Thermoelectric signature of quantum critical phase in a doped spin liquid candidate

Thermoelectric signature of quantum critical phase in a doped spin liquid candidate | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Thermoelectric signature of quantum critical phase in a doped spin liquid candidate Kodai Wakamatsu, Yuji Suzuki, Takenori Fujii, Kazuya Miyagawa, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1239067/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 21 Jun, 2023 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Abstract Quantum spin liquid is a nontrivial magnetic state of longstanding interest, in which spins are strongly correlated and entangled but do not order1, 2; further intriguing is its doped version, which possibly hosts strange metal and unconventional superconductivity3. Promising and currently the only candidate of the doped spin liquid is a triangular-lattice organic conductor, κ-(BEDT-TTF)4Hg2.89Br8, recently found to hold metallicity, spin-liquid-like magnetism and BEC-like superconductivity4-6. The nature of the metallic state with the spin-liquid behaviour is awaiting to be further clarified. Here, we report the thermoelectric signature that mobile holes in the spin liquid background is in a quantum critical state and it pertains to the BEC-like superconductivity. The Seebeck coefficient divided by temperature, S/T, is enhanced on cooling with logarithmic divergence indicative of quantum criticality. Furthermore, the logarithmic enhancement is correlated with the superconducting transition temperature under pressure variation, and the temperature and magnetic field profile of S/T upon the superconducting transition change with pressure in a consistent way with the previously suggested BEC-BCS crossover. The present results reveal that the quantum criticality in a doped spin liquid emerges in a phase, not at a point, and is involved in the unconventional BEC-like nature. Figures Figure 1 Figure 2 Figure 3 Figure 4 full text Strong correlation among electrons brings about various emergent phenomena in solids. Among them, quantum criticality has long been a focus of profound interest since strange metal, unconventional superconductivity and magnetic quantum phase transition all spring from a single point called a quantum critical point as extensively discussed in heavy electron systems and copper oxides 7, 8 . Notably, a recent study of a heavy electron compound with a Kagome lattice, CePdAl, have found a quantum critical phase residing in a range of parameter space, not at a point, suggesting that a non-Fermi liquid phase is stabilised along with a quantum spin liquid (QSL) of f electrons 9 . The relation between the quantum critical phase and frustration has recently attracted intense attention 10, 11 . In this connection, it is notable that the organic conductor, κ-(BEDT-TTF) 4 Hg 2.89 Br 8 (abbreviated as κ-HgBr) is suggested to be a doped QSL that hosts a non-Fermi liquid phase in a finite pressure range. κ-HgBr is a layered compound consisting of conducting BEDT-TTF layers with a nearly isotropic triangular lattice of BEDT-TTF dimers with the transfer integral ratio, t ’/ t , of 1.02 (Fig. 1b) and insulating Hg 2.89 Br 8 layers. The nonstoichiometry of Hg comes from an incommensurate lattice against a BEDT-TTF lattice and the missing content from 3.0 contributes 11% hole doping to a half-filled band 12 . Remarkably, κ-HgBr shows non-Fermi liquidity and spin susceptibility well scaled to that of the spin liquid material, κ-(BEDT-TTF) 2 Cu 2 (CN) 3 , thus suggesting that κ-HgBr hosts a doped QSL 5 . The electronic nature of κ-HgBr is indicated to alter by pressure. The Hall coefficient behaves such that charge carriers are apparently only the doped holes at low pressures due to strong correlation prohibiting double occupancy but is recovered to full band carriers at high pressures 4, 13 . This appears to be a pressure equivalence of the doping-driven p to 1+ p crossover with p a doping content in cuprates 14 . In resistivity, the non-Fermi liquid persists up to around 0.4-0.5 GPa and crosses over or transitions to a Fermi liquid 4, 6, 15 , as theoretically suggested 16 . At low temperatures, superconductivity occurs whose transition temperature, T c , shows dome-like pressure dependence and whose nature changes from BEC-like to BCS condensate 6 . The schematic phase diagram is shown in Fig. 1c. The quantum criticality in a phase instead of at a point and its possible relevance to QSL is an issue of profound significance. The present work aims to verify the quantum critical nature of the electronic state in κ-HgBr with pressure variation through thermoelectric effect which is very susceptible to quantum criticality, exploiting highly compressible feature of organic crystal 17 . Here, we report our observation of the thermoelectric signature of quantum criticality in the doped QSL phase and its possible relevance to superconductivity. Figure 2a and 2b shows the temperature dependence of the Seebeck coefficients divided by temperature, - S / T , under several pressures. Two separate measurements on different κ-HgBr samples give nearly coinciding results. To view the overall profile of - S / T in the pressure-temperature plane, we display the values with a range of colours in Fig. 2c for sample #1 (for sample #2, see Fig. S1 in Supplementary Information). - S / T behaves similarly at every pressure at high temperature above 30-40 K but, below that, shows strong pressure dependence with fan-shaped dispersion; - S / T is highly enhanced at low pressures well below 1 GPa whereas it is progressively reduced with increasing pressure. Sudden decreases in - S / T at low temperatures is due to the superconducting transition, as described in detail later. At high pressures above 1 GPa, where the electron correlation is weakened, - S / T is constant at low temperatures, being consistent with the Fermi liquid behaviour observed in resistivity 4, 6, 15 (see also Fig. S2 in Supplementary Information). For Fermi liquids, S is expected to follow the formula 18 , $$\frac{S}{T}=\pm \frac{{\pi }^{2}}{3}\left(1+\lambda \right)\frac{{k}_{B}}{e}\frac{1}{{T}_{F}} , \left(1\right)$$ where T F is the Fermi temperature and λ is a parameter related to the energy dependence of relaxation time, e.g. λ = 0 in the case of constant (energy-independent) mean free path. The - S / T values at low temperatures are 0.4-0.53 µV/K 2 at 1.50-1.55 GPa, ~0.75 µV/K 2 at 1.2 GPa, and ~1.1 µV/K 2 at 1.0 GPa and, assuming λ = 0, these - S / T values yield T F =530-710, ~380 and ~260 K at 1.50-1.55, 1.2 and 1.0 GPa, respectively. T F decreases with pressure toward zero around 0.5 GPa (Fig. 3a), which is very probably