Breaking Of Translational Invariance Research Articles

Chiral Soliton Lattice turns into 3D crystal

Chiral perturbation theory predicts the chiral anomaly to induce a so-called Chiral Soliton Lattice at sufficiently large magnetic fields and baryon chemical potentials. This state breaks translational invariance in the direction of the magnetic field and was shown to be unstable with respect to charged pion condensation. Improving on previous work by considering a realistic pion mass, we employ methods from type-II superconductivity and construct a three-dimensional pion (and baryon) crystal perturbatively, close to the instability curve of the Chiral Soliton Lattice. We find an analogue of the usual type-I/type-II transition in superconductivity: along the instability curve for magnetic fields eB > 0.12 GeV2 and chemical potentials μ < 910 MeV, this crystal can continuously supersede the Chiral Soliton Lattice. For smaller magnetic fields the instability curve must be preceded by a discontinuous transition.

Adding integrable defects to the Boussinesq equation

The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect contributes to energy and momentum to ensure their conservation, despite the presence of discontinuities and the explicit breaking of translation invariance, leads to sewing conditions relating the two fields and their derivatives on either side of the defect. The manner in which several types of soliton solutions to the Boussinesq equation are affected by the defect is explored and reveals new effects that have not been observed in other integrable systems, such as the possibility of a soliton reflecting from a defect or of a defect decaying into one or two solitons.

A Rigorous Derivation of the Functional Renormalisation Group Equation

The functional renormalisation group equation is derived in a mathematically rigorous fashion in a framework suitable for the Osterwalder–Schrader formulation of quantum field theory. To this end, we devise a very general regularisation scheme which retains reflection positivity as well as the infinite degrees of freedom including smoothness. Furthermore, it is shown how the classical limit is altered by the regularisation process leading to an inevitable breaking of translation invariance. We also give precise conditions for the convergence of the obtained theories upon removal of the regularisation.

Modified Gravity Approaches to the Cosmological Constant Problem

The cosmological constant and its phenomenology remain among the greatest puzzles in theoretical physics. We review how modifications of Einstein’s general relativity could alleviate the different problems associated with it that result from the interplay of classical gravity and quantum field theory. We introduce a modern and concise language to describe the problems associated with its phenomenology, and inspect no-go theorems and their loopholes to motivate the approaches discussed here. Constrained gravity approaches exploit minimal departures from general relativity; massive gravity introduces mass to the graviton; Horndeski theories lead to the breaking of translational invariance of the vacuum; and models with extra dimensions change the symmetries of the vacuum. We also review screening mechanisms that have to be present in some of these theories if they aim to recover the success of general relativity on small scales as well. Finally, we summarize the statuses of these models in their attempts to solve the different cosmological constant problems while being able to account for current astrophysical and cosmological observations.

Thermodynamic geometry and complexity of black holes in theories with broken translational invariance

The relationship between thermodynamics and the Lloyd bound on the holographic complexity for a black hole has been of interest. We consider $D$ dimensional anti-de Sitter black holes with hyperbolic geometry as well as black holes with momentum relaxation that have a minimum for temperature and mass. We show that the singular points of the thermodynamic curvature of the black holes, as thermodynamic systems, correspond to the zero points of the action and volume complexity at the Lloyd bound. For such black holes with a single horizon, the complexity of volume and the complexity of action at minimum mass and minimum temperature are zero, respectively. We show that the thermodynamic curvature diverges at these minimal values. Because of the behaviour of action complexity and thermodynamic curvature at minimum temperature, we propose the action complexity as an order parameter of the black holes as thermodynamic systems. Also, we derive the critical exponent related to the thermodynamic curvature in different dimensions.

On the hydrodynamics of (2 + 1)-dimensional strongly coupled relativistic theories in an external magnetic field

In this paper, we review recent progress on relativistic hydrodynamics in (2 + 1) dimensions with an external magnetic field. We discuss the formalism allowing for momentum loss due to the explicit and spontaneous breaking of translation invariance by scalar operators. We also compare to some results from the gauge/gravity correspondence.

