Tomonaga-Luttinger Liquid State Research Articles

Robust Luttinger Liquid State of 1D Dirac Fermions in a Van der Waals System Nb9Si4Te18.

We report on the Tomonaga-Luttinger liquid (TLL) behavior in fully degenerate 1D Dirac Fermions. A ternary van der Waals material Nb9Si4Te18 incorporates in-plane NbTe2 chains, which produce a 1D Dirac band crossing Fermi energy. Tunneling conductance of electrons confined within NbTe2 chains is found to be substantially suppressed at Fermi energy, which follows a power law with a universal temperature scaling, hallmarking a TLL state. The obtained Luttinger parameter of ∼0.15 indicates a strong electron-electron interaction. The TLL behavior is found to be robust against atomic-scale defects, which might be related to the Dirac electron nature. These findings, combined with the tunability of the compound and the merit of a van der Waals material, offer a robust, tunable, and integrable platform to exploit non-Fermi liquid physics.

Inhomogeneous spin excitations in weakly coupled spin- 12 chains

We present a systematic inelastic neutron scattering and neutron diffraction study on the magnetic structure of the quasi-one-dimensional spin-$\frac{1}{2}$ magnet ${\mathrm{SrCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$, where the interchain coupling in the N\'eel-type antiferromagnetic ground state breaks the static spin lattice into two independent domains. At zero magnetic field, we have observed two new spin excitations with small spectral weights inside the gapped region defined by the spinon bound states. In an external magnetic field along the chain axis, the N\'eel order gets partially destabilized at ${\ensuremath{\mu}}_{0}{H}^{★}=2.0\phantom{\rule{0.16em}{0ex}}\mathrm{T}$ and completely suppressed at ${\ensuremath{\mu}}_{0}{H}_{\mathrm{p}}=3.9\phantom{\rule{0.16em}{0ex}}\mathrm{T}$, above which a quantum disordered Tomonaga--Luttinger liquid (TLL) prevails. The low-energy spin excitations between ${\ensuremath{\mu}}_{0}{H}^{★}$ and ${\ensuremath{\mu}}_{0}{H}_{\mathrm{p}}$ are not homogeneous, containing the dispersionless (or weakly dispersive) spinon bound states excited in the N\'eel phase and the highly dispersive psinon-antipsinon mode characteristic of a TLL. We propose that the two new modes at zero field are spinon excitations inside the domain walls. Since they have a smaller gap than those excited in the N\'eel domains, the underlying spin chains enter the TLL state via a local quantum phase transition at ${\ensuremath{\mu}}_{0}{H}^{★}$, making the N\'eel/TLL coexistence a stable configuration until the excitation gap in the N\'eel domains closes at ${\ensuremath{\mu}}_{0}{H}_{\mathrm{p}}$.

Phase diagram of an interacting staggered Su-Schrieffer-Heeger two-chain ladder close to a quantum critical point

We study the ground-state phase diagram of an interacting staggered Su-Schrieffer-Heeger (SSH) ladder in the vicinity of the Gaussian quantum critical point. The corresponding effective field theory is a double-frequency sine-Gordon (DSG) model which involves two perturbations at the Gaussian fixed point: the deviation from criticality and Umklapp scattering processes. A topological distinction between thermodynamically equivalent phases becomes only feasible when nonlocal fermionic fields, parity and string order parameter, are included into consideration. We prove that a noninteracting fermionic staggered SSH ladder is exactly equivalent to a O(2)-symmetric model of two decoupled Kitaev-Majorana chains, or two 1D p-wave superconductors. Close to the Gaussian fixed point the SSH ladder maps to an Ashkin-Teller like system when interactions are included. Thus, the topological order in the SSH ladder is related to broken-symmetry phases of the associated quantum spin-chain degrees of freedom. The obtained phase diagram includes a Tomonaga-Luttinger liquid state which, due to Umklapp processes, can become unstable against either spontaneous dimerization or the onset of a charge-density wave (CDW). In these gapped phases elementary bulk excitations are quantum kinks carrying the charge QF = 1/2. For sufficiently strong, long-range interactions, the phase diagram of the model exhibits a bifurcation of the Gaussian critical point into two outgoing Ising criticalities. The latter sandwich a mixed phase in which dimerization coexists with a site-diagonal CDW. In this phase elementary bulk excitations are represented by two types of topological solitons carrying different fermionic charges, which continuously interpolate between 0 and 1. This phase has also mixed topological properties with coexisting parity and string order parameters.