ascribable to the progressive renormalisation of the Coulomb interaction. Concomitantly, the temperature dependence of - S / T starts to deviate from the Fermi liquid behaviour of S/T = constant. With further decreasing pressure below 1 GPa, the temperature dependence of - S / T deviates appreciably from the Fermi liquid behaviour and - S / T continues to increase on cooling until superconductivity sets in at T c . The low-temperature value just above T c reaches the values over 2.0 µV/K 2 at 0.5-0.65 GPa and levels off at lower pressures (Fig. 3a), where non-Fermi liquid behaviour of resistivity is observed 4, 6, 15 . The temperature profile of - S / T in the low-pressure region is roughly linear in Fig. 2a, meaning - S / T ∝ln T . Such temperature dependence of the Seebeck coefficient is observed as a signature of quantum criticality in strongly correlated systems such as cuprates 19–21 , iron pnictide 22, 23 , heavy fermion 24–28 and cobalt oxides 29 , and intensively studied theoretically 30–33 . To be quantitative, the temperature dependences of - S / T under pressured below 0.5 GPa were fitted by the form of S/T = γ’ln( T / T 0 ), where T 0 is a parameter of the energy scale of quantum critical fluctuations 30 . The fitting yields T 0 =50-60 K, which is compared to T 0 ~170K for Nd-LSCO and T 0 ~3 K for YbRh 2 Si 2 (ref. 19, 24 ); these values appear consistent with their relative sizes of bandwidths of organic conductors, cuprates and heavy electron systems. Thus, the present observation provides evidence for quantum criticality in the low-pressure region in κ-HgBr. A distinctive feature from the conventional cases is that it is extended in a finite pressure range, namely, in a “critical region” instead of a “critical point”. In the present case, both the magnitude and logarithmic behaviour of - S / T maintain unchanged below 0.5-0.65 GPa as seen in Figs. 2a and 2b. In conjunction with another material that is suggested to host such a phase 9 , spin frustration would be a key to the stabilisation of quantum critical phase, as suggested theoretically 10, 11 . It is noted that the enhanced S / T values are not sharply suppressed upon the crossover from the non-Fermi liquid to the Fermi liquid at around 0.5 GPa. We consider this as a possible manifestation of the strong electron correlation in the marginal Fermi liquid nearby a non-Fermi liquid. The coefficient A in the temperature dependence of resistivity, ρ = ρ 0 + AT 2 , in the Fermi-liquid regime is a measure of correlation strength or quasi-particle dumping rate. The pressure dependence of A measured with a separate sample is displayed in the inset of Fig. 3a, which exhibits its remarkable increase well before entering the non-Fermi liquid regime. In many cases, the quantum critical logarithmic-in-temperature evolution of - S / T appears in the vicinity of magnetic transitions 30 . In κ-HgBr, enhanced spin fluctuations are suggested by NMR studies 34 and thus likely involved in the enhanced S / T albeit in a different way from the magnetic quantum criticality because of no magnetic order in κ-HgBr. It is known that S is empirically well expressed by S ~ C / ne at temperatures, where C is specific heat and n is the density of charge carriers with charge e ; thus, S is roughly an entropy per charge carrier 35 . As indicated by the Hall coefficient and resistivity, the nature of charge carriers is changed with decreasing pressure from the band quasiparticles with Fermi liquidity to emergent holes (that should be called holons) with non-Fermi liquidity, which originate from the prohibition of double occupancy – akin to the Mott localisation in a half-filled system. This drastic change of the carrier nature with no magnetic symmetry breaking should affect the enhanced S / T . The emergent holons may have extraordinary charge fluctuations that are entangled with a QSL having large entropy 36 . The reported values of |γ’|, a possible measure of the strength of quantum fluctuations 30 , are 0.01-0.05 µV/K 2 for electron-doped cuprates 21 , 0.05-0.11 µV/K 2 for hole-doped cuprates 19, 20 , 0.3-0.9 µV/K 2 for ion pnictides (Ba(Fe 1− x Co x ) 2 As 2 ) (ref. 22 ), and 2.3, 4.5, 6.2 µV/K 2 for heavy electron systems (UCoGe, YbRh 2 Si 2 , and CeCu 5.9 Au 0.1 , respectively) 24−26 (see Table S1 in Supplementary Information for |γ’| values of other materials). Thus, the present |γ’| value for κ-HgBr, ~1.2 µV/K 2 , suggests a relatively large coupling of the quantum criticality in the doped QSL to the thermoelectric effect. Remarkably, the superconducting transition temperature T c is well correlated with the low-temperature values of the logarithmically enhanced - S/T , as seen in Figs. 3a and 3b (see Fig. S3 and S4 in Supplementary Information for definitions of T c and onset T * , and the pressure dependences of |γ’| and T c , respectively). Given that |γ’| is an indicator of the strength of critical fluctuation 30 , the correlation suggests that the critical fluctuations mediate or facilitate the electron pairing. Such correlation is also found in cuprates and iron pnictides as well 21, 22 . Figure 4 shows the low-temperature behaviour of - S / T upon superconducting transition under zero and applied magnetic fields perpendicular to the conducting layers. - S / T vanishes in the superconducting state. The transition is quite sharp at high pressures; at lower pressures, however, it becomes rounded with an onset well prior to the bulk transition (Fig. 3b), reserving the possibility of enhanced superconducting fluctuations at lower pressures. Figure 4 also shows that the superconductivity is entirely destroyed by a field of 3 T under 0.9 GPa whereas, under 0.3 GPa, it survives even at a field of 7 T albeit partially very probably as a vortex liquid state. These superconductive features are fully consistent with the previously revealed BEC-to-BCS crossover associated with a non-Fermi liquid to a Fermi liquid crossover in κ-HgBr (ref. 6 ). In the BEC-like regime at low pressures, the superconductivity shows enhanced fluctuations with preformed Cooper pairs and is robust to magnetic field with forming a vortex liquid state 6 . Thus, the present results lend support to the picture of pressure-induced BEC-BCS crossover in κ-HgBr from thermoelectric point of view. The present work is the first thermoelectric investigation of a doped spin liquid candidate. The thermoelectric effect probes the entropy transport by charge carriers, which are influenced by magnetic background and superconductivity if any, and therefore includes information on the surroundings of the doped holes. The logarithmic Seebeck enhancement observed at low pressures signifies that charge carriers that travel, avoiding double occupancies, in the sea of spin liquid suffer from quantum critical fluctuations in charge and/or spin degrees of freedom. It is emphasised that the quantum critical state resides as a phase, not at a point. As pressure is increased, the logarithmic enhancement is suppressed and crosses over to the conventional metallic behaviour, indicating that the doped spin liquid crosses over to a Fermi liquid by reducing the Coulomb interactions. The correlation between the logarithmic Seebeck enhancement and superconductivity suggests that the anomalous quantum critical fluctuations favor the BEC-like electron pairing. It is an issue of further investigation whether spin or charge fluctuations or both mediate the Cooper pairing. References 1. Savary, L. & Balents, L. Quantum spin liquids: A review. Rep. Prog. Phys . 