One-dimensional Critical Systems Research Articles

The operator algebra of cyclic orbifolds

We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.

Eigenstate Entanglement: Crossover from the Ground State to Volume Laws.

For the typical quantum many-body systems that obey the eigenstate thermalization hypothesis (ETH), we argue that the entanglement entropy of (almost) all energy eigenstates is described by a single crossover function. The ETH implies that the crossover functions can be deduced from subsystem entropies of thermal ensembles and have universal properties. These functions capture the full crossover from the ground-state entanglement regime at low energies and small subsystem size (area or log-area law) to the extensive volume-law regime at high energies or large subsystem size. For critical one-dimensional systems, a universal scaling function follows from conformal field theory and can be adapted for nonlinear dispersions. We use it to also deduce the crossover scaling function for Fermi liquids in d>1 dimensions. The analytical results are complemented by numerics for large noninteracting systems of fermions in d≤3 dimensions and have also been confirmed for bosonic systems and nonintegrable spin chains.

Energy transport between critical one-dimensional systems with different central charges

Energy transport can reveal information about interacting many-body systems beyond other transport probes. In particular, in one dimension it has been shown that the energy current is directly proportional to the central charge, thus revealing information about the degrees of freedom of critical systems. In this paper, we explicitly verify this result in two cases for translationally invariant systems based on explicit microscopic calculations. More importantly, we generalize the result to nontranslation invariant setups and use this to study a composite system of two subsystems possessing different central charges. We find a bottleneck effect, meaning the smaller central charge limits the energy transport.

Imbalance entanglement: Symmetry decomposition of negativity

In the presence of symmetry, entanglement measures of quantum many-body states can be decomposed into contributions arising from distinct symmetry sectors. Here we investigate the decomposability of negativity, a measure of entanglement between two parts of a generally open system in a mixed state. While the entanglement entropy of a subsystem within a closed system can be resolved according to its total preserved charge, we find that negativity of two subsystems may be decomposed into contributions associated with their charge imbalance. We show that this charge-imbalance decomposition of the negativity may be measured by employing existing techniques based on creation and manipulation of many-body twin or triple states in cold atomic setups. Next, using a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in a Riemann geometry, we compute this decomposed negativity in critical one-dimensional systems described by conformal field theory. We show that it shares the same distribution as the charge-imbalance between the two subsystems. We numerically confirm our field theory results via an exact calculations for non-interacting particles based on a double-gaussian representation of the partially transposed density matrix.

Entanglement entropies of minimal models from null-vectors

We present a new method to compute Rényi entropies in one-dimensional critical systems. The null-vector conditions on the twist fields in the cyclic orbifold allow us to derive a differential equation for their correlation functions. The latter are then determined by standard bootstrap techniques. We apply this method to the calculation of various Rényi entropies.

Universal Broadening of the Light Cone in Low-Temperature Transport.

We consider the low-temperature transport properties of critical one-dimensional systems that can be described, at equilibrium, by a Luttinger liquid. We focus on the prototypical setting where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. At large distances x and times t, conformal field theory characterizes the energy transport in terms of a single light cone spreading at the sound velocity v. Energy density and current take different constant values inside the light cone, on its left, and on its right, resulting in a three-step form of the corresponding profiles as a function of ζ=x/t. Here, using a nonlinear Luttinger liquid description, we show that for generic observables this picture is spoiled as soon as a nonlinearity in the spectrum is present. In correspondence of the transition points x/t=±v, a novel universal region emerges at infinite times, whose width is proportional to the temperatures on the two sides. In this region, expectation values have a different temperature dependence and show smooth peaks as a function of ζ. We explicitly compute the universal function describing such peaks. In the specific case of interacting integrable models, our predictions are analytically recovered by the generalized hydrodynamic approach.

