%&LaTeX \input{aipcheck} \documentclass[ %final % use final for the camera ready runs draft, % use draft while you are working on the paper %numberedheadings % uncomment this option for numbered sections ]{aipproc} \layoutstyle{8x11double} \begin{document} \title{Microscopic solitons in correlated electronic systems: theory versus experiment.} \classification{ 03.75.Lm, 71.10.Pm, 71.27.+a, 71.45.Lr, 72.15.Nj, 73.20.Qt, 73.40.Gk, 74.81.-g, 74.70.Kn, 74.20.Fg, 75.30.Fv } \keywords{topological defect, soliton, vortex, stripe, tunneling, superconductivity, CDW, SDW, AFM, FFLO, charge ordering.} \author{S. Brazovskii}{address={CNRS UMR 8626 LPTMS, Univ. Paris-Sud, Orsay, France}} \begin{abstract} Symmetry broken electronic states give rise to topological defects: from extended domain walls - "stripes" as solitonic lattices to microscopic solitons as anomalous quasi-particles and instantons in their dynamics. We shall collect and interpret experimental evidences on existence of microscopic solitons, and their determining role in electronic processes of quasi-1D electronic crystals. Thus, the ferroelectric charge ordering in organic conductors gives access to several types of solitons observed in conductivity (holons) and in permittivity (polar kinks), to solitons' bound pairs in optics, to compound charge-spin solitons. In charge density waves, the individual phase solitons have been visually captured in recent STM experiments. The resolved subgap tunneling spectra recover these solitons (in aggregated form of dislocations in statics and as instantons - the phase slips in dynamics), as well as the amplitude kinks - the spinons. The theory relies upon the regime of quantum dissipation provided by soft mode emittance in the course of the soliton creation, and on effects of dimensional crossover. With onset of a 2D or a 3D long range order, the topologically nontrivial solitons experience the confinement resulting in the spin-charge recombination. It originates the symmetry broken spin- or charge- roton configurations with charge- or spin- kinks localized in the core, correspondingly for cases of repulsion and attraction. These complex excitations can be viewed as nucleuses of the melted stripe phases, which appears in doped antiferromagnetic - Mott insulators or in spin-polarized superconductors and charge density waves. \end{abstract} \maketitle \section{ Introduction.} Possibility of constructing elementary particles from continuous matter - fields, is the old paradigm in physics \cite{skyrm}. Such hypothetical objects, known as solitons (or instantons for related transient processes) have been discussed in field theory, high energy physics, cosmology, in conjugated polymers and other quasi one-dimensional (1D) conductors, in biological macro-molecules, etc. Most of strongly correlated electronic systems show various types of symmetry breaking giving rise to degenerate ground states. The degeneracy allows for topologically nontrivial perturbations exploring the possibility of traveling through different allowed ground states. Their most known forms are plain domain walls, vortex lines or dislocations, which are still macroscopic objects extending in two or three dimensions (see \cite{mineev,volovik} for general views and sophisticated applications). Of our special interest, however, are totally localized and truly microscopic objects which energies and quantum numbers are on the one-electron scale. These subgap particles would determine the observable properties, which are usually ascribed to conventional electronic excitations. Role of solitons in electronic properties was appreciated in theories since mid 70's (see reviews \cite{BK84,YuLu}, and also \cite{JOS07,JSSS08} for a more recent development). The solitons were firstly accessed in experiments on conducting polymers of early 80's \cite{Heeger RMP}. New motivations came in early 2000's from discoveries of the ferroelectric