80 , 016502 (2017). 2. Zhou, Y., Kanoda, K. & Ng, T. K. Quantum spin liquid states. Rev. Mod. Phys. 89 , 025003 (2017). 3. Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys . 78 , 17 (2006). 4. Oike, H., Miyagawa, K., Taniguchi, H. & Kanoda, K. Pressure-induced Mott transition in an organic superconductor with a finite doping level. Phys. Rev. Lett . 114 , 067002 (2015). 5. Oike, H. et al. Anomalous metallic behaviour in the doped spin liquid candidate κ-(ET) 4 Hg 2.89 Br 8 . Nat. Commun . 8 , 756 (2017). 6. Suzuki, Y. et al. 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Thermopower as a signature of quantum criticality in heavy fermions. Phys. Rev. B 81 , 205108 (2010). 32. Buhmann, J. M., Ossadnik, M., Rice, T. M. & Sigrist, M. Numerical study of charge transport of overdoped La 2-x Sr x CuO 4 within semiclassical Boltzmann transport theory. Phys. Rev. B 87 , 035129 (2013). 33. Georges, A. & Mravlje, J. Skewed Non-Fermi Liquids and the Seebeck Effect. Phys. Rev. Research 3 , 043132 (2021). 34. Eto, Y., Itaya, M. & Kawamoto, A. Non-fermi-liquid behavior of the organic superconductor κ-(BEDT-TTF) 4 Hg 2.89 Br 8 probed by 13 C NMR. Phys. Rev. B 81 , 212503 (2010). 35. Behnia, K., Jaccard, D. & Flouquet, J. On the thermoelectricity of correlated electrons in the zero-temperature limit. J. Phys. Condens. Matter 16 , 5187–5198 (2004). 36. Furukawa, T., Kobashi, K., Kurosaki, Y., Miyagawa, K. & Kanoda, K. Quasi-continuous transition from a Fermi liquid to a spin liquid in κ-(ET) 2 Cu 2 (CN) 3 . Nat. Commun . 9 , 307 (2018). Declarations Acknowledgments. We thank H. Oike for comments and fruitful discussion. This work was supported by Japan Society for the Promotion of Science (JSPS) under Grant Numbers 18H05225, 19H01846, 20K20894, 20KK0060 and 21K18144, and by JST SPRING under Grant Number JPMJSP2108. Most parts of this work were performed using facilities of the Cryogenic Research Center, the University of Tokyo. Author contributions. K.K. designed the project. H.T. prepared samples. K.W., Y.S, T.F. and K.M. performed experiments and analysed as well as interpreted data with the help of K.K. K.W. and K.K. wrote the manuscript with the input from all authors. Competing financial interests. The authors declare no competing financial interests. Additional information Supplementary information is available for this paper at ***. Correspondence and requests for materials should be addressed to K.K. methods Single crystals of κ-HgBr were grown in the standard electrochemical method. For pressurisation, we used a clamp-type piston-cylinder cell made of CuBe/NiCrAl and Daphne oil 7373 as a pressure-transmitting media. Daphne oil 7373 solidifies on cooling so that the clumped pressure gradually decreases by 0.15-0.2 GPa as the temperature decreases from 300 K to 50 K and then takes a nearly constant value at lower temperatures. To know the internal pressure in the piston-cylinder cell, we used T c of a Sn flake that was mounted in the cell. The pressure values quoted in this article are the internal pressures thus estimated. Thermoelectric effect was measured with a conventional experimental platform where two Cu-plates with the Cernox thermometers attached on both and a heater attached on one plate are bridged by a κ-HgBr crystal. The thermometers were calibrated at each pressure using a reference thermometer. The heater generated a temperature difference, Δ T , between the two Cu-plates, which was maintained less than T /10 throughout the experiments. With measuring thermoelectric potential difference, Δ V , between the plates under the temperature deference, Δ T , the Seebeck coefficient is defined by S =Δ V /Δ T . In the present experiment, temperature gradient was applied along the c-axis in the conduction plane (Fig 1b). The rapid cooling is often detrimental to organic conductors because it may cause crystal cracking and/or conformational disorder of terminal ethylene groups in BEDT-TTF. To minimise these possible faults, we cooled the sample at rates slower than 0.5 K/min. Even with such cautious cooling process, the cracking in κ-HgBr crystal was not avoided at ambient pressure. Thus, the present experiments were performed under pressures, where the sample was free from such a problem. Additional Declarations There is NO Competing Interest. Supplementary Files 220122WakamatsuetalSupplementaryInformation.docx Supplementary Information Cite Share Download PDF Status: Published Journal Publication published 21 Jun, 2023 Read the published version in Nature Communications → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1239067","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":81443097,"identity":"1c795a88-44ab-4aba-86a7-b34ae5505c9d","order_by":0,"name":"Kodai Wakamatsu","email":"","orcid":"https://orcid.org/0000-0002-6191-2455","institution":"University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Kodai","middleName":"","lastName":"Wakamatsu","suffix":""},{"id":81443098,"identity":"cb6f10c0-f108-4ddc-a907-f1f1f4a605fe","order_by":1,"name":"Yuji Suzuki","email":"","orcid":"https://orcid.org/0000-0002-9960-8977","institution":"University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Yuji","middleName":"","lastName":"Suzuki","suffix":""},{"id":81443099,"identity":"67a89cac-764c-4b41-b734-6a8c96a08e10","order_by":2,"name":"Takenori Fujii","email":"","orcid":"","institution":"The University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Takenori","middleName":"","lastName":"Fujii","suffix":""},{"id":81443100,"identity":"10980af6-05ca-4b69-b3b3-ce5e8330fac1","order_by":3,"name":"Kazuya Miyagawa","email":"","orcid":"https://orcid.org/0000-0003-4841-7713","institution":"University of Tokyo","correspondingAuthor":false,"prefix":"","firstName":"Kazuya","middleName":"","lastName":"Miyagawa","suffix":""},{"id":81443101,"identity":"9ee6fec8-2913-40e4-96f7-3806879663d8","order_by":4,"name":"Hiromi