Bipartite fidelity and Loschmidt echo of the bosonic conformal interface

We study the quantum quench problem for a class of bosonic conformal interfaces by computing the Loschmidt echo and the bipartite fidelity. The quench can be viewed as a sudden change of boundary conditions parametrized by $\ensuremath{\theta}$ when connecting two one-dimensional critical systems. They are classified by $S(\ensuremath{\theta})$ matrices associated with the current scattering processes on the interface. The resulting Loschmidt echo of the quench has long time algebraic decay ${t}^{\ensuremath{-}\ensuremath{\alpha}}$, whose exponent also appears in the finite size bipartite fidelity as ${L}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{2}}$. We perform analytic and numerical calculations of the exponent $\ensuremath{\alpha}$, and find that it has a quadratic dependence on the change of $\ensuremath{\theta}$ if the prior and post-quench boundary conditions are of the same type of $S$, while remaining $\frac{1}{4}$ otherwise. Possible physical realizations of these interfaces include, for instance, connecting different quantum wires (Luttinger liquids), quench of the topological phase edge states, etc., and the exponent can be detected in an x-ray edge singularity-type experiment.

Universality of finite-size corrections to geometrical entanglement in one-dimensional quantum critical systems

Recently, the finite-size corrections to the geometrical entanglement per lattice site in the spin-1/2 chain have been numerically shown to scale inversely with system size, and its prefactor b has been suggested to be possibly universal [Q-Q. Shi et al., New J. Phys. 12, 025008 (2010)]. As possible evidence of its universality, the numerical values of the prefactors have been confirmed analytically by using the Affleck-Ludwig boundary entropy with a Neumann boundary condition for a free compactified field [J-M. Stephan et al., Phys. Rev. B 82, 180406(R) (2010)]. However, the Affleck-Ludwig boundary entropy is not unique and does depend on conformally invariant boundary conditions. Here, we show that a unique Affleck-Ludwig boundary entropy corresponding to a finitesize correction to the geometrical entanglement per lattice site exists and show that the ratio of the prefactor b to the corresponding minimum groundstate degeneracy gmin for the Affleck- Ludwig boundary entropy is a constant for any critical region of the spin-1 XXZ system with the single-ion anisotropy, i.e., b/(2 log2 g min ) = −1. Previously studied spin-1/2 systems, including the quantum three-state Potts model, have verified the universal ratio. Hence, the inverse finite-size correction to the geometrical entanglement per lattice site and its prefactor b are universal for one-dimensional critical systems.

Renyi entanglement entropies of descendant states in critical systems with boundaries: conformal field theory and spin chains

We discuss the Renyi entanglement entropies of descendant states in critical one-dimensional systems with boundaries, that map to boundary conformal field theories in the scaling limit. We unify the previous conformal-field-theory approaches to describe primary and descendant states in systems with both open and closed boundaries. We provide universal expressions for the first two descendants in the identity family. We apply our technique to critical systems belonging to different universality classes with non-trivial boundary conditions that preserve conformal invariance, and find excellent agreement with numerical results obtained for finite spin chains. We also demonstrate that entanglement entropies are a powerful tool to resolve degeneracy of higher excited states in critical lattice models.

Conformal field theory out of equilibrium: a review

We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.

Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

We show that infinite matrix product states (MPSs) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an $\mathrm{SU}(n)$ generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.

Practical variational tomography for critical one-dimensional systems

We improve upon a recently introduced efficient quantum state reconstruction procedure targeted to states well-approximated by the multi-scale entanglement renormalization ansatz (MERA), e.g., ground states of critical models. We show how to numerically select a subset of experimentally accessible measurements which maximizes information extraction about renormalized particles, thus dramatically reducing the required number of physical measurements. We numerically estimate the number of measurements required to characterize the ground state of the critical 1D Ising (resp. XX) model and find that MERA tomography on 16-qubit (resp. 24-qubit) systems requires the same experimental effort than brute-force tomography on 8 qubits. We derive a bound computable from experimental data which certifies the distance between the experimental and reconstructed states.

LONG TIME CORRELATIONS OF NONLINEAR LUTTINGER LIQUIDS

An overview is given of the limitations of Luttinger liquid theory in describing the real time equilibrium dynamics of critical one-dimensional systems with nonlinear dispersion relation. After exposing the singularities of perturbation theory in band curvature effects that break the Lorentz invariance of the Tomonaga–Luttinger model, the origin of high frequency oscillations in the long time behaviour of correlation functions is discussed. The notion that correlations decay exponentially at finite temperature is challenged by the effects of diffusion in the density–density correlation due to umklapp scattering in lattice models.

