in both classical and quantum physics, simplifying exact and numerical calculations of different observables. In this thesis, we study different classical and quantum discrete-time models. In the first part, we focus on classical systems through the lens of cellular automata which, despite being simple in construction, exhibit a multitude of differing integrable and chaotic behaviors. We study a specific subset of momentum-conserving parity check automata (MCPCA), which are cellular automata defined on bipartite graphs that conserve momentum and parity along their interaction vertices. We show that these automata describe systems with non-ergodic behavior in the spectra of their autocorrelation functions for different types of lattices. Studying their conserved charges, we demonstrate that the information spreading in such systems undergoes a phase transition between a localized and a delocalized phase. Additionally, we investigate the transport properties of the conserved charges in a specific sector of charge configuration of the MCPCA, we observe that the transport is anomalous in this sector. Moreover, we provide a classification of all possible two-states 3-to-3 cellular automata rules defined on the honeycomb lattice based on their dynamics and spectral properties. In the final part of the thesis, we focus on the quantum dual-unitary gates, which are a special kind of quantum gates that preserve their unitarity condition under space-time exchange. We give exact results on the dynamical correlation functions of dual-unitary circuits defined on random geometries that we dubbed Mikado circuits. We show different random geometries realizations and the exact results for the first two moments of the averaged correlation functions.

Dec 7, 20250 citations

Integrable nonunitary open quantum circuits

Lucas Sá, Pedro Ribeiro, Tomaž Prosen

We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing. In particular, even circuits built from convex combinations of local free-fermion unitaries are nonintegrable. Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetry classes, which are characterized by the behavior under transposition instead of complex conjugation. We confirm all our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipative circuits.We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing. In particular, even circuits built from convex combinations of local free-fermion unitaries are nonintegrable. Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetry classes, which are characterized by the behavior under transposition instead of complex conjugation. We confirm all our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipative circuits.

arXiv:2011.06565Physical review. B./Physical review. BMar 18, 202144 citations

Anomalous transport in non-integrable classical field theories

Matija Koterle, Tomaž Prosen, Tianci Zhou

Anomalous KPZ spin transport is well established in integrable non-Abelian lattice models but has not been investigated in continuum field theories as discretization in numerics generally break the continuum theory's integrability. We show that finite temperature acts as a regulator that can restore anomalous transport over a broad time window. In a family of spin field theories labeled by integer $n$ , the $n = 1$ case is the Landau-Lifshitz model, whose numerical data shows spin superdiffusion with Kardar-Parisi-Zhang (KPZ) scaling and, at lower temperature ballistic energy transport, whereas both observables are diffusive at high temperature. The non-integrable $n = 2$ case shows the same crossover. While Lyapunov analysis confirms the model's non-integrability, the structure of spin-density space-time profiles suggests that long-lived soliton-like trajectories exist at low temperature.Anomalous KPZ spin transport is well established in integrable non-Abelian lattice models but has not been investigated in continuum field theories as discretization in numerics generally break the continuum theory's integrability. We show that finite temperature acts as a regulator that can restore anomalous transport over a broad time window. In a family of spin field theories labeled by integer $n$, the $n = 1$ case is the Landau-Lifshitz model, whose numerical data shows spin superdiffusion with Kardar-Parisi-Zhang (KPZ) scaling and, at lower temperature ballistic energy transport, whereas both observables are diffusive at high temperature. The non-integrable $n = 2$ case shows the same crossover. While Lyapunov analysis confirms the model's non-integrability, the structure of spin-density space-time profiles suggests that long-lived soliton-like trajectories exist at low temperature.

arXiv:2601.19894Jan 27, 20260 citations

\sigma

n

, the

n = 1

case is the Landau-Lifshitz model, whose numerical data shows spin superdiffusion with Kardar-Parisi-Zhang (KPZ) scaling and, at lower temperature ballistic energy transport, whereas both observables are diffusive at high temperature. The non-integrable

n = 2

case shows the same crossover. While Lyapunov analysis confirms the model's non-integrability, the structure of spin-density space-time profiles suggests that long-lived soliton-like trajectories exist at low temperature.