Taniguchi","email":"","orcid":"","institution":"Saitama University","correspondingAuthor":false,"prefix":"","firstName":"Hiromi","middleName":"","lastName":"Taniguchi","suffix":""},{"id":81443102,"identity":"6a127d81-54c8-4801-9a23-9af3bb6345ce","order_by":5,"name":"Kazushi Kanoda","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVRIiWNgGAWjYBACCRDBw2CDJgIEzAS0pMHUGBCt5TCmFpxAsv3swQdvas7LmzPwH2DmYfgjJzkj+QDDjxoGdnMcWqR58pIN5xy7bbizgZkBqMXAWFoiLYGx5xgDs2UDdi1yDDlm0jxstxk3HABq4f1nkDhPIseAgbeBgdngAA4t/G+AWv6dswdrAdoC1sL4F48WaQmgLbxtBxLhWmYDtTDjs0Vyxhtjw7l9yck7m5kNDs5hMDaW7HmWcFjmmAROv0iczzF88Oabne129saHD94wyMlJHE8++PBNjU0yrhCDAwNgtEBcIpAAYkgkGxDUAmfxQ3TaEdQyCkbBKBgFIwUAAEtvTEeY/qN3AAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0001-5992-0125","institution":"University of Tokyo","correspondingAuthor":true,"prefix":"","firstName":"Kazushi","middleName":"","lastName":"Kanoda","suffix":""}],"badges":[],"createdAt":"2022-01-07 15:36:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1239067/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1239067/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41467-023-39217-7","type":"published","date":"2023-06-21T04:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":17996878,"identity":"d381716a-0656-484a-98ca-eb3acf079a73","added_by":"auto","created_at":"2022-02-07 15:40:08","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":123607,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eCrystal structure and schematic phase diagram of κ-HgBr.\u003c/strong\u003e\u003c/p\u003e\u003cp\u003e\u003cstrong\u003ea\u003c/strong\u003e, Layered crystal structure of κ-HgBr. The orange and blue spheres indicate Br and Hg ions in the insulating layers, respectively. \u003cstrong\u003eb\u003c/strong\u003e, In-plane molecular arrangement in the conducting layer of κ-HgBr. The BEDT-TTF molecules form dimers (circled by dotted lines), which construct an isosceles triangular lattice, characterised by two kinds of transfer integrals of \u003cem\u003et\u003c/em\u003e and \u003cem\u003et\u003c/em\u003e’ between the adjacent antibonding dimer orbitals; the ratio, \u003cem\u003et’\u003c/em\u003e/\u003cem\u003et\u003c/em\u003e, is 1.02 according to the molecular orbital calculations (see Supplementary Information). \u003cstrong\u003ec\u003c/strong\u003e, Schematic pressure-temperature phase diagram of κ-HgBr drawn with reference to the previous studies\u003csup\u003e4, 6, 15\u003c/sup\u003e. The NFL, FL, SC, and \u003cem\u003eP\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e stand for non-Fermi liquid, Fermi liquid, superconductivity, and critical or crossover pressure between NFL and FL, respectively. The red bold line indicates the possible critical phase inferred from ref.\u003csup\u003e4, 6, 15\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"floatimage11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/c4bfdcd0994c3184183e5770.jpg"},{"id":17996881,"identity":"3d444973-2fe0-4569-b70c-86117c4e2cff","added_by":"auto","created_at":"2022-02-07 15:40:09","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":86782,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTemperature and pressure profiles of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e in κ-HgBr.\u003c/strong\u003e\u003c/p\u003e\u003cp\u003e\u003cstrong\u003ea and b\u003c/strong\u003e, Temperature dependence of the Seebeck coefficient divided by temperature, -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e, in the samples #1 and #2, respectively. The essential features of the results coincide with each other. Thin straight lines indicate the dependence of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003eµln\u003cem\u003eT\u003c/em\u003e.\u003cstrong\u003e c\u003c/strong\u003e, Contour plot of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e in the pressure-temperature plane for the sample #1. The contour plot for the sample #2 is shown in Fig. S1 in Supplementary Information. The open circles indicate the superconducting transition temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e determined from -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e (see Fig. 4 and Supplementary Information for definition of \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e).\u0026nbsp;\u003c/p\u003e","description":"","filename":"groupimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/c4f332729f6b9c57eb460db0.png"},{"id":17996879,"identity":"95c0daa9-dce6-4cae-8779-bb4da3150245","added_by":"auto","created_at":"2022-02-07 15:40:09","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":65537,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePressure dependences of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e and superconducting transition temperature\u003c/strong\u003e\u003csub\u003e\u003cstrong\u003e \u003c/strong\u003e\u003c/sub\u003e\u003cstrong\u003ein κ-HgBr.\u003c/strong\u003e\u003c/p\u003e\u003cp\u003e\u003cstrong\u003ea,\u003c/strong\u003e Pressure dependences of the -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e value at 8 K, just above \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e, and the Fermi temperature, \u003cem\u003eT\u003c/em\u003e\u003csub\u003eF\u003c/sub\u003e. The square and circle markers correspond to the samples #1 and #2, respectively. The blue and red markers are the -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e and \u003cem\u003eT\u003c/em\u003e\u003csub\u003eF\u003c/sub\u003e values, respectively. Inset shows the pressure dependence of the coefficient, \u003cem\u003eA\u003c/em\u003e, in the fit of the form, \u003cem\u003eρ\u003c/em\u003e=\u003cem\u003eρ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e+\u003cem\u003eAT\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e, to the separately measured resistivity. \u003cstrong\u003eb\u003c/strong\u003e, Pressure dependences of the superconducting transition temperature, \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e, and its onset, \u003cem\u003eT\u003c/em\u003e\u003csup\u003e*\u003c/sup\u003e. The error bars of \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e indicate the widths of the bulk superconducting transition. \u003cem\u003eT\u003c/em\u003e\u003csup\u003e*\u003c/sup\u003e is defined as the temperature at which -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e starts to deviate from the normal-state behaviour. The definitions of \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e, its error bar and \u003cem\u003eT\u003c/em\u003e\u003csup\u003e*\u003c/sup\u003e are described in Supplementary Information.