charge ordering (\cite{monceau:01,chow}, and a review \cite{sb-cofe}) in organic conductors \cite{lebed-springer} , from new accesses via nano-scale experiments \cite{lat:05,lat:06} in materials with Charge Density Waves (CDW) \cite{gorkov:89}, from optics of new conducting polymers \cite{vardeni}. Today, various solitons show up in conductivity, tunneling spectroscopy, optical absorption. Instantons - the corresponding dynamical processes - are responsible for subgap transitions leading to a pseudogap formation \cite{SB+SM:03,SB+SM:05,SB+SM:08}. The content of this review will be mostly endorsed by events in Electronic Crystals, see \cite{ECRYS:05,ECRYS:08}, especially cases of Charge/Spin Density Waves, Charge Ordering/Disproportionation, Wigner crystals, and stripes. We shall review the last decade experiments confirming the solitons in organic conductors with the ferroelectric charge ordering. Three different types of solitons show up in experiments: a single-charged amplitude king - the "holon" in a general terminology of strongly repulsive electronic systems, the polarization kink carrying a fractional charge, and the topologically bound charge-spin soliton. We shall interpret new experiments on direct observations of microscopic solitons in processes of the coherent internal tunneling in CDWs \cite{lat:05,lat:06}. There are amplitude solitons corresponding to the long sought special quasi-particle - the spinon. The same experiment also gives access to the junction reconstruction via spontaneous creation of the grid of phase solitons. The low energy tunneling recovers the quantum phase slips at adjacent chains. We shall recall the theory \cite{SB+SM:08} of internal coherent tunneling in the pseudogap region. The instanton approach allows to calculate the interchain tunneling current both in single electron (amplitude solitons) and bi-electron (phase solitons) channels. \section{ Observations of solitons in charge-gap systems} \subsection{ Ferroelectricity and charge ordering in quasi 1D organic conductors.} This decade, a true workshop on solitons was opened in organic conductors from the family (TMTCF)$_{2}$X, see \cite{lebed-springer} - chapter \cite{sb-cofe}. The facility is provided by the discovery of the ferroelectricity endorsed by the charge ordering \cite{monceau:01,chow,nad-co}. The zoo of solitons is largely accessed thanks to possibility of switching on/off of the Mott state by means of the charge ordering. This is the state with no magnetic long range order: spin degrees of freedom are gapless and split-off, charge degrees of freedom are described by the chiral phase $\varphi(x,t)$ which we define as for a conventional $2K_{F}$ CDW $\sim\cos(\varphi+2K_Fx)$, and the phase Hamiltonian becomes \[ H=(\hbar/4\pi\gamma)[v(\partial_{x}\varphi)^{2}+(\partial_{t}\varphi)^{2}/v] -U\cos(2\varphi-2\alpha) \label{H-CO} \] Here $v$ is a phase velocity, and the spontaneous phase center shift $\alpha$ is the signature of the ferroelectricity, $U$ is the Umklapp scattering amplitude \cite{DL+EL} which is responsible for the charge gap formation. The constant $\gamma$ (usually called the Luttinger liquid parameter - $K_{\rho }$) determines several regimes: $\gamma<1$ at any repulsion, then $U$ is preserved if it is already build-in, - this is the generic Mott state case for a half-filled band; $\gamma<1/2$ at a stronger repulsion, then $U$ is spontaneously generated even away from the bare half-filling, - this is our Charge Ordering case. The multiple ground state degeneracy gives rise to a variety of solitons. Even at the given charge ordering amplitude, the ground state energy $H_{U}=-U\cos (2\varphi -2\alpha)$ is doubly degenerate between $\varphi=\alpha$ and $\varphi=\alpha\pm\pi $ (we assume $ U>0$). It allows for phase $\mp\pi$ solitons - "holons" with the charge $e$ and no spin. The quantum Sine-Gordon model is exactly solvable; the excitation spectrum and even