One-dimensional Field Theory Research Articles

Continuous matrix product states with periodic boundary conditions and an application to atomtronics

We introduce a time evolution algorithm for one-dimensional quantum field theories with periodic boundary conditions. This is done by applying the Dirac-Frenkel time-dependent variational principle to the set of translational invariant continuous matrix product states with periodic boundary conditions. Moreover, the ansatz is accompanied with additional boundary degrees of freedom to study quantum impurity problems. The algorithm allows for a cutoff in the spectrum of the transfer matrix and thus has an efficient computational scaling. In particular we study the prototypical example of an atomtronic system - an interacting Bose gas rotating in a ring shaped trap in the presence of a localised barrier potential.

Negativity spectrum of one-dimensional conformal field theories

The partial transpose $\rho_A^{T_2}$ of the reduced density matrix $\rho_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of $\rho_A^{T_2}$, that we dub negativity spectrum, in the ground sate of gapless one-dimensional systems described by a Conformal Field Theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of $\rho_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.

Dark-soliton-like excitations in the Yang–Gaudin gas of attractively interacting fermions

Yrast states are the lowest energy states at given non-zero momentum and provide a natural extension of the concept of dark solitons to strongly interacting one-dimensional quantum gases. Here we study the yrast states of the balanced spin- Fermi gas with attractive delta-function interactions in one-dimension with the exactly solvable Yang–Gaudin model. The corresponding Bethe-ansatz equations are solved for finite particle number and in the thermodynamic limit. Properties corresponding to the soliton-like nature of the yrast excitations are calculated including the missing particle number, phase step, and inertial and physical masses. The inertial to physical mass ratio, which is related to the frequency of oscillations in a trapped gas, is found to be unity in the limits of strong and weak attraction and falls to in the crossover regime. This result is contrasted by one-dimensional mean field theory, which predicts a divergent mass ratio in the weakly attractive limit. By means of an exact mapping our results also predict the existence and properties of dark-soliton-like excitations in the super Tonks–Girardeau gas. The prospects for experimental observations are briefly discussed.

Entanglement entropy from the truncated conformal space

A new numerical approach to entanglement entropies of the Renyi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced already in very small system sizes. The new method treats ground states and excited states on the same footing, and the applicability for excited states is illustrated by reproducing Renyi entropies of low-lying states in the transverse field Ising model.

A hydrodynamic approach to non-equilibrium conformal field theories

We develop a hydrodynamic approach to non-equilibrium conformal field theory. We study non-equilibrium steady states in the context of one-dimensional conformal field theory perturbed by the irrelevant operator. By direct quantum computation, we show, to first order in the coupling, that a relativistic hydrodynamic emerges, which is a simple modification of one-dimensional conformal fluids. We show that it describes the steady state and its approach, and we provide the main characteristics of the steady state, which lies between two shock waves. The velocities of these shocks are modified by the perturbation and equal the sound velocities of the asymptotic baths. Pushing this approach further, we are led to conjecture that the approach to the steady state is generically controlled by the power law t−1/2, and that the widths of the shocks increase with time according to t1/3.

On the Study of Oscillons in Scalar Field Theories: A New Approach

We study configurations in one-dimensional scalar field theory, which are time-dependent, localized in space, and extremely long-lived, called oscillons. How the action of changing the minimum value of the field configuration representing the oscillon affects its behavior is investigated. We find that one of the consequences of this procedure is the appearance of a pair of oscillon-like structures presenting different amplitudes and frequencies of oscillation. We also compare our analytical results to numerical ones, showing excellent agreement.

Nonlinear Fluctuating Hydrodynamics in One Dimension: The Case of Two Conserved Fields

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has exactly two locally conserved fields. Through numerical simulations the peak structure of the steady state space-time correlations is determined and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the completely asymmetric Levy distribution with parameter $\alpha = 5/3$. As a by-product, we completely classify the universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients.

Casimir effect from a scattering approach

The Casimir force is a spectacular consequence of the existence of vacuum fluctuations and thus deserves a place in courses on quantum theory. We argue that the scattering approach within a one-dimensional field theory is well suited to a discussion of the Casimir effect. It avoids in a transparent way divergences appearing in the evaluation of the vacuum energy. Furthermore, the scattering approach connects in a natural manner to the standard discussion of one-dimensional scattering problems in a quantum theory course. Finally, it allows for the introduction to students of the methods employed in the current research literature to determine the Casimir force in real-world systems.