\u003c/p\u003e","description":"","filename":"groupimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/175869c390850711bd725ffc.png"},{"id":17996880,"identity":"311a85fd-47e9-4bbc-976c-7e25784884d8","added_by":"auto","created_at":"2022-02-07 15:40:09","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":61347,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTemperature dependence of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e at zero and applied magnetic fields.\u003c/strong\u003e\u003c/p\u003e\u003cp\u003e\u003cstrong\u003ea-d\u003c/strong\u003e, Temperature dependence of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e (for the sample #1) under the pressures of 0.3 (a), 0.5 (b), 0.65 (c) and 0.9 GPa (d). Magnetic field was applied perpendicular to the conducting plane.\u003c/p\u003e","description":"","filename":"groupimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/55c09263d76c610c6398b25d.png"},{"id":38910657,"identity":"91e72009-a7d4-4847-8c3f-79d82105b2b4","added_by":"auto","created_at":"2023-06-22 07:09:44","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":560051,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/3be48bcb-761d-4a39-aa35-3007065679fa.pdf"},{"id":17996882,"identity":"26f083a3-d5be-4040-a5b4-44b05ae224ef","added_by":"auto","created_at":"2022-02-07 15:40:09","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":316934,"visible":true,"origin":"","legend":"\u003cp\u003eSupplementary Information\u003c/p\u003e","description":"","filename":"220122WakamatsuetalSupplementaryInformation.docx","url":"https://assets-eu.researchsquare.com/files/rs-1239067/v1/3e339823e35513d3b96b01a4.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Thermoelectric signature of quantum critical phase \r\nin a doped spin liquid candidate","fulltext":[{"header":"full text","content":"\u003cp\u003eStrong correlation among electrons brings about various emergent phenomena in solids. Among them, quantum criticality has long been a focus of profound interest since strange metal, unconventional superconductivity and magnetic quantum phase transition all spring from a single point called a quantum critical point as extensively discussed in heavy electron systems and copper oxides\u003csup\u003e7, 8\u003c/sup\u003e. Notably, a recent study of a heavy electron compound with a Kagome lattice, CePdAl, have found a quantum critical phase residing in a range of parameter space, not at a point, suggesting that a non-Fermi liquid phase is stabilised along with a quantum spin liquid (QSL) of \u003cem\u003ef\u003c/em\u003e electrons\u003csup\u003e9\u003c/sup\u003e. The relation between the quantum critical phase and frustration has recently attracted intense attention\u003csup\u003e10, 11\u003c/sup\u003e. In this connection, it is notable that the organic conductor, \u0026kappa;-(BEDT-TTF)\u003csub\u003e4\u003c/sub\u003eHg\u003csub\u003e2.89\u003c/sub\u003eBr\u003csub\u003e8\u003c/sub\u003e (abbreviated as \u0026kappa;-HgBr) is suggested to be a doped QSL that hosts a non-Fermi liquid phase in a finite pressure range.\u003c/p\u003e\n\u003cp\u003e\u0026kappa;-HgBr is a layered compound consisting of conducting BEDT-TTF layers with a nearly isotropic triangular lattice of BEDT-TTF dimers with the transfer integral ratio, \u003cem\u003et\u003c/em\u003e\u0026rsquo;/\u003cem\u003et\u003c/em\u003e, of 1.02 (Fig.\u0026nbsp;1b) and insulating Hg\u003csub\u003e2.89\u003c/sub\u003eBr\u003csub\u003e8\u003c/sub\u003e layers. The nonstoichiometry of Hg comes from an incommensurate lattice against a BEDT-TTF lattice and the missing content from 3.0 contributes 11% hole doping to a half-filled band\u003csup\u003e12\u003c/sup\u003e. Remarkably, \u0026kappa;-HgBr shows non-Fermi liquidity and spin susceptibility well scaled to that of the spin liquid material, \u0026kappa;-(BEDT-TTF)\u003csub\u003e2\u003c/sub\u003eCu\u003csub\u003e2\u003c/sub\u003e(CN)\u003csub\u003e3\u003c/sub\u003e, thus suggesting that \u0026kappa;-HgBr hosts a doped QSL\u003csup\u003e5\u003c/sup\u003e. The electronic nature of \u0026kappa;-HgBr is indicated to alter by pressure. The Hall coefficient behaves such that charge carriers are apparently only the doped holes at low pressures due to strong correlation prohibiting double occupancy but is recovered to full band carriers at high pressures\u003csup\u003e4, 13\u003c/sup\u003e. This appears to be a pressure equivalence of the doping-driven \u003cem\u003ep\u003c/em\u003e to 1+\u003cem\u003ep\u003c/em\u003e crossover with \u003cem\u003ep\u003c/em\u003e a doping content in cuprates\u003csup\u003e14\u003c/sup\u003e. In resistivity, the non-Fermi liquid persists up to around 0.4-0.5 GPa and crosses over or transitions to a Fermi liquid\u003csup\u003e4, 6, 15\u003c/sup\u003e, as theoretically suggested\u003csup\u003e16\u003c/sup\u003e. At low temperatures, superconductivity occurs whose transition temperature, \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e, shows dome-like pressure dependence and whose nature changes from BEC-like to BCS condensate\u003csup\u003e6\u003c/sup\u003e. The schematic phase diagram is shown in Fig.\u0026nbsp;1c.\u003c/p\u003e\n\u003cp\u003eThe quantum criticality in a \u003cem\u003ephase\u003c/em\u003e instead of at a point and its possible relevance to QSL is an issue of profound significance. The present work aims to verify the quantum critical nature of the electronic state in \u0026kappa;-HgBr with pressure variation through thermoelectric effect which is very susceptible to quantum criticality, exploiting highly compressible feature of organic crystal\u003csup\u003e17\u003c/sup\u003e. Here, we report our observation of the thermoelectric signature of quantum criticality in the doped QSL phase and its possible relevance to superconductivity.\u003c/p\u003e\n\u003cp\u003eFigure 2a and 2b shows the temperature dependence of the Seebeck coefficients divided by temperature, -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e, under several pressures. Two separate measurements on different \u0026kappa;-HgBr samples give nearly coinciding results. To view the overall profile of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e in the pressure-temperature plane, we display the values with a range of colours in Fig. 2c for sample #1 (for sample #2, see Fig. S1 in Supplementary Information). -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e behaves similarly at every pressure at high temperature above 30-40 K but, below that, shows strong pressure dependence with fan-shaped dispersion; -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e is highly enhanced at low pressures well below 1 GPa whereas it is progressively reduced with increasing pressure. Sudden decreases in -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e at low temperatures is due to the superconducting transition, as described in detail later.