Superconducting Correlations in the One-Dimensional Kondo Lattice Models under Magnetic Fields

We analyze superconducting correlations in the one-dimensional Kondo lattice models with Ising anisotropy under transverse magnetic fields, using the density matrix renormalization group. For the spin-1/2 local spin model, the Ising anisotropy is introduced by the ferromagnetic Ising interaction between the local spins, while for the spin-1 model, it is taken by the single-ion anisotropy. The magnetic properties under the transverse fields for the spin-1/2 model are very similar to those for the spin-1 model [K. Suzuki and K. Hattori, J. Phys. Soc. Jpn. 88, 024707 (2019).]. For the superconducting correlations, we analyze various Cooper pairs within nearest-neighbor pairs including composite ones between the local spins and the electrons. We find that, for the spin-1/2 model, the superconducting correlations are highly enhanced in the Tomonaga-Luttinger liquid state near the Kondo-plateau phase, where the conduction electrons and the local spins are strongly coupled with a finite spin gap for the Ising axis. This is a clear contrast to the model under the longitudinal magnetic fields, where there are no noticeable superconducting correlations. Competitions between the transverse magnetic field and the Kondo singlet formation lead to this enhanced superconducting correlations. For the spin-1 model, the single-ion anisotropy suppresses the superconducting correlation and there is no noticeable enhancement. We also examine the large Kondo exchange coupling limit. For the moderate ferromagnetic Ising interaction between the local spins, we find that another type of superconducting correlation is enhanced inside the ferromagnetic phase. We discuss a possible relation between our results and reentrant superconductivity in U-based ferromagnetic superconductors under transverse magnetic fields.

Why it is so hard to detect Luttinger liquids in angle resolved photo-emission spectroscopy?

The problem of photoemission from a quasi-1D material is studied. We identify two issues that play a key role in the detection of gapless Tomonaga–Luttinger liquid (TLL) phase. Firstly, we show how a disorder—backward scattering as well as forward scattering component—is able to significantly obscure the TLL states, hence the initial state of angle resolved photo-emission spectroscopy (ARPES). Secondly, we investigate the photo-electron propagation towards a sample’s surface. We focus on the scattering path operator contribution to the final state of ARPES. We show that, in the particular conditions set by the 1D states, one can derive an exact analytical solution for this intermediate stage of ARPES. The solution shows that for particular energies of incoming photons the intensity of photo-current may be substantially reduced. Finally, we put together the two aspects (the disorder and the scattering path operator) to show the full, disruptive force of any inhomogeneities on the ARPES amplitude.

133Cs-NMR study on aligned powder of competing spin chain compound Cs2Cu2Mo3O12

S = 1/2 competing spin chain compound Cs2Cu2Mo3O12 has two dominant exchange interactions of the nearest neighbouring ferromagnetic J1 = 93 K and the second nearest neighbouring antiferromagnetic J2 = +33 K, and is expected to show the nematic Tomonaga-Luttinger liquid (TLL) state under high magnetic field region. The recent theoretical study by Sato et al. has shown that in the nematic TLL state, the spin fluctuations are expected to be highly anisotropic, that is, its transverse component is suppressed. Our previous NMR study on the present system showed that the dominant contribution to nuclear spin relaxation comes from the longitudinal component. In order to conclude that the transverse component of spin fluctuations is suppressed, the knowledge of hyperfine coupling is indispensable. This article is solely devoted to investigate the hyperfine coupling of 133Cs-NMR site to prove that the anisotropic part of hyperfine coupling, which connects the nuclear spin relaxation with the transverse spin fluctuations is considerably large to be Aan = +770 Oe/μB.

Rb-NMR study of the quasi-one-dimensional competing spin-chain compound Rb2Cu2Mo3O12

Rb-NMR study has been performed on the quasi-one dimensional competing spin chain Rb2Cu2Mo3O12 with ferromagnetic and antiferromagnetic exchange interactions on nearest neighboring and next nearest neighboring spins, respectively. The system changes from a gapped ground state at zero field to the gapless state at H_C \simeq 2 T, where the existence of magnetic order below 1 K was demonstrated by a broadening of NMR spectrum, associated with a critical divergence of 1/T_1. In higher temperature region, 1/T_1 showed a power-law type temperature dependence, from which the field dependence of Luttinger parameter K was obtained and compared with theoretical calculations based on the spin nematic Tomonaga Luttinger Liquid (TLL) state.

