Lifshitz Field Theories Research Articles

Information theoretic measures for Lifshitz system

In this work, we have studied various mixed state information theoretic quantities for an excited state of Lifshitz spacetime in 3 + 1-dimensions. This geometry is the gravity dual to a class of 2 + 1-dimensional quantum field theories having Lifshitz symmetry. We have holographically calculated mutual information, entanglement wedge cross section, entanglement negativity and mutual complexity for strip like subsystems at the boundary. For this we have used the results of holographic entanglement entropy and complexity present in the literature. We first calculate all of these mentioned quantities for the pure state of Lifshitz spacetime. Then we have moved on to calculate all these quantities for excited state of the Lifshitz spacetime. The gravity dual of excited state of Lifshitz systems in field theory can be obtained by applying constant perturbations along the boundary direction. Further, we would like to mention that for the simplicity of calculation we are only considering results up to the first order in perturbation. The change in the obtained holographic information theoretic quantities are then related to entanglement entropy, entanglement pressure, entanglement chemical potential and charge using the stress tensor complex. These relations are analogous to the first law of entanglement thermodynamics given earlier in the literature. All the calculations are carried out for both values of dynamical scaling exponent (z) present in the Lifshitz field theory.

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Finite-size versus finite-temperature effects in the critical long-range O(N) model

In this paper we consider classical and quantum versions of the critical long-range O(N) model, for which we study finite-size and finite-temperature effects, respectively, at large N. First, we consider the classical (isotropic) model, which is conformally invariant at criticality, and we introduce one compact spatial direction. We show that the finite size dynamically induces an effective mass and we compute the one-point functions for bilinear primary operators with arbitrary spin and twist. Second, we study the quantum model, mapped to a Euclidean anisotropic field theory, local in Euclidean time and long-range in space, which we dub fractional Lifshitz field theory. We show that this model admits a fixed point at zero temperature, where it displays anisotropic Lifshitz scaling, and show that at finite temperature a thermal mass is induced. We then compute the one-point functions for an infinite family of bilinear scaling operators.In both the classical and quantum model, we find that, as previously noted for the short-range O(N) model in [1], the large-N two-point function contains information about the one-point functions, not only of the bilinear operators, but also of operators that appear in the operator product expansion of two fundamental fields only at subleading order in 1/N, namely powers of the Hubbard-Stratonovich intermediate field.

Reflected entropy and Markov gap in Lifshitz theories

We study the reflected entropy in (1+1)-dimensional Lifshitz field theory whose groundstate is described by a quantum mechanical model. Starting from tripartite Lifshitz groundstates, both critical and gapped, we derive explicit formulas for the Rényi reflected entropies reduced to two adjacent or disjoint intervals, directly in the continuum. We show that the reflected entropy in Lifshitz theory does not satisfy monotonicity, in contrast to what is observed for free relativistic fields. We analytically compute the full reflected entanglement spectrum for two disjoint intervals, finding a discrete set of eigenvalues which is that of a thermal density matrix. Furthermore, we investigate the Markov gap, defined as the difference between reflected entropy and mutual information, and find it to be universal and nonvanishing, signaling irreducible tripartite entanglement in Lifshitz groundstates. We also obtain analytical results for the reflected entropies and the Markov gap in 2 + 1 dimensions. Finally, as a byproduct of our results on reflected entropy, we provide exact formulas for two other entanglement-related quantities, namely the computable cross-norm negativity and the operator entanglement entropy.

Holographic two-point functions in a disorder system

We study the holographic dual of two-point correlation functions for nonconformal field theories. We first take into account a Lifshitz geometry as the dual of a Lifshitz field theory which may appear at a critical or IR fixed point. We explicitly show the holographic relation between a Lifshitz geometry and a Lifshitz field theory by calculating two-point correlators and equation of state parameter on both sides. We also investigate a disorder deformation, which allows a UV conformal field theory to flow into a new IR Lifshitz field theory. In this deformed theory, we investigate an anomalous dimension representing the change of an operator's scaling dimension along the RG flow.

Higher Rank 1 + 1 Integrable Landau–Lifshitz Field Theories from the Associative Yang–Baxter Equation

We propose a construction of 1+1 integrable Heisenberg-Landau-Lifshitz type equations in the ${\rm gl}_N$ case. The dynamical variables are matrix elements of $N\times N$ matrix $S$ with the property $S^2={\rm const}\cdot S$. The Lax pair with spectral parameter is constructed by means of a quantum $R$-matrix satisfying the associative Yang-Baxter equation. Equations of motion for ${\rm gl}_N$ Landau-Lifshitz model are derived from the Zakharov-Shabat equations. The model is simplified when ${\rm rank}(S)=1$. In this case the Hamiltonian description is suggested. The described family of models includes the elliptic model coming from ${\rm GL}_N$ Baxter-Belavin elliptic $R$-matrix. In $N=2$ case the widely known Sklyanin's elliptic Lax pair for XYZ Landau-Lifshitz equation is reproduced. Our construction is also valid for trigonometric and rational degenerations of the elliptic $R$-matrix.