\u003c/p\u003e\n\u003cp\u003eAt high pressures above 1 GPa, where the electron correlation is weakened, -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e is constant at low temperatures, being consistent with the Fermi liquid behaviour observed in resistivity\u003csup\u003e4, 6, 15\u003c/sup\u003e (see also Fig. S2 in Supplementary Information). For Fermi liquids, \u003cem\u003eS\u003c/em\u003e is expected to follow the formula\u003csup\u003e18\u003c/sup\u003e,\u003c/p\u003e\n\u003cdiv class=\"Equation\" id=\"Equa\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$\\frac{S}{T}=\\pm \\frac{{\\pi }^{2}}{3}\\left(1+\\lambda \\right)\\frac{{k}_{B}}{e}\\frac{1}{{T}_{F}} , \\left(1\\right)$$\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eT\u003c/em\u003e\u003csub\u003eF\u003c/sub\u003e is the Fermi temperature and \u003cem\u003e\u0026lambda;\u003c/em\u003e is a parameter related to the energy dependence of relaxation time, e.g. \u003cem\u003e\u0026lambda;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0 in the case of constant (energy-independent) mean free path. The -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e values at low temperatures are 0.4-0.53 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e at 1.50-1.55 GPa, ~0.75 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e at 1.2 GPa, and ~1.1 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e at 1.0 GPa and, assuming \u003cem\u003e\u0026lambda;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0, these -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e values yield \u003cem\u003eT\u003c/em\u003e\u003csub\u003eF\u003c/sub\u003e =530-710, ~380 and ~260 K at 1.50-1.55, 1.2 and 1.0 GPa, respectively. \u003cem\u003eT\u003c/em\u003e\u003csub\u003eF\u003c/sub\u003e decreases with pressure toward zero around 0.5 GPa (Fig. 3a), which is very probably ascribable to the progressive renormalisation of the Coulomb interaction. Concomitantly, the temperature dependence of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e starts to deviate from the Fermi liquid behaviour of \u003cem\u003eS/T\u003c/em\u003e = constant.\u003c/p\u003e\n\u003cp\u003eWith further decreasing pressure below 1 GPa, the temperature dependence of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e deviates appreciably from the Fermi liquid behaviour and -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e continues to increase on cooling until superconductivity sets in at \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e. The low-temperature value just above \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e reaches the values over 2.0 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e at 0.5-0.65 GPa and levels off at lower pressures (Fig. 3a), where non-Fermi liquid behaviour of resistivity is observed\u003csup\u003e4, 6, 15\u003c/sup\u003e. The temperature profile of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e in the low-pressure region is roughly linear in Fig. 2a, meaning -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e\u0026prop;ln\u003cem\u003eT\u003c/em\u003e. Such temperature dependence of the Seebeck coefficient is observed as a signature of quantum criticality in strongly correlated systems such as cuprates\u003csup\u003e19\u0026ndash;21\u003c/sup\u003e, iron pnictide\u003csup\u003e22, 23\u003c/sup\u003e, heavy fermion\u003csup\u003e24\u0026ndash;28\u003c/sup\u003e and cobalt oxides\u003csup\u003e29\u003c/sup\u003e, and intensively studied theoretically\u003csup\u003e30\u0026ndash;33\u003c/sup\u003e. To be quantitative, the temperature dependences of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e under pressured below 0.5 GPa were fitted by the form of \u003cem\u003eS/T\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u0026gamma;\u0026rsquo;ln(\u003cem\u003eT\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e), where \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is a parameter of the energy scale of quantum critical fluctuations\u003csup\u003e30\u003c/sup\u003e. The fitting yields \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e=50-60 K, which is compared to \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e~170K for Nd-LSCO and \u003cem\u003eT\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e~3 K for YbRh\u003csub\u003e2\u003c/sub\u003eSi\u003csub\u003e2\u003c/sub\u003e (ref.\u003csup\u003e19, 24\u003c/sup\u003e); these values appear consistent with their relative sizes of bandwidths of organic conductors, cuprates and heavy electron systems.\u003c/p\u003e\n\u003cp\u003eThus, the present observation provides evidence for quantum criticality in the low-pressure region in \u0026kappa;-HgBr. A distinctive feature from the conventional cases is that it is extended in a finite pressure range, namely, in a \u0026ldquo;critical region\u0026rdquo; instead of a \u0026ldquo;critical point\u0026rdquo;. In the present case, both the magnitude and logarithmic behaviour of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e maintain unchanged below 0.5-0.65 GPa as seen in Figs. 2a and 2b. In conjunction with another material that is suggested to host such a phase\u003csup\u003e9\u003c/sup\u003e, spin frustration would be a key to the stabilisation of quantum critical phase, as suggested theoretically\u003csup\u003e10, 11\u003c/sup\u003e. It is noted that the enhanced \u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e values are not sharply suppressed upon the crossover from the non-Fermi liquid to the Fermi liquid at around 0.5 GPa. We consider this as a possible manifestation of the strong electron correlation in the marginal Fermi liquid nearby a non-Fermi liquid. The coefficient \u003cem\u003eA\u003c/em\u003e in the temperature dependence of resistivity, \u003cem\u003e\u0026rho;\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003e\u0026rho;\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003eAT\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e, in the Fermi-liquid regime is a measure of correlation strength or quasi-particle dumping rate. The pressure dependence of \u003cem\u003eA\u003c/em\u003e measured with a separate sample is displayed in the inset of Fig. 3a, which exhibits its remarkable increase well before entering the non-Fermi liquid regime.