Sine-square deformation of solvable spin chains and conformal field theories

We study solvable spin chains, one-dimensional massless Dirac fermions and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function f(x) = sin 2(πx/ℓ), where x is the position and ℓ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac–Moody) algebras, including c = 1 Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.

Sine-square deformation of free fermion systems in one and higher dimensions

We study free fermion systems with the sine-square deformation (SSD), in which the energy scale of local Hamiltonians is modified according to the scaling function f(x)=sin^2[\pi(x-1/2)/L], where x is the position of the local Hamiltonian and L is the length of the system in the x direction. It has been revealed that when applied to one-dimensional critical systems the SSD realizes the translationally-invariant ground state which is the same as that of the uniform periodic system. In this paper, we propose a simple theory to explain how the SSD maintains the translational invariance in the ground-state wave function. In particular, for a certain one-dimensional system with SSD, it is shown that the ground state is exactly identical with the Fermi sea of the uniform periodic chain. We also apply the SSD to two-dimensional systems and show that the SSD is able to suppress the boundary modulations from the open edges extremely well, demonstrating that the SSD works in any dimensions and in any directions.

Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidtecho, and light-cone effects

We study a particular type of local quench in a generic quantum critical one-dimensionalsystem, using conformal field theory (CFT) techniques, and providing numerical checks ofthe results in free fermion systems. The system is initially cut into two subsystemsA andB which are gluedtogether at time t = 0. We study the entropy of entanglement (EE) between the two partsA andB, using previous results obtained by Calabrese and Cardy (2004 J. Stat. Mech. P06002; 2007J. Stat. Mech. P10004), and further extending them. We also study in detail the(logarithmic) Loschmidt echo (LLE). For finite size systems both quantities turn out to be(almost) periodic in the scaling limit, and exhibit striking light-cone effects. Whilethese two quantities behave similarly immediately after the quench—namely asc/3logt for theEE and c/4logt for the LLE—we observe some discrepancy once the excitations emitted bythe quench bounce on the boundary and evolve within the same subsystemA (orB). The decay of the EE is then non-universal, as noticed by Eisler andPeschel (2007 J. Stat. Mech. P06005). In contrast, we find that the evolution of the LLE is lesssensitive than the EE to non-universal details of the model, and is still accurately describedby our CFT prediction. To further probe these light-cone effects, we also introduce avariant of the Loschmidt echo specifically constructed to detect the excitations emitted justafter the quench.

Connecting distant ends of one-dimensional critical systems by a sine-square deformation

We study one-dimensional quantum critical spin systems with sine-square deformation, in which the energy scale in the Hamiltonian at position $x$ is modified by the function ${f}_{x}=\mathrm{sin}{}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where $L$ is the length of the system. By investigating the entanglement entropy, spin correlation functions, and wave-function overlap, we show that the sine-square deformation changes the topology of the geometrical connection of the ground state drastically: Although the system apparently has open edges, the sine-square deformation links those ends and realizes the periodic ground state at the level of the wave function. Our results propose a method to control the topology of quantum states by energy-scale deformation.

Boundary multifractality in critical one-dimensional systems with long-range hopping

Boundary multifractality of electronic wave functions is studied analytically and numerically for the power-law random banded matrix (PRBM) model, describing a critical one-dimensional system with long-range hopping. The peculiarity of the Anderson localization transition in this model is the existence of a line of fixed points describing the critical system in the bulk. We demonstrate that the boundary critical theory of the PRBM model is not uniquely determined by the bulk properties. Instead, the boundary criticality is controlled by an additional parameter characterizing the hopping amplitudes of particles reflected by the boundary.

The Scalar Flux and Current for Small Critical One-Dimensional Systems

We consider the scalar flux and current distributions for vanishingly small one-speed bare homogeneous critical systems in the three standard one-dimensional geometries. The similarities and differences between geometries is quite different from the corresponding results recently discussed by Karlsson and Pázsit, who considered large (diffusive) systems.