Inhomogeneities in the two-flavor chiral Gross-Neveu model

We investigate the finite-temperature and -density chiral Gross-Neveu model with an axial U$_A$(1) symmetry in $1+1$ dimensions on the lattice. In the limit where the number of flavors $N_\mathrm{f}$ tends to infinity the continuum model has been solved analytically and shows two phases: a symmetric high-temperature phase with a vanishing condensate and a low-temperature phase in which the complex condensate forms a chiral spiral which breaks translation invariance. In the lattice simulations we employ chiral SLAC fermions with exact axial symmetry. Similarly to $N_\mathrm{f}\to\infty$, we find for $8$ flavors, where quantum and thermal fluctuations are suppressed, two distinct regimes in the $(T,\mu)$ phase diagram, characterized by qualitatively different behavior of the two-point functions of the condensate fields. More surprisingly, at $N_\mathrm{f}=2$, where fluctuations are no longer suppressed, the model still behaves similarly to the $N_\mathrm{f}\to\infty$ model and we conclude that the chiral spiral leaves its footprints even on systems with a small number of flavors. For example, at low temperature the two-point functions are still dominated by chiral spirals with pitches proportional to the inverse chemical potential, although in contrast to large-$N_\mathrm{f}$ their amplitudes decrease with distance. We argue that these results should not be interpreted as the spontaneous breaking of a continuous symmetry, which is forbidden in two dimensions. Finally, using Dyson-Schwinger equations we calculate the decay of the U$_A$(1)-invariant fermion four-point function in search for a BKT phase at zero temperature.

Higher-order topological quantum paramagnets

Quantum paramagnets are strongly-correlated phases of matter where competing interactions frustrate magnetic order down to zero temperature. In certain cases, quantum fluctuations induce instead topological order, supporting, in particular, fractionalized quasi-particle excitations. In this work, we investigate paradigmatic spin models and show how magnetic frustration can also give rise to higher-order topological properties. We first study the frustrated Heisenberg model in a square lattice, where a plaquette valence bond solid appears through the spontaneous breaking of translational invariance. Despite the amount of effort that has been devoted to study this phase, its topological nature has so far been overlooked. By means of tensor network simulations, we establish how such state belongs to a higher-order symmetry-protected topological phase, where long-range plaquette order and non-trivial topology coexist. This interplay allows the system to support excitations that would be absent otherwise, such as corner-like states in the bulk attached to dynamical topological defects. Finally, we demonstrate how this higher-order topological quantum paramagnet can also be induced by dipolar interactions, indicating the possibility to directly observe this phase using atomic quantum simulators.

Inhomogeneous Conformal Field Theory Out of Equilibrium

We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity v(x) explicitly breaking translation invariance. Such inhomogeneous CFT is argued to effectively describe 1+1-dimensional quantum many-body systems with certain inhomogeneities varying on mesoscopic scales. Both heat and charge transport are studied, where, for concreteness, we suppose that our CFT has a conserved U(1) current. Based on projective unitary representations of diffeomorphisms and smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states defined by smooth inverse-temperature and chemical-potential profiles β(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (x)$$\\end{document} and μ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu (x)$$\\end{document}. Using this recipe, the following exact analytical results are obtained: (i) the full time evolution of densities and currents for heat and charge transport, (ii) correlation functions for components of the energy–momentum tensor and the U(1) current as well as for any primary field, and (iii) the thermal and electrical conductivities. The latter are computed by direct dynamical considerations and alternatively using a Green–Kubo formula. Both give the same explicit expressions for the conductivities, which reveal how inhomogeneous dynamics opens up the possibility for diffusion as well as implies a generalization of the Wiedemann–Franz law to finite times within CFT.