\u003c/p\u003e\n\u003cp\u003eIn many cases, the quantum critical logarithmic-in-temperature evolution of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e appears in the vicinity of magnetic transitions\u003csup\u003e30\u003c/sup\u003e. In \u0026kappa;-HgBr, enhanced spin fluctuations are suggested by NMR studies\u003csup\u003e34\u003c/sup\u003e and thus likely involved in the enhanced \u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e albeit in a different way from the magnetic quantum criticality because of no magnetic order in \u0026kappa;-HgBr. It is known that \u003cem\u003eS\u003c/em\u003e is empirically well expressed by \u003cem\u003eS\u003c/em\u003e~\u003cem\u003eC\u003c/em\u003e/\u003cem\u003ene\u003c/em\u003e at temperatures, where \u003cem\u003eC\u003c/em\u003e is specific heat and \u003cem\u003en\u003c/em\u003e is the density of charge carriers with charge \u003cem\u003ee\u003c/em\u003e; thus, \u003cem\u003eS\u003c/em\u003e is roughly an entropy per charge carrier\u003csup\u003e35\u003c/sup\u003e. As indicated by the Hall coefficient and resistivity, the nature of charge carriers is changed with decreasing pressure from the band quasiparticles with Fermi liquidity to emergent holes (that should be called holons) with non-Fermi liquidity, which originate from the prohibition of double occupancy \u0026ndash; akin to the Mott localisation in a half-filled system. This drastic change of the carrier nature with no magnetic symmetry breaking should affect the enhanced \u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e. The emergent holons may have extraordinary charge fluctuations that are entangled with a QSL having large entropy\u003csup\u003e36\u003c/sup\u003e. The reported values of |\u0026gamma;\u0026rsquo;|, a possible measure of the strength of quantum fluctuations\u003csup\u003e30\u003c/sup\u003e, are 0.01-0.05 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e for electron-doped cuprates\u003csup\u003e21\u003c/sup\u003e, 0.05-0.11 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e for hole-doped cuprates\u003csup\u003e19, 20\u003c/sup\u003e, 0.3-0.9 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e for ion pnictides (Ba(Fe\u003csub\u003e1\u0026minus;\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003eCo\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e)\u003csub\u003e2\u003c/sub\u003eAs\u003csub\u003e2\u003c/sub\u003e) (ref.\u003csup\u003e22\u003c/sup\u003e), and 2.3, 4.5, 6.2 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e for heavy electron systems (UCoGe, YbRh\u003csub\u003e2\u003c/sub\u003eSi\u003csub\u003e2\u003c/sub\u003e, and CeCu\u003csub\u003e5.9\u003c/sub\u003eAu\u003csub\u003e0.1\u003c/sub\u003e, respectively)\u003csup\u003e24\u0026minus;26\u003c/sup\u003e (see Table S1 in Supplementary Information for |\u0026gamma;\u0026rsquo;| values of other materials). Thus, the present |\u0026gamma;\u0026rsquo;| value for \u0026kappa;-HgBr, ~1.2 \u0026micro;V/K\u003csup\u003e2\u003c/sup\u003e, suggests a relatively large coupling of the quantum criticality in the doped QSL to the thermoelectric effect.\u003c/p\u003e\n\u003cp\u003eRemarkably, the superconducting transition temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e is well correlated with the low-temperature values of the logarithmically enhanced -\u003cem\u003eS/T\u003c/em\u003e, as seen in Figs. 3a and 3b (see Fig. S3 and S4 in Supplementary Information for definitions of \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e and onset \u003cem\u003eT\u003c/em\u003e\u003csup\u003e*\u003c/sup\u003e, and the pressure dependences of |\u0026gamma;\u0026rsquo;| and \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e, respectively). Given that |\u0026gamma;\u0026rsquo;| is an indicator of the strength of critical fluctuation\u003csup\u003e30\u003c/sup\u003e, the correlation suggests that the critical fluctuations mediate or facilitate the electron pairing. Such correlation is also found in cuprates and iron pnictides as well\u003csup\u003e21, 22\u003c/sup\u003e. Figure 4 shows the low-temperature behaviour of -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e upon superconducting transition under zero and applied magnetic fields perpendicular to the conducting layers. -\u003cem\u003eS\u003c/em\u003e/\u003cem\u003eT\u003c/em\u003e vanishes in the superconducting state. The transition is quite sharp at high pressures; at lower pressures, however, it becomes rounded with an onset well prior to the bulk transition (Fig. 3b), reserving the possibility of enhanced superconducting fluctuations at lower pressures. Figure 4 also shows that the superconductivity is entirely destroyed by a field of 3 T under 0.9 GPa whereas, under 0.3 GPa, it survives even at a field of 7 T albeit partially very probably as a vortex liquid state. These superconductive features are fully consistent with the previously revealed BEC-to-BCS crossover associated with a non-Fermi liquid to a Fermi liquid crossover in \u0026kappa;-HgBr (ref.\u003csup\u003e6\u003c/sup\u003e). In the BEC-like regime at low pressures, the superconductivity shows enhanced fluctuations with preformed Cooper pairs and is robust to magnetic field with forming a vortex liquid state\u003csup\u003e6\u003c/sup\u003e. Thus, the present results lend support to the picture of pressure-induced BEC-BCS crossover in \u0026kappa;-HgBr from thermoelectric point of view.\u003c/p\u003e\n\u003cp\u003eThe present work is the first thermoelectric investigation of a doped spin liquid candidate. The thermoelectric effect probes the entropy transport by charge carriers, which are influenced by magnetic background and superconductivity if any, and therefore includes information on the surroundings of the doped holes. The logarithmic Seebeck enhancement observed at low pressures signifies that charge carriers that travel, avoiding double occupancies, in the sea of spin liquid suffer from quantum critical fluctuations in charge and/or spin degrees of freedom. It is emphasised that the quantum critical state resides as a phase, not at a point. As pressure is increased, the logarithmic enhancement is suppressed and crosses over to the conventional metallic behaviour, indicating that the doped spin liquid crosses over to a Fermi liquid by reducing the Coulomb interactions. The correlation between the logarithmic Seebeck enhancement and superconductivity suggests that the anomalous quantum critical fluctuations favor the BEC-like electron pairing. It is an issue of further investigation whether spin or charge fluctuations or both mediate the Cooper pairing.\u003c/p\u003e"},{"header":"References","content":"\u003cp\u003e1. Savary, L. \u0026amp; Balents, L. Quantum spin liquids: A review. \u003cem\u003eRep. Prog. Phys\u003c/em\u003e. \u003cstrong\u003e80\u003c/strong\u003e, 016502 (2017).\u003c/p\u003e\n\u003cp\u003e2. Zhou, Y., Kanoda, K. \u0026amp; Ng, T. K. Quantum spin liquid states. \u003cem\u003eRev. Mod. Phys.\u003c/em\u003e \u003cstrong\u003e89\u003c/strong\u003e, 025003 (2017).