Bootstrapping the 3d Ising twist defect

Recent numerical results point to the existence of a conformally invariant twist defect in the critical 3d Ising model. In this note we show that this fact is supported by both epsilon expansion and conformal bootstrap calculations. We find that our results are in good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects.

PRINCIPLE ACTION FOR PARTICLES FASTER THAN LIGHT

A general action for particles faster than light is presented. It is demonstrated that this action is invariant under reparametrizations. For several cases, it is shown that in the high velocity regime, the action is invariant under anisotropic spacetime transformation and at quantum level the system has fractal behavior. For those cases, it is shown that the action describes a particle in Finsler geometry and equivalent to one-dimensional field theory in a curved space, where the metric depends on temporal derivatives.

Double-Delta Potentials: One Dimensional Scattering

The path is explored between one-dimensional scattering through Dirac-$\delta$ walls and one-dimensional quantum field theories defined on a finite length interval with Dirichlet boundary conditions. It is found that two $\delta$'s are related to the Casimir effect whereas two $\delta$'s plus the first transparent P$\ddot{\rm o}$sch-Teller well arise in the context of the sine-Gordon kink fluctuations, both phenomena subjected to Dirichlet boundary conditions. One or two delta wells will be also explored in order to describe absorbent plates, even though the wells lead to non unitary Quantum Field Theories.

Vacuum energy and renormalization on the edge

The vacuum dependence on boundary conditions in quantum field theories is analysed from a very general viewpoint. From this perspective, the renormalization prescriptions not only imply the renormalization of the couplings of the theory in the bulk but also the appearance of a flow in the space of boundary conditions. For regular boundaries, this flow has a large variety of fixed points and no cyclic orbit. The family of fixed points includes Neumann and Dirichlet boundary conditions. In one-dimensional field theories, pseudo-periodic and quasi-periodic boundary conditions are also RG fixed points. Under these conditions massless bosonic free field theories are conformally invariant. Among all fixed points only Neumann boundary conditions are infrared stable fixed points. All other conformal invariant boundary conditions become unstable under some relevant perturbations. In finite volumes, we analyse the dependence of the vacuum energy along the trajectories of the renormalization group flow providing an interesting framework for dark energy evolution. In contrast, the renormalization group flow on the boundary does not affect the leading behaviour of the entanglement entropy of the vacuum in one-dimensional conformally invariant bosonic theories.

Higher Derivative Operators as loop counterterms in one-dimensional field theory orbifolds.

Using a 5D N=1 supersymmetric toy-model compactified on S_1/(Z_2 x Z_2'), with a ``brane-localised'' superpotential, it is shown that higher (dimension) derivative operators are generated as one-loop counterterms to the (mass)^2 of the zero-mode scalar field, to ensure the quantum consistency of the model. Such operators are just a result of the compactification and integration of the bulk modes. They are relevant for the UV momentum scale dependence of the (mass)^2 of the zero-mode scalar field, regarded as a Higgs field in more realistic models. While suppressed for a small compactification radius R, these operators can affect the predictive power of models with a large value for R. A general method is also provided for a careful evaluation of infinite sums of 4D divergent loop-integrals (of Feynman diagrams) present in field theory orbifolds. With minimal changes, this method can be applied to specific orbifold models for a simple evaluation of their radiative corrections and the overall divergences.

