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Magnon Bose–Einstein Condensation and Superconductivity in A Magnon Bose–Einstein condensation and superconductivity in a frustrated Kondo lattice Pavel A. Volkova,1 , Snir Gazitb,c , and Jedediah H. Pixleya aDepartment of Physics and Astronomy, Center for Materials Theory, Rutgers, The State University of New Jersey, Piscataway, NJ 08854; bRacah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel; and cThe Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel Edited by Y. B. Kim, University of Toronto, Toronto, ON, Canada, and accepted by Editorial Board Member Zachary Fisk July 6, 2020 (received for review January 13, 2020) Motivated by recent experiments on magnetically frustrated glets to produce an ordered state of spin triplets. The measured heavy fermion metals, we theoretically study the phase diagram critical properties of this transition agree well with the BEC uni- of the Kondo lattice model with a nonmagnetic valence bond solid versality class. There are several material candidates to host the ground state on a ladder. A similar physical setting may be natu- magnon BEC phenomena in a metal: YbAl3C3 (13), CeAgBi2 rally occurring in YbAl3C3, CeAgBi2, and TmB4 compounds. In the (20), and TmB4 (21). Each of these compounds has either sup- insulating limit, the application of a magnetic field drives a quan- porting experimental evidence of a VBS ground state in the tum phase transition to an easy-plane antiferromagnet, which is absence of a field or magnetization plateaus that acquire a finite described by a Bose–Einstein condensation of magnons. Using a slope that we expect is due to the Zeeman coupling to the con- combination of field theoretical techniques and density matrix duction band. Moreover, in YbAl3C3 the field tuned transition renormalization group calculations we demonstrate that in one to a magnetically ordered phase (22) is accompanied by a loga- dimension this transition is stable in the presence of a metallic rithmic behavior of specific heat that has been interpreted as a Fermi sea, and its universality class in the local magnetic response signature of a non-Fermi liquid (13). is unaffected by the itinerant gapless fermions. Moreover, we Currently, it is unknown if the magnon BEC transition can find that fluctuations about the valence bond solid ground state take place in a metal. To address this question, we study a lattice can mediate an attractive interaction that drives unconventional model (23) exhibiting a magnon BEC transition to an easy-plane superconducting correlations. We discuss the extensions of our (XY) antiferromagnetic (AFM) phase in its insulating limit and PHYSICS findings to higher dimensions and argue that depending on the solve it in the presence of a metallic conduction band (Fig. 1 A filling of conduction electrons, the magnon Bose–Einstein conden- and B). Using a combination of a low-energy field theoretical sation transition can remain stable in a metal also in dimensions analysis and density matrix renormalization group (DMRG) cal- two and three. culations we present a comprehensive solution to the problem in one dimension and demonstrate that the magnon BEC transition Kondo lattice j frustrated magnetism j strongly correlated electrons is stable in a metal. Our main results are summarized in Fig. 1C. We find that orrelated metals with closely competing quantum ground the VBS state and the BEC transition survive in the presence Cstates provide an important platform for studying a range of a metallic conduction band. For the case with two partially of fascinating phenomena such as strange metallicity (1, 2), filled bands we find that superconducting correlations induced by unconventional superconductivity (SC) (3), and fractionalized excitations (4, 5). An important example thereof is the Kondo Significance lattice model realized in heavy fermion materials (6), where the competition between magnetic order and screening of local Magnetically frustrated Kondo lattices are correlated met- moments induces a non-Fermi liquid in the vicinity of a quan- als not governed by symmetry-breaking quantum criticality, tum critical point (4, 7). More recently, it has been pointed offering a new perspective on the puzzling phenomenology out (8, 9) that Kondo systems with frustrated local moments of strange metal behavior and unconventional superconduc- open a largely unexplored avenue of quantum criticality beyond tivity. While this concept is likely to be realized in certain the Ginzburg–Landau–Wilson paradigm, where screening com- heavy-fermion compounds, the corresponding models are petes with a quantum disordered spin state, such as a spin notoriously difficult to solve, hindering robust theoretical pre- liquid (10) or a static-crystalline pattern of local singlets, i.e., dictions. Here we