Gravitational quasinormal modes for Lifshitz black branes

We study the scalar and vector channels of gravitational quasinormal modes for Lifshitz black branes emerging in Einstein-Maxwell-Dilaton and Einstein-Proca theories in four and five dimensions, finding significant differences between the two models. In particular, rather surprisingly, in the Einstein-Maxwell-Dilaton model the dispersion relations for the shear and sound modes are given by ωshear ∼ −i k4 and ωsound ∼ −i k2, while in the Einstein-Proca model they take the more conventional form ωshear ∼ −i k2 and ωsound ∼ k —the proportionality constants depend on the dynamical exponent and the appropriate factors of temperature. Through the holographic duality, this calculation provides information about the relaxation of the momentum and energy flux operators in a putative dual Lifshitz field theory. Comparing with the dispersion relations obtained directly by considering Lifshitz hydrodynamics suggest that the mass density of the equilibrium state in the Einstein-Maxwell-Dilaton model is infinite.

Emergent geometry and path integral optimization for a Lifshitz action

Extending the background metric optimization procedure for Euclidean path integrals of two-dimensional conformal field theories, introduced by Caputa et al. [Phys. Rev. Lett. 119, 071602 (2017), J. High Energy Phys. 11 (2017) 097], to a $z=2$ anisotropically scale-invariant $(2+1)$-dimensional Lifshitz field theory of a free massless scalar field, we find optimal geometries for static and dynamic correlation functions. For the static correlation functions, the optimal background metric is equivalent to an AdS metric on a Poincar\'e patch, while for dynamical correlation functions, we find Lifshitz like metric. This results suggest that a MERA-like tensor network, perhaps without unitarity, would still be considered an optimal background spacetime configuration for the numerical description of this system, even though the classical action we start with is not a conformal field theory.

Informational properties of holographic Lifshitz field theory * *Supported by the Natural Science Foundation of China (11775036, 11905083, 11847055, 11705161) and Fok Ying Tung Education Foundation (171006). Guoyang Fu is supported by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_2973). Top Talent Support Program from Yangzhou University

In this paper, we explore the properties of holographic entanglement entropy (HEE), mutual information (MI) and entanglement of purification (EoP) in holographic Lifshitz theory. These informational quantities exhibit some universal properties of holographic dual field theory. For most configuration parameters and temperatures, these informational quantities change monotonically with the Lifshitz dynamical critical exponent z. However, we also observe some non-monotonic behaviors for these informational quantities in some specific spaces of configuration parameters and temperatures. A particularly interesting phenomenon is that a dome-shaped diagram emerges in the behavior of MI vs z, and correspondingly a trapezoid-shaped profile appears in that of EoP vs z. This means that for some specific configuration parameters and temperatures, the system measured in terms of MI and EoP is entangled only in a certain intermediate range of z.

An exact solution to asymptotic Bethe equation

We present an exact solution to the asymptotic Bethe equation of weakly anisotropic Heisenberg spin chain, which is a set of non-linear algebraic equations. The solution describes the low-energy excitations above ferromagnetic ground state with fixed magnetisation, and it has a close relation to generalised Jacobi polynomial. It is equivalent to a generalised Stieltjes problem and in the continuous limit, it becomes a Riemann–Hilbert problem closely related to the finite-gap solutions of classical Landau–Lifshitz field theory.

Energy-entropy relation for asymptotically Lifshitz spacetimes with universal horizons

We numerically solve for 2+1 asymptotically Lifshitz universal horizon solutions in Horava-Lifshitz gravity for dynamical exponents $z=2$ through $z=8$. We find that for all $z$ there is a thermodynamical first law and Smarr formula. Furthermore, we find that the energy-entropy relation expected for a thermal state in a two dimensional Lifshitz field theory, $E=\frac{2}{z+2}TS$, is also satisfied for universal horizons, including the correct $z$ scaling.

New heat kernel method in Lifshitz theories

We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Hořava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.

Integrable matrix models in discrete space-time

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 \sigmaσ-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.