Spectral and transport properties of quantum wires with bond disorder

Systems with bond disorder are defined through lattice Hamiltonians that are of pure nearest neighbour hopping type, i.e., do not contain on-site contributions. They stand representative for the general family of disordered systems with chiral symmetries. Application of the Dorokhov–Mello–Pereyra–Kumar transfer matrix technique P.W. Brouwer et al. [Phys. Rev. Lett 81 (1998) 862; Phys. Rev. Lett. 84 (2000) 1913] has shown that both spectral and transport properties of quasi one-dimensional systems belonging to this category are highly unusual. Most notably, regimes with absence of exponential Anderson localization are observed, the single particle density of states exhibits singular structure in the vicinity of the band centre, and the manifestation of these phenomena depends in an apparently topological manner on the even- or oddness of the channel number. In this paper we re-consider the problem from the complementary perspective of the nonlinear σ -model. Relying on the standard analogy between one-dimensional statistical field theories and zero-dimensional quantum mechanics, we will relate the problem to the behaviour of a quantum point particle subject to an Aharonov–Bohm flux. We will build on this analogy to re-derive earlier DMPK results, identify a new class of even/odd staggering phenomena (now dependent on the total number of sites in the system) and trace back the anomalous behaviour of the bond disordered system to a simple physical mechanism, viz. the flux periodicity of the quantum Aharonov–Bohm system. We will also touch upon connections to the low energy physics of other lattice systems, notably disordered chiral systems in 0 and 2 dimensions and antiferromagnetic spin chains.

ADIABATIC SPECTRAL FLOW AND CHANGE OF MORSE INDICES

The concept of the Morse index characterizes the behavior near the critical points of the action functional that governs physical systems. We study the specific feature of Morse indices for the case that there are several number of critical points; on each of these critical points the Morse indices are defined. As a concrete example, we consider a model of one-dimensional field theory model. By using the idea of the adiabatic spectral flow, it is shown that the change of the Morse indices is correlated with the structure change inherent in the solutions of field equation that is controlled by the parameters built in the field action.

Gauging of SL(2,R) WZNW models and the Liouville field

We consider the gauging of theSL(2, R) WZNW model by its nilpotent subgroupE(1). The resulting space-time of the corresponding sigma model is seen to collapse to the one-dimensional field theory of Liouville.

Approximate calculation of the free energy of Fermi systems with topological solitons: soliton creation energy of polyacetylene

The authors investigate an approximate technique for the calculation of the free energy of fermionic systems that are coupled to topological solitons. They apply this to a solvable one-dimensional model field theory: polyacetylene in the static dimerization approximation. They estimate the free energy of solitons with various scales of spatial extension and make comparisons with exact results. In an appendix, the authors make the connection between the high-temperature limit of their work and Ginzburg-Landau theory.

Some instanton solutions in string theory

In recent work Rey presented an exterior instanton solution for the massless fields in the low-energy limit of heterotic string theory to lowest order in α′. We extend this solution to O ( α′) in bosonic string theory and present several related solutions. For the spherically symmetric solution, the corresponding Weyl invariant sigma model is the product of a Wess-Zumino-Witten model and a one-dimensional conformal field theory. An exact solution is then obtained by fixing the metric and antisymmetric tensor field in their lowest order form and rescaling the dilation order by order in α′.

Ground-state structure of supersymmetric Yang-Mills theory

A supersymmetric one-dimensional field theory is obtained from SU(2) supersymmetric pure Yang-Mills theory in 3 + 1 dimensions using an improved Born-Oppenheimer approximation. The gauge-fixing procedure employed involves a mixing of spin and isospin degrees of freedom. Despite the fact that supersymmetry imposes an apparently overdetermined set of equations on the ground-state wave functions, we find their asymptotic behavior. They decay exponentially. By straightforward modifications, we determine the asymptotic behavior of the ground-state wave functional of the original supersymmetric Yang-Mills theoy. A few related issues are discussed.

A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model

LetH=−Δ+V onl 2(ℤ), whereV(x),x∈ℤ, are i.i.d.r.v.'s with common probability distributionv. Leth(t)=∫e −itv dv(v) and letk(E) be the integrated density of states. It is proven: (i) Ifh isn-times differentiable withh (j)(t)=O((1+|t|)−α) for some α>0,j=0, 1, ...,n, thenk(E) is aC n function. In particular, ifv has compact support andh(t)=O((1+|t|)−α) with α>0, thenk(E) isC ∞. This allowsv to be singular continuous. (ii) Ifh(t)=O(e −α|t|) for some α>0 thenk(E) is analytic in a strip about the real axis. The proof uses the supersymmetric replica trick to rewrite the averaged Green's function as a two-point function of a one-dimensional supersymmetric field theory which is studied by the transfer matrix method.