provide a comprehensive solution for a 1D a valence bond solid (VBS) (11). The recent discoveries of Kondo ladder with a frustrated valence bond solid ground heavy fermion metals with local moments residing on geomet- state in the presence of a magnetic field. We prove the stabil- rically frustrated lattices [e.g., the Shastry–Sutherland lattice in ity of the field-induced magnon Bose–Einstein condensation Yb2Pt2Pb, Ce2Pt2Pb, and Ce2Ge2Mg (12); a distorted trian- and demonstrate the emergence of unconventional supercon- gular lattice in YbAl3C3 (13); and a distorted Kagome lattice ductivity. We also present strong evidence that our results in CeRhSn (14) and CePd1−x Nix Al (15)], provide an excel- carry over to two and three dimensions. These findings anchor lent platform to study the interplay of magnetic frustration and future studies of frustrated heavy fermion systems. metallicity. From a theoretical perspective, it is necessary to establish what Author contributions: P.A.V., S.G., and J.H.P. designed research, performed research, properties of frustrated magnetism, including quantum critical analyzed data, and wrote the paper.y phenomena, are stable in the presence of a metallic band. This The authors declare no competing interest.y question is delicate as both the magnetic fluctuations and the This article is a PNAS Direct Submission. Y.K. is a guest editor invited by the Editorial electronic excitations are gapless at the critical point. In this Board.y work, we consider one of the best understood transitions in insu- Published under the PNAS license.y lating quantum magnets, the so-called magnon Bose–Einstein 1 To whom correspondence may be addressed. Email: [email protected] condensation (BEC) (11, 16). It occurs in insulating frustrated This article contains supporting information online at https://www.pnas.org/lookup/suppl/ VBS magnets, such as TlCuCl3 (17, 18) and SrCu2 (BO3)2 (19), doi:10.1073/pnas.2000501117/-/DCSupplemental.y where the application of a magnetic field destroys the local sin- www.pnas.org/cgi/doi/10.1073/pnas.2000501117 PNAS Latest Articles j 1 of 6 Downloaded by guest on September 23, 2021 P The conduction electron Hamiltonian reads Hc = A C k,p=± h y Ep (k) , where the dispersion is given by E±(k) = J ,t k,p k,p ┴ ┴ −2t cos k ∓ t − µ, for a chemical potential µ, the lattice con- h' k ? p J ,t g stant is set to unity, and k,± = ( k,1 ± k,2)= 2 are two- | | | | component spinors in the bonding/antibonding basis. The result- B AFM metall ing band structure is presented in Fig. 1B. For the low-energy 2t “-” properties of the system, it is important whether the Fermi E h energy crosses both bands (as in Fig. 1B) or only one, which ┴ F g h we will refer to as two- and one-band cases, respectively. As 4t “+” c the localized spins are usually due to f electrons with a large ║ VBS total angular momentum (27) as compared to the conduc- tion electrons (often from s or d states) we have omitted the k metal SC Zeeman term in Hc . Below we will argue that relaxing this -k+ -k- k - k + approximation does not fundamentally change any of our main F F F J F K conclusions. Finally, the conduction electrons interact with the localized Fig. 1. (A) Schematic depiction of the model studied. Each site hosts a local- P spins via an AFM Kondo coupling HK = JK Sr,α · sr,α ized spin and a conduction electron site coupled to their nearest neighbors. r,α y Shaded ovals represent the localized spin singlets in the VBS phase. (B) The where sr,α = r,α(σ=2) r,α and JK > 0. To make headway ana- band structure of conduction electrons for t? < 2tk. Depending on the fill- lytically, we project the Sr,α operators in HK onto the low-energy ing, one or two bands can cross the Fermi level. (C) Schematic phase diagram sector of Eq. 1 and obtain in the hard-core boson representation of the model. The VBS–AFM transition occurs at hc and is of the BEC univer- 0 sality class. In the AFM phase, partial gaps open near hg and hg determined JK X y y y H ≈ (a a )( σ + σ ) by the filling at nonzero JK (shaded green areas). For two bands crossing the K 4 r r r,+ z r,+ r,− z r,− Fermi level SC emerges for sufficiently large JK . r [2] JK X y + y + − p [ar ( σ r,− + σ r,+) + h:c:]: 2 2 r,+ r,− spin fluctuations develop. In the AFM phase, the Kondo inter- r action induces partially gapped regimes for certain values of the magnetic field. Finally, we show that for a single partially One sees that the spin-flip term acts only between the two filled band, the stability of the magnon BEC transition carries fermion bands. As is shown below, this has important conse- over to two-dimensional (2D) and three-dimensional (3D) gen- quences, namely, stabilizing the BEC transition against Kondo eralizations of the model. This allows us to provide a clear-cut screening.
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