Decoherent quench dynamics across quantum phase transitions

We present a formulation for investigating quench dynamics across quantum phase transitions in the presence of decoherence. We formulate decoherent dynamics induced by continuous quantum non-demolition measurements of the instantaneous Hamiltonian. We generalize the well-studied universal Kibble-Zurek behavior for linear temporal drive across the critical point. We identify a strong decoherence regime wherein the decoherence time is shorter than the standard correlation time, which varies as the inverse gap above the groundstate. In this regime, we find that the freeze-out time \bar{t}\sim\tau^{{2\nu z}/({1+2\nu z})}t-∼τ2νz/(1+2νz) for when the system falls out of equilibrium and the associated freeze-out length \bar{\xi}\sim\tau^{\nu/({1+2\nu z})}ξ‾∼τν/(1+2νz) show power-law scaling with respect to the quench rate 1/\tau1/τ, where the exponents depend on the correlation length exponent \nuν and the dynamical exponent zz associated with the transition. The universal exponents differ from those of standard Kibble-Zurek scaling. We explicitly demonstrate this scaling behavior in the instance of a topological transition in a Chern insulator system. We show that the freeze-out time scale can be probed from the relaxation of the Hall conductivity. Furthermore, on introducing disorder to break translational invariance, we demonstrate how quenching results in regions of imbalanced excitation density characterized by an emergent length scale which also shows universal scaling. We perform numerical simulations to confirm our analytical predictions and corroborate the scaling arguments that we postulate as universal to a host of systems.

Normal-ordering approximations and translational (non)invariance

Normal-ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this work we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.

Holographic axion model: A simple gravitational tool for quantum matter

This is a complete and exhaustive review on the so-called holographic axion model -- a bottom-up holographic system characterized by the presence of a set of shift symmetric scalar bulk fields whose profiles are taken to be linear in the spatial coordinates. This simple model implements the breaking of translational invariance of the dual field theory by retaining the homogeneity of the background geometry and therefore allowing for controllable and fast computations. The usages of this model are very vast and they are a proof of the spectacular versatility of the framework. In this review, we touch upon all the up-to-date aspects of this model from its connection with massive gravity and effective field theories, to its role in modeling momentum dissipation and elastic properties ending with all the phenomenological features and its hydrodynamic description. In summary, this is a complete guide to one of the most used model in Applied Holography and a must-read for any researcher entering this field.

On the universality of AdS2 diffusion bounds and the breakdown of linearized hydrodynamics

The chase of universal bounds on diffusivities in strongly coupled systems and holographic models has a long track record. The identification of a universal velocity scale, independent of the presence of well-defined quasiparticle excitations, is one of the major challenges of this program. A recent analysis, valid for emergent IR fixed points exhibiting local quantum criticality, and dual to IR AdS2 geometries, suggests to identify such a velocity using the time and length scales at which hydrodynamics breaks down — the equilibration velocity. The latter relates to the radius of convergence of the hydrodynamic expansion and it is extracted from a collision between a hydrodynamic diffusive mode and a non-hydrodynamic mode associated to the IR AdS2 region. In this short note, we confirm this picture for holographic systems displaying the spontaneous breaking of translational invariance. Moreover, we find that, at zero temperature, the lower bound set by quantum chaos and the upper one defined by causality and hydrodynamics exactly coincide, determining uniquely the diffusion constant. Finally, we comment on the meaning and universality of this newly proposed prescription.

Disorder-induced enhancement of entanglement growth in one dimension: Information leakage at the scale of the localization length

When a group of compactly packed free fermions is allowed to spread over an empty one-dimensional lattice, the spreading particles can create entanglement between different parts of the lattice. We show, though breaking of translational invariance (TI) of the lattice by disorder slows down the spreading of local observables, the entanglement entropy of a subsystem can nonetheless receive a remarkable enhancement as long as the subsystem lies within the single-particle localization length. We show, the main mechanism behind this enhancement is the re-entrant exchange of particles between the subparts due to transport of mutual information due to back scattering. We discuss the length and time scales relevant to the phenomenon. We study the phenomenon for breaking of TI by both quasi-periodic and random potentials. We further explore the effect of randomness only in the initial state. This also exhibits a similar enhancement effect even in a TI lattice. We also touch upon the special case of periodic potential, where qualitatively similar phenomenology emerges, though the coherence in the back scattering in this case leads to effects not captured by our simple yet generic picture.