transitions intensities are known, see \cite{klainert:79,essler:01}. Just at the border line $\gamma=1/2$ of the Charge Ordering regime, the spectrum changes giving rise to kink's bound states - breathers. Indirect evidences on solitons come already from conductivity and thermodynamics: separation of charge from spin; enhanced activation energy for interchain transport. But particularly intriguing is the observation of solitons in dynamics - via optics for organics (and conducting polymers) and via tunneling (by now only in CDWs, see the next section). \begin{figure}[tbh] \includegraphics[width=0.3\textwidth,height=4cm]{optics.eps}\\ \caption{Optical absorption in two organic conductors from the family $(TMTSF)_2X$ showing a coexistence of the high frequency features inherent to the 1D Mott insulator state, and the low energy metallic rise. We interpret the major peak as the excitation of the two-kink bound state, while the shoulder above it should be the threshold for creating unbound pairs. The lower curve, for $X=AsF_6$, shows an almost ideal example, while the upper, for $X=PF_6$, curve is complicated by an additionally symmetry breaking (the spin-Peierls transition) at lower energies; it signifies the processes of spin-charge topological reconfinement described in the next subsection. Optical conductivity is replotted from data of M. Dressel group (unpublished, see \cite{TMTCF optics, vescoli-98} for complementary information).} \label{fig:optics} \end{figure} The interpretation of the Fig.\ref{fig:optics} corresponds to detailed theory for optics of solitons \cite{klainert:79, essler:01} based upon the exact solution of the quantum sine-Gordon model. A general feature of the whole gapful regime $\gamma<1$ is the two-particle gap $E_{g}=2\Delta$ for creation of a pair of $\pm{\pi}$ solitons. But contrary to the common sense intuition, and to the elementary theory of semiconductors, the absorption intensity $I(\Omega)$ has no singularity at the threshold $E_{g}$. Optical density of states law $DOS\sim(\Omega-E_{g})^{-1/2}$ is compensated by vanishing of the matrix element, then the absorption starts smoothly as $I\sim(\Omega-E_{g})^{1/2}$. Moreover, for the generic Mott state without the charge ordering, $1/2<\gamma<1$, there would be no bound states, hence no absorption below $E_{g}$. But our case of the Mott state due to the spontaneous charge ordering requires $\gamma<1/2$, hence a number of sharp peaks in absorption even below the two-particle gap $\Omega<2\Delta$ - at a sequence of quantum breathers which are bound states of two solitons. This is what seems to be observed in experiment, Fig.\ref{fig:optics}. \subsubsection{Further symmetry breaking: spin-charge reconfinement and compound solitons.} Some compounds from the $(TMTTF)_{2}X$ family show a subsequent, at lower temperature $T_2$ phase transition of the tetramerization which can be identified as the spin-Peierls state; it gains the energy from opening of the spin gap $\Delta_s$. We argue that this transition brings to life a special state of two topologically coupled solitons which explore both the charge and the spin sectors, thus realizing the spin-charge reconfinement. Within the reduced symmetry, the potential part of the Hamiltonian becomes \[ H_{U}=-Ucos(2\varphi -2\alpha)-Vcos(\varphi -\beta)\cos\theta \] Here $\theta$ is the spin chiral phase, such that $\theta^{\prime }/\pi$ is the smooth spin density. The new, at $TT_2$). Its formation destroys the spin liquid which existed at $T>T_2$ on top of the charge ordering. Major effects of the $V$-term are a) to open the spin gap $2\Delta_s$ corresponding to creation of the triplet excitation as a new \{$\delta\theta=2\pi$, $\delta\varphi=0$\} purely spin solitons; b) to prohibit former $\delta\varphi =\pi$ charged solitons - the holons, now they are confined in pairs bound by spin strings; c) to allow for compound spin-charge topologically bound solitons \{$\delta\varphi=\pi$, $\delta\theta=\pi$\} which leave the Hamiltonian invariant. This particle bears the normal electron's quantum numbers: the charge $e$ and the spin $1/2$, but their localization is different: sharply within $\hbar v/\Delta$ for the charge $e$, while loosely within $\hbar v/\Delta_s$ for the spin $1/2$. \section{Spin-gap cases with continuous symmetry: incommensurate CDW and superconductors.