Intertube effects on one-dimensional correlated state of metallic single-wall carbon nanotubes probed by C13 NMR

The electronic states in isolated single-wall carbon nanotubes (SWCNTs) have been considered as an ideal realization of a Tomonaga-Luttinger liquid (TLL). However, it remains unclear whether one-dimensional correlated states are realized under local environmental effects such as the formation of a bundle structure. Intertube effects originating from other adjacent SWCNTs within a bundle may drastically alter the one-dimensional correlated state. In order to test the validity of the TLL model in bundled SWCNTs, low-energy spin excitation is investigated by nuclear magnetic resonance (NMR). The NMR relaxation rate in bundled mixtures of metallic and semiconducting SWCNTs shows a power-law temperature dependence with a theoretically predicted exponent. This demonstrates that a TLL state with the same strength as that for effective Coulomb interactions is realized in a bundled sample, as in isolated SWCNTs. In bundled metallic SWCNTs, we found a power-law temperature dependence of the relaxation rate, but the magnitude of the relaxation rate is one order of magnitude smaller than that predicted by theory. Furthermore, we found an almost doubled magnitude of the Luttinger parameter. These results indicate suppressed spin excitations with reduced Coulomb interactions in bundled metallic SWCNTs, which are attributable to intertube interactions originating from adjacent metallic SWCNTs within a bundle. Our findings give direct evidence that bundling reduces the effective Coulomb interactions via intertube interactions within bundled metallic SWCNTs.

Angular dependence of electron spin resonance for detecting the quadrupolar liquid state of frustrated spin chains

Spin nematic phase is a phase of frustrated quantum magnets with a quadrupolar order of electron spins. Since the spin nematic order is usually masked in experimentally accessible quantities, it is important to develop a methodology for detecting the spin nematic order experimentally. In this paper we propose a convenient method for detecting quasi-long-range spin nematic correlations of a quadrupolar Tomonaga-Luttinger liquid state of $S=1/2$ frustrated ferromagnetic spin chain compounds, using electron spin resonance (ESR). We focus on linewidth of a so-called paramagnetic resonance peak in ESR absorption spectrum. We show that a characteristic angular dependence of the linewidth on the direction of magnetic field arises in the spin nematic phase. Measurments of the angular dependence give a signature of the quadrupolar Tomonaga-Luttinger liquid state. In our method we change only the direction of the magnetic field, keeping the magnitude of the magnetic field and the temperature. Therefore, our method is advantageous for investigating the one-dimensional quadrupolar liquid phase that usually occupies only a narrow region of the phase diagram.

Dichotomy between Attractive and Repulsive Tomonaga-Luttinger Liquids in Spin Ladders.

We present a direct NMR method to determine whether the interactions in a Tomonaga-Luttinger liquid (TLL) state of a spin-1/2 Heisenberg antiferromagnetic ladder are attractive or repulsive. For the strong-leg spin ladder compound (C_{7}H_{10}N)_{2}CuBr_{4} we find that the isothermal magnetic field dependence of the NMR relaxation rate T_{1}^{-1}(H) displays a concave curve between the two critical fields bounding the TLL regime. This is in sharp contrast to the convex curve previously reported for a strong-rung ladder, (C_{5}H_{12}N)_{2}CuBr_{4}. We show that the concavity and the convexity of T_{1}^{-1}(H), which is a fingerprint of spin fluctuations, directly reflect the attractive and repulsive fermionic interactions in the TLL, respectively. The interaction sign is alternatively determined from an indirect method combining bulk magnetization and specific heat data.

NMR study on the quasi one-dimensional quantum spin magnet with ladder structure

The two-legged spin ladder Cu(CO3)0.5(ClO4)(H2O)0.5(NH3)2.5 consists of a rung formed by two Cu(II)’s and of a spacing molecule CO $_{3}^{2-}$ between each two rungs. The non-centrosymmetric shape of CO $_{3}^{2-}$ molecule brings a slight bond alternation along the leg, and hence the system can be considered as an alternating spin chain, which is confirmed so far by the temperature dependence of magnetic susceptibility. In order to investigate its spin state at low temperatures, we have performed experiments of 1H-NMR, magnetization and specific heat under wide range of magnetic field, and have found the critical diverging of longitudinal relaxation rate 1/T 1, the spectral broadening and the lambda-type anomaly in specific heat at T N≃ 3.4 K, indicating the existence of long range magnetic order. In paramagnetic state well above T N, 1/T 1 showed a power-law temperature dependence, suggesting the realization of Tomonaga Luttinger liquid state.

Absence of Luttinger liquid behavior in Au-Ge wires: A high-resolution scanning tunneling microscopy and spectroscopy study

Au-induced atomic wires on the Ge(001) surface were recently claimed to be an ideal one-dimensional (1D) metal and their tunneling spectra were analyzed as the manifestation of a Tomonaga-Luttinger liquid (TLL) state. We reinvestigate this system for atomically well-ordered areas of the surface with high-resolution scanning tunneling microscopy and spectroscopy (STS). The local density-of-states maps do not provide any evidence of a metallic 1D electron channel along the wires. Moreover, the atomically resolved tunneling spectra near the Fermi energy are dominated by local density-of-states features, deviating qualitatively from the power-law behavior. On the other hand, the defects strongly affect the tunneling spectra near the Fermi level. These results do not support the possibility of a TLL state for this system. A 1D metallic system with well-defined 1D bands and without defects are required for the STS study of a TLL state.