Inhomogeneous phases in the Gross-Neveu model in 1+1 dimensions at finite number of flavors

We explore the thermodynamics of the 1+1-dimensional Gross-Neveu (GN) model at finite number of fermion flavors $N_f$, finite temperature and finite chemical potential using lattice field theory. In the limit $N_f \rightarrow \infty$ the model has been solved analytically in the continuum. In this limit three phases exist: a massive phase, in which a homogeneous chiral condensate breaks chiral symmetry spontaneously, a massless symmetric phase with vanishing condensate and most interestingly an inhomogeneous phase with a condensate, which oscillates in the spatial direction. In the present work we use chiral lattice fermions (naive fermions and SLAC fermions) to simulate the GN model with 2, 8 and 16 flavors. The results obtained with both discretizations are in agreement. Similarly as for $N_f \rightarrow \infty$ we find three distinct regimes in the phase diagram, characterized by a qualitatively different behavior of the two-point function of the condensate field. For $N_f = 8$ we map out the phase diagram in detail and obtain an inhomogeneous region smaller as in the limit $N_f \rightarrow \infty$, where quantum fluctuations are suppressed. We also comment on the existence or absence of Goldstone bosons related to the breaking of translation invariance in 1+1 dimensions.

Dimer description of the SU(4) antiferromagnet on the triangular lattice

In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of their properties throughout the phase diagram. In systems with both spin and orbital (or valley) degrees of freedom such a starting point gives rise to SU(4)-symmetric models. Here we consider SU(4)-symmetric "spin'' models, corresponding to Mott phases at half-filling, i.e. the six-dimensional representation of SU(4). This may be relevant to twisted multilayer graphene. In particular, we study the SU(4) antiferromagnetic "Heisenberg'' model on the triangular lattice, both in the classical limit and in the quantum regime. Carrying out a numerical study using the density matrix renormalization group (DMRG), we argue that the ground state is non-magnetic. We then derive a dimer expansion of the SU(4) spin model. An exact diagonalization (ED) study of the effective dimer model suggests that the ground state breaks translation invariance, forming a valence bond solid (VBS) with a 12-site unit cell. Finally, we consider the effect of SU(4)-symmetry breaking interactions due to Hund's coupling, and argue for a possible phase transition between a VBS and a magnetically ordered state.

Partial Equilibration of the Anti-Pfaffian Edge due to Majorana Disorder.

We consider electrical and thermal equilibration of the edge modes of the anti-Pfaffian quantum Hall state at ν=5/2 due to tunneling of the Majorana edge mode to trapped Majorana zero modes in the bulk. Such tunneling breaks translational invariance and allows scattering between Majorana and other edge modes in such a way that there is a parametric difference between the length scales for equilibration of charge and heat transport between integer and Bose mode, on the one hand, and for thermal equilibration of the Majorana edge mode, on the other hand. We discuss a parameter regime in which this mechanism could explain the recent observation of quantized heat transport [M. Banerjee etal., Nature (London) 559, 205 (2018)NATUAS0028-083610.1038/s41586-018-0184-1].