} \subsection{Common problems of excitations and stripes in CDWs and superconductors} Both CDWs and superconductors (SC) are the spin-singlet states, which are subject to deparing under the Zeeman splitting effect of the magnetic field \cite{hmf rvs}. Above a certain critical field, the ground state develops a periodic superstructure (see a short review and references in \cite{JOS07,JSSS08}). In superconductors it is known as the FFLO state named after \cite{FF,LO}, which has attracted a great deal of attention recently (see e.g. \cite{yuji,singleton}) because of events in organic and heavy-fermion superconductors, in cold atoms. Its formation is expected to be a very week effect in conventional superconductors, unless it is endorsed by strong coupling $\Delta\sim E_{f}$, or by flattened Fermi surfaces - particularly in quasi-1D case. Then the theories of early 80's (see \cite{BGK:80} and the review \cite{BK84} for CDWs, and \cite{buzdin} for superconductors), predict formation of solitonic lattices with unpaired spins localized at midgap states near the order parameter nodes. For CDWs there are convincing theoretical and experimental evidences, that beyond the coherent walls, also the separate amplitude solitons (the walls building blocks) exist as quasi particles - the spinons. Based upon the CDW notion, in superconductors we expect to find a tightly bound pair of half-integer vortices sharing one unpaired spin, as we shall describe below. We start to show most typical and convincing evidences for existence of the gap in the excitation spectrum of singlet-ground-state electronic systems: superconductors and CDWs, Fig.\ref{fig:tunnels}. Within the standard BCS - Bogolubov view, the excited states are the linear combinations of: electrons and holes at $\pm p$ for a SC, or of electrons or holes at $-p$ and $p+2p_{f}$ for CDWs. Corresponding spectra are $E(k)=\pm(\Delta^{2}+(v_{f}k)^{2})^{1/2}$, $k=p-p_{f}$. But is it always true? It is proved for typical SCs - Fig.\ref{fig:tunnels}a; left questionable for strong coupling SC cases - Fig.\ref{fig:tunnels}b: high-T$_{c}$, real space pairs, cold atoms, bi-polarons; stays clearly incomplete for CDWs - Fig.\ref{fig:tunnels}c, as proved by modern experiments ; is certainly inconsistent for 1D and even quasi 1D systems as proved theoretically - Fig.\ref{fig:tunnels}d, \cite{JOS07,JSSS08}. There are the instantons, the solitons and their arrays which are responsible for these confusions. A related complex of well established facts for incommensurate CDWs (ICDW) which are symmetrically equivalent to SCs is summarized in \cite{SB-ECRYS:08}. \begin{figure}[tbh] {\label{fig:Nb}\includegraphics[width=0.2\textwidth, height=3cm]{tunnels-Nb.eps}} {\label{fig:CaC}\includegraphics[width=0.2\textwidth, height=3cm]{tunnels-CaC.eps}} {\label{fig:NbSe3}\includegraphics[width=0.2\textwidth, height=3cm]{tunnels-NbSe3.eps}} {\label{fig:I(U)}\includegraphics[width=0.2\textwidth, height=3cm]{U-T.eps}} \caption{Deparing gaps from tunneling experiments - plots of the tunneling conductance $dI/dU$ versus voltage $U$. Superconductors: $Nb$ (a) and $CaC_{6}$ (b); CDW in $NbSe_{3}$ (c). The panel (d) - gives the theoretical prediction \cite{SB+SM:08} for the instanton-mediated tunneling current $I(U)$ in the subgap region $|U|<2\Delta$ (U is shown in units of $\Delta$; the plots correspond to temperatures $T/\Delta=1/4,1/6,1/8$).