Stability of the Tomonaga–Luttinger liquid state in gamma-irradiated carbon nanotube bundles

We report experimental results for the changes in conductivity of single-wall carbon nanotube bundles when irradiated by 60Co γ-rays in various environments. In the current study the samples investigated were irradiated in hermetic cells, either evacuated (0.1 Pa) or filled with hydrogen or deuterium at atmospheric pressure. In situ measurements of the resistance change as a function of irradiation dose at room temperature are presented. It was found that, for all irradiation conditions, the normalized resistance versus irradiation dose demonstrates a logarithmic behaviour. A phenomenological model for the observed dependence is derived. The current–voltage characteristics of the irradiated samples were measured in the temperature range from 4.5 to 300 K using short (10 ns) electric pulses, and the results demonstrate a scaling behaviour. This scaling occurs in the universal coordinates that correspond to the Tomonaga–Luttinger liquid concept. Our results confirm the existence of the Tomonaga–Luttinger liquid phase up to room temperature in carbon nanotubes after γ-irradiation to a dose of 5 × 107 rad in vacuum, 1.7 × 107 rad in hydrogen and 1.24 × 108 rad in deuterium.

Tunability of Electronic States in Ultrathin Gold Nanowires

The electronic state in ultrathin gold nanowires is tuned by careful engineering of the device architecture via a chemical methodology. The electrons are localized to an insulating state (showing variable range hopping transport) by simply bringing them close to the substrate, while the insertion of an interlayer leads to a Tomonaga Luttinger liquid state.

One dimensional bosons: From condensed matter systems to ultracold gases

The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

Manipulating the Tomonaga-Luttinger exponent by electric field modulation

We establish a theoretical framework for artificial control of the power-law singularities in Tomonaga-Luttinger liquid states. The exponent governing the power-law behaviors is found to increase significantly with an increase in the amplitude of the periodic electric field modulation applied externally to the system. This field-induced shift in the exponent indicates the tunability of the transport properties of quasi-one-dimensional electron systems.

Correlation function for the one-dimensional extended Hubbard model at quarter filling

We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion $U$ and the intersite repulsion $V$ at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator by utilizing the renormalization-group technique based on the bosonization method, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly-accurate numerical density-matrix renormalization group method.

Geometry-driven shift in the Tomonaga-Luttinger exponent of deformed cylinders

We demonstrate the effects of geometric perturbation on the Tomonaga-Luttinger liquid (TLL) states in a long, thin, and hollow cylinder whose radius varies periodically. The variation in the surface curvature inherent to the system gives rise to a significant increase in the power-law exponent of the single- particle density of states. The increase in the TLL exponent is caused by a curvature-induced potential that attracts low-energy electrons to region that has large curvature.

Probing the intrinsic state of a one-dimensional quantum well with photon-assisted tunneling

The photon-assisted tunneling (PAT) through a single wall carbon nanotube quantum well (QW) under influence an external electromagnetic field for probing of the Tomonaga Luttinger liquid (TLL) state is suggested. The elementary TLL excitations inside the quantum well are density ($\rho_{\pm}$) and spin ($\sigma_{\pm} $) bosons. The bosons populate the quantized energy levels $\epsilon^{\rho +}_n =\Delta n/ g$ and $\epsilon^{\rho -(\sigma \pm)}_n = \Delta n$ where $\Delta = h v_F /L $ is the interlevel spacing, $n$ is an integer number, $L$ is the tube length, $g$ is the TLL parameter. Since the electromagnetic field acts on the $\rho_{+}$ bosons only while the neutral $\rho_{-}$ and $\sigma_{\pm} $ bosons remain unaffected, the PAT spectroscopy is able of identifying the $\rho_{+}$ levels in the QW setup. The spin $\epsilon_n^{\sigma+} $ boson levels in the same QW are recognized from Zeeman splitting when applying a d.c. magnetic field $H \neq 0$ field. Basic TLL parameters are readily extracted from the differential conductivity curves.

Direct observation of transition from Tomonaga–Luttinger liquid states to superconductive phase in carbon nanotubes

We report the direct observation of the transition from the Tomonaga–Luttinger liquid (TLL) states to the superconductive (SC) phase in partially end-bonded multi-walled carbon nanotubes, in which the SC phase and the TLL states interplay. Two different transition behaviors are observed when the temperature decreases: (1) a drastic increase in the Luttinger parameter g toward g=1 in the power laws on non-normalized differential conductance (d l/d v) vs. voltages and (2) a deviation from the TLL formula in the low voltage-energy region of normalized d l/d v vs. eV/ kT. These findings shed light on understanding of one-dimensional electron correlations.