\u003c/p\u003e\n\u003cp\u003e3. Lee, P. A., Nagaosa, N. \u0026amp; Wen, X. G. Doping a Mott insulator: Physics of high-temperature superconductivity. \u003cem\u003eRev. Mod. Phys\u003c/em\u003e. \u003cstrong\u003e78\u003c/strong\u003e, 17 (2006).\u003c/p\u003e\n\u003cp\u003e4. Oike, H., Miyagawa, K., Taniguchi, H. \u0026amp; Kanoda, K. Pressure-induced Mott transition in an organic superconductor with a finite doping level. \u003cem\u003ePhys. Rev. 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Commun\u003c/em\u003e. \u003cstrong\u003e9\u003c/strong\u003e, 307 (2018).\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgments.\u003c/strong\u003e We thank H. Oike for comments and fruitful discussion. This work was supported by Japan Society for the Promotion of Science (JSPS) under Grant Numbers 18H05225, 19H01846, 20K20894, 20KK0060 and 21K18144, and by JST SPRING under Grant Number JPMJSP2108. Most parts of this work were performed using facilities of the Cryogenic Research Center, the University of Tokyo.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions.\u0026nbsp;\u003c/strong\u003eK.K. designed the project. H.T. prepared samples. K.W., Y.S, T.F. and K.M. performed experiments and analysed as well as interpreted data with the help of K.K. K.W. and K.K. wrote the manuscript with the input from all authors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting financial interests.\u003c/strong\u003e The authors declare no competing financial interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAdditional information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSupplementary information is available for this paper at ***.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCorrespondence and requests for materials should be addressed to K.K.\u003c/strong\u003e\u003c/p\u003e"},{"header":"methods","content":"\u003cp\u003eSingle crystals of \u0026kappa;-HgBr were grown in the\u0026nbsp;standard\u0026nbsp;electrochemical method. For pressurisation, we used\u0026nbsp;a\u0026nbsp;clamp-type piston-cylinder cell made of CuBe/NiCrAl and Daphne oil 7373 as a pressure-transmitting media. Daphne oil 7373 solidifies on cooling so that the clumped pressure gradually decreases by 0.15-0.2 GPa as the temperature decreases from 300 K to 50 K and then takes a nearly constant value at lower temperatures. To know the internal pressure in the piston-cylinder cell, we used \u003cem\u003eT\u003c/em\u003e\u003csub\u003ec\u003c/sub\u003e of a Sn flake that was mounted in the cell. The pressure values quoted in this article are the internal pressures thus estimated.\u003c/p\u003e\n\u003cp\u003eThermoelectric effect was measured with a conventional experimental platform where two Cu-plates with the Cernox thermometers attached on both and a heater attached on one plate are bridged by a \u0026kappa;-HgBr crystal. The thermometers were calibrated at each pressure using a reference thermometer.\u0026nbsp;The heater generated a\u0026nbsp;temperature difference, \u0026Delta;\u003cem\u003eT\u003c/em\u003e, between the two Cu-plates, which was maintained less than \u003cem\u003eT\u003c/em\u003e/10 throughout the experiments.\u0026nbsp;With measuring\u0026nbsp;thermoelectric potential difference, \u0026Delta;\u003cem\u003eV\u003c/em\u003e, between the plates under the temperature deference, \u0026Delta;\u003cem\u003eT\u003c/em\u003e, the Seebeck coefficient is defined by \u003cem\u003eS\u003c/em\u003e=\u0026Delta;\u003cem\u003eV\u003c/em\u003e/\u0026Delta;\u003cem\u003eT\u003c/em\u003e. In the present experiment, temperature gradient was applied along the c-axis in the conduction plane (Fig 1b).\u003c/p\u003e\n\u003cp\u003eThe rapid cooling is often detrimental to organic conductors because it may cause crystal cracking and/or conformational disorder of terminal ethylene groups in BEDT-TTF. To minimise these possible faults, we cooled the sample at rates slower than 0.5 K/min. Even with such cautious cooling process, the cracking in \u0026kappa;-HgBr crystal was not avoided at ambient pressure. Thus, the present experiments were performed under pressures, where the sample was free from such a problem.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-1239067/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1239067/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Quantum spin liquid is a nontrivial magnetic state of longstanding interest, in which spins are strongly correlated and entangled but do not order1, 2; further intriguing is its doped version, which possibly hosts strange metal and unconventional superconductivity3. Promising and currently the only candidate of the doped spin liquid is a triangular-lattice organic conductor, κ-(BEDT-TTF)4Hg2.89Br8, recently found to hold metallicity, spin-liquid-like magnetism and BEC-like superconductivity4-6. The nature of the metallic state with the spin-liquid behaviour is awaiting to be further clarified. Here, we report the thermoelectric signature that mobile holes in the spin liquid background is in a quantum critical state and it pertains to the BEC-like superconductivity. The Seebeck coefficient divided by temperature, S/T, is enhanced on cooling with logarithmic divergence indicative of quantum criticality. Furthermore, the logarithmic enhancement is correlated with the superconducting transition temperature under pressure variation, and the temperature and magnetic field profile of S/T upon the superconducting transition change with pressure in a consistent way with the previously suggested BEC-BCS crossover. The present results reveal that the quantum criticality in a doped spin liquid emerges in a phase, not at a point, and is involved in the unconventional BEC-like nature.","manuscriptTitle":"Thermoelectric signature of quantum critical phase \nin a doped spin liquid candidate","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-02-07 15:40:07","doi":"10.21203/rs.3.rs-1239067/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"8e7a66f4-2a40-457d-a326-6b27e3c59eb0","owner":[],"postedDate":"February 7th, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2023-06-22T07:09:38+00:00","versionOfRecord":{"articleIdentity":"rs-1239067","link":"https://doi.org/10.1038/s41467-023-39217-7","journal":{"identity":"nature-communications","isVorOnly":false,"title":"Nature Communications"},"publishedOn":"2023-06-21 04:00:00","publishedOnDateReadable":"June 21st, 2023"},"versionCreatedAt":"2022-02-07 15:40:07","video":"","vorDoi":"10.1038/s41467-023-39217-7","vorDoiUrl":"https://doi.org/10.1038/s41467-023-39217-7","workflowStages":[]},"version":"v1","identity":"rs-1239067","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1239067","identity":"rs-1239067","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}