Phase crystals

Superconductivity owes its properties to the phase of the electron pair condensate that breaks the $U(1)$ symmetry. In the most traditional ground state, the phase is uniform and rigid. The normal state can be unstable towards special inhomogeneous superconducting states: the Abrikosov vortex state, and the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform superconducting state can go into a fundamentally different and more ordered non-uniform ground state, that we denote as a phase crystal. The new state breaks translational invariance through formation of a spatially periodic modulation of the phase, manifested by unusual superflow patterns and circulating currents, that also break time-reversal symmetry. We list the general conditions needed for realization of phase crystals. Using microscopic theory we then derive an analytic expression for the superfluid density tensor for the case of a non-uniform environment in a semi-infinite superconductor. We demonstrate how the surface quasiparticle states enter the superfluid density and identify phase crystallization as the main player in several previous numerical observations in unconventional superconductors, and predict existence of a similar phenomenon in superconductor-ferromagnetic structures. This analytic approach provides a new unifying aspect for the exploration of boundary-induced quasiparticles and collective excitations in superconductors. More generally, we trace the origin of phase crystallization to non-local properties of the gradient energy, which implies existence of similar pattern-forming instabilities in many other contexts.

Current phases in Hofstadter ladder with staggered hopping

Hofstadter ladder describes a Boson ladder under a uniform magnetic field and supports nontrivial energy band and fractional quantum Hall states. Staggered hopping is illuminated from the SSH model and proved to have non-trivial effects on current phases. We introduce staggered hopping on Hofstadter ladder to study the novel current phases. Exact diagonalization (ED) and density matrix renormalization group (DMRG) methods have been employed to study the current phases of the ladder in noninteraction and strong interaction (hard core boson) cases. By observing energy singularities and the new flux patterns when increasing the staggered hopping strength, we extend Meissner and vortex phase to horizontal current phase, vertical current phase and vortex phase. The horizontal current phase has stronger chiral currents in horizontal direction, which is the long direction of the ladder. The vertical current phase has stronger chiral currents in vertical direction. The above two phases do not break translational invariance while the vortex phase does. The current patterns of horizontal current phase are proved to be continuously deformed form the Meissner phase, and the vortex phase has similar signatures. The vertical current phase is only visible when the hopping is staggered. These phases generally exist in noninteraction regimes and interacting superfluid regimes. We have defined new quantities (i.e. current inhomogeneity and nearest overlap) to characterize different quantum phases. In noninteraction case, the horizontal current phase go through the vortex phase to enter the vertical current phase by second order phase transitions, but in strong interaction case such a change can be directly made in a first order phase transition. The direct transition is made in higher fillings with almost identical flux. Surprisingly, the three phases turn into only two phases in Mott regimes, and the phase transition between the horizontal current phase and the vertical current phase has disappeared. We call the new phase as Mott-homogenous phase. The staggered hopping has exotic effects in strong interaction case. For &lt;i&gt;n&lt;/i&gt; = 0.25 filling, the staggered hopping shrinks the region of vortex phases and produces Mott-SF transition. When the staggered hopping is weak, the system achieves Mott-SF transition just by varying the flux. This research can enrich current phases in lattice systems and illuminate further studies on chiral currents.

Intrinsic cluster-shaped density waves in cellular dynamical mean-field theory

It is well known that cellular dynamical mean-field theory (CDMFT) leads to the artificial breaking of translation invariance. In spite of this, it is one of the most successful methods to treat strongly correlated electrons systems. Here, we investigate in more detail how this broken translation invariance manifests itself. This allows us to disentangle artificial broken translation invariance effects from the genuine strongly correlated effects captured by CDMFT. We report artificial density waves taking the shape of the cluster---cluster density waves---in all our zero temperature CDMFT solutions, including pair density waves in the superconducting state. We discuss the limitations of periodization regarding this phenomenon, and we present mean-field density-wave models that reproduce CDMFT results at low energy in the superconducting state. We then discuss how these artificial density waves help the agreement of CDMFT with high temperature superconducting cuprates regarding the low-energy spectrum, in particular for subgap structures observed in tunneling microscopy. We relate these subgap structures to nodal and antinodal gaps in our results, similar to those observed in photoemission experiments. This fortuitous agreement suggests that spatial inhomogeneity may be a key ingredient to explain some features of the low-energy underdoped spectrum of cuprates with strongly correlated methods. This work deepens our understanding of CDMFT and clearly identifies signatures of broken translation invariance in the presence of strong correlations.