Towards a Unitary, Renormalizable, and Ultraviolet-Complete Quantum Theory of Gravity

For any fundamental quantum field theory, unitarity, renormalizability, and relativistic invariance are considered to be essential properties. Unitarity is inevitably connected to the probabilistic interpretation of the quantum theory, while renormalizability guarantees its completeness. Relativistic invariance, in turn, is a symmetry which derives from the structure of spacetime. So far, the perturbative attempt to formulate a fundamental local quantum field theory of gravity based on the metric field seems to be in conflict with at least one of these properties. In quantum Ho\v{r}ava gravity, a quantum Lifshitz field theory of gravity characterized by an anisotropic scaling between space and time, unitarity and renormalizability can be retained while Lorentz invariance is sacrificed at high energies and must emerge only as approximate symmetry at low energies. I review various approaches to perturbative quantum gravity with a particular focus on recent progress in the quantization of Ho\v{r}ava gravity, supporting its theoretical status as a unitary, renormalizable and ultraviolet-complete quantum theory of gravity.

Relating bulk to boundary entanglement

Quantum many-body systems have a rich structure in the presence of boundaries. We study the groundstates of conformal field theories (CFTs) and Lifshitz field theories in the presence of a boundary through the lens of the entanglement entropy. For a family of theories in general dimensions, we relate the universal terms in the entanglement entropy of the bulk theory with the corresponding terms for the theory with a boundary. This relation imposes a condition on certain boundary central charges. For example, in $2+1$ dimensions, we show that the corner-induced logarithmic terms of free CFTs and certain Lifshitz theories are simply related to those that arise when the corner touches the boundary. We test our findings on the lattice, including a numerical implementation of Neumann boundary conditions. We also propose an ansatz, the boundary Extensive Mutual Information model, for a CFT with a boundary whose entanglement entropy is purely geometrical. This model shows the same bulk-boundary connection as Dirac fermions and certain supersymmetric CFTs that have a holographic dual. Finally, we discuss how our results can be generalized to all dimensions as well as to massive quantum field theories.

Holomorphic structure and quantum critical points in supersymmetric Lifshitz field theories

We construct supersymmetric Lifshitz field theories with four real supercharges in a general number of space dimensions. The theories consist of complex bosons and fermions and exhibit a holomorphic structure and non-renormalization properties of the superpotential. We study the theories in a diverse number of space dimensions and for various choices of marginal interactions. We show that there are lines of quantum critical points with an exact Lifshitz scale invariance and a dynamical critical exponent that depends on the coupling constants.

Holographic geometry for nonrelativistic systems emerging from generalized flow equations

An intriguing result presented by two of the present authors is that an anti de Sitter space can be derived from a conformal field theory by considering a flow equation. A natural expectation is that given a certain data on the boundary system, the associated geometry would be able to emerge from a flow, even beyond the conformal case. As a step along this line, we examine this scenario for non-relativistic systems with anisotropic scaling symmetries, such as Lifshitz field theories and Schr\"odinger invariant theories. In consequence we obtain a new hybrid geometry of Lifshitz and Schr\"odinger spacetimes as a general holographic geometry in this framework. We confirm that this geometry reduces to each of them by considering special non-relativistic models.

Broken Lifshitz Invariance, Spin Waves, and Hydrodynamics

In this paper, based on the basic principles of thermodynamics, we explore the hydrodynamic regime of interacting Lifshitz field theories in the presence of broken rotational invariance. We compute the entropy current and discover new dissipative effects which are consistent with the principle of local entropy production in the fluid. In our analysis, we consider both the parity even and the parity odd sector upto first order in the derivative expansion. Finally, we argue that the present construction of the paper could be systematically identified as that of the hydrodynamic description associated with spin waves (away from the domain of quantum criticality) under certain limiting conditions.

Domain-wall dynamics in the Landau-Lifshitz magnet and the classical-quantum correspondence for spin transport

We investigate the dynamics of spin in the axially anisotropic Landau--Lifshitz field theory with a magnetic domain wall initial condition. Employing the analytic scattering technique, we obtain the exact scattering data and reconstruct the time-evolved profile. We identify three qualitatively distinct regimes of spin transport, ranging from ballistic expansion in the easy-plane regime, absence of transport in the easy-axis regime and log-modified diffusion for the isotropic interaction. Our results are in perfect qualitative agreement with those found in the anisotropic quantum Heisenberg spin-$1/2$ chain, indicating a remarkable classical-quantum correspondence for macroscopic spin transport.

Holographic mutual and tripartite information in a symmetry breaking quench

We study the time evolution of holographic mutual and tripartite information for a zero temperature CFT, derives to a non-relativistic thermal Lifshitz field theory by a quantum quench. We observe that the symmetry breaking does not play any role in the phase space, phase of parameters of sub-systems, and the length of disentangling transition. Nevertheless, mutual and tripartite information indeed depend on the rate of symmetry breaking. We also find that for large enough values of δt the quantity teqδt−1, where δt and teq are injection time and equilibration time respectively, behaves adiabatically, i.e. its value is independent of length of separation between sub-systems. We also show that tripartite information is always non-positive during the process indicates that mutual information is monogamous.