} \label{fig:tunnels} \end{figure} \subsection{Solitons in incommensurate CDWs} While the charge ordering was a crystal of electrons, the ICDW is a crystal of electron pairs. Its lowest energy current carrier may be the charge-$2e$ defect of adding/missing one period at the defected chain. It is the $\pm 2\pi$ soliton of the ICDW order parameter $\mathcal{O}_{ICDW}=A\cos (2K_{f}x+\varphi)$, which has been recently captured and visualized in STM experiments \cite{brun}, Fig.\ref{fig:brun}. \begin{figure}[tbh] \includegraphics[width=0.2\textwidth,height=3cm]{2Pi.eps} \includegraphics[width=0.2\textwidth,height=3cm]{NiPdBr.eps} \caption{Left: Visualization of the $2\pi$ soliton - the prefabricated pair of electrons, by the STM on $NbSe_{3}$ \cite{brun}. At the (red) front line the defected chain is displaced by half of the period. Along the defected chain the whole period $\pm2\pi$ is missed or gained. Right: STM visualization of the half-period soliton (the amplitude kink) in a quasi1D complex Ni$_{0.05}$Pd$_{0.95}$Br \cite{takaishi}. White arrow: 1D chains direction, blue arrow: the defected chain.} \label{fig:brun} \end{figure} The singlet pair can be broken into spin $1/2$ components, but it will not be an expectedly liberated electron-hole pair at $\pm\Delta_{0}$. Rather, there will be two spin carrying "amplitude solitons" (AS) - zeros of the order parameter distributed over the length $\xi_{0}=\hbar v_F/\Delta_{0}$. (\cite{brazov:89}, see the Appendix). The unpaired electron is trapped at the midgap state associated to the amplitude soliton, with the energy $\approx2\Delta_{0}/3$, the total charge $0$, and the spin $1/2$, so this is the CDW realization of the spinon. The AS has been visualized, see Fig.\ref{fig:brun}b, by the STM as a half-period defect \cite{takaishi}, similarly to the full-period phase soliton of Fig.\ref{fig:brun}a. A regular lattice of ASs should appear in high magnetic field \cite{BDK:81} as it might have been observed as the CDW superstructure \cite{hmf rvs} and in spin-Peierls systems (it was clearly seen by the NMR experiments \cite{horvatic}). Generalization of the spin-soliton lattice from the ICDW to the superconductor is the FFLO \cite{FF,LO} phase in spin-polarized superconductors. Then the same AS becomes an elementary stripe fragment in both cases of the CDW and FFLO, with a similarity \cite{JSSS08} to holons versus stripes in doped AFM insulator. Fig.\ref{fig:lat} quotes some results from a new type of internal interlayer tunneling experiments (see \cite{lat:05,lat:06,lat:07} for the CDW applications). Together with the expected sharp feature at the threshold $2\Delta$ for free electron-hole pairs, we see the lower energy peak positioned close to the theoretical value $E_{AS}=2/\pi\Delta$ for the amplitude soliton energy \cite{lat:05}. Moreover, at even lower energies forbidden for pair-breaking processes, we see the threshold $V_{t}$ which terminates the bi-electron tunneling processes leading to formation of solitons captured in the Fig.\ref{fig:brun}. Even the oscillating fine structure within the gap is not a noise - it records events of sequential entering into the junction of dislocation lines which are aggregates of $2\pi$ solitons \cite{lat:06}. \begin{figure}[tbh] {\label{fig:CDW1}\includegraphics[width=3cm,height=3cm]{CDW1.eps}} {\label{fig:CDW2}\includegraphics[width=4cm,height=3.5cm]{CDW2.eps}} \caption{Tunneling in mesa-junctions of $NbSe_{3}$ \cite{lat:05,lat:06} - the material with two CDWs, shows a coexistence of several spectral feature: the peak at $2\Delta_0$ for the inter-gap creation of free e-h pairs, creation of the amplitude soliton at $E_{as}\approx 2\Delta_0/3$, the bi-particle spinless charge injection threshold at $V_{t}\ll\Delta_0$. a) the high $T$ CDW1; b) the low $T$ CDW2 \cite{lat:07}.} \label{fig:lat} \end{figure} \section{Solitons confinement and the long range order.} A major puzzle, as well as the inspiration, coming from the described experiments is that the amplitude solitons were observed within the low temperature ($T